programs that work: mathematics …nh394rd7950/hedrick...my interview transcripts when she helped me...
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PROGRAMS THAT WORK: MATHEMATICS TEACHER PREPARATION
PROGRAMS IN JAPAN, THE UNITED STATES, AND FINLAND
A DISSERTATION
SUBMITTED TO THE GRADUATE SCHOOL OF EDUCATION
AND THE COMMITTEE ON GRADUATE STUDIES
OF STANFORD UNIVERSITY
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
Benjamin Hedrick
June 2015
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This dissertation is online at: http://purl.stanford.edu/nh394rd7950
© 2015 by Benjamin James Hedrick. All Rights Reserved.
Re-distributed by Stanford University under license with the author.
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I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.
Shelley Goldman, Primary Adviser
I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.
Jo Boaler
I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.
Rachel Lotan
I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.
Raymond McDermott
Approved for the Stanford University Committee on Graduate Studies.
Patricia J. Gumport, Vice Provost for Graduate Education
This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file inUniversity Archives.
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Abstract
This study examines successful and internationally renowned mathematics
teacher training programs in Japan, the United States, and Finland. Using interview
data from program instructors, prospective mathematics teachers, and program
directors, as well as from my own experiences as a teacher and researcher in each of
these countries, I examine themes in mathematics teacher education are consistent
across these three cultural contexts, namely the relationship between theory and
practice, authenticity, and the goals of mathematics teaching. In addition, I examine
themes unique to each context, such as juku (Japan), English Language Learners (the
United States), and equity versus elitism (Finland). I also look at the themes of
relationships and expectations that are reflected in my personal account of learning to
teach in Japan, the United States, and Finland.
Despite recent emphasis on reform teaching methods, such as those proposed
by the National Council of Teachers of Mathematics, and drastic reforms in the last
decade through policies such as No Child Left Behind and Race to the Top, the United
States continues to fall further behind in international comparisons of mathematics
achievement. Believing that problems do exist within the American educational
system, educational research leaders have pointed to the continued educational success
of nations such as Finland and Japan in order to learn what features of these high-
performing systems might be components for success. While there is great value in
international comparisons, this viewpoint ignores high-performing programs in the
United States and ignores the fact that not all overseas programs are equal. I argue
that the exemplary cases are the most valuable, including the case of well-established
programs in the United States as well.
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Acknowledgements
The path that has brought me to writing this dissertation has been full of ups
and downs, uncertainties and setbacks, but the one constant has been the love and
support of many people. Mere thanks on a page of paper are not sufficient for all that
they have given, but hopefully it’s a good start.
First and foremost, I want to thank Shelley Goldman, my advisor. Shelley
adopted me as her student when I was at one of my lowest points, and most of the
successes I have experienced since then are directly attributable to her care and
guidance. Shelley has been with me every step of the way to support me
educationally, emotionally, and personally. Sometimes fate simply takes you to where
you need to be, and that place was with her. Thank you for all the patience, support,
guidance, and patience (yes, I said it twice on purpose). I would not be here without
you.
I also want to thank the members of my dissertation committee, all of whom
have been of tremendous support through the years. Ray McDermott helped me to
look at things from so many different points of view, one of which was seeing the
value in my own experiences and opinions. I’ll never forget your visit to our
Qualitative Methods class during my second quarter at Stanford when you talked
about your experiences in Japan, nor will I forget our many conversations about Japan
in the years that followed. Thank you to Jo Boaler, whose passion for mathematics
education and student learning is inspirational. I still have Jo’s handwritten notes in
my interview transcripts when she helped me code them and see patterns, and the
lesson stays with me today. Rachel Lotan has been with me since day one at Stanford
and gave me my first job with the STEP program, which gave me incredible joy and
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purpose. In the years that followed Rachel provided me with support and guidance,
not the least of which was being a part of this committee and helping me see past
PISA tests to the heart of education. Finally, a thank you to Brad Osgood for agreeing
to chair this committee. I have heard good things about Brad for years, and I’m sorry
that it took so long to finally meet.
This dissertation also would not have been possible without the help of the
CATE project, the Institute of International Education (IIE), the Finnish Fulbright
Center, and the Center for International Mobility (CIMO). Many thanks to all the
people in these organizations who funded or supported this project.
There are also many people in many countries who took care of me and
supported me as I was learning to teach, and any good skills that I have developed are
a result of your friendship. From my days in Japan, Fumishige Takeyama, Toshiyuki
Yoshida, and Nobuyuki Morimasa were phenomenal teachers and great friends. I
never would have become a teacher without you as examples. Yoko and Hiroshi
Takeyama were also great friends who opened their home to me and adopted a stray
American. In the United States, Jared Rashford, Jane Bruner, and Brian Wynne were
great friends and teachers who continue to help students even after I have left the
classroom. In Finland, Miika Lehtovaara and Eero Ropo were the people who helped
me survive the Finnish winter and find a home in a northern land. I learned so much
about teaching and research from you both.
My time at Stanford also led to many friendships with people who have helped
me in any number of ways through the dissertation and life in general. My cohort
friends Judy Hicks, Erin Baldinger, Brent Evans, Matt Kasman, Tom Lascher,
Arghavan Salles, Tara Chiatovich, Larry Samuels, and Dan Mindich (we miss you,
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Dan) will be with me always. Thanks also to Stanford friends Robin Starr, Rachel
Baker, and Brian Holtzman for all the fun and shop talk. Through Stanford I also met
Dawn, Gary, and Sara Foster, who have become a second family.
I also have to thank my family, who have supported me not only through this
dissertation process, but also through all my overseas adventures and the craziness that
those have caused. Thanks to my brother, Joe, who set an early high bar for education
and for setting an intellectual standard that I always try to live up to. Thanks to my
father, Ben, who always expected the best of me and who has always been proud of
me. No matter what strange choices I made, you’ve always had confidence in me and
supported me, and your unwavering love is always with me. Thanks to my mother,
Marcy, who has had the misfortune of putting up with the aforementioned group of
stubborn men for so many decades. You’ve always been my model of patience, hard
work, and positivity, and I wish I could be more like you. And thanks to Aseem Giri,
who can only be listed with the rest of the family. No matter the ups and downs, you
are the one person in the whole world I know I can call (and have on more than one
occasion).
Last thanks go to the most important person, Jesse Foster, who at the very
moment of writing this section is helping me with formatting. One of the happiest
days of my life was when you said that you would marry me, and I look forward to
many decades together with you. Thank you for your patience with everything from
my dissertation tantrums to my long absences for work travel. Thank you for your
support and constant reminders that anything is possible. Thank you for the home-
cooked meals after weeks on the road. Thank you for the long conversations about the
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little things that helped me keep everything in perspective. I love you, and I cannot
thank you enough.
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Table of Contents
LIST OF TABLES .................................................................................................................................XI
TABLE 1: THE CATE TEACHERS AND STUDENTS . . . . . . . . . . . . . . . . . . . . . . . . 146 .............XI
CHAPTER 1: INTRODUCTION ......................................................................................................... 1
CHAPTER 2: LITERATURE REVIEW ............................................................................................. 7 TEACHER PREPARATION AND EDUCATION IN FINLAND .......................................................... 7 TEACHER PREPARATION AND EDUCATION IN JAPAN ................................................................ 8 TEACHER PREPARATION AND EDUCATION IN THE UNITED STATES ................................ 10 INTERNATIONAL COMPARATIVE STUDIES .................................................................................... 11 TRACKING AND ABILITY-GROUPING ................................................................................................ 13 CONCEPTUAL FRAMEWORK ................................................................................................................. 15 RESEARCH QUESTIONS ........................................................................................................................... 17
CHAPTER 3: METHODS .................................................................................................................. 18 DATA SOURCES ........................................................................................................................................... 18 DATA COLLECTION ................................................................................................................................... 19 DATA ANALYSIS .......................................................................................................................................... 23 CHALLENGES AND LIMITATIONS ....................................................................................................... 24
CHAPTER 4: JAPAN (PERSONAL) ................................................................................................ 27 THE AMERICAN QUESTION ................................................................................................................... 30
A. Relationships......................................................................................................................................... 32 1. Student and Teacher .................................................................................................................................................... 33 2. Student and Student ..................................................................................................................................................... 34 3. Teacher and Teacher ................................................................................................................................................... 36
B. Expectations .......................................................................................................................................... 37 1. A Focus on Students ..................................................................................................................................................... 37 2. The Role of the Teacher .............................................................................................................................................. 38 3. Differences by Grade Level ....................................................................................................................................... 39
CONCLUSION ................................................................................................................................................ 41
CHAPTER 5: JAPAN (TEACHER TRAINING) ............................................................................. 42 TAKEYAMA-SENSEI’S CLASS ................................................................................................................. 45 YOSHIDA-SENSEI’S CLASS ...................................................................................................................... 50 A PLACE FOR THEORY AND A PLACE FOR PRACTICE ................................................................. 56 A FOCUS ON AUTHENTICITY ................................................................................................................. 58 THE GOALS OF MATHEMATICS TEACHING ..................................................................................... 59 A FOCUSING CULTURAL ISSUE: TEACHING IN CRAM SCHOOLS VERSUS CLASSROOMS ............................................................................................................................................................................ 62 CONCLUSION ................................................................................................................................................ 66
CHAPTER 6: UNITED STATES (PERSONAL) ............................................................................ 71 A. Relationships......................................................................................................................................... 76
1. Student and Teacher .................................................................................................................................................... 77 2. Student and Student ..................................................................................................................................................... 79 3. Teacher and Teacher ................................................................................................................................................... 81
B. Expectations .......................................................................................................................................... 82 1. Do what the teacher tells you to do ...................................................................................................................... 83 2. The Role of the Teacher .............................................................................................................................................. 84 3. Equality and equity ....................................................................................................................................................... 85
CONCLUSION ................................................................................................................................................ 89
CHAPTER 7: UNITED STATES (TEACHER TRAINING) ......................................................... 91 MATHEMATICS CURRICULUM AND INSTRUCTION CLASS 1 ................................................... 93
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MATHEMATICS CURRICULUM AND INSTRUCTION CLASS 2 ................................................... 97 MATHEMATICS CURRICULUM AND INSTRUCTION CLASS 3 ................................................ 102 A PLACE FOR BOTH THEORY AND PRACTICE ............................................................................. 108 A FOCUS ON AUTHENTICITY .............................................................................................................. 111 THE GOALS OF MATHEMATICS TEACHING .................................................................................. 114 A FOCUSING CULTURAL ISSUE: ENGLISH LANGUAGE LEARNERS ..................................... 116 CONCLUSION ............................................................................................................................................. 119
CHAPTER 8: FINLAND (PERSONAL) ....................................................................................... 124 THE AMERICAN (PISA) QUESTIONS ................................................................................................ 129
A. Relationships...................................................................................................................................... 131 1. Student and Teacher ................................................................................................................................................. 132 2. Student and Student .................................................................................................................................................. 134 3. Teacher and Teacher ................................................................................................................................................ 135
B. Expectations ....................................................................................................................................... 136 1. Everyone Participates .............................................................................................................................................. 136 2. The Role of the Teacher ........................................................................................................................................... 139 3. Teachers are Qualified ............................................................................................................................................. 140
CONCLUSION ............................................................................................................................................. 141
CHAPTER 9: FINLAND (TEACHER TRAINING) .................................................................... 143 THE FULBRIGHT TEACHERS AND STUDENTS ............................................................................ 148 A PLACE FOR THEORY BUT MOSTLY PRACTICE ........................................................................ 151 A FOCUS ON AUTHENTICITY .............................................................................................................. 153 THE GOALS OF MATHEMATICS TEACHING .................................................................................. 157 A FOCUSING CULTURAL ISSUE: EQUITY VERSUS ELITISM .................................................... 159 CONCLUSION ............................................................................................................................................. 167
CHAPTER 10: CONCLUSION ....................................................................................................... 171 MY NEW AMERICAN QUESTION ....................................................................................................... 172 RELATIONSHIPS WITH TEACHERS MATTER .............................................................................. 174 PRACTICE, SUPPORTED BY THEORY, IS KEY ............................................................................... 178 WHERE PATHS DIVERGE ..................................................................................................................... 182 TEXTBOOKS ............................................................................................................................................... 182 TESTING ...................................................................................................................................................... 185 AREAS FOR FUTURE RESEARCH ....................................................................................................... 187
REFERENCES ................................................................................................................................... 190
APPENDIX ........................................................................................................................................ 200 APPENDIX A: CATE INTERVIEW PROTOCOL, FACULTY.......................................................... 200 APPENDIX B: CATE INTERVIEW PROTOCOL, TEACHER CANDIDATES ............................ 202 APPENDIX C: CATE INTERVIEW PROTOCOL, PROGRAM DIRECTOR ................................. 204 APPENDIX D: FULBRIGHT INTERVIEW PROTOCOL, PROSPECTIVE TEACHERS .......... 207 APPENDIX E: FULBRIGHT INTERVIEW PROTOCOL, PROFESSORS .................................... 209 APPENDIX F: CATE DATA CODES ...................................................................................................... 211 APPENDIX G: FULBRIGHT DATA CODES ........................................................................................ 212 APPENDIX H: THE PAINTED CUBES PROBLEM .......................................................................... 214 APPENDIX I: THE TV ANTENNA PROBLEM .................................................................................. 217
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List of Tables
Table 1: The CATE Teachers and Students . . . . . . . . . . . . . . . . . . . . . . . . 146
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Chapter 1: Introduction
I never wanted to be a teacher when I was growing up, and I never wanted to
leave the United States. That’s a particularly odd first sentence for a doctoral
dissertation about my life as a teacher and my research in three different countries, but
it’s true. There was nothing about teaching or living abroad that was in any way
abhorrent; I simply had no interest in teaching (I wanted to be a lawyer), and traveling
was not very fun. In retrospect I did spend an unusual amount of time teaching my
friends, particularly in mathematics, and for the service requirement of high school
graduation I volunteered as a mathematics tutor at the local community college. The
decision later in life to become a teacher surprised me, though it did not surprise my
parents.
The origin of my interest in teaching is obvious: I learned to teach in Japan,
and once involved in the profession, I fell in love. But why Japan, and why Finland?
The answer to both of these questions is also quite easy: I have no idea. To be fair, I
took two years of Japanese language classes while I was an undergraduate student,
which is where I distinctly remember my first semester instructor saying that she
thought I would very much enjoy the Japanese Exchange and Teaching (JET)
Programme. I smiled, nodded, and then told a friend in class that I would never live
outside of the United States. Less than two years later I was living in Japan as a
teacher for JET. As to how I ended up in the Japanese classes, the answer to that
question is also easy: martial arts. As a freshman I promised myself that I would learn
a martial art, and that I would get involved with one new hobby that was completely
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and totally different from anything I had ever done (that turned out to be ballroom
dancing, which is another long story). The group that I joined was led by a former
member of the US military who learned karate in Japan and who occasionally brought
Japanese masters to visit us at the university. Since none of the masters spoke
English, we were required to learn limited Japanese vocabulary in order to
communicate with them. In addition, as we ascended in the ranks, we were required
to write papers about the history of martial arts and the history of Japan. These things
naturally led to the Japanese classes, which led naturally to moving to Japan.
I have no idea why I decided that I had to study a martial art in college. Even
more interestingly, I have no idea what would have happened had I joined, say, a
Korean martial art. I remember going to demonstrations and choosing the Japanese
one because it was closer to my dormitory. Small things make a large difference.
And as for Finland, I have absolutely no idea. I suppose it just sounded cool. Thus, it
is somewhat surprising to say that I ended up in two countries that are known for
strong education almost by sheer dumb luck.
Finland has been of particular interest for mathematics educators based on the
country’s consistently strong mathematics performance on the 2000, 2003, 2006, and
2009 PISA examinations. Although Finland primarily gained international attention
on the 2000 PISA due to its #1 ranking in literacy rather than its #4 ranking in
mathematics among the 28 participating OECD countries, the 2003 PISA results,
which focused on mathematics, showed Finland to have the highest levels of
mathematics achievement of all 30 OECD countries (OECD 2001, 2004). Finnish
students continued to excel in the 2006 and 2009 PISA examinations with Finland
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ranked second amongst OECD countries to Korea in mathematics with a non-
statistically significant difference in scores (OECD, 2010).
Japan has also maintained a steady and strong performance in international
assessments such as PISA, though in recent years focus has shifted to higher-scoring
Asian countries such as Korea and Singapore (a non-OECD country). When Finland
was ranked as the number one country mathematically in the 2003 PISA, Japan was
fourth, and in 2009 PISA results, Japan maintained a 4th
place ranking (OECD 2004,
2010). Both Japan and Finland were amongst the five countries that reported over
20% of students achieving the top scores (a 5 or a 6) on the PISA examination, and
both also reported no statistical difference in scores between males and females
(OECD 2010).
In stark comparison, the United States consistently maintains a position below
and statistically different from the OECD average score in mathematics on the PISA.
Despite recent emphasis on reform teaching methods (NCTM, 2000) and drastic
reforms in the last decade through policies such as No Child Left Behind and Race to
the Top, the United States continues to fall further behind in international comparisons
of mathematics achievement. Students’ scores on recognized measures such as the
Programme for International Student Assessment (PISA) and the Trends in
International Mathematics and Science Study (TIMSS) are slowly dropping relative to
other competitive, industrialized nations (Organization for Economic Co-operation
and Development, 2004, 2007, 2010; Institute of Educational Sciences, 2004, 2008,
2009).In 2003, the United States ranked number 24 of 29, and in the most recent
PISA, the United States ranked 25th
out of 34 countries (in a three-way tie). US scores
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have been statistically lower than the OECD average, though the United States is one
of the top scorers in the categories of gender differences and spread between the
highest and lowest performers (OECD 2010). Believing that problems do exist within
the American educational system, educational research leaders have pointed to the
continued educational success of nations such as Finland and Japan in order to call for
a dramatic overhaul of outdated methods in the US that are no longer serving the
needs of 21st century students (Darling-Hammond, 2008a, 2008b, 2009).
Revitalization of the American system of education, however, is not as simple
as copying systems that work in other countries. Differences in governmental
structure, population demographics, and culture make many reforms and systems that
are successful in Finland and Japan unfeasible in the United States (Darling-
Hammond, 2009; Kupiainen, Sirkku, & Pehkonen, 2008; OECD, 2011a). One of the
keys to improving the American system and providing effective and equitable
education might be identifying those aspects of the educational systems in other
countries that could be transferable and practical for adoption in American classrooms
(Darling-Hammond, 2008).
The chapters in this dissertation are organized to tell a story, supported by
observations and formal data collection, that takes place over the course of almost
exactly 10 years. I have decided to tell this story chronologically rather than
thematically in part because of my experiences: I taught in Japan first, the United
States second, and Finland third. My interest in teaching developed along this path,
and in fact the impetus for this dissertation also developed along the same path. The
main disadvantage to this organization is that it necessarily postpones deeper
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comparisons amongst the countries and experiences until the end, once whole story
has played out and all three countries have been described. To write the story
thematically, however, would require an incredible amount of exposition concerning
the relevant cultural contexts and issues of all three countries only then to be followed
by analysis. Culture and relevance cannot easily be separated into distinct pieces,
which would make the story choppy and disjointed. As such, I decided that the
chronological approach would be best as I could slowly build a parallel story followed
by a summary analysis of the experiences and data as a whole as a final chapter.
Taking this organizational challenge into account, in order to make these
comparisons make more sense as the chapters progress, I have organized the general
structure of each set of chapters to be as similar to the others as possible. Chapters 4,
6, and 8 deal with my personal experiences and observations in the classroom in
Japan, the United States, and Finland respectively. These chapters will tell the story of
life inside the classroom, supported in large part by my personal journal entries during
this time. Prior to my trip to Japan in 2000, I began to keep a daily journal in which I
recorded the facts of the day, often with accompanying details of what I was thinking
and feeling. While my memory of the events is imperfect, these journals provide
accurate reflections on these events and also serve to pinpoint exact dates and times of
specific events. Given that I spent three years teaching in Japan, over four years
teaching in the United States, and one year teaching in Finland, I literally have
thousands of journal entries from which to draw relevant information. Chapters 5, 7,
and 9 focus on the teacher training data collected in Japan, the United States, and
Finland, and each of these chapters follows the corresponding personal chapter for that
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country. Thus the data collected in Japan form the basis for Chapter 5, the United
States Chapter 7, and Finland Chapter 9.
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Chapter 2: Literature Review
TEACHER PREPARATION AND EDUCATION IN FINLAND
In Finland, teaching is a profession that has traditionally enjoyed high respect
and admiration (Simola, 2005). In a recent national poll, over 26% of upper secondary
school graduates rated teaching as the most desirable profession (Sahlberg, 2007). As
a result, admission into teacher education programs in Finland is highly competitive,
in some cases with only 10% of applicants accepted (Westbury, Hansen, Kansanen, &
Björkvist, 2005) and an overall acceptance rate into all programs of about 25%
(Sahlberg, 2007). Selection into these schools of education are based on matriculation
examination scores, the type of high school diploma they earned, out-of-school
accomplishments, a written examination on a prescribed list of books on teaching
pedagogy, a mock teaching event, and an interview where they must explain why they
want to become teachers (Sahlberg, 2011).
There are only eleven teacher training programs in Finland (Finnish Teacher
Training Schools, 2014), and they are exclusively at the Masters level; there are no
alternate methods to receive a teaching certificate (Sahlberg, 2007). Finnish pre-
service secondary school teachers must first study their major subject before being
accepted into the Faculty of Education. Additionally, Finnish teachers (both
elementary and secondary) commonly pursue certification in multiple subjects and are
responsible for teaching multiple subjects once they obtain a teaching position.
Finnish prospective teachers also, as part of their degree, engage in educational
research and eventually produce a Master’s thesis based on this research.
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In terms of national policy and practice, Finland has an unusual mix of
autonomy and strict regulations. Finland has a national core curriculum, and teachers
are expected to assess students using guidelines from it. Teachers, however, are given
broad latitude for selecting the textbooks they wish to use and pedagogical strategies
they wish to employ. The curriculum is also highly un-prescriptive, and the 10 page
document that details basic school mathematics directly mentions that different
curricula will incorporate both local priorities and community aspirations and values
(OECD 2011a). Although each of the eleven teaching training programs in Finland
has full pedagogical autonomy, the departments have detailed standards for improving
the quality of their teacher-education programs (Sahlberg, 2007). Two commonalities
amongst all schools of education are that programs are grounded in research-based
theories while simultaneously requiring the pre-service teachers to learn through
practice-based teaching, developing high levels of content knowledge (Sahlberg, 2007;
OECD 2011a).
TEACHER PREPARATION AND EDUCATION IN JAPAN
Much like in Finland, teachers in Japan traditionally enjoy a great deal of
respect, and it is a difficult profession to enter. According to data compiled for 2009
by the Japanese Ministry of Education (called Monbukagakusho, often abbreviated
MEXT), there were 166,729 applications for all types of teaching positions, and
26,910 were hired (approximately 16%). Looking specifically at upper secondary
school teachers (equivalent to the last three years of high school in the United States),
there were 34,732 applications and 4,289 hires (approximately 12%). Admission into
teacher training programs, following the requirements for all other majors, requires
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completion of the National Entrance Exam, which tests students in Japanese, foreign
language, mathematics, science, and social studies (Wang et al, 2003). Each
university, however, might also have its own additional entrance exams in some of all
of these areas. There are also different levels of certificate available depending on the
institution: the “advanced level” is conferred with a Master’s degree or higher, the first
level is conferred with a Bachelor’s degree, and the second level is conferred with a
junior college degree, though this last level is only a temporary certificate (Ingersoll,
2007). Teaching is also considered an attractive career option with high autonomy
and good pay, and retention rates have been traditionally very high (Ingersoll, 2007).
There are currently about 850 institutions in Japan that offer courses on teacher
education (Ingersoll, 2007). Although programs differ, there are minimum
requirements set by the Ministry of Education for a secondary teaching credential in
terms of credit for classes about teaching and classes in a content area. Pre-service
teachers are also required to complete a minimum of three weeks of student teaching
as part of their training (Wang, 2003). Simply graduating from a credentialing
program, however, is not sufficient to teach. Each prefecture (more or less equivalent
to a state in the United States) decides what tests an applicant must pass in order to be
granted a license. Such tests could be written tests, interviews, essays, or proficiency
tests, and topics could range from content knowledge to pedagogical theory to student
guidance and counseling to educational laws and regulations, to name a few topics
(Wang, 2003).
Japan has a national curriculum, and much like Finland, it is relatively brief in
nature. For example, the Grade 9 mathematics curriculum consists of 4 objectives, 10
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content goals for Numbers and Mathematics Expressions, 9 content goals for
Geometric Figures, 4 content goals for Functions, and 2 for Data Handling that take
less than 3 pages (MEXT, 2009, in Japanese only). Detailed versions of the
curriculum, however, are also published that are quite long and are designed to be
more step-by-step guides. Although technically following the curriculum is not
mandatory, prefectures receive their funding from the Ministry of Education and thus
follow it fairly rigidly (OECD, 2011a). Textbooks, which tend to be much smaller
than their American counterparts, are selected at the prefectural level, and although
prefectural choices differ, there is not a wide variety of texts. All course textbooks
must also be approved by the Ministry of Education.
TEACHER PREPARATION AND EDUCATION IN THE UNITED STATES
Teaching in the United States is not considered a rigorous profession, and it is
considered a relatively “non-competitive, easy-in” occupation (Ingersoll, 2007, p. 10).
Lortie described teaching as having a low entry bar and a wide entry gate (1975).
When comparing college entrance exam scores, such as the SAT, scores for pre-
service teachers tend to be in the low range compared with other college graduates
(Henke, Chen, & Geis, 2000). There is also a wide variety of standards for acceptance
into a teacher training programs in the United States. In general, the majority of
programs have requirements such as a minimum grade point average, interviews,
experience working with children, or a basic skills test such as the GRE or the
PRAXIS. There are over 1500 teacher training programs in the United States, and
most are undergraduate 4-year programs or programs that add a 5th
year (American
11
Council on Education, 1999). Others are Masters programs, which may take one to
two years to complete and require a Bachelor’s degree for admission.
Programs of teacher education differ from institution to institution, though all
follow the relevant state accreditation policies (Wang et al, 2003). Requirements
include some degree of content training, though emphasis is often places on teaching
strategies and classroom management. Content knowledge is assumed though other
coursework and occasionally assessed though subject matter tests. As entrance
requirements differ, so do exit requirements. An adequate grade point average,
completion of required courses, and often a content examination are standard, though
the types of tests and minimum scores can vary widely. The number of hours of
student teaching also varies, though it is required of all programs.
The United States does not have a national curriculum, though a recent
movement with the Common Core Standards has been an attempt to create something
akin to a national curriculum. Curriculum decisions are made at both the state and
local levels, and as such, a great deal of diversity of curricula and textbooks exists
even within states. There is, however, great similarity as well, due in part to
organizations such as the National Council for Accreditation of Teacher Education
(NCATE). Although NCATE accreditation is voluntary, a survey of teacher education
programs found that about 1400 of 1500 received approval or accreditation based on
local, state, or NCATE standards (NASDTEC, 2002).
INTERNATIONAL COMPARATIVE STUDIES
Laukkanen writes that “Individual countries can use [international
comparisons] as mirrors in which to reflect their own performance and policies.
12
Although it is not wise to import policies from other countries as such, countries can
benchmark their own products with products from elsewhere. Countries can also learn
from each others' good practices” (2007, p. 319). Though adopting or even adapting
practices from other countries with different cultures, values, educational laws, and/or
ethnic/linguistic compositions would be challenging at best, as well as only a partial
solution to many educational challenges, there is much that can be learned from
international comparisons and studies (Darling-Hammond, 2009). Believing that
problems do exist within the American educational system, educational research
leaders have pointed to the continued educational success of nations such as Finland
and Japan in order to call for a dramatic overhaul of outdated methods in the US that
are no longer serving the needs of 21st century students (Darling-Hammond, 2008,
2009). In addition, scholars are calling for increased international comparative
research (Hudson & Zgaga, 2008).
Shulman famously referred to what he called “visions of the possible,”
meaning that it is of high value to have an idea of what might be attainable (1998). In
the case of international education, it is of value to see how other countries structure
their systems in order to better understand what is possible for our own. In the OECD
Report “PISA 2009 Results: What Students Know and Can Do” the authors write:
Last but not least, the most impressive outcome of world-class education
systems is perhaps that they deliver high-quality learning consistently across
the entire education system, such that every student benefits from excellent
learning opportunities. To achieve this, they invest educational resources
where they can make the greatest difference, they attract the most talented
teachers into the most challenging classrooms, and they establish effective
spending choices that prioritise the quality of teachers. (OECD, 2010)
13
In the 2009 PISA, over 20% of Finnish and Japanese students performed at the top
two levels (5 and 6), two of the five OECD countries to do so, while only 10% in the
United States reached these levels (Fleischman, 2010; OECD, 2010). Similarly, 23%
of U.S. students perform at level 1 or below, compared to about 8% in Finland and
14% in Japan. Once again Finland and Japan are leaders in this area: only four
countries can claim rates this low, and both are well below the OECD average of 19%
(OECD, 2010). In addition, the United States produces one of the largest gaps
between the bottom 5% and top 5% of its students in terms of raw score and has one
of the largest gender differences of all OECD countries; Finland consistently has the
lowest gap between the bottom and top 5% and is one of the few countries with no
statistically significant gender gap.
TRACKING AND ABILITY-GROUPING
Tracking, as defined by Oakes, is “the process whereby students are divided
into categories so that they can be assigned in groups to various kinds of classes”
(1985). As practiced in the United States, students are grouped into performance
groups by perceived ability, though student self-selection or parental request may also
be deciding factors (Carey, Farris, & Carpenter, 1994). Typically students are
grouped into high performance classes (sometimes called “honors” classes), middle
performance classes (often referred to as “on-level” classes), or low performance
classes (called by names ranging from “remedial” classes to “vocational track”
classes) (Oakes, 1985). Proponents of the tracking system (hereafter referred to as
ability-grouping or same performance grouping) argue that the arranging of students
by ability makes for more homogeneous classrooms with similar learning needs
14
(Hallinan, 1994; Lubienski, 2000). In principle, these classes can thus focus on the
specified learning goals of the group, leading to increased knowledge and achievement
for all involved (Turney, 1931).
Opponents, however, note that ability-grouping does not often work as well as
it should in theory and favor non-tracked classrooms (which will hereafter be referred
to as mixed performance grouping). While some studies have shown that ability-
grouping does not have a positive learning effect for any ability-group level (Slavin,
1990), others have shown positive effects of mixed performance grouping that range
from increased critical thinking skills to positive mathematical identity (Oakes, 1985;
Linchevski & Kutscher, 1998; Boaler, 2002; Boaler, 2006) as well as long term job
prospects and social mobility (Boaler, 2008). Studies have also shown that ability-
grouping influences everything from academic achievement and future career
selection to choices of friends (Kubitschek & Hallinan, 1998; Hallinan & Oakes,
1994). In addition, student perceptions of their academic identity are formed early in
the tracking process. Students in low performance classes realize that there is a strong
negative stigma attached to being in these tracks, which in turn negatively affects their
self-perceptions and self-value (Oakes, 1985). Low performing students also feel
these values are reinforced by teachers as well as their peers (Hallinan & Oakes,
1994). In writing about educational stratification, Collins wrote that “individuals may
struggle with each other, [and] individual identity is derived primarily from
membership in status group” (Collins, 1971, p 102). Whether that status group is low
status or high status is irrelevant; students know where they belong, and they will meet
the educational expectations for that track. Additionally, once a student has been
15
placed in a low performance track, the subsequent lack of attainment of higher level
skills effectively prevents upward movement, dooming the vast majority of students to
more limited educational opportunities and choices (Oakes, 1985). This lack of
upward movement is also problematic in the context of educational equity and
equality since same performance classrooms are far less homogeneous than teachers
and administrators believe (Hallinan & Oakes, 1994). Disproportionate percentages of
high performance students are Asian and Caucasian, while low performance students
are predominantly Hispanic and African-American (Oakes, 1985; Gamoran, 1987;
Delpit, 1995).
Tracking is a fairly ubiquitous practice in the United States, though there are
many schools that are “detracked” or where students are homogeneously grouped.
While tracking does not typically occur in Japan, there are rare cases of schools with
some ability grouping. Schools in Finland have been detracked by national law since
the 1980s.
CONCEPTUAL FRAMEWORK
The value of teachers and strong teacher training programs has been cited as
one of the main reasons for success on the PISA (OECD, 2005). The beliefs about the
teaching and learning of mathematics that pre-service teachers bring to the classroom
also have a profound effect on student learning, which has been well-documented by
researchers (Cooney, Shealy, & Arvold, 1998; Battista, 1994; Stipek, Givvin, Salmon,
& MacGyvers, 2001; Handal, 2003). Research also suggests that teachers, given the
opportunity to learn about teaching in the context of practice, have stronger impact on
16
student learning (Boyd et al, 2009; NRC, 2010). Studies are also beginning to explore
what particular program features might make a positive difference in the learning of
pre-service teachers as well as, ultimately, student learning (Brouwer & Korthagen,
2005; Boyd et al., 2006, 2009). Despite these strong indicators, research into teacher
preparation programs is still not well developed, and there is a clear need for more
(Cochran-Smith & Zeichner, 2005; NRC, 2010).
All three of these countries in this study are featured in the series “Strong
Performers and Successful Reformers” published by PISA (OECD, 2011a). The
United States edition features Chapter 5 Finland: Slow and Steady Reform for
Consistently High Results and Chapter 6 Japan: A Story of Sustained Excellence.
Both countries cite a strong focus on teachers and teacher education as top reasons for
continued success. Though the United States is far from a strong performer, it has a
reputation as a strong reformer based on current or proposed educational overhauls. A
case study analysis of strong programs from each of these three countries will allow us
to understand better how these programs function and what connects them to the
production of internationally well-regarded teachers. The addition of first-person
accounts of teaching in all three of these systems provides context and details of the
daily interactions in the classroom and also serves as a vehicle for describing
similarities, differences, and salient features of each system. Ultimately we will learn
more from this process as to what features of these three distinct countries and cultures
are similar or different, and the process could conceivably be applied to additional
countries to continue the process of understanding the strengths of these programs
from a cultural perspective.
17
RESEARCH QUESTIONS
Building both on previous knowledge in the field as well as the current work of
the CATE (Coherence and Assignment Study in Teacher Education) project, this
project seeks to apply a cultural lens to examine the nature of the preparation of future
teachers of mathematics.
While all programs aim to prepare teachers to be successful in the classroom,
differences in cultural values will both directly and subtly alter the nature of what is
emphasized and how it is emphasized. Given this hypothesis, this study will answer
the following questions:
1. What themes in mathematics teacher preparation are consistent across Japanese,
American, and Finnish contexts? What themes are emphasized but unique to each
context?
2. What themes in mathematics teacher preparation can be seen in how mathematics
pedagogy courses are taught? How do students and teacher educators describe these
themes?
3. What themes are reflected in my personal account of learning to teach in Japan, the
United States, and Finland?
18
Chapter 3: Methods
DATA SOURCES
Data from the Finland and United States CATE project sites were collected in
the fall of 2012 and the winter of 2013 respectively. Data for the Japan site were
collected during the spring of 2013. Different collection dates reflect the differing
structure of the universities (Japan’s school year, for example, begins in the spring,
while the Finland programs begin in the fall) as well as the availability of and access
to classes.
The two sites in Finland were the Catherine University, for the CATE project
data, and Lapinkaari University, for the Fulbright data1. Teacher preparation courses
at both universities are at the Masters level, as they are for all teacher education
programs across the country. It should be noted that students in Finland do not pursue
a Bachelor’s degree followed by a Master’s degree; they graduate with only the
Masters. Both universities are considered excellent centers of teacher training, and
both are located in major metropolitan centers.
The United States site was Foster University, located in a western state near a
major metropolitan center. Foster University is a private institution, known for having
very strong teacher preparation program, but somewhat smaller than the norm for
teacher preparation programs in the United States. Foster University is also known for
1 Data for this study were collected as part of two different studies. Some data were collected through the CATE project or using the CATE project protocols, and such data are identified throughout as CATE data. Other data were collected through my Fulbright research project using different protocols, and such data are idenitfied as Fulbright data.
19
having a strong social justice view of teacher education. Although many teacher
preparation programs in the United States train teachers as part of a Bachelor’s degree,
the program at Foster University is at the Masters level.
Yamato University is considered one of the strongest teacher preparation
programs in all of Japan. A National University, Yamato University was officially
established in the last century as four separate teacher preparation institutions merged.
Although students can pursue a Masters degree in education, the vast majority of
students are at the undergraduate level and are pursuing a Bachelor’s degree. Students
take education courses in their 2nd
, 3rd
, and 4th
years.
DATA COLLECTION
The study investigates the mathematics teacher training programs at two major
universities in Finland, one in the United States, and one in Japan. One of the
programs in Finland and the program in the United States were chosen by a group of
researchers on the CATE (Coherence and Assignment Study in Teacher Education)
project, centered at the University of Oslo, based on the reputations the universities
have for effectiveness, their selectivity, and the preparation of teachers for grades 8-
13. The University in Japan was selected based on the same criteria. Data collection
at the Japan site was conducted by me and precisely followed the same methods as
data collection at the other sites. In addition, data at the US site was collected by me
again, though data from the Finland site was collected by a graduate student who had
received the same training and instructions for data collection. Personal data for the
study come partially from memory, but completely supported by daily journal entries
that began a few months before I went to Japan and continue to this day. In addition,
20
data from another university in Finland comes from the spring of 2010 when I was
conducting research as part of a Fulbright program. For ease, I will sometimes refer to
the two Finnish data collections as either the CATE data collection or the Fulbright
data collection.
Data collection at Yamato University, Foster University, and Catherine
University followed the procedures set forth by the CATE project. Trained
researchers attended three consecutive mathematics methods courses at the university
and took field notes. A period of three weeks was selected in that it was long enough
to see patterns emerge and to observe more than one course in isolation, yet short
enough that data collection was feasible. The researchers collected all handouts and
other daily artifacts from the classes, and field notes were labeled with time stamps
and direct quotes whenever possible. These field notes were then edited and
elaborated into a summary document. At the end of the third observation, students
were given a three page survey and asked to complete it. These surveys were
completed anonymously and contained similar questions to the rubric, such as to rate
how often they had the opportunity to watch or analyze videos of teacher training or to
connect ideas in one class to ideas in another. Unfortunately the CATE project later
rescinded permission to use the survey data, so it is not part of this project.
After the data collection was complete, the teacher or teachers of the course
were interviewed using a structured interview protocol (see Appendix A). Samples
questions included “Can you describe any opportunities the candidates have to analyze
and reflect on their own field work?” and “Can you tell me about one or two
assignments or activities in the class that require the candidates to make connections
21
between theory and classroom practice? “. In addition, two students from each class
were also randomly selected and interviewed using a similar structured interview
protocol (see Appendix B). Interviews with multiple teachers were conducted with
both teachers present, and interviews with student pairs were always conducted in the
form of a group interview rather than as separate individual interviews. For Foster
University, there was also a structured interview with the program director with
similar questions (see Appendix C), though there was no corresponding interview for
the Finnish program or the Japanese program.
All data in Finland were collected in the native language of the country by a
native speaker of Finnish. The data were translated into English by a native speaker in
Finland, who was not part of the CATE project. As data collection in Japan was not
officially part of the CATE project, but rather an extension of the project by me, the
data collection instruments were translated into Japanese by a professional translator
who was not part of the CATE project, and the interviews and observations were
completed by me (an imperfect Japanese speaker). As such, the field notes were
written in English, and the observations were audio-taped and translated into English
by the same translator. Interviews were conducted completely in Japanese, audio-
taped, and transcribed directly into English.
Data collection from Lapinkaari University took place during my Fulbright
year and thus follows a different set of rules. Interview subjects for this study were
selected from the 2009-2010 cohort of teacher trainees at the University. I first visited
two mathematics pedagogy classes, taught by two different professors, to solicit the
help of volunteers. Following these visits, I used the student mathematics/science
22
education email listserv to contact all of the students in the program2. A combined
total of 16 prospective teachers were recruited through this process, and 13 were
subsequently interviewed over a six-week period. One volunteer withdrew from the
study due to time constraints, and two volunteers never replied to the follow-up email.
One student – the only one over the age of 30 – was a high school student when
Finnish schools were still ability-grouped. In addition, both of the aforementioned
professors were also interviewed in the same manner.
The interview protocols (see Appendix D and Appendix E) for this study were
designed as a semi-structured interview, with opportunities to ask participants to
expand on or clarify their answers. Since I am unable to speak Finnish, interviews
were conducted exclusively in English. Though levels of English ability varied, all
interviewees were sufficiently fluent in English to make specific questions with fairly
detailed answers possible, and non-leading follow-up questions for clarification were
often utilized in order to obtain more salient responses. Interviewees did occasionally
have difficulty finding English words or expressing their thoughts as clearly as they
would have liked. Field notes were taken during the interview for resolution of any
linguistic issues through the help of a member of the faculty of Education. To make
participants more comfortable, interviews took place primarily at Lapinkaari
University and lasted approximately 30 minutes. The interview protocol included
several questions allowing the prospective teachers to hypothesize directly about same
performance classrooms as well as to contrast learning effectiveness with mixed
performance classrooms. Other questions asked about how they decided to become
2 Since Finnish prospective teachers choose to become certified in many subjects, there is no official
count of how many mathematics education students there are. The number is less than 50.
23
teachers and what they were learning in their classes. The professors had similar
questions but were also asked about their feelings regarding the PISA studies and what
they would want me to tell my American colleagues about teacher training in Finland.
DATA ANALYSIS
The analysis of interview data was qualitative, following grounded theory
techniques (Glaser and Strauss, 1967; Charmaz, 1995). The Japanese interview
transcripts, observation notes, and class transcripts were first holistically open-coded
in idea-unit segments, looking for emerging themes. For example, the professor
would often talk about his own experiences in the classroom, which was coded as
“authenticity.” At other times he would talk about how students in different grade
levels might approach different mathematical concepts and how tasks and teaching
strategies would need to be adapted, which was coded as “practice.” A list of the
codes developed in this manner and used in the analysis are listed in Appendix F.
The class observation field notes, professor interview, student interview, and
program director interview for the Foster University data were also coded in the same
way, though the codes used for the Japan open coding were deliberately applied as
well to find areas of overlap. Many of the codes, such as the ones previously
mentioned, appeared in this data as well, though some new codes also appeared. In
this manner, it was possible to determine which codes appeared for multiple contexts,
as the same process was applied to the observation notes, professor interview, and
student interview for Catherine University in Finland. This process also allowed for
the appearance of codes unique to each context, which allowed for differentiation as to
what aspects of the programs might also be unique. The Japan data and US data were
24
once again coded to make sure that codes that appeared in the Finnish coding did not
apply to the previously coded materials.
In a similar manner, the interview transcripts for the Finnish Fulbright data
were first holistically open-coded in idea-unit segments, looking for emerging themes.
Open-coding identified many statements focused on the systems (both mixed
performance and same performance) as a whole, and initial codes reflected both
positive and negative attitudes toward these systems. For example, Student 4 reported
that “in a way it’s interesting to have mixed performance classes, because it’s, it gives
a teaching fresh, because you always have to consider everyone who is in the class,”
which was recorded as a positive statement for mixed performance classes. A list of
the codes developed in this manner and used in the analysis are listed in Appendix G.
CHALLENGES AND LIMITATIONS
German military strategist Helmuth Von Moltke famously said, “no battle plan
survives contact with the enemy.” Similarly, no research plan ever ends up
proceeding exactly as planned (and I imagine that many researchers would find this
analogy a little too fitting). The data collection in the United States went quite
smoothly, in large part because of my familiarity with the program and the people and
because English is my native language. The data collection in Finland, which was
conducted by someone else, appears to have gone quite poorly, as evidenced by the
quality of the data and by the lamentations of the CATE project principal
investigators. The data collection process in Japan also had some hiccups, as I had
originally planned to collect data from two universities, though in the end I was only
able to collect data from one. I also was unable to interview one professor I wanted to
25
interview, and the program director was not available for the entirety of my time in
Japan. The result was still a great deal of quality data and information, though gaps
here and there made it less parallel than I would have liked.
The addition of the Finnish data from the Fulbright study went a long way
toward filling gaps, though this data collection was not specifically designed to go
along with the CATE interview protocols. Still, many of the questions asked
supplemented the CATE instruments, and from these data I was able to find patterns
that matched the Japan and US data. That there were also student and professor
interviews, and so many more of them, was also incredibly useful (and serendipitous).
This study examines teacher preparation programs recognized as being
excellent, and this study does not speak to (nor does it attempt to speak to) teacher
training programs in general for these three countries. It does, however, look at what
trends could be identified for these high-performing universities in an attempt to
understand features they may have in common, or what features may be specific to a
particular program, country, or culture. The addition of personal experiences and
reflections in many ways helps to support the data with an “insider” view of the
systems, but such a view as well is subject to observation bias or generalization of my
experiences to the experiences across the country. The unofficial motto of the JET
Programme in Japan was ESID: Every Situation Is Different. Though all 60 or 70 of
us in my prefecture were there under the same program, our schools and duties and
opportunities were wildly different. My Japan was different from my colleague’s
Japan, and our reflections would be quite different. Still, these experiences did
26
happen, and combined with data collection, the combination of personal observation
and formal data collection is a way to tell at least a part of the story.
27
CHAPTER 4: Japan (personal)
When this dissertation is finally published, it will be almost exactly 15 years
since I departed for Japan and began a life dominated by the concerns of education.
Of course, at the time I did not know this fact. To be completely honest, I had no
interest in education either; going to Japan was an excuse to live in a new place and
travel before returning home a year later to enroll in law school. Much of what I know
now I learned in retrospect, not at the time, and I have years of daily journal entries to
prove my naivety. I would love to be able to write that my experiences in some of the
finest educational settings in the world were the product of careful planning and
deliberation. That would be a lie. What is true is that my experiences, teachers, and
friends in Japan forever changed how I look at education, and I came to learn how to
experience, identify, and take away as much as possible the best features from the
teachers, systems, or countries I encountered.
On July 15, 2000, I left New York City on a non-stop flight to Tokyo to begin
a year of teaching English as second language as part of the Japan Exchange and
Teaching (JET) Programme, the largest teacher exchange program in the world
(McConnell, 2000). Prior to departure I received not one but two journals as presents
from friends, and I decided to try my hand at writing my experiences in this new
country every day (a habit I still have, well over 5000 journal entries later). The plane
landed on July 16, and after 2 days of orientation, I arrived in Yamaguchi-prefecture
on July 19. This was to be my home for the next 3 years. On July 20 I wrote about
how lonely I was.
28
I was horribly unprepared for pretty much everything. From a cultural
standpoint, I had limited Japanese proficiency, having studied with little earnest for
two years because I was taking the classes for sheer fun, not for practical purposes. I
also had no teaching background. True, I had tutored formally and informally for
many years, but I had no idea how to prepare a class or manage students. Worse, I had
no idea there was anything I needed to know. The town to which I was headed was so
small that it did not appear on any map (2000 was long before Google maps or the
equivalent made finding even remote places simple), and for several days I literally
had no idea where exactly I was until someone showed me a map of the prefecture.
With one exception, no one I met spoke English at all, and when I was dropped off at
my new apartment on July 19, they told me that school started on September 1. Until
then, I was free. Then they left.
Being lonely and free, I spent much of my first month trying to find ways to
amuse myself and was ultimately fairly successful. The other JETs were interesting,
and I made some friends who I still see regularly. I also wanted to know more about
my upcoming job and school. I learned that the PTA was holding a “soft volleyball”
tournament and that my school, Nippon Junior High School3, had a team. On August
10 I showed up unexpectedly at practice, and the volleyball coach, Tanaka-sensei,
took me under his wing to show me how to play. On September 3 we played against
the four elementary school PTA teams, and afterwards my principal, Fujii-sensei, had
me give an introductory speech to the whole crowd. Both of these small events
tremendously changed my life as they directly led to the experiences that began to
3 All places and names of people throughout this dissertation are pseudonyms.
29
shape my understanding of Japanese education. These experiences were linked with
Tanaka-sensei, who also happened to be the head teacher and a mathematics teacher,
and our friendship led to me being involved with his mathematics classes. One of the
elementary school principals asked Fujii-sensei if he would be OK sharing me from
time to time to visit her school, which led to numerous visits to all four elementary
schools over the following years.
These small things were important because, as I learned over the course of
three years, my job was not taxing. While the JET Program does an excellent job of
bringing native English speakers to Japan to assist in the learning of English, there are
no set job requirements, and few of the Japanese teachers of English had any idea
what to do with us. In retrospect, the situation was much like today where the
government buys laptops for a whole school and declares that they now have
technology, which will make the students smarter. My school now had a native
English speaker, so everyone was going to be speaking fluent English.
Much of my first year was spent acting as a human tape recorder; I spoke
words, the students repeated them. As I had little Japanese ability and no training as a
teacher, no one wanted me teaching their classes. This was fairly typical, and most
years about 50% of the first-year JETs in our prefecture left the Programme, many of
them because of boredom and a feeling of being ineffective. But the freedom to watch
classes without responsibility, to observe the students and the teachers and how they
engaged with students, was tremendously instructive. As my language abilities grew I
was able to interact more, and as such I picked up more responsibility. I was able to
join mathematics classes where language was something often unnecessary. I was
30
able to teach at the elementary schools where teaching language was secondary to
teaching culture and learning culture in return.
It took almost two years before I decided that I wanted to become a teacher
when I returned to the United States. By then I was fully integrated into my school,
and indirectly I was being taught how to teach by my colleagues and my environment.
If anyone had asked me what I had learned, I probably would not have been able to
explain, for I myself did not know. These lessons and how they shaped me will
become more obvious in later chapters as I contrast them with experiences in the
United States and Finland, just as they became more obvious and clear to me.
THE AMERICAN QUESTION
Several years into my graduate program at Stanford, I was helping facilitate a
conversation between current students in Mathematics Education and some visiting
Japanese teachers. The Americans were in awe of what the Japanese visitors were
saying about their school administrators, particularly how supportive and
understanding they were. The Japanese teachers explained that, in Japan, to become a
principal you have to have been a teacher first. The Americans replied that we have
much the same rule here, yet we do not produce administrators who matched the norm
that the Japanese teachers described. The conversation was about to move on, but I
was able to realize there was a huge disconnect that neither side understood.
I asked the Japanese teachers how long a principal served as a teacher before
becoming a principal. They replied with the answer I already knew: about 20 or 25
years. I then asked them how long someone served as principal, and they replied that,
because the person needed to be a teacher for so long, he or she served as principal
31
typically for less than a decade. I then asked the Americans to explain how people
become principals in this country, which typically requires only 3 years of classroom
teaching, and at the end of the explanation one Japanese teacher asked the simple
question: “That’s how you do it in America?”
Anyone who travels knows that one of the greatest values of such an
experience is the challenging of assumptions and beliefs, particularly the ones that you
never had cause to question previously. The idea that things can be done a different
way, or that different cultural contexts can often define the way practices take place,
becomes apparent when away from your own cultural norms. My own upbringing
took place in the United States, so I am inherently biased to believe that what I
experienced and what is part of my cultural is “normal,” while things that happen in
other places are the exceptions and therefore unusual. Fortunately I travel frequently,
and I am just as frequently reminded that this ethnocentric mode is a particularly
limited way of thinking. I often find myself asking questions that puzzle the listener,
and I realize that, in their context, I’m not asking a “normal” question. I’m asking an
“American Question.”
Over the years I have asked many American Questions, and the first one that I
clearly recall was a direct result of my visits to elementary schools. It was a Sunday
morning, and I heard someone knocking on my door, which was an unusual thing. I
opened the door to find three of the elementary school girls standing outside, and they
asked if I could come out and play. Caught completely off guard, I said yes, and we
spent about an hour playing tag and other games at the nearby park.
32
The next morning I went straight to the principal’s office to report the event
and make sure that everything was OK. He was confused and asked if I was upset that
they interfered with my vacation time. I said, no, that I actually had a lot of fun, but I
was worried about being a male teacher, living alone, having young girls come to my
apartment. The principal was even more confused. “You’re a teacher; of course it’s
OK” was his reply. Japanese culture holds teachers to a high standard, and teachers
are regarded with trust and respect. In the principal’s mind, it was natural that
students would want to spend time with a teacher and that they would feel comfortable
doing so. In the United States, any male who wants to be around elementary school
aged girls is a suspect and not to be trusted. Even now I marvel that I met more male
elementary school teachers in three years in rural Japan than I have in my entire life in
the United States. Different cultures produce different norms.
This American Question also highlights two of the areas where I found
profound differences in cultures that shape how we teach: relationships and
expectations. I will illustrate these differences with stories of how I perceived the
differences and how they were made real to me.
A. Relationships
As the above story about the interactions between teachers and students
demonstrates, there are different cultural understandings of the relationship between
teachers and students. There are also differences in how students relate to and have
relationships with other students, as well as and how teachers relate to and have
interactions with other teachers. This topic of relationships in Japan is dramatically
different from what I have experienced in the United States, particularly at the
33
elementary and secondary educational levels, and to some extent at the tertiary level as
well. Perhaps I now have a “reverse bias” and am more inclined to favor something
dramatically opposed to the norm in the United States. I found understanding of these
relationships to be instructive and valuable for me later in my teaching career.
1. Student and Teacher
Often when I see video clips of classrooms in high performing countries, I see
a somewhat skewed version of what I experienced in my own classroom. In these
videos I see well-behaved, uniformed students sitting in neat rows and raising their
hands when the teacher asks a question. While this situation of course happens, it is
not necessarily indicative of what education actually looks like (see Differences by
Grade Level in the following section) over the course of an entire class period, or over
the course of a longer educational unit. It also gives a potentially false picture of the
separation between the teacher as expert and the students as recipients of knowledge
and expertise, with the teacher at the front of the room, and the students isolated and
not working together in groups.
Under the surface of the scene, the relationship between students and teachers
was characterized by respect. I mean this as a generalization rather than a description
of every student-teacher relationship; generally speaking, respect was obvious.
Respect is different from deference. We had one student teacher (who later joined the
school full time) who bore a small resemblance to a TV character on a new and
popular TV show who was, by local accounts, not the most attractive woman. The
character’s name became a popular, slang synonym for “ugly,” and the students used
that name for her. Students would often use nicknames for teachers to their faces, pull
34
on their clothes, and playfully smack them when they made a joke – clearly things that
Americans would not typically associate with respect. But respect it was. These same
students would stay after school to talk about their lives and their work and ask the
teachers for help and advice. Teachers were not put on a pedestal for their skill and
knowledge; they were viewed as likeable people and strong role models for what
students wanted to be. Teachers were not treated with awe, but rather with
friendliness. Students were comfortable with their teachers and eager to learn from
them.
In a similar way, it was clear that teachers had respect for the students.
Teachers were of course harsh when harshness was appropriate, but in general,
teachers saw students as people who were learning and growing, and for whom
making mistakes was normal and common. I rarely witnessed teachers condescending
to students. More often teachers applauded students for their good intentions and
good efforts and helped guide them on paths that accentuated the good and corrected
the less good. The Japanese word ganbatte, which we translate into English as good
luck (as in “good luck on your test”), more accurately translates to “do your best” or
“persevere,” which is more reflective of the meaning behind the phrase and indicative
of effort over result. When students acted out or struggled, teachers met to discuss
why the problem might be happening and what they might do to help the student,
which is in stark contrast to the American style where we simply remove the student
from the classroom. Students were given time and space to reflect and to grow, and
teachers knew that no model or rule worked for all students
2. Student and Student
35
Much has been made of the bullying problems in Japan, and frankly, there is
truth to these stereotypes. Bullying is indeed a big problem (and a word that I learned
early in my time in schools) and in some ways a natural result of a culture that tries to
include everyone into a harmonious whole. When a student does not fit into the
pattern, and it is obvious to everyone around, the student can be ostracized and treated
poorly. I witnessed that in Japan on several occasions.
What is often ignored is the positive side of this situation where students are
trained to take care of each other, not by their teachers, but by their fellow students.
Being a senior student implies responsibility to train and help the junior students, and
junior students accept their “inferior” status as they learn the knowledge and skills
necessary for advancement. This system is fairly well known (called the sempai-
kohai, or senior-junior, relationship) and probably most visible in sports settings. It
carries over into the academic and personal settings as well as the older students
mentor the younger ones, and the younger ones know that they will subsequently help
younger students when their turn comes. I would often hear a student refer to another
student simply as “sempai” or call them that directly instead of using their name.
Interestingly I remember this most because of an opposite example, yet this
example proved the rule to me. I was assisting one of the Japanese English teachers
one day, and she had created an activity where students would draw sentences from a
hat to read out loud. The sentences came directly from the reading they were doing,
so none of them were a surprise. One of the students, who I will call Kenji, was
extremely weak in English (as well as other subjects, including Japanese) and
happened to draw the hardest sentence of the lot. He read poorly and with many
36
mistakes, and some of the students in the class began to laugh at him. I got angry and
asked Kenji to the front of the room (much to the horror of the teacher, who had no
idea what I was doing). Opening the textbook, I asked him to read page 1. He looked
at me and said that it was in Japanese. I agreed, and he read the entire first paragraph
with relative ease. I then read the second paragraph, and I had to ask him numerous
times how to read the characters. The class was silent, and I told them how hard it
was for me to speak their language, and how no one would ever make fun of me for
making mistakes. I didn’t learn until months later that this exercise prompted several
of the students to begin working with Kenji after class on English, as well as other
subjects, to help him out. They had failed to help him, and as his classmates, they had
a responsibility to do so.
3. Teacher and Teacher
While many of the teacher obligations in Japan mystified and frustrated me, I
cannot deny that teachers treated each other with respect as well. It was fairly well
known who did not like whom, but one never saw a teacher treat another poorly.
Indeed, though I heard teachers talk about each other, I never once heard a teacher
complain about another’s teaching. In our small school the teachers were organized
into grade level teams, and when the new teacher had to stay late to work on lesson
plans or projects, all the teachers stayed late to continue to work and help when
needed. A mentor teacher system simply did not exist – it was everyone’s
responsibility to help.
37
B. Expectations
The idea of expectations is the second part of where I found differences
between what I experienced in Japan as compared to what I experienced growing up in
the United States. Teachers performed different functions and duties than a typical
American teacher would ever be required to do, but in Japan it was normal and
expected. There were also ways that teachers deliberately and systematically
interacted differently with different grade levels of students that make perfect sense in
the Japanese educational setting but would make no sense or simply not be applicable
in the United States.
1. A Focus on Students
Perhaps one of the most important lessons I learned from teachers in Japan is a
strong concern for the well-being of their students. Japanese teachers were not
responsible solely for, say, the mathematical development of their students, but for the
student as a whole, called zenjin kyoiku or “whole person education” in Japanese
(Shimahara & Sakai, 1995). I was astounded that teachers knew where their students
lived, what their parents did, what school clubs they belonged to, and all sorts of other
personal details. Teachers were responsible for visiting the homes and families of
every student (a policy that I sadly never attempted in the United States, nor was
allowed to join in Japan), which made conversations with parents easier and more
meaningful. Students saw how involved teachers were in their lives and looked to
them for help and support due to that level of involvement.
I also remember one evening when I stopped by the school quite late to help
out with a volleyball game, and all of the 9th
grade teachers were still there working.
38
One of their students was having anxiety issues and had stopped coming to school.
Rather than this being an issue for the parents and the family, it was a school issue,
and the teachers were working and talking late into the night to try to figure out a
solution for the student. Not only did they stay late that evening, but for many
evenings before and after the day that I saw them there. Even though the issue was
not purely academic, as we might consider it here in the United States, for these
teachers it was a student issue, and therefore it was their responsibility to help resolve
it. The issue actually persisted for well over a year, with teachers visiting the student’s
house numerous times, and materials being sent home for the student to work on (or
not work on). The student did eventually graduate, though I unfortunately have no
knowledge of what happened to him in high school. I would imagine that the teachers
there, also concerned about the student’s well-being, began to meet much the same
way the junior high school teachers did.
2. The Role of the Teacher
The above example also shows that the role of the teacher in Japan is quite
different from the role of the typical American teacher. To be fair, I have learned that
some of the things that were normal in my little rural school were handled differently
in bigger, more urban schools. For example, teachers and students were responsible
for the cleaning of the school since there was no janitorial staff. During the summer I
would often see the principal outside pruning the bushes, as it was something he loved
doing, and once a year there would be no classes as we waxed and sealed all of the
school’s floors. Much is made of this in newspapers here and in other countries to
show the level of dedication and connection that the students have to the school
39
(which is true), but I have never seen one that describes how vital the teachers are as
well in this process. The teacher fills the roles of janitor, guidance counselor, coach,
surrogate parent as well as their nominal role as instructor. Teachers do have the
elevated respect and authority that gets described in the press, but they also have an
extremely heightened list of responsibilities as well.
3. Differences by Grade Level
The final observation I would like to make is perhaps the one that fills my
mind the most: differences by grade level. This is the area where I believe I learned
the most, and when people ask about what impressed me the most about the Japanese
educational system, my answers are linked to what happens in certain grades.
Similarly, when people ask me what I liked the least and which qualities are least
worthy of emulation, I once again cite what happens in certain grades.
My day-to-day role in Japan was as a junior high school teacher, which is the
equivalent of grades 7, 8, and 9 in the United States. 7th
grade students are called
ichinensei (first year students), 8th
graders are called ninensei (second year students),
and 9th
graders are called sannensei (third year students). The same titles are also
given to students in grades 10, 11, and 12 in high school, so it is important to delineate
which school level in addition to the “year” of the students. The bulk of my time was
spent with the junior high school ichinensei and ninensei, and very little was spent
with the sannensei. The reasoning was astoundingly simple.
Ichinensei are technically learning English for the first time. Although by the
time they reach 7th
grade they have had at least some exposure to English (television,
cram schools, informal elementary school programs, and such), junior high school is
40
the first time that they are formally learning the English language. Native speakers are
quite helpful at this point in time to help with pronunciation, rhythm, and other verbal
skills that might otherwise develop oddly or incorrectly. Lessons also tend to be more
fun at the lower levels, and JET Programme teachers were frequently asked to plan
and execute different linguistic games for the students to play. Sannensei, on the other
hand, were either mostly or completely off-limits. Ninth grade students are almost
constantly preparing for their high school entrance exams, which are intensely high-
stakes tests that determine which schools will accept them, and in turn, affect their
chances of going to a good college. Every high school produces its own test, and
while English is always one of the required subjects, no part of the exam requires
speaking (and typically only about 5% would involve listening). Native speakers, by
sheer definition of being speakers, were seen to detract from the “learning” process
and were a waste of valuable test preparation time.
This sannensei rigidity contrasted greatly with the freedom I experienced in all
of the elementary schools I visited (and indeed, all of the elementary schools I visited
in years after the JET Programme). While sannensei in many high school classrooms
do look like the rigid videos I mentioned earlier, elementary schools often feel like
uncontrolled chaos. Students are running and screaming and shouting, and little is
done by teachers to impose order and control. I remember during my first elementary
school visit that the teachers suggested that I go outside and play with the students,
since the students were quite interested in me (for many I was the first foreigner they
had ever seen, and even for some of the teachers I was the first foreigner they had ever
spoken with). About 10 minutes into playing with the students I noticed I was the
41
only adult, and I immediately felt a surge of panic. Was I responsible for them? I
could barely even understand them! Later I learned that teachers as a general rule do
not go out to supervise students the way American teachers do. While the teachers
were nearby and available in case of an emergency, the students were responsible for
themselves. If there were an argument or disagreement, the students had to figure
things out for themselves without the adults getting involved. This system was
deliberately designed so that students learned to work together and figure things out
for themselves, skills that later translated well into classroom learning.
I did see the lessons learned in play enacted in the classroom as well, both in
the elementary school classes and the middle school classes. Students when young
were encouraged to be curious, to explore, and to ask questions. Too much order was
seen to be stifling to creativity, and behavior that would have resulted in punishment
in most American classrooms was completely tolerated in the Japanese ones.
Mathematics classrooms were open, free places where students played games and
talked about problems. For example, I recall one class where students were making up
their own problems in context, which led to many strange, interesting, and often
comical story problems. There was of course some practice, but that was a minor
ending activity rather than the focus activity. I saw much the same thing in the middle
school for the ichinensei and ninensei (though with far less running around and
screaming), but not for the sannensei. At some point learning stopped being fun and
became about passing an exam.
CONCLUSION
42
These experiences form the basis of many of my beliefs about education. I
saw the benefit of collaborative student work, the value of knowing your students and
forming a meaningful relationship with them, and the need for teachers to dedicate
themselves to their profession. Of course, I did not know at the time how powerful
these experiences would be, nor did I know how these experiences would later be
analyzed and used when I became a licensed teacher in the United States. I did not
have a classroom of my own to try new things or to experiment with different ways of
learning, and I still lacked any kind of formal teacher training. In the next chapter I
begin to integrate data I collected in Japan as part of the CATE study where I was an
observer and interviewer, but not an active participant chronicling my own
experiences. Chapter 5 explores the data I collected from a highly regarded teacher
training program in Japan, and a pattern to the organization of teacher training across
three cultures will begin to emerge. The sixth chapter looks at my teaching
experiences in the United States, and connections will be made to how these
observations and my take-aways from them affected my teaching in the United States,
as well as the new lessons I learned in this setting. Chapter 8 will continue with my
experiences teaching in Finland. Similarly, Chapter 7 will examine the teaching
training data collected from the United States, and Chapter 9 will examine the teacher
training data collected in Finland as well as data collected from students who were at
the time enrolled in secondary mathematics teacher training.
CHAPTER 5: Japan (teacher training)
43
While Japan was chronologically my first experience teaching, it was
chronologically last in terms of data collection at a university renowned for its teacher
training program. Yamato University, located near a major Japanese metropolitan
center, is nationally known for its high quality teacher education program at all levels,
including the preparation of high school mathematics teachers. Like most programs in
Japan, teacher training at Yamato is a four-year program, with the first year being
general studies, and the later three years focusing on education in general as well as
subject-specific education. While students have access to a nearby school that is used
for observing lessons with current teachers and real students, actual practice in
classrooms for prospective teachers does not begin until later in the third year at the
earliest. Student teaching follows a pattern similar to many American programs in
that the student teachers are placed directly into classrooms for an approximately four
week period full time. In terms of teacher education coursework, there are a wide
variety of classes that students are required to take, ranging from educational history
and educational law to subject-specific pedagogy classes.
Data were collected in May of 2012 over a period of three consecutive weeks.
I observed two different classes during this period: a mathematical pedagogy class
(Mathematics Methods) for prospective junior high school and high school
mathematics teachers, taught by professor Yoshida (Yoshida-sensei), and a
Mathematics Curriculum Theory class, taught by Professor Takeyama (Takeyama-
sensei), an internationally known teacher and researcher at Yamato University. In
addition to attending these classes, the lessons themselves were audio-recorded, and
field notes were taken, following the protocols created through the CATE project. To
44
supplement these notes and add detail where words alone would not suffice, I took
photographs of the chalkboard and described in my field notes when the photographs
were taken. The audio recordings were transcribed directly into English without
translation by a professional translator. Two willing students, both male, were
randomly selected from Yoshida-sensei’s class to be interviewed at the end of the
three weeks, and the professor was also interviewed using the CATE project’s
interview protocol (see Appendix A). In the student interview, some interesting
information regarding the students’ perspectives of teaching in relation to Japanese
cram schools was shared, and the professor was then asked a follow-up question based
on this information that was not part of the original protocol4. Interviews and field
notes were open coded for emerging themes, and codes such as “authenticity,”
“textbooks,” and “purpose of mathematics teaching” were several that occurred
consistently across data sources (see Appendix F for a list of codes and sample
statements). All these codes were emergent rather than imposed, though the nature of
the CATE study made codes such as “theory/practice,” for example, automatically
present.
Yoshida-sensei’s class was characterized as the equivalent of the mathematics
Curriculum and Instruction class at Foster University (the US University) or the
Didactics class at Catherine University (the Finland University), but the nature of the
Yamato University Program made this match somewhat imperfect. This Mathematics
Methods class more heavily focused on the mathematics and less on what we might
traditionally think of as pedagogy, and Takeyama-sensei’s Mathematics Curriculum
4 The students were also given the survey instrument created by the CATE project, but permission to
use this instrument (and thus the data) was subsequently revoked by the project.
45
Theory class, on the other hand, was more focused on teaching and learning, though
less focused on the mathematics. The CATE study assumed that the mathematics
pedagogy class would be a single class, whereas at Yamato University there were two
important and complementary classes that filled this role. As such, it was highly
valuable to be able to observe both of these classes.
TAKEYAMA-SENSEI’S CLASS
Takeyama-sensei’s class begins promptly at 8:50 every morning, with the majority of
the students in the room and seated by 8:45. The classroom is set up in rows of tables;
two two-person tables are placed next to each other on either side of the room and
separated by a walkway down the center of the classroom. There are 7 such rows for
only about 20 students (16 male, 4 female), so many seats are empty. Students tend to
sit in clusters near the back of the room rather than close to the front. There are also 4
televisions in the room, and Takeyama-sensei is able to show videos to the whole class
using these televisions.
46
Photo: Takeyama-sensei’s classroom
Takeyama-sensei has been a faculty member at Yamato University for over a
decade and has written scholarly articles in Japanese and presented at international
conferences, including the National Council of Teachers of Mathematics (NCTM) in
English. Takeyama-sensei himself is an animated and friendly man, often smiling and
telling jokes to make the students laugh. Having students work together is a directly
stated goal of the class, as he says to his students near the end of my first classroom
observation, which takes place near the end of the academic semester. During the
class period students sometimes answer questions from the professor, talk with
neighbors, or shift seats around to form groups of about 4 to 5 to discuss the topics of
the day. At the beginning of the third observed class, Takeyama-sensei forms the
students into groups before the lesson begins. During discussion Takeyama-sensei
47
visits these groups to listen in and occasionally comment, but mostly he leaves them
alone to have their own, natural conversations. I am able to sit with the students on
multiple occasions during my observations, and the dialogue is always relaxed and
focused. These observations come during the first and third visits, as the second of my
observations takes place at the nearby practice school, a junior high school. Students
are prepared during my first observed class to observe a lesson for 7th
grade students
about multiplying positive and negative numbers. According to Takeyama-sensei, this
information is supposed to be new for the students, but he predicts that many of the
students will have already learned this information outside the regular classroom. The
assumption turns out to be correct: many of the students in the class are able to
complete the activity before the lesson had really begun. Even more interestingly,
when the classroom teachers asks the students the “why” of their answers, the
responses are similar to those presented by the prospective mathematics teachers in
Yoshida-sensei’s class; namely, the students “know” the right answer but struggle to
explain why the answers are correct. More specific details are provided in the
following section describing Yoshida-sensei’s classes.
Takeyama-sensei’s classes are clearly structured to build on each other and to
build on the work being done in Yoshida-sensei’s class. The first lesson I observed
centered around a 50 minute teaching video, and the focus of the class is
deconstructing teacher moves with the goal of paying attention to the teacher moves
that they will observe the following week at the practice school. Instructions for the
visit and the video take up the majority of the class time, so there is little else to note
about the class structure or topics at this point in time.
48
During the second observed class, with all of the students dressed formally for
the official visit, Takeyama-sensei shows the students the teacher’s lesson plan for the
day and gives them a handout to use for the teaching they are to observe. Takeyama-
sensei asks them to pay attention to the key question for the lesson:
九九表のきまりをもとにして負の数も含めた乗法の積を求めましょう (Let’s
use the basics of the 9x9 multiplication table to determine the products of negative
numbers). He asks them to pay attention to how the teacher is using the familiar
information to scaffold the introduction of negative numbers in the “easy” case of
positive times negative and negative times positive, but also in the “difficult” case of
negative times negative. Other instructions include asking the students to take notes
on how the teacher is instructing, and noting what students appear to be understanding
versus not understanding. Takeyama-sensei also instructs the students on how to take
field notes with time stamps so they can better discuss the class the following week.
Most of the class time is spent at the school itself, which is a standard classroom,
though filled with about 25 observers, making it quite cramped and crowded.
Ostensibly the students are accustomed to this type of observation, as it is normal for
TGU to come to the school, but I can only guess on the effect that such an observation
has on the students or on the lesson planning of the teacher himself. From my own
personal experience being observed in these formal settings as part of jyugyo kenkyu
(lesson study), I would say this lesson does not, in fact, represent a typical one.
Private conversations with Takeyama-sensei and his graduate students following the
class reveal that they also do not feel that the class is representative, nor do they feel
that it is an overly successful class period. One major note is that the class is
49
extremely teacher-centered, and the students are not given as much time to work in
groups and discover the patterns as they normally would be given. Perhaps part of the
reason is that so many students already know the answers to the worksheet, and the
lesson plan is structured to have that understanding as the learning outcome. Students
who do not previously know the information are in groups with students who do, and
they also quickly “know” the answers without exploring the topic. During the lesson I
observe one group where one student correctly fills out his handout (a grid listing the
numbers from -4 to +4 across the top and -4 to +4 along the side, creating a space to
fill in all 81 multiplications) and then another student simply copies the answers. It is
difficult to draw any generalizable conclusions from a single observation, and for
these reasons I believe that this one observation might not be the best example of an
exemplary mathematics class.
In the third and final class, the prospective teachers spend a large portion of the
class time debriefing with each other and sharing the notes they made the previous
week, pointing out the things that stood out to each individual. Examples include the
fact that many students already knew the answers and that the class was teacher-
focused. Takeyama-sensei takes time near the end of the class period to point out
some salient notes of his own (such as the fact that some students were giving answers
to questions on a worksheet before the lesson had even begun, and noting that the
students were extremely fast with their ku-ku (the multiplication tables up to 9 times 9)
but less adept when multiplying by 11 or 12, which they had not memorized. At the
end of class Takeyama-sensei also briefly talks about the Match Stick problem, which
50
will be detailed in a moment as it is a focal topic of Yoshida-sensei’s classes,
primarily the first class.
YOSHIDA-SENSEI’S CLASS
Yoshida’s sensei’s class takes place on the same day as Takeyama-sensei’s, though at
the end of the day from 4:10 to 5:30. Many of the same students are in this class as
well, though it is not a precise overlap. Yoshida-sensei was a junior high school
mathematics teacher for 18 years and has been a member of the Yamato University
faculty for an additional 7 years. While Yoshida-sensei still engages students with
questions and probed thinking, his class is clearly more teacher-centered, and his fast-
paced lessons put an incredible amount of information into the class period. The
layout of Yoshida-sensei’s class is almost identical to that of Takeyama-sensei, but
with a larger room and more rows (and no televisions). The 23 students enrolled in
the class (5 female, 18 male, most in their third year though at least one of them in the
fourth year) sit in clusters as before, but without very much communication between
or amongst themselves.
Yoshida-sensei’s first class begins with him welcoming the students back from
the Golden Week vacation, noting that it has been two weeks since he last saw them
and refreshing them on their previous topic: junior high school mathematics lessons.
Yoshida’s class focuses on the contents of a typical junior high school lesson on
multiplying negative numbers, and he also notes that the students will be observing an
actual lesson on this content the following week (i.e. the practice lesson observed
during Takeyama-sensei’s class). The first 25 minutes of class are exposition leading
up to questions and discussion. Early in the class, for example, Yoshida-sensei
51
explains the following. Note that junior high school in Japan is a three year system, so
“first-grade students,” for example, would be the equivalent of American 7th
graders,
and “third-grade students” would be the equivalent of American 9th
grade students:
Second-grade students don’t expand their knowledge of actual numbers, but
the target is to deepen their understanding of linking calculation to whatever it
is they are trying to solve – again deepening understanding of equations.
Third-grade students aim to understand averages and numerical concepts. This
is not all that different from that of the 1st-grade. Contents include square
roots, irrational numbers and absolute values. So, particularly in the 2nd grade,
they learn about linking calculation to whatever it is they are trying to solve,
and in the 3rd grade they use that ability. Finally they focus on quadratic
equations.
Yoshida-sensei has tailored this lesson to delve into the content of where a
first-grade junior high school mathematics class would be right now in the curriculum,
and indeed where the observed class will be: positive and negative numbers. He
begins by talking about how the textbooks typically introduce the topic with the
following direct example: Sapporo has a high temperature of 5 degrees and a low of
minus 3 degrees [the fact that it was Centigrade was of course understood], and what
does that minus actually mean? Getting students to think about real-life situations is
the intentional first step, and addition/subtraction and multiplication/division will
follow later in the month. At the end of his introduction, his task for the class is
simple. “Let’s … think about how you – third-year university students – understand
the reason for the multiplication of two minuses making a plus. So your
understanding of it has to be put in a way that a first-grade junior high school student
can understand it.” Then, in what was not typical for the classes I observed, he opens
the floor for volunteers.
52
One student observes that multiplying a positive and a negative is easy: 3 x -2
was simply three -2s, making negative 6. So -3 x -2 was just (-1 x 3) x -2, or -1 x (3 x
-2), though his explanation for the last step makes the assumption that the students
would understand a “direction change” when multiplying by negative 1. Another
student gives an answer that negatives are like changing direction or reversing
direction, so reversing direction twice is the same as not reversing direction at all.
Yoshida-sensei spends about 10 minutes getting more answers, offering little
commentary himself, but quickly reverting to cold-calling students for replies. He
allows a total of 20 minutes of student-supplied answers before saying, about 46
minutes into the 90 minutes of class, “So how do they handle it in the textbooks?” He
then spends the next 20 minutes going through textbook examples, which deal with
fictional students explaining their reasoning.
Yoshida-sensei ends his first class by moving on to what is called the Match
Stick problem, and he has created magnetic “match sticks” from magnet tape and a
cardboard box. The problem goes like this: imagine you are making squares with the
match sticks, so that it takes 4 match sticks to make one square. You then continue to
make contiguous squares so that two squares look like a sideways digital 8 and use 7
match sticks, and three squares use 10 match sticks. How many match sticks will be
needed to make 10 squares? A photo of this lesson and his magnetic match sticks is
below.
53
Photo: the Match Stick problem
Here again Yoshida-sensei opens up the floor for answers, which range from deriving
an algebraic formula (Y = 3n + 1) to simply adding up 4+3+3+3+3+3+3+3+3+3 to get
31 match sticks. Yoshida-sensei ends class by asking the students to think of other
ways students might answer, and he supplies some suggestions. Some students might
multiply 4 x 10 and then subtract off the 9 “overlaps” to get the right answer. Other
students might think about adding up all the vertical sticks and all the horizontal sticks
as the pattern.
Yoshida-sensei’s class the following week is even more teacher-centered, but
at the same time it addresses multiple relevant themes in a detailed and
comprehensible way. He begins class by showing some photographs of the class they
had observed earlier that morning, noting that those students were trying to make
sense out of multiplying negative numbers based on their own experiences, with one
student saying that a negative number was like being in debt. “The way the kids are
thinking is connected to the teacher’s way of teaching. How you include the kids’
54
way of thinking into the lesson – how you expand the conversation to get them to
think about why they think that way – is important in solving the problems.”
After about 20 minutes, Yoshida-sensei abruptly transitions back to the Match
Stick problem, handing out copies of textbooks showing how this lesson is taught
(noting that they use straw instead of match sticks, and joking that giving matches to
junior high school students is not a very good idea). He notes that getting the right
answer is not the point of the lesson: “So when the textbook asks how many matches
does it take to make 5 squares, how you respond to the kids’ way of thinking is
important. For example, if one student gives the answer and everyone else just says,
yes, that’s right, the lesson is over! The more examples are given, the more interest is
created and the problem can be expanded – so it’s important to find ways of expanding
the problem to create a wider response.” Yoshida-sensei then details multiple ways
the problem can be expanded or changed: asking for 100 squares, asking for n squares,
changing the problem to triangles or hexagons instead of squares, or pinning pieces of
paper to the board to make the problem 3 dimensional. He also phrases the question
from the opposite direction: if you had 100 match sticks, how many squares could you
make? Will there be any matches left. “This is what the teacher usually does – thinks
about how to approach teaching the topic; but what about if we ask the students to
think about how to do this? Instead of just asking them to think of a problem, ask
them to alter one of the conditions of the existing problem to make another problem.
Get them to think about how the problem can be changed.” Yoshida-sensei then opens
up the floor for thoughts from the students on how to solve the original 100 square
problem, which generates a little over 10 minutes of discussion. Most students who
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volunteered answers favor an algebraic approach (noting that each additional square
added three additional match sticks), while others are more visual and devise methods
of counting horizontal and vertical sticks.
Yoshida-sensei ends class by delving even more theoretically into how the
problem can be adapted for higher grade levels. Another problem he gives, which is
also fairly standard, involves drawing squares with dots. A 2x2 square has 4 dots in
two rows of two dots. A 3x3 square has 8 dots, looking like 3 rows of three dots,
except that the middle dot is not there (i.e. a hollow square). Similarly a 4x4 square
has 12 dots: 4 in the first row, 2 rows of two dots, forming the sides, and a last row of
4 dots. A photo of this lesson is included below, where you can see from the “n” on
the far right that Yoshida-sensei is setting up the question of how many dots appear in
an n by n square.
Photo: Drawing squares with dots
Class does not end with any discussion of the problem, as Yoshida-sensei says that the
pattern here is quadratic and would be much more involved than the previous problem,
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and he also tells the students to finish some homework for the next week, as the topic
will be changing to geometry.
The third class is similar to the second class, though the topic is completely
different. Yoshida-sensei continues to run a highly teacher-centered class, though the
focus is still mainly on the mathematics and the connection to student thinking and
problem solving. The focus of this class is also on proofs, a topic that is traditionally
challenging for students. He says that “Whatever the subject – be it Japanese or
physics or social studies or whatever – having the ability to consider and judge and
express is very important.” The focus here is on explanations, again rather than
simply getting correct answers. “From elementary school to 1st-grade junior high
school, an intuitive, manipulative understanding is sufficient but from 2nd grade of
junior high school to high school the way of thinking is a more deductive, explanative
way of thinking. That’s the big difference.”
A PLACE FOR THEORY AND A PLACE FOR PRACTICE
In terms of how classes were run, Yoshida-sensei’s class is heavily lecture-
based, which is an intentional choice on his part. The students who were interviewed
responded negatively to questions about presenting in front of the class, doing mock
lessons, or really in any way engaging in the practice of teaching. Yoshida-sensei
does bring into the class many of his own personal experiences, and he often asks
students how a lesson might change based on the targeted students’ level of
understanding. For example, in the first observation, Yoshida-sensei asks students how
they would explain the idea that a negative times a negative is a positive to different
grades of students, exemplifying where certain ideas that the prospective teachers take
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for granted break down. He says, “In this class I don’t get them [the prospective
teachers] to role-play so much. However, like in today’s class, I get them to talk, and
the class as a whole is structured so that it gives an image of a junior high school
lesson.” In answering another interview question, he shows that his methods are quite
deliberate, saying that practice will come for the students in September (when they
typically begin their student teaching period), and that “I try to make it so that the
contents of the class link up well with those of [Takeyama-sensei].”
In the classes, students are encouraged to contribute their thoughts on the
theory behind the concepts Yoshida-sensei was teaching, and he constantly probes to
see what the class believes and how they theoretically would approach topics. There
is not, however, any type of practice teaching. Takeyama-sensei’s class, on the other
hand, is mostly group-focused, and students are presenting projects and discussing
what they saw in the practice lesson (recall that the second observation of Takeyama-
sensei’s class was actually an observation of an actual lesson). Takeyama-sensei’s
classes are focused on the art of teaching rather than the mathematics; though
mathematics is discussed, it is more of the background than the focus as it is in
Yoshida-sensei’s class. On the day when Yoshida-sensei was talking about analytic
geometry and pattern recognition in the afternoon, Takeyama-sensei was showing
classroom videos, having students discuss the teacher moves in groups, and having
students discuss the role of a “why” question in a mathematics class.
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A FOCUS ON AUTHENTICITY
“In Yoshida-sensei’s lesson, for example, [he provides] copies of old
elementary school and junior high school textbooks, and he uses his own experiences
as examples – actual examples of students’ responses and the like. He provides
accounts of actual experiences and different materials. I think that’s the greatest
help.”
One of the interview students said this when asked what he thought Yoshida-
sensei was trying to teach him about teaching and learning, and the interview with
Yoshida-sensei and the transcripts of his classes support this idea. I believe the word
“authentic” fits well here based on these sources, and though the word “authentic”
does not appear in any of the transcripts, the emphasis on actual examples, including
textbooks, is striking. In the first class I observed, Yoshida-sensei begins class by
talking about multiplying two negative numbers and spent over 20 minutes eliciting
student responses on how they might explain the idea to students. Immediately
following that, Yoshida-sensei asks, “So how do they actually handle it in textbooks?”
and spends the next 15 minutes exploring the actual sections in textbooks (note that
there are only 6 companies authorized to publish textbooks in Japan, and he compared
the lessons not only among textbooks, but also between elementary level textbooks
and junior high school textbooks). Yoshida-sensei himself says in the interview that
he often explains to the students “the relationship between the curriculum and the
textbooks.” Here I would note that Japan has a national curriculum, which makes
such discussions far easier than they might be in the United States. More will be said
about this in later sections describing teaching in the United States.
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Yoshida-sensei also demonstrates in his interview that this authenticity is
shared rather than something he does all on his own. When asked if he showed videos
of lessons, he says he has in the past but is probably not going to do so this year.
Takeyama-sensei would be doing more of that, he says, and also taking the
prospective teachers to see actual classes and actual students, so he [Yoshida] would
not. The students mention this as well in the interview, talking about the observation
they did in Takeyama-sensei’s class and how they were able to “verify” some of what
they had learned through the authentic experience. Both the students and Yoshida-
sensei say that this class is not the place for any kind of practice teaching, as that is
something they will do later, but no one seems to be bothered by this fact. Each part
of the learning experience has a place, and the separation is simply normal. In
Yoshida-sensei’s class, the prospective teachers are able to think about the lessons and
how different grades of students might understand concepts in different ways, and
what actual teaching might look like.
THE GOALS OF MATHEMATICS TEACHING
“We want them to become teachers who can teach lessons in which importance
is placed on the process and the way of thinking.” – Yoshida-sensei
“If you ask what are the goals [of this course], it’s probably to nurture better
teachers.” – Interview student
Despite the fact that Yoshida-sensei’s class is mathematically heavy, the basis
of the problems is more about finding different ways to solve (and to explain) the
mathematics than simply a rigorous teaching of mathematics. When looking at the
Match Stick problem, as described earlier, Yoshida-sensei goes through the process of
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having his students work through the exercise. He also explains why he would ask for
the number of match sticks needed to make 10 squares instead of 5 (too easy to count)
and has students offer different explanations of how they are able to see the pattern,
mimicking what would happen in an actual classroom. This is particularly important
to note in a class that is heavily teacher-centered. To make his point about what
students would actually do, and to make this clear to his students, Yoshida-sensei
takes a large chunk of his class for simply listening to different opinions and solution
methods, as he would want them to do in their own classes. Yoshida-sensei finishes
this class with the following words:
So the teacher has to be prepared and imagine the different ways of thinking
that the students can be expected to come up with in order to put this text to
good use. It’s not just teaching the method that’s in the textbook, but listening
to all the students’ ways of thinking and linking it with the theory. It’s
important that students make their own conclusions and that everyone makes
mistakes and that the teachers facilitate those mistakes and ideas and
discussions to develop a network of learning.
There are multiple takeaways just from this brief quote. First, we note that
Yoshida-sensei begins by talking about the teacher being prepared to understand
student thinking, not simply the mathematics involved. Through the entire lesson,
Yoshida-sensei never once talks about “mastery” on the teacher’s part, but rather
about seeing things from the students’ perspectives. I unfortunately did not ask
Yoshida-sensei if he has ever read any of the work by Deborah Ball, but this seems to
be a classic example of Pedagogical Content Knowledge (PCK) (Shulman, 1986; Ball
et al., 2001). Believing that teaching pedagogy and content knowledge were being
treated as mutually exclusive, Shulman believed that teacher education programs
should combine these two areas.
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In Shulman’s view, pedagogical content knowledge is a form of practical
knowledge that is used by teachers to guide their actions in highly
contextualized classroom settings. This form of practical knowledge entails,
among other things: (a) knowledge of how to structure and represent academic
content for direct teaching to students; (b) knowledge of the common
conceptions, misconceptions, and difficulties that students encounter when
learning particular content; and (c) knowledge of the specific teaching
strategies that can be used to address students’ learning needs in particular
classroom circumstances. (Rowan et al., 2001. p. 2).
Also, in Yoshida-sensei’s quote, we once again see a reference to linking theory with
practice, making the theory visible and almost tangible for the prospective teachers.
Here we see the emphasis on student agency – i.e. letting students make sense of
mathematics rather than learning it as a set of rote facts – as well as the well-
documented case of letting mistakes be acceptable tools for learning instead of reasons
for shame and criticism (Hatano & Inagaki, 1998; Stigler & Hiebert, 1999). Finally
we see the theme of ideas and discussions, promoting a student-focused idea of
learning and reinforcing a sense of agency.
Both the students I interviewed and Yoshida-sensei also mention the idea of
“problem-solving” several times. While on occasion they use this phrase to talk about
the actual process of solving a problem, more often it was meant in the deeper sense of
conceptual mathematics instead of procedural mathematics. I began this section with
Yoshida-sensei’s quote on what he sees as the goal of his class – importance on the
process and way of thinking – and he later says he wants his future teachers to “teach
lessons that focus on problem solving.” The students agree with this, using the word
“nurture” to describe the way Yoshida-sensei is preparing them. They talk about how
the course emphasizes the “importance of logical thinking” and the “mathematical
way of thinking.” One of the students directly states that “in order to do [convey the
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fun of mathematical thinking] – I learned this in a lesson but I forgot which one – the
kids have to know what they are doing, not whether they can answer the question or
not.” This demonstrates how Yoshida-sensei focuses his lessons not on the finding of
an answer, but the process of finding an answer.
A FOCUSING CULTURAL ISSUE: TEACHING IN CRAM SCHOOLS
VERSUS CLASSROOMS
Cram schools – called juku in Japanese – are special schools attended by
students on weekends and in the evenings and designed to give students a competitive
edge in the entrance exams for high school and college (and, to a lesser degree, to elite
elementary and junior high schools). Juku have existed for decades and are a standard
part of the Japanese educational landscape (Rohlen, 1980). Bray refers to juku and
similar institutions in other countries as “shadow education” as they operate away
from the light of the mainstream educational systems and exist on the periphery
(2007). Dierkes estimates that there were over 50,000 juku in operation in Japan
(2010), and Bray estimates that approximately 70% of all Japanese students attend
juku at some point during their regular schooling (2007). A recent study estimates that
about 15.9% of Japanese elementary school students attend juku, and 62.5% of third
year junior high school students (the equivalent of American 9th
graders, the year
before entering high school in Japan) are regularly attending juku for tutoring after
school (Bray & Lykins, 2012).
When I asked the two student teachers if they had any other thoughts or
opinions they wanted to share, despite the fact we had been talking exclusively about
Yoshida-sensei’s class and their work at Yamato University, they wanted to talk about
juku. As it turned out, both students have part-time jobs as instructors at a local cram
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school, and both have been doing this job for quite a while. They found a dramatic
disconnect between what is expected of classroom teachers and what is expected of
juku teachers. One of them says almost immediately, “I feel what the school teachers
focus on and what cram school teachers focus on is different. What I’m told in cram
school teacher training and what I’m told here in educational training at the university
as a school teacher is completely different – it’s almost as if they are completely
opposite. It’s confusing.”
The first area of difference the students talk about is the perception of teachers
in these two different areas. Both say that cram school teachers are generally viewed
as better teachers than those in the regular classroom, both by parents and by students.
In fact, they say students find cram school teachers easier to understand and classes
much more useful or practical than what they experience in the daytime. The
interviewees clearly see there are different purposes for these schools, and the way
they talk about this difference shows they place higher value on the skills being taught
to them as “normal” teachers versus the skills being taught at juku for students to be
successful on exams. Still, there is definite confusion on their part to understand why
there is such a gap in teaching theory, and both say that they are going to bring up this
topic with both Yoshida-sensei and Takeyama-sensei when they have the opportunity.
The obvious difference in purpose is that cram schools aim solely at solving
problems on a test, and the students summed up the difference and their confusion in
the following long quote:
That’s right; the aim of schools and cram schools are different. This is a
personal view, but results of periodic tests in Japan’s schools are important and
the test results of junior high school students’ studying to enter and high school
are important; almost half of the entrance process for university is based on
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high school results. Cram schools can improve those results and that’s by
improving test results, and improving test results means being able to solve the
problems in the tests. That’s why the aim of crams schools is to solve the
answers. I’ll not go as far as saying that should be the aim of the teachers, but
problem solving is important. In that respect, the reason there is a gap in what
the schools and cram schools place importance on and focus on is that there are
problems regarding the questions in the periodic tests. As long as you can
answer the questions, it’s okay. It’s not good that you only have to have
enough knowledge to be able to calculate the correct answer. For example,
when we went to observe the lesson last week, it was about multiplying
positive and negative numbers and why multiplying two minuses makes a plus.
When I was thinking about my own explanation, I thought that a cram school
teacher would say ‘because it does! That’s the way it is.’ But a school teacher
would ask why and the focus of the lesson would naturally shift to that. I think
that’s because there are problems with the way the questions are asked in the
tests. If you know that multiplying two minuses makes a plus, you can answer
the questions, and that’s what’s causing the differences in the aims of the
schools and cram schools. So it’s how the tests are created – we need to do
something about that. If we can do that then maybe the difference between the
schools and the cram schools would not be so great. But I think that both types
of schools have their good points; they should learn from each other.
This quote shows multiple important insights into how the students view the
purpose of teaching and the role of teachers. First, though their view of cram schools
is generally more negative, perhaps mostly because of their view of entrance exams,
they do see positive features of them. Even entrance exams themselves are not
necessarily bad; changing the nature of the questions – possibly to something more
than rote memorization or basic problem solving – fits his need for something better.
But his point that “It’s not good that you only have to have enough knowledge to be
able to calculate the correct answer” shows that, at best, the teaching of cram schools
needs to be supported by the work done by regular classroom teachers. Here again we
see the dichotomy between theory and practice and how the students are trying to
make sense of both parts. One of the students says, “But when you think about what
we’re learning at the university about schools … rather than being able to solve
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problems, the university tries to get [students] to think about why they’re solving the
problems. It’s difficult to put into words. Maybe it’s the difference between theory
and actual reality.”
Fortunately the interview with the Yamato University students took place
before the interview with Yoshida-sensei, and I was able to ask him about his views of
the difference between school teachers and cram school teachers. He replies with
ideas much the same as those of the students – that the purpose of cram school is
answering the problem, while the focus of the classroom teacher is on problem
solving, or on the process of problem solving. Yoshida-sensei also confirms that
normal school teachers also have a need to focus on answering problems, but that, “…
the process of problem solving should be emphasized. While doing that, students will
acquire different ways of thinking and begin to understand what is different and what
is similar – they’ll start to understand what is a good way of thinking, and I think
that’s one of the most interesting parts of mathematics. I try to convey the importance
of that.”
Yoshida-sensei actually does make one direct reference to cram schools near
the middle of the third lesson I observed, which happened before my interview with
him. While it is a passing reference, it is a somewhat telling one. The third
observation deals with geometric proofs, and Yoshida-sensei is almost 30 minutes into
a long lecture about the nature of proofs, how and where they appeared in textbooks,
and the intuitive nature of mathematics compared to the need to be able to prove
theorems concretely once at a higher mathematical level. Then he says, “Even if you
go to cram school and learn about right triangles first, you realize that you need to
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proceed one by one,” where “proceed one by one” means going through the steps of
the proof rather than shortcutting the proof by simply saying that the Pythagorean
Theorem applied at that point. In the context of his lessons and his interview, it seems
clear that he is saying that a cram school would simply tell a student that, at this point
in the proof, you could conclude that the result would be true and stop. In learning
mathematics, however, a logical flow needs to be established, which is his point in
teaching this class and his theory for the teaching and learning of mathematics. The
contrast of views and goals of the teacher education program versus cram school can
be encapsulated in that small “even if” statement.
CONCLUSION
Despite the demand from the public for teachers who teach basic skills and
teach to the test and the value that the public places on such teachers, the professors at
Yamato University believe that strong thinking skills and process trump memorization
and rote problem solving. The practical need to learn the skills of teaching and for
students to arrive at correct mathematical answers is acknowledged and given what
they believe to be the proper place, but the emphasis is on developing students as
problem solvers. Yoshida-sensei accomplishes this task by using authentic materials
and experiences, and he pushes his students to in turn think about their students: how
they learn, how they think, and what explanations might be best suited for a particular
developmental level.
Alan Schoenfeld has stated that mathematics is an inherently social activity
(1992), and Takeyama-sensei and Yoshida-sensei show this to be one of their
principles of both teaching and learning as they engage their prospective teachers in
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class and highly encourage these teachers to do the same thing to an even higher
degree with their own future students. Schoenfeld also describes the purpose of
problem solving as “seeking solutions, not just memorizing procedures” and
“exploring patterns, not just doing exercises” (p. 4), which is also evident in the
classes I observed. These professors advocate for inquiry-oriented classrooms,
focusing on student work rather than teacher exposition and using student ideas as the
motivator for discussion. Smith and Stein describe five practices for facilitating
inquiry-based classrooms – anticipating what students will do--what strategies they
will use--in solving a problem; monitoring their work as they approach the problem in
class; selecting students whose strategies are worth discussing in class; sequencing
those students' presentations to maximize their potential to increase students' learning;
and connecting the strategies and ideas in a way that helps students understand the
mathematics learned – and we can see evidence of all 5 of these in the classes
observed (2011).
Yoshida-sensei also displays a typical Japanese practice of allowing students to
present their ideas and solutions to problems rather than simply telling them the
correct answer (or acknowledging an answer as correct once given and then ending
further discussion). When asking about multiplying two negative numbers, for
example, Yoshida-sensei allowed many students to reply with little to no input as to
the “rightness” of the answers. Japanese mathematics educators tend to embrace error
and have a tendency to use student errors as a focal point of their lessons, which in
part comes from the viewpoint that knowledge is constructed by the student rather
than a set of facts and skills presented by the teacher (Hatano & Inagaki, 1998).
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Struggling to find a solution, making mistakes, and learning how and where these
mistakes originated are considered essential in learning mathematics in Japan (Stigler
& Hiebert, 1999). Although we were only able to see a glimpse of this teaching and
learning strategy in Yoshida-sensei’s class, the concepts of participation, creating your
own mathematical ideas, and learning from mistakes are strong theoretical beliefs in
the Japanese education system.
So what can we learn about mathematics teacher training that is strongly
Japanese from these observations and interviews? The clearest take-away is the
contrast between what occurs in the juku versus what occurs in the typical classroom.
In some respects this may seem not distinct at all, as one can read about conflict
between similar ideas in almost any editorial focusing on the Common Core and
traditional algorithmic computational practices versus creative thinking and numerical
fluency5. What makes this particular issue so interesting is that Japan has embraced
the exam culture, which has a history going back more than a thousand years to China.
Yet, while success on an exam is a clear and unambiguous factor in entrance to
prestigious high schools and universities, it still is not seen as the most important
focus in public education. Contrast this with high stakes tests in the United States,
such as the SAT, where a perfect or near-perfect score does not equate to automatic
acceptance into the most elite of schools. Even with a visibly strong exam culture, the
national choice is to value conceptual understanding over test-taking skills. In short,
the Japanese educational system, including the mathematics teacher training program,
5 A comparison of “old standards” versus Common Core for various levels of mathematics can be found
at http://excelined.org/common-core-toolkit/old-standards-v-common-core-a-side-by-side-comparison-
of-math-expectations/, and an example of anti-CCSS math backlash can be found at
http://www.huffingtonpost.com/2014/03/28/viral-common-core-homework_n_5049829.html
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believes that true mathematical learning comes not from exam scores, but from
understanding how and why things work. Even after schools use examination scores
for placement, these institutions still believe that the business of education is
accomplished through conceptual understanding.
Another take-away, though far less visible from this data, is the strong focus on
mathematics and classes devoted to learning mathematics. This distinction will
become clearer once we look at the United States and Finland, and we see that subject-
specific education actually falls on either side of the Japanese model. In the Japan
program there is a strong focus here on content, but importantly, there is also a strong
focus on theory. The numerous textbook examples highlighted by Yoshida-sensei
show several approaches to the problems, rather than a single approach. Emphasis is
placed on the thinking and thus on the learning of the mathematics and on the
transferability of these thinking skills to later mathematics. As Yoshida-sensei
mentions in his example with positive and negative numbers, you begin by referencing
what students already know and move from there.
The level of alignment between these two classes (and within the classes) was
also quite clear, which provided support from one lesson to the next. Looking at the
practice lesson observed in the middle of the three observations, one can see how both
classes prepared the prospective teachers for the observation – both in terms of the
content and the pedagogy – and how both classes took time to follow up on the lesson
and extend the learning. Though Yoshida-sensei says in his interview that regular
faculty meetings were not common, he clearly knows what is happening in Takeyama-
sensei’s class and tailors his instruction deliberately and appropriately.
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In the next section we move away from Japan and begin to look at comparable
data from the United States. Here I examine an exemplary teacher education program
with notable teachers and look at interviews with teachers and students, as well as an
interview with the program director. I also look at my own experiences inside a
teacher training program and in the classroom, and begin to compare and contrast
directly how these successful programs operate in distinctly different cultures and
contexts.
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CHAPTER 6: United States (personal)
My return from Japan back to the United States took almost a year while I took
advantage of being overseas to travel. I had already identified several teacher training
programs that interested me before I left Japan, and I completed my applications from
various internet cafes across South Eastern Asia. The program that interested me the
most was one offered at Duke University, which combined excellent teacher training
with a requirement to take graduate level mathematics classes. There were also
significant scholarships available that almost completely paid for the opportunity. On
June 7, 2004, I moved to Durham, North Carolina, and on June 14 I officially began
the path to becoming a certified high school mathematics teacher.
Duke University’s two year program was compressed into a very hectic year,
comprised of summer, fall, spring, and summer semesters. During the summer we
would attend classes all day, followed by additional classes almost every weekday
evening. The fall and spring semesters were filled with regular classes through the
morning, followed by a mad rush to local high schools to teach every afternoon,
followed by evening classes several times a week. The program was aimed primarily
at students who had just finished their undergraduate work so that there was little
wasted time between graduation and enrollment, and the summer finish was designed
so that there was also little wasted time between graduation and starting a teaching
job. One of the other appeals of the Duke program was that the practice teaching was
not forced into one or two long, full-time months, as many teacher training programs
in the US tend to do. Instead, I began the school year with the regular teachers during
the full week of teacher in-service (before the students arrived) and attended every day
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of class for the entire fall semester, including final exams. This pattern was repeated
for the spring semester at a different school and with a different mentor teacher.
My fall semester placement was at a low SES school in Durham where I taught
Math Money Management (a basic math course with mostly seniors and second-year
seniors who needed one more math credit to graduate), Integrated Math 3 (the content
equivalent of pre-calculus, but designed to be learned by students in an exploratory,
self-directed way), and AP Calculus AB. My mentor teacher had been in the
classroom for 8 years, and she had previously worked with student teachers in her
classroom. The Duke program director chose this placement for me primarily because
of the Integrated Math class, as those were a fairly new addition to the course
offerings at the school and in the school district, and this particular mentor teacher
philosophically believed in the value of such a class. Integrated Math is a high school
math series that emphasizes mathematical problem solving and understanding through
complex, real-world problem solving. It differs from traditional approaches in that the
chapters and sections are designed to be explored by students through questions and
activities that lead them to formula and big ideas. Students in the school had mixed
opinions regarding Integrated Math, as did parents who pressured the school into also
offering “traditional” teacher-led mathematics classes. Still, many teachers, including
the mathematics department chair and my mentor teacher, were supporters of
Integrated Math. At my next placement, I learned that the teachers at that school did
not support the Integrated Math approach, and they taught the class exactly in the
same way as the traditional classes despite the Integrated Math name. Still, I am sad
to say that I was underwhelmed by the teaching practices in the Integrated 3 class
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where I was the student teacher. The textbook for the class was decent in that it
guided and scaffolded students through lessons to discover formula and concepts. The
teacher facilitated these explorations as students worked in groups and helped each
other through the lessons. At the end, however, the teacher then “taught” the lesson to
the whole class, just in case anyone did not quite understand the lesson, and as a result
students learned that they did not need to fully embrace the Integrated Math
philosophy. If they were patient, the teacher would tell them what they needed to
know.
My second placement, on the other hand, was at a different school with a
veteran teacher of over 20 years who had absolutely no interest in doing things any
other way than the way she had been doing them all her career. Her classes were
completely scripted: students would put solutions to homework problems on the board
at the beginning of class, the teacher would go through these solutions with the
students (who were all sitting in orderly rows), and then the teacher would briefly
teach new material and give a homework assignment for the next day. As I slowly
took over the class as part of my student teaching, I was not allowed to deviate from
this pattern in the slightest. Later at Stanford when I read Stigler and Hiebert, and to
this day, I am reminded of the pattern they found in American classrooms of
homework review, show an example of something new, have students do an example,
and then assign more homework (1999). Teaching ideas to the contrary were met with
severe skepticism at best and outright rejection at worst. The program director at
Duke, who was quite competent and whose opinion I always valued, stated that being
with this mentor teacher would allow me to experience a traditional style of instruction
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to contrast what I learned in my first placement. I learned that traditional teaching was
not for me. I also made a promise that I would never subject my students to long
discussions of answers to homework problems, which for my later students was a
fairly radical idea (at least for them) that freed up a great deal of instructional time for
discussion and exploration.
During this second semester of teaching I received an email from an alumnus
of the Duke masters program who was looking for a mathematics teacher to fill a
somewhat unusual position at Washington High School in Georgia. They were
looking for someone to work not in the mathematics department, but in the “talented
and gifted” department as a mathematics teacher. Specifically they wanted someone
who could teach AP Calculus BC (their current teacher, in his opinion, was doing an
exceptionally poor job6) and who could also teach non-traditional seminar classes and
mentor students doing directed studies classes of their own design. I happened to
interview while a group of three students were working on an independent Japanese
language class, and I sat and talked with them in Japanese for about 10 minutes. Less
than an hour after leaving the school, my answering machine recorded the job offer.
My classes at Duke finished on June 24, 2005, and almost immediately I
moved to Georgia to attend summer training for teaching AP Calculus. Once that was
done, I took a much needed vacation and began my teaching career on August 15,
2005. Looking back at my journal entries, it is very clear that although my first few
days (and indeed, years) of teaching were a learning process, I was incredibly happy.
6 This opinion unfortunately proved to be quite correct. Most students failed the AP examination, most
with a score of 1, though the lowest grade of any student in the class was a 93. I also worked with the
few students who took Calculus as juniors as they worked on advanced math, and there were many
important gaps in their knowledge and understanding of the principles of Calculus.
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I noted on many days in August and September that I felt I had made the right career
decision. That is most certainly not to say that it was without challenges. Here is the
beginning of my journal entry for September 28, 2005, coming back to school after
classes were canceled on Monday and Tuesday for “gas days” in Georgia due to
concerns about having enough fuel:
As expected, coming back to school today was rather tough. When I got into
school I could honestly feel the gloom and confusion, so I knew it would be a
tough day. Calculus class was OK, though. Lots of petty questions about the
writing project (as well as whining), and my enthusiasm for the class was not
high. But things went OK, I guess. I wonder if I’m just setting my
expectations too high.
While it was a challenging year, I also had the freedom to develop my own teaching
style and system. The above quote comes from teaching a group of high achieving
students who were unaccustomed to having to work hard in a mathematics class, and
even less interested in writing in a mathematics class. From my student teaching
experiences I learned that I did not want my class to follow a routine that I felt to be
stagnant and counter-productive to learning, so I employed techniques that I learned in
Japan: asking questions, having discussions, and listening to what the students thought
and had to say. I cannot claim to have been particularly good at this right away, and
even now I feel that I still have quite a lot to learn about how to help students guide
and own their own learning. This style certainly was a novel experience for my
students in a mathematics class, and one which many of them appreciated and
enjoyed. For others it was scary and bewildering, and they rebelled at what they
considered “not being taught.”
While I found my teacher training program at Duke to be quite helpful in terms
of exposing me to varied and often valuable ideas (which I later learned was not how
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many people felt about their own experiences), I found myself relying more and more
heavily on the lessons I learned in Japan. Perhaps that is not the correct way to state
things, though. Many of the lessons I “learned” in Japan I only learned in retrospect
once I had the experience of seeing things from another perspective. I realized that I
had become accustomed to doing things in a certain way, which also in retrospect is
not all that unusual. It is often said that people teach in the way in which they
themselves were taught, and I found myself in some ways reverting to my own
experiences as a student and as a student teacher in terms of being the teacher in front
of the class and the content authority. Some of the cultural norms that I had learned to
appreciate in Japan, such as closer relationships with students, were mildly or severely
frowned upon. Ostensibly I was engaged in the same profession as I had been in
Japan, but things were demonstrably different.
A. Relationships
One of the defining characteristics of education in Japan was the nature of the
different types of relationships established in the school. As a foreigner, it was
sometimes challenging to determine what was “normal” and what was special because
of my non-Japanese background. After years of talking and watching, these things
became clear, but experiencing different norms in a different culture helped make
things clearer in retrospect. This perspective in turn helped me better understand
relationships in the United States.
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1. Student and Teacher
To say that the close relationship that teachers foster with their students in
Japan is frowned upon in the United States would be an example of extreme
understatement. With my first year of students at Washington High School I took the
time to get to know them not only as learners, but also as people. I wanted to know
who they were, what they were passionate about, and what their dreams were so that I
could help them as much as possible. Some of them wanted to be doctors and lawyers
and engineers. I asked them why and made them think more deeply about their
choices so that, when they did make their choices, they could do so with as much
knowledge as possible. Some of my students wanted to do things that their school and
their teachers (and often their parents as well) found odd. One student in particular
wanted to be a musician, and he composed and performed his own music while in high
school. His parents wanted him to be an engineer, and as he rebelled, his grades began
to drop in my Calculus class. We sat down and talked about his goals, and I let him
know that doing well in Calculus – or in any class that he did not necessarily see the
future value in – could help him. Getting a good score on the AP exam would mean
no more math classes at the University of Georgia for a music major, which could
mean more time for other classes. His grades improved almost immediately.
Students would join me for lunch, or stop by my office after school, or even
join the math team just because they felt like they were in an environment where they
could learn, be themselves, and have fun. My colleagues were concerned about me
and warned me often not to become too involved with my students. This was what I
referred to in Chapter 1 with the American Question: isn’t becoming too closely
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involved with students a concern? In Japan the answer was a resounding no, but in the
United States the answer is clearly yes. As if to prove the point, another teacher
directly across the hall from me and the other teachers in my office developed a
relationship with a student that was too friendly, and one day he simply did not come
back to work. Later details were sketchy, but he was fired pending a full
investigation. I have no idea what happened to him, but his removal from the school
was used as a lesson for us all. Several months later one of my students came into my
office crying because she was having some personal issues, and she needed to talk
with someone she trusted. I listened attentively, and I made sure that the door was
open and that both of us were visible. I gave her advice and help, but I didn’t give her
the hug she so clearly needed because American teachers cannot do such things. The
irony that this would not have been a problem for a teacher in Japan, a country where
you do not even touch a family member at an airport after they have been gone for a
long time, was not lost on me. We needed to maintain a “professional distance.”
Similarly, I was also struck by the diversity of teacher-student relationships
with regard to my colleagues. Lest I seem overly critical, many of my colleagues had
extremely good relationships with students. I was fortunate to work in an office full of
teachers rather than in my own separate classroom, and I could watch and learn as
gifted teachers (and gifted people) worked within the range of my sight and hearing.
But other teachers were not in the same situation, and the stories that I heard from
students often surprised me. While some of the teachers I worked with stressed the
value of creativity and thinking, others (particularly in mathematics) were draconian
rule-followers.
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In my second year of teaching, one the first day of pre-calculus class, one of
the students raised his hand and asked if he had to do things “your way.” I had known
the student for a year, and we had developed a good relationship, so I assumed that he
was just having fun with me. I was wrong. In the previous year, his mathematics
teacher told all her students that they had to solve problems precisely the same way
that she demonstrated. Alternate but correct methods were marked as incorrect.
Slight deviations in method or style resulted in points being deducted. This one line
turned out to be a wonderful talking point for the year as students would share their
solutions to problems that did not match the way that I or other students solved them.
In most cases they were creative and brilliantly correct. In other cases the methods
worked, but only in certain cases, and we were able to explore the how and why of
their methods to see the underlying mathematical principles. In this manner the
classes felt Japanese to me in that mistakes were teaching tools rather than problems to
be corrected.
2. Student and Student
While teachers may be isolated and have their own rules guiding relationships
with students, students have their own rules and practices for interacting with each
other. Here again I was able to experience the full range of possibilities in my
American classrooms.
Aside from teaching mathematics classes, I was also responsible for “teaching”
directed studies classes. Students created their own curriculum and assessments, and
my responsibility was to help them along. Sometimes interests meshed, as with some
students who wanted to study Japanese or the probability of casino games. Others,
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like the student who studied ancient Greek, were more challenging. The students
knew that they were on their own, and as a result paradoxically took time to work with
other students who were on their own to help motivate and push them. Though they
could easily have spent the entire class period silently working alone (and admittedly
sometimes did), there was far more conversation and discussion than one might
imagine.
Core classes themselves were often quite different. Though some of my
colleagues did an excellent job of leading discussion-based, interactive, group-oriented
classes, these were again the exception rather than the rule. I would often have to
walk into other classes to borrow a student for one reason or another, and most often
the students would be in rows and the teacher would be at the front of the room. I
found myself teaching more teacher-directed classes than I would have liked, and even
though there was much discussion and debate in my classes, I realize in retrospect that
I had in some ways regressed to teach the way that I was taught. Since that time I
have seen more examples of excellent group work and teaching, but I also see teachers
with backgrounds like my own following the fold into more “traditional” modes of
teaching.
While I wish I had been truer to the lessons I learned in Japanese classrooms in
this regard, I do have at least some evidence that the philosophy of students working
with students was at least moderately successful. I used to write dozens of letters of
recommendation for my students each year as they applied to college, and I had them
answer several questions for me in order to understand what they needed. Here is an
excerpt from one student answering the question of why she was asking me for a
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letter. It also briefly touches on the teacher-student relationship I mentioned
previously and how effective that was in helping this particular student.
I really appreciate having had you as a teacher because I walked away from the
class not only knowing how to take a derivative, but also understanding the
value of the help of my classmates. The first day of class you told us that it
would be very important for us to form study groups because the content of the
class would not be easy. I laughed in my head because, first of all, I wasn’t
friends with any of the kids in the class so I thought it was highly unlikely I
would every end up at their house to study, and second, every math class
leading up to yours was a cakewalk. Two weeks later, after an incredibly
frustrating night of calc homework, I begged my parents, on the verge of tears,
to help me switch out of the class, yet a brief meeting with you soon let me
know that you had no intentions of letting me drop the class. Fast forward to a
few weeks ago [almost a year after my class ended]; I’m at [Neal’s] house with
four other students from the Tech class [the college class subsequent to my
Calculus class], and we’re all sitting on his bed, studying together for the next
math quiz. I realized you had been right all along; it’s ok to swallow your
pride at times and ask for help because now some of the kids in that class are
among my closest friends and I would never hesitate for a second to ask them a
question.
The student who wrote this paragraph was a junior when she first started in my
Calculus class, which means that, for the first two years of high school, the idea of
working in groups for math class was completely foreign. Asking questions or asking
for help was not OK.
3. Teacher and Teacher
Teacher isolation in American classrooms is well documented (Fantilli &
McDougall, 2009; Schlichte, Yssel, & Merbler, 2005; Flinders, 1988; Lortie, 1975),
and I found this characterization to be generally accurate. At the beginning of my
teaching, perhaps for the first month, I made a point of eating lunch in the teacher’s
lounge in the cafeteria. After much listening to badmouthing of students and
administrators, I decided that I would be better off in my office with the other teachers
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in my department. As previously mentioned, students would also drop by as well, and
the discussions were interesting and fun.
For the other teachers, though, the lounge time was one of their only chances
to see other teachers. I never had my own classroom and instead “floated” to other
rooms throughout the day. Sometimes, though not often, those teachers would stay in
the room while I taught, as I didn’t mind, and when students were working in groups,
we could talk. They tended to enjoy this and often commented that it was their only
time aside from lunch to talk with another teacher during the day. Relationships
between teachers are much harder to qualify here because, simply, often the
relationships did not occur during the school day and occurred only in sporadic, short
bursts. To put it in another way, they were not “relationships” in any deep sense of the
word, but rather intermittent conversations with colleagues. This description is much
shorter than the corresponding section in the Japan chapters as there is comparatively
little more to say.
B. Expectations
As with Japan, there are expectations of what happens in school that connect
closely with the relationships discussed previously, but also extend these ideas into
how teachers and students “do school” in the United States.
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1. Do what the teacher tells you to do
Stereotypically, outsiders might think that this heading would have been
reversed with the counterpart in the Japan section (namely, A Focus on Students). By
no means do I mean to imply that American teachers or schools do not focus on
students, but rather that the focus is on the teacher as the ultimate source of knowledge
and authority (Boud & Feletti, 1991). While the typical Japanese model is to guide
students to develop their own knowledge in conjunction with their fellow students, the
typical American model is for students to mimic the information provided by the
teacher. In the case of mathematics classes, this often results in the familiar pattern of
seeing an example problem, working through a similar problem, and then doing
homework of many more repetitions of the same basic pattern (Stigler & Hiebert,
1999). I must preface again that this is a general trend, as Stigler and Hiebert noted,
and by no means represents every mathematics teacher in the country.
After leaving the high school classroom for graduate school, I encountered far
more examples of collaborative learning and “reform” teaching that more closely
resembled the patterns I experienced in Japan. Working with prospective mathematics
teachers, we discussed the models and theories they learned in class, and we compared
and contrasted them with what they were experiencing and observing in their
placement schools. I noticed that, much like me, they did not fully import all of their
practices into their own classrooms, falling back on traditional methods rather than the
more challenging and time consuming methods they had just learned. The time
constraints of new teaching resulted in cutting back on what they wanted, focusing
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instead on survival teaching. And telling students what to do in the traditional
American sense is far more time efficient for the novice teacher.
I reference again several examples mentioned previously. My own student
teaching placement at the second school could have been the sample case for this style
of teaching. At Washington High School, the “do we have to do things your way”
question exemplifies the idea that the teacher is always right and the student, even
when right, can still be wrong. As a final example of this, an example I clearly
remember, is the day several of my math team students brought their tests to me from
an Algebra II class where they were asked to multiply a 2x2 matrix with a 3x2 matrix.
This cannot be done, and the students of course got the “wrong” answer on the test.
When the students approached the teacher and asked what the correct answer was, she
told the students that they were wrong and should figure out the correct answer
themselves. While this example is extreme, it always reminds me that the teacher is
not always right, and that admitting errors is as good for teachers as it is for students.
2. The Role of the Teacher
As the above stories and descriptions show, there is a far greater degree of
variability in role of the teacher than was present in Japan. In some ways the role of
the teacher in Japan is more defined in the sense that, while there are the normal
differences in teachers as different people, the role of the teacher is more precise.
Teachers in Japan are expected to sponsor a club; teachers in the United States are not,
and as such the amount of time spent after school varies tremendously. It is also worth
noting that, as a standard responsibility, Japanese teachers earn no extra pay for
coaching, whereas American teachers typically do, at least for some clubs or sports.
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There is no expectation for American teachers to stay late at work; some leave at the
first possible moment, while others can be found at the school late at night on
weekdays and on weekends. I remember a student coming into my office one day at
7pm saying, “Hi Mr. Hedrick! I knew you’d be here, and I have a math question.” No
one forced me to stay, and there was no peer pressure to stay. And even at 7pm, there
were often a few other cars still in the parking lot.
As also mentioned previously, the teacher’s role includes that of a professional
distance. Teachers are expected to be masters of their subject area, but not counselors
or surrogate parents. Japanese parents often leave some of what in the United States
would be considered familial responsibilities to teachers, and classes such as moral
education are required courses. Aside from private schools, no school in the United
States would even attempt such a class as it would tread on the rights and privileges of
the family. While teachers are protected against being fired for religious or political
beliefs, sharing them with students or showing that the teacher is also an independent
person is frowned upon. That is not the role of the teacher.
3. Equality and equity
My teacher training program and my teaching situation immediately following
it perhaps led me to see this particular issue more strikingly, but I also believe that my
experiences in Japan made me far more aware of equality and equity issues in the
United States. I of course thought about the issue in more simplistic ways when I was
in Japan, as I was the only non-Japanese person at my school, and indeed the only
non-Japanese person in my town of over 8,000 people. As a person in the majority in
almost every category, I had almost no experience being an outsider, and being in
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Japan gave me the opportunity to gain some insight into that state, though to be fair
being White and male in Japan is both a blessing and a challenge, where being in the
minority in the United States is not so balanced for many. It made me more aware of
others, as well as aware of my privileges.
The first school I worked at as part of my teacher training was primarily
African-American, though the composition of the classes depended greatly on which
class it was. The AP Calculus class was predominantly White, while the Math Money
Management class was predominantly African-American with a few Hispanic or
White students as well. At Washington High School I never taught a single African-
American student in my Calculus classes, though there was a decent representation of
African-American students at the school. In fact, in most cases my Calculus classes
were majority Asian, with White students being a slim minority. After three years of
teaching classes of 100% Japanese students (figures from the CIA World Factbook put
the racial percentage of Japan at about 98% to 99% Japanese; 2014), I became much
more aware of these differences. Japanese classes are not tracked the way that
American classes are, so all students end up taking the same classes. While this is
powerful on the surface, it is also a little disingenuous, as there are more subtle
features of the Japanese educational system that serve as proxies for tracking. The
need to apply to high school, and the stratifying that occurs as a result, is one quick
and easy example.
The fact that these differences are so apparent means that, even if in a shallow
way, we have these conversations as a part of American education. I found out during
my third year of teaching in Japan that some of my students came from a fairly
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impoverished part of the town, and one that I never really had a chance to see. I was
never able to learn if any of the students were burakumin – the outcast group of Japan
in some ways comparable to the untouchable caste in India – and to be honest, I am
not sure one way or another if they even might have been. The situation simply was
not talked about. In the northern part of Japan are what remains of the Ainu people,
the indigenous population of Hokkaido. I have had the opportunity to work with this
group in a minor way over the past decade, and their culture has almost completely
disappeared. There are also the descendents of Korean workers (of both the voluntary
and involuntary type) living in Japan, and the third generation, who has never set foot
in Korea or spoken a word of Korean, is still denied citizenship.
It may seem odd considering how far we have to go in terms of equity and
equality in American classrooms, but I see this as a defining and in many ways
positive part of American teaching culture. In the United States we have tracking and
stratification that occurs for many reasons, socio-economic status and race being two
of the most dominant, but we know this is a problem, admit it (for the most part), and
try to do something about it. Making education equitable is part of the conversation,
as we will see in more detail in the following chapter, and some schools of education
and teachers actively work to make things better. In fact, it was an issue that I tackled
directly while I was a teacher.
As I mentioned previously, the school where I taught was fairly affluent, and
the students in my Calculus class were exclusively Asian and White. The county that
I worked in, however, was not so racially uniform. The north part of the county,
where I worked, was fairly well represented by my students. The south part of the
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county, however, was predominantly African-American. On paper the two halves
were equal, and in fact the south part of the county received additional funding
because of the high proportion of low SES students. The truth of the matter is that
money is by no means a substitute for real resources. One of the schools there was a
mathematics magnet school, and students from many other high schools in the south
would take buses every morning to go there for higher quality mathematics classes.
Some of these students would ride 2 hours each direction to get there – I know this
because I worked with one of them who made this trip 5 days a week for 4 years. But
the Calculus class that I taught at my “normal” north high school was not offered at
the mathematics magnet school. In fact, it had not been offered in 7 years. After my
first year of teaching I was made aware of this school, and I was asked by the county
mathematics coordinator if I could do something about it.
As part of my work at Washington High School, I managed to create a
partnership with this mathematics magnet high school 40 miles away, and I offered
my Calculus course to them in real time with cameras, microphones, and televisions. I
established a relationship with the other school’s faculty, who were primarily people
of color, and brought advanced Calculus to students of color. To this day I ascribe my
willingness to lead this project – which required many, many extra hours of work for
no compensation – to the lessons I learned in another country. Teachers are expected
to work hard because that is what teachers do. Students are taught the same regardless
of where they come from. And being an outsider is challenging and scary, but
sometimes the kindness of one mathematics teacher can change an entire experience.
It did for me, and it did for my students at the other high school. At the end of the first
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year of this project, one of my students selected me as “the teacher who most
influenced his school career,” and I was invited to join a ceremony at the school. I
was presented with a certificate, written by the student. It reads in part:
Mr. Hedrick is one of the greatest teachers that I have ever had the honor of
learning from. I never thought I could feel that way about any teacher that
gave me an “F” on a math test. Being in your class was a real reality check
that it was time to start working harder if I wanted to be successful. … You
never gave up on any of your students. … Thanks for the “F”s Mr. Hedrick and
thank you for being such an excellent teacher and challenging me.
I do want to note that the above student actually earned an “A” in the class
overall and easily scored a 5 on the AP examination at the end of the year – the first
passing grade on that exam that anyone in the school had ever earned. The student
was African-American, born to a teenage mother who I met for the first time at the
ceremony, and he was raised as an only child and without a father. But he had talent
and worked hard, and he earned not only a full scholarship to college, but also full
funding all the way through his PhD, if he wanted to pursue one.
CONCLUSION
The teaching profession in the United States is a flexible one, and for that
reason alone difficult to encapsulate in simple, pithy descriptions. Though I learned a
great deal from Japan about teaching and try to import those ideas into my own
American classrooms, I think that flexibility is what makes me prefer to teach here
rather than Japan. Practices that are standard and good in Japan can be found here
(e.g. focus on problem solving, group-oriented activities); similarly, practices that are
standard and bad can be found here as well (e.g. focusing on testing, systemic tracking
of students). I believe that the country could benefit from some of the expectations
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and supports that Japan has for its teachers, such as a strong role in communities and
automatic assumptions of ability and professionalism (and from a national curriculum,
but that debate gets very political very quickly these days), but our culture does not
allow for it. Teaching is often cited as a “prestigious” job – a 2014 Harris poll showed
that 60% of those surveyed believed teaching to be prestigious, and 81% would
encourage a child to pursue that profession (Harris 2014). Yet ask any teacher, and
likely you will not hear the same thing, as the teaching profession is both romanticized
and disdained (Ball & Forzani, 2011). Teachers in Japan are truly respected and the
title of sensei is used for teachers at the elementary and secondary level, as well as for
professors and doctors. When the list of “best” careers in the United States naturally
includes doctors, lawyers, engineers, and teachers, then perhaps this view will change.
Although there is a great variety in teaching, there is also a great deal we can
learn about what makes for great teaching here. Though my own experiences were
varied, recall that I have deliberately chosen for this study exemplary schools of
education that are renowned for their level of excellence in training future teachers.
Here we see visions of the possible (Shulman, 1998) where theory and practice meet,
and where teachers are taught not how to teach as they themselves were taught, but to
teach in the methods that have been shown to work. We see highly successful
teachers in the making, and we see that there are more similarities with the Japanese
system than a brief first glance might show.
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CHAPTER 7: United States (teacher training)
Foster University was selected by the CATE Project as one of their two study
sites in the United States, and as I was the research assistant responsible for the data
collection and had personal knowledge of the teacher training program, it made for an
ideal sample for this study as well. Internationally known for many programs,
including education, Foster University is a major research university located in the
western United States and is near a major metropolitan center. Unlike Japan, where
teacher training is conducted as part of the undergraduate experience over a four year
period, the teacher training program at Foster University takes place over a full
calendar year, beginning in the summer and finishing in the spring, and is part of a
Masters degree program. Teaching certification programs in the United States can be
found at the undergraduate level as well the graduate level, so this system is not at all
unusual. This is usually determined by each state’s policy. Unlike many American
universities, however, the student teaching component of the program takes place over
the full academic year rather than one period of time. Prospective secondary teachers
at Foster University work at a single placement from the first teacher workday to the
final day of the school year, and every evening they attend their academic classes at
the University. In addition, prospective teachers also attend classes and work in a
non-school year placement in the summer prior to the full academic year. Similar to
Yamato University, there are a variety of required classes, ranging from subject-
specific pedagogy classes to Teaching and Learning in Heterogeneous Classrooms.
Observation and interview data were collected over a 2 month period
beginning in February of 2013, and classroom data were collected over a period of
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three consecutive weeks in February of 2013. Following the CATE protocols, I
observed a mathematics Curriculum and Instruction course taught by two graduate
students: Emily and Samantha (pseudonyms). Both Emily and Samantha were in the
process of completing their PhDs in Mathematics Education, and both had experience
teaching at the secondary level; Emily had been a middle school math teacher for three
years, and Samantha had been a high school math teacher for five years and an
elementary school teacher for one year. Both had experience supervising student
teachers, leading professional development workshops, and teaching a similar course
at another local university teacher preparation program for the prior two years.
As this data collection fell under the CATE project IRB, audio-recordings of
the classes and interviews could not be made. In many cases, however, I was able to
record direct quotations as part of the field notes, and I was also able to supplement
class observation notes with photographs of the classroom and by obtaining copies of
the instructors’ Power Point presentations. Two willing students, both female, were
randomly selected from the class to be interviewed after the three weeks of
observation were complete, and the instructors were also interviewed, all using the
CATE project interview protocols (see Appendix A and Appendix B)7. In addition, I
was able to interview the program director for Foster University’s teacher training
program. These classes and interviews were all conducted in English, so no
translation was necessary. Interviews and field notes were open-coded, though
attention was paid to the codes that were created when open-coding the Japanese data
(e.g. Theory/Practice, Authenticity).
7 The students were also given the survey instrument created by the CATE project, but permission to
use this instrument (and thus the data) was subsequently revoked by the project.
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MATHEMATICS CURRICULUM AND INSTRUCTION CLASS 1
Emily and Samantha’s
8 Mathematics Curriculum and Instruction class begins
promptly at 3:15pm on Tuesday, February 5, 2013. This class is the first of three that
I am observing, but already the fifth of the term for them and their students. Before
class even begins, the two instructors have their PowerPoint presentation displayed at
the front of the room, a hand-written agenda on the front board (listing Painted Cubes,
Textbook Analysis, Learning Segment Calendar, and Assessment & Rubric
workshop), and student name cards on tables. During the interview with the
instructors I learn that they pseudo-randomly assign their 15 students (6 male, 9
female) to collaborative groups in different ways every class period, though they
always have three groups of four students and one group of three students.
The students walk into class slowly, find their names, and talk with their fellow
students about their placements (the term for the school at which they are completing
their student teaching). Many of the students see the “Painted Cube” problem on the
agenda and note that they have already seen and completed this problem (a copy of the
problem is provided in Appendix H). The instructors laugh, as they provided this
example to another professor in the program, and note that having solved the problem
previously is actually not an issue since the focus of the discussion is academic
language. The instructors then proceed to explain what academic language means
with general examples from mathematics. A student interrupts to ask a question,
which is common in all three observations, wondering what an idiom in mathematics
would be. Samantha quickly gives the example of “for what values of x does this
8 Graduate students often call professors in the School of Education at Foster University by their first
names.
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hold?” and notes that “hold” in this case functions as a mathematical idiom. Students
ask a few more minor questions before beginning the task, which is to read through
the Painted Cubes problem and code it for academic language.
As this activity is supposed to be a warm-up rather than an in-depth exercise,
the students are only given about 10 minutes to complete this task. The instructors
then wrap up, and on the Power Point screen are the four categories of Academic
Demands: Vocabulary, Syntax, Idioms, and Peer Communication. For the next ten
minutes, by category, students volunteer their answers, and the instructors record them
on the Power Point. To conclude, the instructors note that they made this activity
deliberately to talk about language learning demands, which will be the focus later of
the learning segment. The question is how to scaffold the language, in particular for
English language learners (ELLs), so that students can acquire the necessary
mathematical academic language.
Emily then tells the students that they are slightly shifting gears to look at
textbooks and how different ones might be beneficial or useful to them as teachers.
Emily and Samantha gathered three different textbooks and gave samples to the
students before class so that they could look over them in groups. Samantha makes
explicit the other part of the question: What is this textbook not useful for? What
might it not accomplish?
As the students work in groups, Samantha explains to me that these three
samples are all the same topic, and one textbook is an exploratory activity, one is very
language heavy, and the last is very traditional in style. The instructors circulate from
group to group during the 20 minutes that the activity takes place. Emily and
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Samantha then gather the class back together, announce that they had planned to do a
gallery walk of the work but changed their minds, as the discussions were very
productive. Instead they immediately shift to work on learning calendars. The
learning calendar is a section of the learning segment, a multi-day, in-depth lesson
plan that will be used in actual teaching in the students’ placements. In particular, a
learning calendar is a detailed description of lessons and pacing that, if given to
another teacher, he or she could follow the intent of the lesson, though without enough
detail to qualify as an actual set of lesson plans. Emily and Samantha remind the
students that the learning calendar should:
Identify the course and grade level
Include the length of each period you teach
Identify the learning goals/topics addressed each day
Identify possible activities for each lesson
Identify two possible teacher questions to be used at some point in the
lesson
Identify how you will incorporate formative assessment throughout the
lessons (exit ticket, journal prompt, class work, presentations, peer and
self assessment, etc.)
Designate topic(s) and structure of homework assignments
Include the use of technology in one segment of a lesson
The instructors leave the students with this before taking a break, just a little
over an hour into the class. They resume 15 minutes later with an hour and a half left
in the lesson to begin the actual work of the learning calendar, which they call a
“rehearsal” of the learning calendar. The students have many questions about this
assignment, ranging from questions on terminology (Foster University’s classes use an
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Understanding by Design model (Wiggins & McTighe, 2005), and questions related to
length in days of a Big Idea and how they connect to Learning Objectives) to how
backward design comes into play in this particular context. To make the process more
concrete, the instructors present several learning objectives associated with the
Pythagorean Theorem and multiple textbooks, and they ask the groups to structure the
activities they would use to develop the idea as part of a mock learning calendar.
Samantha notes that the students will be given about half an hour to work, and the
students are also given dry erase markers, post-it notes, and everything they need to
start drawing a rough calendar. The walls of the classroom function as dry erase
boards, so students begin writing proofs and in one case the words “Monday,”
“Wednesday,” and “Friday” on the walls as they structure the lessons. As before, the
instructors walk around the room and offer help.
At the end of the activity, Emily and Samantha begin by asking the students to
continue to think about what they have put together. In addition, Samantha says, “We
encourage you to take advantage of your peers – this is part of being a teacher.” The
drafts of learning calendars are due the following week. The class concludes with
work on rubrics, which appears to be a topic from an earlier class that is ongoing. It
sounds as though they have taken some form of assessments – one student has created
one and another has taken it – and there is definitely peer feedback going on. As time
runs out, Emily concludes by detailing the specifics of what is due next week, and she
reminds the students of the readings that need to be completed: a Jackson et al reading
on launching complex tasks and reducing cognitive demand (2012) and chapter 4 from
a book by Jo Boaler (Boaler & Humphries, 2012).
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MATHEMATICS CURRICULUM AND INSTRUCTION CLASS 2
The second week of observations shows the patterns that have been described
previously: tables with name cards are set up, an agenda is on the board (listed are
Equations?, Setting up Complex Tasks, Telling in Instruction, and Workshop). The
class is being held in a different room because of people coming in to observe, but
otherwise all is the same. Emily and Samantha begin class promptly at 3:15 by
passing out strips of paper to pairs of students (instructing the group of three to work
together). Students are asked to sort the papers into one group of equations and one
group of non-equations (thus the “Equations?” on the agenda). Before they begin
sorting, students are given the instruction that they are only to consider real numbers,
and that they will be asked to explain their answers. I observe the group sitting next to
me, who sort their papers into piles they name “equation,” “non-equation,” and
“maybe.” Afterwards there is debate about x2 = -4 (not an equation: no solutions since
they are only working with real numbers) and x + 3 < 5 (not an equation: no equals
sign). This leads to a discussion of “well-defined,” as slide 4 of the presentation
describes that an equals sign is necessary for an equation. Students are then asked to
sort the equations based on truth, which is another sticking point in the discussion.
Eventually students land on the fact that the necessary categories here are not just
“true” and “false,” but also “sometimes true.”
This activity takes only about 20 minutes to complete, followed by about 10
minutes of discussing how this activity is relevant for teaching, from how to introduce
a topic using tasks to how to structure as assessment with a range of understandings
and misconceptions. Students ask about when to tell students what the definition of an
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equation is, and one student notices that the instructors did not give the categories of
“true,” “false,” and “sometimes” to them and asks if this would be appropriate for high
school students. Emily says yes, as she built this activity deliberately to encourage
discussion.
Following this is a brief discussion of academic language, carrying over from
the previous week’s work. Samantha identifies some common “trouble spots” for
teachers: differentiating among the words simplify, evaluate, and solve, using the word
“canceling” instead of “simplify,” using “FOIL” as a verb instead of as a tool for the
distributive property, labeling geometric figures (e.g. trying to draw a triangle with
sides 2, 3, and 7), and precise definitions of words. As usual, the students interrupt
with clarifying questions. Why is simplify the wrong word for something like 3 + 7?
Is PEMDAS the same thing as FOIL? At what point do you tell students that there is a
difference between “no real solutions” and “undefined” for equations when students
do not know about imaginary numbers?
This discussion leads into the antenna problem (see Appendix I), where the
students are given a small amount of time to work the problem and then much more
time to examine the scaffolds in the task and apply the Jackson et al. reading to
determine how the problem might be rewritten for a less mathematically experienced
class (e.g. students taking geometry, the course prior to the one this task was designed
for) without reducing cognitive load. Discussion of the task follows with students
offering ideas of what they might do (provide a picture, making sure that students
understand that the antenna is perpendicular to the roof), including ideas that are
disagreed upon (pointing out right triangles, as doing so would significantly reduce the
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cognitive load). The students point out that there are also ideas that they think would
be good to share with some students, but not all students. Samantha says that there are
indeed some things you should “keep in your metaphorical back pocket” to help
students if an additional scaffold turns out to be needed. Samantha further explained
that she always trusted her high school students to do something rather than rob them
of the opportunity to do so. She could always scaffold later.
This idea of “telling” has come up a few times in the discussion today, perhaps
in part because it is included on the day’s agenda by Emily and Samantha. They then
move to address this topic more directly and in more detail. The Boaler and
Humphries book also includes video of Humphries teaching, and one of the sections is
specifically devoted to the idea of telling. Emily and Samantha show only a short clip,
where a student named Sam draws and shades 2/3 of a circle and explains how he
arrived at his answer for the area of the shaded region. His answer is incorrect, yet
another student, in the spirit of the class, chimes in to say that Sam’s answer is
different from hers, but that Sam’s explanation has convinced her to change. Emily
and Samantha then provides transcripts of the lesson to the class and ask them to code
for instances of telling. Discussion was abbreviated, but the students came to the
consensus that the teacher’s choice to eventually “tell” the students something factual
or recall-based was not really relevant to the lesson being taught and was therefore an
instance where telling rather than exploring was the better teacher move.
After a break, the students are asked to form two groups rather than four
groups and think of an instance where they as student teachers struggled with the issue
of telling versus not telling. I join Emily’s group, and one prospective teacher shares
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an example of a lesson on similarity where the point was to show a similarity rule for
triangles. In this particular class she showed the students pairs of polygons – triangles,
quadrilaterals, hexagons, and more – with all sides congruent to each other without
marking angles. An animated discussion of what could be similar and what had to be
similar followed, but there was never clear consensus. With 7 minutes left in the
class, the prospective teacher basically gave the answer and ended class. There was
not much debate on this instance as all agreed it was a choice made by time
constraints. Another example followed where the prospective teacher tried to show
students where the number e comes from (they were doing compound interest). One
of the other prospective teachers did not follow, so the other prospective teacher put
the formula on the board and explained how, for larger and larger values of n, (1 +
r/n)(n/r)
would approach e. The problem was that the students were not savvy enough
with their calculators to get the answer, and the whole class got frustrated. The
prospective teacher ended up telling the students what the answer was, which defeated
the purpose of the exploration. The affordance was that it saved time (or would have,
had that been the initial choice), but the negative impact was that the exploration did
not happen. Students suggest scaffolding the button pushing part of the assignment, as
that part is not essential to the learning of the lesson, or possibly using something like
Excel to set up the problem in another way.
The class comes back together again at 5:20 to talk about the learning
calendars from before and the next step: elaborating 90 minutes of the calendar (i.e.
one block period or about 2 “regular” periods) into what they call an elaborated lesson
plan, or a description of exactly what happens for those 90 minutes. Inherent in this
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plan is what should be told and what should not be told, following in the footsteps of
today’s lesson. The lesson should develop procedural fluency, conceptual
understanding, mathematical reasoning, and productive disposition, as detailed on the
Power Point slide. As such, there should also be formative assessments, and it should
be obvious that instruction should change after the assessment, else they are not
formative. Samantha acknowledges that the students have requested more examples
to look at (though she does not reference when or how these requests were made), and
the instructors provide more example that they have used while teaching at another
university and that show both a narrative style and a plan in a bulleted list, just to give
different options. They stress that it is the content, not the format, that is most
important.
Emily then announces that the remaining 35 minutes of class for Workshop
time, which clearly has meaning to the students. They are allowed to work on their
learning calendar, the next assignment, or any other parts of the learning segment that
they choose to work on. One student next to me immediately begins looking at the
sample elaborated lessons, while in another group one person leaves and another joins
in as they talk about their calendar and schedule. Across the room two students sitting
near each other work individually and silently on their computers. Emily and
Samantha walk around the room, talking with the groups and individuals, and during
the 35 minutes sometimes group dissolve and sometimes silently working individuals
begin talking with a neighbor. The time is being used productively and constructively.
Emily ends the class and says that they have not been able to answer all
questions, but that the students should feel free to email anything they want to ask. A
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draft of the elaborated lesson plan is due next week, and Emily will happily email a
template to anyone who wants it, though she does not want anyone to feel forced to
use that particular style. The reading assignment next week is a Gutstein article about
social justice and connecting that to math tasks and student interests (2007). The
syllabus also states that students should pick another reading, but Emily cancels that
part of the assignment.
MATHEMATICS CURRICULUM AND INSTRUCTION CLASS 3
The third week of observing the Curriculum and Instruction class takes us back
to the same room as the first week, which is much more spacious and comfortable.
Things are much the same as the previous two weeks: name cards are randomly
assigned again to form three groups of four and one group of three, and the agenda for
the day is on the board (Gutstein discussion; Connecting math tasks to student
interests & experiences; Workshop). The only difference is that, at the beginning of
class, 3 of the 15 students are missing. Samantha asks where they are, and another
student replies that they are all in the same carpool for their placement. Samantha and
Emily begin the class right on time, and the missing students show up together about 5
minutes late.
Samantha notes that the choice of wording, i.e. the use of “interests,” is a
deliberate choice to align to the wording needed in the end-of-year portfolio
assessment that all of the teacher candidates must complete and pass in order to
receive their teaching credentials. The relevant wording relates to connecting to
students’ interests and prior knowledge. Emily makes the point immediately that,
while Gutstein talks about what is important in the community in which he teaches
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and models his lessons around those interests, it is also possible and sometimes
necessary to start with relevant standards or curricular requirements and then find
ways to address student interests.
The warm-up activity around the Gutstein reading is for the students to choose
a quote from the reading that they found “important, or interesting, or confusing, or
provocative, or some other adjective” and then talk about it in the group. This activity
lasts a little over 15 minutes, and the group closest to me is having a heated debate.
When Emily asks students to share with the whole group, this particular smaller group
immediately volunteers to share. The debate is the politicization of education. One
student says that Gutstein is biased throughout his lesson development, and that while
it is beneficial for him to do so, a neutral position is better. Another student
immediately asks if it isn’t better to give opinions and state that they are opinions
rather than try to be completely neutral, which is not only impossible but possibly not
beneficial. Another student chimes in to say that Gutstein himself says that he failed
to stimulate contrary discourse, even citing page 437 as the source of the quote.
Discussion then moves to the idea of student comfort in class, with yet another sharing
a personal story of having a contradictory political view to the one being presented by
the teacher. The situation was uncomfortable, and she felt that same discomfort once
again when reading the article. Another student shares a personal example of a
government teacher who never told students her political affiliation, and even at the
end of the year the students had no idea. That created a safe space for discussion and
exploration of ideas.
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At this point Samantha moves into the conversation, saying, “So, I’m
interested in this issue of whether classrooms are political, and I’m trying to figure out
where the math is. What problems are we solving? Is it the problems we choose that
make it political?” Samantha gently steers the discussion back to the mathematics and
the role of math, directly acknowledging that the politics are important. From this
point the conversation stills deals a little with the politics, but the mathematics move
to the forefront. The Gutstein article talks about maps and projections of Greenland
and the like, and the students all believe that the context is important. The issue would
not be of as much interest if the maps were of, for example, a foreign planet. One
student mentions that the student responses in the Gutstein article are almost devoid of
math, so the challenge is to enter the context of the problem while remaining “true to
the mathematics.” Another student ties this back to the idea of relevance and
anchoring the mathematics in the students’ interests.
Samantha uses this moment to introduce the next stage of class. Emily and
Samantha give each group a particular math task, and their job is to become experts at
the task (in 15 minutes), then take 15 minutes to answer the questions, and then
prepare to share their task with the rest of the class. The students immediately begin
working, some quite loudly and heatedly. Emily and Samantha walk around the room
to listen and assist, and they keep the students on task and on time. 15 minutes later
Emily tells the students to begin working on the questions if they have not done so
already and not to worry if they have not finished the task. The tasks are designed to
take multiple periods, so this is an abbreviated version. The loud and passionate group
is still loud and passionate, but the two most active students are also smiling and
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nodding their heads as they work. Clearly there is disagreement, but it is still friendly
and productive. At the end of the time period, Samantha calls for a quick 10 minute
break.
After the break, each group is given about 10 minutes to present their task and
thoughts. Group One has a task about malaria drug levels in the bloodstream and how
potent the drug is. Students need to model the amount of the drug in the bloodstream
and figure out how often someone needs to take the drug. The students believe that
the task fits well into something like an Algebra II curriculum, but they also believe
that it could be modified to be a good algebra lesson with the removal of concepts like
inverse functions and some other vocabulary. They say that, in terms of student
interests, hopefully none of them have had malaria, but probably many of them have
taken medicine. They suggest possibly showing a video about malaria and what parts
of the world it affects to add context and boost interest. She also says that you could
modify the task to look at how long it takes a drug to leave your system and relate it to
the recent Lance Armstrong affair – if you want to dope, how long before you test
negative for the drug? The students laugh, and they point out that such a thing is
current and relevant for high school students.
The second group presents a task that involves the Future Farmers of America
and the creation of a miniature golf course. Part of the interest of this task is that it
was an actual problem: different FFA groups designed each of the 18 holes, and a
school principal was required to figure out how to set up the course such that the end
of the first hole was adjacent to the beginning of the second hole, and so on all the way
to the 18th
hole. The lesson is a geometry one in fitting everything together in a way
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that minimizes the space taken up by all the different pieces. The students say that the
relevant student interests might be miniature golf because it is fun, there is potentially
interest in the Future Farmers of America, or perhaps a similar idea of arranging a
room or something else that might be more personally relevant. On the other hand, the
students note that they completed the task without using much of what was supposed
to be relevant, namely area and perimeter formulas.
The third group’s task is to install a security camera in their shop in order to
catch shoplifters. The room is somewhat U-shaped, not a quadrilateral, and the first
part of the task is to put the camera in a certain corner and show using geometry and
area formulas that the camera covers 15% of the shop. The task is then to find the best
place for the camera and to justify that the place is the best, which uses more area
formulas and requires the use of mathematical justifications. The group explains that
all students have had the experience of shopping, and perhaps some of them can relate
to the idea of shoplifting as well (the class laughs – it is clearly a joke). They believe
that the task is appropriate for lower grade students, though they suggest that a
limitation like a 4-foot radius on the camera view could also allow students to
calculate areas of sectors. Another way to alter complexity is to make the shop 3-
dimensional. Overall, like the previous task, they say that the math is ultimately pretty
easy.
The fourth group is the one that was the most animated and vocal, and once
they present their topic, is it easy to see why. Their task is called “Driving While
Brown,” and it talks about racial profiling in Chicago. The students are given cubes
that represent the racial population of Chicago without being told the actual numbers
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or percentages, and you randomly choose cubes to see what the population looks like.
Then the students are given statistics for random discretionary traffic stops to see how
the numbers do or do not match up. The students say that this is clearly a probability
and proportions lesson, and it gets into the ideas of independence and the Law of
Large Numbers. The original target population for the task was a classroom that was
99% Latino, and the students theorize that at least some of the students or their friends
or parents had been pulled over and might relate to the topic personally. The students
are critical of the task, as they say the conclusions that the task leads students to draw
are not valid, and the task needs more ways to approach the topic to draw conclusions
that are more quantifiably valid. There is no discussion in the task of confounding
factors, which is a necessary element of such a task. The students also mention that
the cubes in the original task were colored, which they found to be somewhat
offensive and to perpetuate stereotypes. Ultimately their main criticism is that the task
leads students to one and only one conclusion, and there is no room for exploration.
At this point Emily and Samantha take a moment to talk about the tasks, and
they share where they found them. While they say that they do not advocate for any
particular site – some examples are good, some are bad – it is good to have options
and ideas for planning good lessons that connect to student interests. One of the sites
also is technology-focused, which is a good resource. Emily and Samantha take a
moment to let the students debrief in their groups about the tasks and issues, and then
they give the students the remainder of class (about 30 minutes) as workshop time.
This time is used in the same manner as the previous week, and about 5 minutes
before the end of class, Emily again gives a preview of the coming week. The next
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topic will be math task debriefs, with two readings about facilitating discussions in
classrooms. She warns the students that one of the readings is 80 pages long, but that
they only need to read pages 171-206.
A PLACE FOR BOTH THEORY AND PRACTICE
Linking theory and practice is a deliberate choice at Foster University’s teacher
training program. When asked the question, “What are some of the main opportunities
that your program provides to help candidates learn about the relationship between
theory and real classroom teaching,” the program director replies that opportunities are
“abundant.” According to her, the Curriculum and Instruction courses are all about
that “incredible emphasis on clinical practice,” which makes up 50% of the curriculum
if not more. She also states that, in her opinion, this is one of the biggest strengths of
this particular program as opposed to others, both nationally and globally.
The classroom observations and interviews strongly support this view of the
relationship between theory and practice here: they are not elements in isolation, but
rather pieces in conjunction. All three observations begin with a teaching-related
activity directly linked to the theoretical readings assigned in the previous week. In
the first observation, for example, the students had just read an article about academic
language demands, and the students were given the Painted Cubes math task and
asked not only to complete the relevant mathematical part of the activity, but also to
use the knowledge of the reading (plus clarifying information presented by the
instructors and clarified through examples and answers to questions) to think about
what the reading meant in the context of an actual, true mathematical assignment for
high school students. It is important to note as well that in this example, as in all the
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examples that follow, the prospective teachers were intentionally forced the assume
the role of the student learner, not in the sense of play-acting, but in the true sense of
being the people for whom the learning activity was designed. In this manner, every
class was a model of practice as well as the theory as the prospective teachers could
engage in the practice of learning. The first observed class ended with the instructors
assigning two readings – one on launching complex tasks and reducing cognitive
demand, the other on designing and implementing instruction – and the second class
observed began with students working on a custom-made assignment to look at
“equations” to determine what were or were not equations, and then further to decide
which equations were true, not true, or maybe true. This activity as well was modeled
authentically, and the prospective teachers themselves noted that certain outcomes,
such as the creation of the “maybe” category, were not scripted by the instructors but
rather left to the students themselves to figure out. As such, one question from the
students was what to do if the “real” students did not recognize the need for this
category. The discussion that followed about launching complex tasks and cognitive
demand was thus grounded in practice rather than simply theoretical, and the students
experienced both facets simultaneously. Similarly in the third observation, although
the mathematical side of the task followed slightly later than in the previous two
examples, the students were asked to relate the reading to their own experiences either
as students or as teachers, and conversation and discussion flowed from this direction.
The Gutstein reading, although theoretical, was also heavily grounded in practice,
which perhaps made this task the easiest of the three, though no less focused on
linking theory and practice as a result.
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Of course, the introductory tasks of each class were not the only instance of
this clear and ongoing link between theory and practice. As the students themselves
said in the interview, the instructors had a goal of helping the students to be analytical
at all times, looking at classes not just in terms of “well/not well,” but perhaps even
more so in terms of what happened and why. They said that classes always begin with
guiding questions, which are an integral part of the Understanding by Design model of
instruction (Wiggins & McTighe, 2005). The instructors themselves talked about the
Understanding by Design model in the interview as well, at one point talking about
teaching the students how to write high quality learning objectives, which is
something that seems quite easy in theory but is quite difficult in practice. The
instructors as well as the students also talked about the main focus of the class: the
culminating learning segment assignment. The students talked about specific pieces
of the process that were challenging, sometimes expectedly and sometimes not, from
multiple rounds of revisions to self-assessment to crafting the learning objectives. For
the instructors, the focus was connecting theory to practice at every step of the
development of the learning segment, and they said that some of these connections
were implicit as part of their instruction, while others were made explicit in writing or
in class discussion. For example, they cited the National Research Council (NRC) 5
Strands of Mathematical Proficiency (i.e. conceptual understanding, procedural
fluency, strategic competence, adaptive reasoning, and productive disposition) to talk
about multi-dimensional work and multi-dimensional learning objectives, where
“multi-dimensional” means addressing more than one of the strands in any given
instance (Kilpatrick, Swafford, & Findell, 2001). As one of the students said in the
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interview, “…it helped me consider how I was going to teach the lessons, but more
about how my students were going to learn from the lessons.”
Examples from the classes are abundant, as described above, but perhaps the
most telling example of how comprehensively theory and practice were linked came
from the instructors very firm refusal to answer the question, “Of all those elements—
planning, role play, looking at pupil work, examining video, or national curriculum—
could you pick one of them that you use the most and talk me through a lesson that
uses those opportunities?”. The instructors said that they use all of these strategies in
different ways in every class and could not – and should not – be able to identify a
class that used only one of those strategies or an instance where using one of them
“most” would make any theoretical or practical sense. Though they did not give the
citations, they actually responded to this question by saying that “research shows that
attention to just one of these things is insufficient for preparing future teachers.” In
enacting their own practice, the instructors rely on the relevant theory.
A FOCUS ON AUTHENTICITY
As previously mentioned by the program director at Foster University, over
50% of the curriculum focuses on clinical practice, so a focus on authenticity and
work related to the practice of teaching is extremely explicit in the program. Over the
summer portion of the program, the prospective teachers are in the school setting for
19 days, and they are in schools for the full 180 academic days of the year. The
prospective teachers are required to make lessons plans, then enact the lessons and
provide evidence of teaching and analysis of student work. The program director
mentioned directly that you see evidence of this in the project for the heterogeneous
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classrooms course, the language policies and practice course, their unit plan project,
and finally the performance assessment at the end of the year. Included in the
curriculum are 6-8 full weeks of independent student teaching, which takes place
during the 180 academic days. The program director estimated that the total time
spent in actual classrooms over one calendar year was over 700 hours, based on a rate
of about 20 hours per week on average.
While there is of course a great deal of authenticity in the enactment of
practice, there is also a deliberate focus on authenticity even in the theoretically-based
Curriculum and Instruction classes. We have already discussed how the theoretical
warm-up activities of each of the three observed classes is in itself an example of
authenticity: the warm-up activities model group instruction, exploration of conceptual
ideas, and student-focused collaboration that the prospective teachers themselves are
taught to value and reproduce in their own placements and future classrooms. Many
more examples of such practice occurred during these observations. In the first class,
for example, when the students were discussing academic language, when Samantha
was answering a student question about what an idiom in math would look like, she
mentioned that the instructors would be asking the students to debrief in their learning
tasks, so “we’ll be practicing this lot.” Given the debriefing was an essential
component of the course (the instructors as well as the program director mentioned
this in their interviews as well, in the sense of wanting the program to produce
reflective teachers), the idea of practicing the skill to be “assessed” was imperative.
Later in the same lesson, Samantha talked about the task that the prospective teachers
had just completed and how it might enact itself in a room full of different learners,
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i.e. not adults with mathematics degrees. Asking the students to think about the fact
that their example worked perfectly for one audience, but as enacted would of course
not work well for a different audience, acknowledged that there are differences
depending on different audiences. The instructors wanted the students to think about
what would happen in a different setting in a different, real classroom. As another
example of authenticity in a different manner, after the equations warm-up in the
second observation, Samantha asked the class directly, “Let’s reflect on why we did
this, besides the fact that equations are fun.” Lessons from Emily and Samantha were
not delivered without the same level of deliberate analysis that they, and the program,
expected from the students, and they wanted to make sure that the students were able
to reflect on their own education as it related to the education of their current and
future students.
Looking at authenticity in yet another way, both Emily and Samantha were
experienced teachers, and it was clear that they were creating a safe and comfortable
learning space. Students asked questions constantly, not afraid to interrupt or ask for
clarification when something was not clear to them. Neither Emily nor Samantha
dodged or put off a single question during the entire 9 hours of the three observations,
and every question was answered thoroughly and thoughtfully. The students
themselves seemed very comfortable admitting when they did not understand
something, such as the third observation when one prospective teacher was talking
about a derivation of the number e, and another student did not understand and wanted
to see how it was done. Another way to see this creation of a safe learning space was
in the prevalence of laughter. When coding my observation notes, it became apparent
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to me that I had noted in multiple places that there was laughter in the class. Though I
am not sure that I wrote down every instance of laughter in the class, I counted 7
instances in the first class, 11 in the second, and 6 in the third. While I cannot draw
any strong conclusions simply from the presence of laughter, based on the interviews
and observations, I would posit that they are another sign of creating an authentic
classroom setting where the students and teachers develop a strong rapport that is
dramatically different from the “professional distance” that one typically sees in
American secondary classrooms.
Finally, as another example of authenticity, there was a constant focus on the
learning segment project at the end of the course, and even though there were different
themes to each and every class, they were all tied together in one authentic
assessment. The learning segment was constantly referenced, and in each class a large
block of time – on average about 30 minutes – was devoted to workshop time for the
project. As one student said during the interview, the whole learning segment
assignment was of great value, particularly the assessment component, “because of all
the theories of using feedback and formative assessment and trying that out in class. It
was cool.” The other student concurred, saying that “it gave me something to use in
the classroom, which really helped.”
THE GOALS OF MATHEMATICS TEACHING
When asked about the type of teacher she would like to see graduating from
the Foster University program (not specifically a mathematics teacher), the program
director said that a graduate should always put the interests of the student first, make
pedagogical decisions that are in the best interest of the student, and get to know
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students and how students learn best. A graduate should be “an ultimate professional,
meaning collegial, responsible, and making decision from a solid knowledge base.”
Graduates should also know that the job is not so much about teaching as it is about
the learning of the student, and as such graduates must always have a theory of
learning. She also describes an activity she calls “the quintessence of teaching.”
During orientation day (their second day in the program), student are asked to do a
drawing of “teaching.” At the end of the program year, she gives the drawing back,
and asks the students if they would want to change it. At the beginning, she says,
many of them draw a traditional classroom where the teacher is the focal point and
there is rarely a notion of subject matter. When the students do a new drawing, many
times the teacher disappears, there are many students in the picture, and the subject
matter becomes salient. There is also a lot of symbolism that expresses relationships:
hearts, arrows, thought bubbles, and word bubbles.
The instructors of the course offer a much more mathematics-specific answer
to this question, but also a strong general answer as well. This particular course
focused on the ability to design and plan a sequence of lessons, to think through
phases of design, and to learn different instructional techniques. They also strive to
create a strong link between theory and practice. Looking at the mathematics in
particular, this course and the sequence of courses focus on the meaning of
mathematics classes. They said that questions they seek to answer are what does it
mean to do or learn math? What could a mathematics classroom look like? What is
math proficiency? They also seek to build on what the candidates are learning in their
placements. The students supplemented this answer by saying that the instructors
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wanted them to see how everything is connected. For example, one student said that
this meant connecting ideas within the lesson, learning about connecting content to
prior content, and how the 5 strands of mathematical proficiency [from the NRC
reading] are connected to the content. “In other words, not learning mathematical
topics in isolation.”
In the classroom observations we see many of these ideas made explicit,
particularly in the warm-up activities. All of the activities – and the vast majority of
class – are student-centered, and students are making sense of the mathematics
themselves. Instructions for the math tasks in the third observation, for example, were
to focus on adaptation of the task and how the task would relate to student interests.
Students were asked to think about where the task might fit into a school’s curriculum
or how it could be adapted for other levels rather than being told precisely what the
“right answer” was. The students were also critical of the tasks for being a little too
straight-forward and limiting creativity, or on the other side for being too general and
avoiding the relevant mathematics concepts. The instructors never ventured whether
these opinions were right or wrong, and they never took a stand to defend the tasks.
Discussion, debate, and sense-making were the standard procedure. These ideas,
which closely follow the 5 strands, were the stated and demonstrated goal of
mathematics teaching.
A FOCUSING CULTURAL ISSUE: ENGLISH LANGUAGE LEARNERS
An idea that appears often in the classroom observation notes and the interview
notes is that of equity and language learning, not solely in the sense of learning
mathematical language, but in the sense of language learning for those who do not
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speak English as a native language. There are various ways to describe these students,
and the common terminology at Foster University is English Language Learner, or
ELL for short. Though ELLs of course come from many different nations and speak
many different languages, the majority of ELLs in this region are from Spanish-
speaking countries (Larson, 2004; Capps, Fix, Murray, Passel, & Herwantoro, 2005).
Many of these students also tend to come from lower socioeconomic backgrounds (the
percentage of students eligible for free or reduced lunch in the state is around 58%),
creating strong inequity in their opportunities. The program director mentioned equity
in several places during her interview, and her response to question 6 (“Does your
program also have a vision of the role you hope your teachers will play in terms of
service, for example, serving a particular community, or promoting national values?”)
was almost exclusively about working with ELLs. To be fair, she did admit that she
had recently been reading a book on the subject, and it was foremost in her mind
during the time of the interview. Foster University’s teacher training program
regularly conducts its own research on its own program, and one study showed that
students learned quite a lot about working with ELLs and came to the university with
the desire to work with ELLs. It also turned out that their placements were a strong
predictor: if the school and cooperating teacher were not successful with ELLs, many
of the students said that they did not want to work in a school with a majority of ELLs.
She also said that the majority of graduates who are still teaching (which she notes
with pride is well above the national average), a majority teach at Title I schools (a
Title I school is one with a high number of high percentage of children from low
income families, which in the United States correlates strongly with ethnicity).
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Issues regarding teaching students whose first language is not English came up
directly and frequently in the first observation, which was very natural for a class
segment devoted to academic language. Interestingly, some of the language used by
the instructors themselves was very complex and at times challenging for the
prospective teachers, and I believe they did so to identify those features that make
language challenging in order to make them obvious and explicit to native English
speakers. In the discussion of academic language demands, the first of the four
categories was Vocabulary, which they clarified with “important mathematical words
as well as word related to the context of the problem” as well as “words that mean
different things in ‘English’ and ‘math’ (e.g. table).” The second was
Syntax/Language Structure, which includes grammatical structures unique to math.
One of the students commented on this directly in class, saying, “It might be confusing
for English learners that [examples 9, 10, and 11] have no verbs.” One of those
examples was the phrase “side length 3 cm.” Idioms was perhaps the most clear
example of an area where language learners in general have difficulty, and the
instructors supplied examples like “rain cats and dogs” where the meaning is not
deducible from the individual words. Examples identified in the class activity were
phrases like “the pattern holds” and “x by x.” The fourth and final category was Peer
Communication Demands, which was defined as the language demands of working
together on a mathematical task. The students also requested an example of this, and
Emily gave examples of using sentence starters, knowing proper mathematical
language when going to the board, or being able to explain what you don’t understand
from what another student said. Non-specific pronouns were identified as a major
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challenge (e.g. when students say something like “it’s five,” other students or the
teachers might not know what “it” refers to), and for the Painted Cubes task, finding
ways to refer to the cubes with one side painted or two sides painted. One student
mentioned that, in an earlier class where they worked on the Painted Cubes problem,
one student referred to the cubes as “that one,” “that one,” and “that one.”
The takeaway from the lesson, for both native English speakers and non-native
English speakers, was that academic language is important and has to be built.
Mathematical language and discourse does not simply happen, and the actions,
meaning, focus, and goals are embedded in mathematical practices (Moschkovich,
2004, 2007). This is true in any mathematics class with new vocabulary and content,
but it is even truer and more challenging when the students do not yet have the basic
vocabulary to build one to create the academic vocabulary. Scaffolding language in
these tasks to emphasize what is important and necessary is a skill that is vital for
teaching ELLs and a skill that students at Foster University learn. This is in direct
contrast with classes in Japan where, though not completely homogeneous, the
population is about 98.5% ethnically Japanese, and where people almost all speak the
same language (Central Intelligence Agency, 2014).
CONCLUSION
It is important to remember that, when reading this description of the work that
goes on at Foster University, this institution was not chosen for this study as an
example of a typical teacher training program in the United States. Rather, Foster
University was chosen specifically because it is an exemplary example. As such,
some of the seeming contradictions between the previous chapter and this chapter are
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immediately recognizable as the difference between the highest standard and a few
examples from the field. “Student teachers’ learning opportunities reflect the
orientations and expertise of their instructors and cooperating teacher, and the
knowledge and skills they develop do not reflect common agreements about the
preparation required for initial practice” (Ball, Sleep, Boerst, & Bass, 2009, p. 459).
Mathematics teachers trained at Foster University receive a comprehensive education
in how to teach from highly qualified and experienced instructors, as well as in how
students learn. Group instruction is the norm, and discussion flows about how to teach
non-tracked classes versus the standard tracked classes in many American high
schools. The program director noted that Foster University students, in their
applications to join the program, note that they want to teach diverse populations and
work at diverse schools. Not many actually do, she notes, because schools are
tracked, whether inter-school or intra-school. The strength of her belief in this
principle were the words that followed – in her opinion, this separating of students is
the end of democracy.
At Foster University we see a strong blending of the theory of teaching with
the practice of teaching. While practice forms the core of the experience, theory is
strongly interwoven and forms the backbone for making informed pedagogical
choices. Most clear in this program, from the interviews to the observations, is the
strong emphasis on practice. Ball and Forzani write that learning depends
fundamentally on what happens in the classroom, and as such practice ought to form
the core of prospective teacher’s preparation (2009, 2011). They follow Ball and
Forzani’s recommendation to center education on tasks and activities, as evidenced in
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the observation, rather than simply teaching knowledge (2011). There is also a focus
around core practices – in this case the 5 Strands of Mathematical Proficiency – that
take precedence over simply providing knowledge of teaching or of mathematics
(McDonald, Kazemi, & Kavanagh, 2013). Putting together the ideas of focus on
practice, connecting teacher training programs with schools, and centering around
pedagogies and principles, is the additional idea of adding reflection and investigation
into the curricular methods class, which Emily and Samantha demonstrated numerous
times (Grossman, Hammerness, & McDonald, 2009). We also saw work during the
second observation on telling versus not telling, focusing on the situation and context
(Chazan & Ball, 1999) and students working to provide explanations of tasks, theories,
and practice as part of their required course (Inoue, 2009; Charalambous, Hill, & Ball,
2011). What was not as apparent in these observations was the use of rehearsal, which
the instructors mentioned in their interview and which was briefly mentioned in
Observation 3 in planning for the following class. Rehearsal formed an integral part
of the course as a way to help prospective teachers develop necessary skills (Lampert,
Franke, Kazemi, Ghousseini, Turrou, Beasley, Crowe, 2013). As a favor to the
instructors for allowing me into their classroom and for all their time and help, I
assisted them the following week as they videotaped the aforementioned rehearsal. I
have no observational notes from this time (as I was behind the camera), but I saw the
instructors engage in rehearsal with students and discuss how important it was as a
tool to help them learn, and how they themselves needed to engage in rehearsal in
order to teach it (Kazemi, Franke, & Lampert, 2009).
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Examining these principles and practices, we see some strong connections with
the program at Yamato University, but we also see some important distinctions as
well. Both programs have a strong sense of connection amongst classes; just as the
Yamato University professors knew what their colleagues were doing and teaching, so
too did the Foster University instructors know what was going on in previous and
concurrent classes, and they directly demonstrated that they supported each other in
their teaching. Both programs also had a strong focus on students. Yoshida-sensei
frequently asked his students to think about what different levels of students would
think, do, or know, and the Foster University instructors devoted much time to
thinking about student interests, engagement, and learning. Direct connections to
practice were also extremely visible, with Yamato University’s connection with the
practice school and Foster University’s incredible commitment to having its
prospective teachers in the classroom as much as possible. Textbooks were also a
topic in both programs, though the connection in the United States was less focused
than the case in Japan. Textbooks in the United States are not as regulated as they are
in Japan, and as such the focus in the Foster University classes was examination of
what textbooks had to offer, as well as what the deficiencies in each textbook could
be.
Also like Yamato University, we saw an issue that is culturally significant and
unique amongst this sample of universities: English Language Learners. Japan is a
Japanese-only educational system, at least in public education, and as we will see
later, Finland is a bilingual country where native Finnish-speakers (approximately
90% of the population) learn Swedish as a required language, and Swedish native
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speakers (approximately 10% of the population) learn Finnish as a required language.
Speakers of other languages are an extreme minority. The United States is thus
unusual in its linguistic (as well as ethnic) variation, and Foster University
acknowledges and even embraces this challenge. While the Curriculum and
Instruction class touched on these ideas, the program itself devotes time and resources
to making sure that its future teachers are prepared to teach and succeed in these
classrooms.
The subsequent section follows in the footsteps of the previous chapters to
describe my personal experiences teaching and learning in Finland. My experiences
there were of course informed by my teaching experiences in Japan and the United
States, as well as a full year of doctoral work at Stanford University. My role in
Finland was also somewhat different, as half of my time there was spent as a
researcher, and another half of the time as a volunteer in a bilingual Finnish and
English school. Unlike my other teaching roles, I was not in the classroom every day,
but in this case I had the opportunity to see the same students for the entire day since
this was an elementary school rather than a middle school or high school. By this time
I also had a stronger comparative lens for schools, which affected the way that I
viewed the teaching and learning activities. There is also a discrepancy in data
collection that I detail more in the final section on Finland.
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CHAPTER 8: Finland (personal)
I arrived in Finland on August 24, 2009, almost exactly 9 years after I first
arrived in Japan. For better or for worse I brought that Japan lens along with me, and I
found myself comparing my experiences more to those in Japan than to those in the
United States. I also brought along a strong sense of overconfidence: if I could thrive
in Japan, then surely Finland could not be as hard. I was wrong. It was harder.
But first, how did I end up living, teaching, and researching in Finland in 2009,
which was what should have been my second year of graduate school at Stanford?
The answer to that question very nicely connects this chapter to the third chapter. In
the 2007-2008 school year, my third and what was to be final year at Washington
High School, I began feeling restless and went looking for a challenge. Even though
there was still much to learn about teaching, I could already feel myself falling into a
rut in my role in the school. I devoted a tremendous amount of time to teaching and to
running the school math team (which I also founded my first year there), and after two
full years of this pace, I had almost no personal life to speak of. My Calculus students
had performed incredibly well for 2 years, and my distance students at the math
magnet school had also performed incredibly well and were poised to do so again. An
attempt to expand the distance program that year went incredibly poorly (the physics
teacher at my school was pressured into the program by the administration, and she
had no success at all), and I started to feel the urge to either go abroad again or
continue my education in a more formal way. In the end, I chose both, applying to
graduate school and also applying for the Fulbright Teacher Exchange Program, where
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I requested Finland as my top choice. Interestingly I do not recall knowing a great
deal about Finland at the time. It simply seemed like a fun and different place to go.
In the end I was accepted by both programs, but the Fulbright offer was for
India, not Finland, which made graduate school the easy choice. But because of my
application the Fulbright program had my name in their system, and in late 2008 they
announced a brand new program called the Fulbright Distinguished Award in
Teaching. The program was designed for current teachers of any subject with at least
5 years of experience who were interested in both teaching and research in another
country. I was eligible because of my involvement with supervising prospective
mathematics teachers, a loophole that no longer exists in the application process, and
the program was so new and announced so late that my odds were very good. The
Fulbright Program offered only six possible destination countries, and Finland was
one of them. The application was supposed to be general so that you could be sent to
any one of the six (the others being India, Singapore, South Africa, Argentina, and
Israel), but I put all my eggs in one basket and tried again for Finland. This time, I
succeeded. I took a temporary leave of absence from the university, spent the summer
taking high school students around Japan, and returned to the United States on August
3, 2009, to prepare to leave for Finland. I quickly submitted my visa application,
which was returned on August 19, and attended an introductory meeting for this
program in Washington DC from August 19-22. I was the first of the cohort to leave
for the program two days later.
All of this information not only sets the stage for how I ended up in Finland,
but also why my time there played out as it did. I left with high ideals of studying
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high school mathematics classes and with a distorted sense of my own knowledge and
capabilities. With the program being completely new, there was no structure in place
to assist me, and the Fulbright program in Finland did not know how to categorize me.
Sometimes I was oriented or grouped with the Fulbright professors, and sometimes
with the Fulbright students. When I arrived in my new town and new home, the room
that I was given was a tiny, cramped dorm room that inexplicably was occupied upon
my arrival, and I had to find a place to sleep for my first two nights. When I met my
advisor, Teemu, I was given some introductions to a couple local schools, but the
teachers there did not have any interest in someone observing them. Repeated efforts
to get into classes failed, aside from a few one-time visits, so my teaching and
researching efforts went nowhere for several months. When Teemu asked me if I
would be interested in visiting an elementary school instead of a high school, my
journal records tell me that I was not interested and almost turned it down (and indeed
almost just did not show up, which would have been incredibly poor manners on my
part). But on November 6, 2009, I attended a few classes at Kalevala Elementary
School and met a teacher named Janne. Suddenly I had a place to teach in the most
unlikely place: a sixth grade classroom.
Kalevala Elementary School is a fairly typical Finnish elementary school
except that it has a bilingual English/Finnish program as well as a bilingual
German/Finnish program. Students starting in first grade and continuing through sixth
grade learn the standard Finnish curriculum, except they learn it in both languages.
Classes are conducted primarily in the “second” language, though sometimes more
complicated English grammar lessons or other challenging points were explained by
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Janne in Finnish, just to make sure that the students understood completely and in both
languages (as often in classes like science the students needed to learn vocabulary in
both languages). Janne is Finnish and a fluent English speaker with only a trace of an
accent, as he also learned English as part of his regular schooling rather than by living
abroad or attending a specialized school. His father is a fairly well-known Finnish
educational researcher, so teaching and learning were a substantial part of his
childhood. Janne had been teaching at Kalevala for the past 7 years; two years with a
different class of students, and five years with the current class of students. In
Finland, elementary school students are often kept together in the same group with the
same teacher for 2-3 years. At Kalevala, the dominant idea was to keep the students
together for as many years as possible. Janne’s group of sixth graders had been
together since first grade, and Janne had been their one and only teacher9. The
teachers at Kalevala had an excellent working relationship, however, so the students
did on occasion see other teachers. Janne was musically talented and would often
teach music or other classes for other teachers, and in return the other teachers would
teach subjects where they were especially skilled. But for the most part, the students
worked with Janne all day, every day. I was able to join his classes regularly, develop
a relationship with his students, assist with classes, and later teach classes on my own
in English and mathematics.
At the same time, I also needed to figure out my research plan, which was
much more challenging. Working with my advisor, Teemu, at the university, we
9 Teaching the same cohort of students for multiple years is not unique to Finland. A study in rural
China, where this practice also exists, showed that less-experienced teachers experienced more positive
results in mathematics for their students as a potential result of having the same cohort of students (Park
& Hannum, 2001).
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developed a plan for me to conduct a study with some of the current prospective
mathematics teachers. Finnish teachers almost always are certified in multiple
subjects, so some of the students were most interested in mathematics, while others
were more interested in subjects like chemistry or physics but also needed to be able to
teach mathematics. As all of the theory classes were taught exclusively in Finnish,
and I could barely introduce myself in the language, the research consisted mainly of
interviews with these students, and we focused on ability grouping in classrooms.
Details from that portion of my time in Finland will be described in great detail in the
following chapter.
The Fulbright grant only covered 6 months of time in Finland, and I wanted to
stay through the academic year to finish with my students at Kalevala, so I applied for
and received a research grant through the Center for International Mobility (CIMO).
Interestingly, the headquarters for CIMO is next door to the Fulbright headquarters,
and the good relationship between the two programs probably helped my application
to be accepted, which provided just enough funding for me to stay in Finland through
the end of May of 2010. I also later learned that the director of CIMO was Pasi
Sahlberg, whose 2011 book Finnish Lessons and whose lectures on Finnish education
are world-renowned (and in a later conversation with him, he remembered my
application). As both Fulbright and CIMO made this experience and learning
possible, I take every possible opportunity to thank them for it. With all due respect to
Pasi, my Finnish Lessons would never have been learned without their financial
support.
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THE AMERICAN (PISA) QUESTIONS
When I was first getting to know Janne, who is now a wonderful friend and a
model for many of the qualities I want to possess as a teacher, I spent most of my time
observing, learning, and asking basic questions. He was patient with them, and as I
learned later, many of them were questions he had answered numerous times before. I
was not the only person Teemu had sent to visit Janne’s class, and over the course of
the year I saw many other groups from many other countries observe for an hour or
even a full day. One of the things that strikes you in the class is how well Janne
knows his students, which is not surprising considering that he has known most of
them for over 5 years. He knows their parents, what clubs they belong to outside of
school, what subjects they like, and what subjects they struggled or struggle in. Janne
is also an amazing dedicated teacher. So one day I asked the American Question:
what happens if students are “stuck” with a less gifted teacher, or if there is a student
that you really cannot get along with?
Janne laughed and explained that Finnish students are expected to learn to cope
with challenges, and it is only by being in such situations that they learn how to
succeed. In life, there will always be people you don’t like but have to work with, so
why should students at such a young age be allowed to give up so easily? And the
same is true for teachers as well. Teachers need to be able to help students of all
types, and they cannot simply pick and choose who learns and who does not. And
Janne was quite honest about the reality of the situation. He said that there were still a
couple students that he did not like, and that they did not like him much either. But
they worked together, and the students were learning; dislike was no excuse for not
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doing his job. Even after careful observation of the students, I could never really
notice anything that showed dislike on either side, as both the students and Janne were
always engrossed in daily activities. Janne also challenged the American idea of only
having a student for a single year. In that short time, the student or the teacher can
simply count down the days until the school year is over, and no one is really required
to make the effort to get along. But when you know that you will be together for two
years – or three years, or six years – you are more inclined to fix problems before they
develop. Janne did also admit that this rule, like any rule in the real world, does have
caveats. If there really is a bad match with a student and teacher, then a student can be
moved to another class. It is very rare, though.
The other part of the American Question might be better phrased as the
International Question: what makes Finland do so well so consistently on the PISA, a
measure of international comparison? Later in our relationship, when we could easily
talk over a beer after work, Janne rolled his eyes. He basically said this:
Everyone talks about PISA. PISA this and PISA that. People come to my
classroom all the time looking for secrets, and I want to ask them for help.
Student X is not doing his homework. Student Y is starting to get interested in
boys and pretending to be dumb. What people do not understand is that this is
a real class with real students. We have problems every day, and every day I
have to figure out what to do.
I spoke with other teachers who said much the same thing, though more
obliquely, to say that test scores really do not matter to them. Teemu himself said
much the same thing, and he does work with PISA. For him, the good PISA results
benefitted him and his university as people would constantly come to visit, and
universities abroad would want to hear more about what happens in Finland. I saw
Teemu give a presentation about the Finnish education system many times, and I met
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several translators who then interpreted for their group (Teemu had a stock
presentation that he always gave in Finnish, though he said that he also had an English
one somewhere. The translators were always Finns, though, so he stuck with the
Finnish one). The message was overwhelmingly clear: yes, the PISA results are good.
But that has nothing to do with what is happening in my classroom today.
A. Relationships
One of the interesting things that I noticed in Finland concerning relationships
was the titles used to address people of differing status. In Japan, even though
students at my school would sometimes use familiar nicknames for teachers, most
often they would refer to teachers and professors as sensei, or loosely, teacher. The
term has much more significance, however, as sensei is also the term used for doctors,
for example. In the United States, at Foster University, relationships with professors
were much more casual, though in many cases in universities across the United States,
professors are referred to as “Professor” or “Doctor” followed by a last name. Primary
and secondary schools are much the same way, and teachers are usually addressed as
“Mister” or “Miss” and their last name. In Finnish schools, however, every teacher
that I met, whether at elementary school, high school, or university, was called by his
or her first name. Though this may seem like a small detail, it was a clear signal to me
that school relationships were different than what I had encountered in Japan or in the
United States.
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1. Student and Teacher
The day before I first visited Janne’s class at Kalevala Elementary School, a
student named Clara became a new student in his class. Clara was from the United
Kingdom, and her parents had just moved to Finland for work. The main reason that
Clara’s parents chose to enroll her at Kalevala was the fact that Clara did not speak a
word on Finnish, yet they wanted her to be enrolled in a Finnish school rather than an
international school.
Clara clearly did not want to be in Finland or in this class, and she was
painfully and almost aggressively shy. When Janne went over to speak with her at one
point, she audibly and rudely said “don’t look at me” while staring at her shoes. This
behavior persisted for well over a month. But Janne was patient, and he gave her
space to become calm, to adjust, and to slowly acclimate to the class. By the end of
the year, when I would come to Janne’s class, Clara’s voice was the one I could hear
from the hallway before I entered the room.
In some ways, Janne did not do anything special in his treatment of Clara; the
relationship between teachers and students in this school was characterized by respect
for each other as different human beings. As mentioned earlier in the American
Question, the natural assumption was that people are different, and it is necessary for
everyone to learn how to get along. The most frequent scene in Janne’s classroom
was one of friendship and laughter. Janne would often tell jokes to the students
(whose knowledge and understanding of humor in English, including puns, was
remarkable), and they would often make fun of him and tease him back. Although
Janne was skilled at playing the guitar and other instruments, he could not sing, and
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the students would often complain about having to listen to his awful voice. But Janne
would also chastise students when it was necessary, and he would not mince words or
try to find a “soft” way to tell the students that something was wrong. At those times
the room, aside from Janne, would get very silent, but the students always seemed to
understand that Janne was there to help them. The respect and friendship, even with
the different status levels, was still evident. Part of this relationship comes in part
from the fact that the relationships were so long-term, but another part comes from
Finnish culture where directness and honesty are part of the normal way of interacting
with others.
Janne was also willing to do almost anything for his students, and the extra
hours that he put in after school and on his own time helped establish this relationship.
In an effort to make English more fun and more relevant, Janne wrote his own
modified version of Hamlet, and he was responsible for casting, directing, set design,
and all the aspects of the production that was performed at the end of the year for the
parents of the students. Practices went late into the night some days, and Janne even
arranged for the students to have a sleep-over at the school several days before the
performance. The students loved it, and Janne used the time during these evenings
mainly to work on the play, but he also would let students work on their own while he
helped them with homework or other issues. Even though the purpose of the time was
play practice, his devotion to his students was not limited to one task at a time.
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2. Student and Student
Just like students and teachers must learn to work with each other over the
many years of being in the same classroom, so too must the students learn to work
with each other. It was easy to see in Janne’s classroom which students had strong
friendships, though all of the students were able to work well together. Janne would
often mix up groups so that everyone had a chance to work with everyone. The idea
of separating the students so that some would never have to work together did of
course occur to him, and on occasion he would deliberately not put two students
together when it was warranted. The students were typical youngsters, and as such
sometimes close friendships would degenerate into bitter fights. Janne was
surprisingly in tune with these events and would know the background and causes of
these fights, which enabled him to talk with the students both individually and
together to repair the relationships. Towards the end of the school year, when the
students viewed me also as a teacher and a full member of the class, I was able to
work with Janne in a case when two best friends were no longer speaking to each
other. One of them had just gotten her first boyfriend, and the changed relationship
and jealousy were hard for the 6th
grade students to manage. With help from Janne,
they talked and worked things out.
Relationships between students who were not as close were still very strong,
and the students were very supportive of each other and their educations. Such an
attitude was the norm, and it was clearly expected by both Janne and by the other
students. I did not have the chance to see how this relationship crossed over into other
classes and other grades – during lunch time the class always ate together at the same
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long table in the cafeteria, and during the multiple recess periods each day, the
students broke into smaller groups, but they still stayed with their own class.
3. Teacher and Teacher
While the schools I visited in Finland had areas for teachers to meet and
congregate, overall the structure of the schools themselves was much like an American
school: each teacher had his or her own room, which also served as the teacher’s
office. More like in Japanese schools, however, the Finnish teachers I met had far
more interactions and shared responsibilities more. In one school I visited, the math
and science teachers had their own separate “lounge” area where they met for
meetings after school and during off periods. At another school, while there were no
lounges by discipline (as it was a much smaller school), there was a similar lounge
area for all teachers that served the same purpose. Janne’s school had a similar large
lounge where most of the teachers could be found before the school day began and
where some, though not all, could be found at the end of the school day. Though
teachers were sometimes more isolated during the day, there were more opportunities
for collegial interaction than in an American school.
Also, following what I mentioned in the American Question section, there is a
well-understood idea in Finnish schools that teachers who are struggling will receive
help. I had only a couple opportunities to talk with principals in Finland (once in
Janne’s school, and once at the local normal school), and the principals confirmed that
teachers do struggle at times, and if and when that happens, they organize other
teachers to work with and assist them. Janne’s principal also mentioned that he
himself has taken that role with teachers, though he admitted that doing so was easier
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for him as he used to be an elementary school teacher than it might be for a high
school principal with more specialized teachers. For obvious reasons I was not able to
identify and talk with a struggling teacher, but the impression I received from other
teachers and principals is that such help would be, in general, welcomed and
understood rather than fought. In other words, working with other teachers in this
context seemed to be an acceptable and understood practice.
B. Expectations
As with both Japan and the United States, there are expectations of what
happens in school that connect closely with the relationships discussed previously.
Here again we see some cultural and systemic differences (as well as similarities)
regarding how teachers and students work together in the school environment.
1. Everyone Participates
One of the most interesting things I observed in Finland was a standing belief
among teachers, which bore out in practice, that all students needed to participate in
classes and lessons. Before I say more about this, I must make two notes. First, this
observation comes from a limited number of discussions with a variety of teachers and
a great number of talks with some exemplary teachers. I cannot make a claim as to
what happens in all classrooms with all teachers, but at the same time, I was struck by
this pattern. Second, “participation” has a very specific meaning. Take for example
the case of Clara, who at first could not even handle having the teacher notice her
presence in the classroom, let alone join in an all-class discussion. Janne worked hard
to increase her participation to levels that he considered normal in his classroom, and
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the increase in active participation occurred over a period of months. Yet he was
always conscious of her presence and encouraging her participation in increments.
Perhaps participation for Clara meant that she was able to talk with a neighbor or even
engage in the day’s task, but he always expected some level of participation. In some
ways I think the term engagements also works well here, but perhaps participation is a
better choice of words because it allows for a lower level of connection with the work
and is more fluid than engagement. To clarify, on some days a particular student
might have been having an off day or no feeling well, and the student was not engaged
in the work. Janne always expected the student to participate, though, on some level.
One other area where I saw interesting participation was in the woodworking
and home economics classes. For one semester, all the students in the 6th
grade class I
worked with took a woodworking class (led, as always, by Janne – teachers had to
have a very wide set of skills), and the other semester the students took a home
economics class that involved skills such sewing and cooking. When I took these
classes myself as a student, the boys typically liked the woodworking and participated
more, while the reverse was true for the home economics class. For Janne’s class, and
in fact for all the classes at Kalevala, the groups were separated by gender. During the
second semester, when my involvement with the class was at its highest, the girls were
all in the woodworking class. I was surprised when I saw this (and even more so at
the level of sophistication of the work and the potentially dangerous tools being used),
but I was even more surprised by how every single girl in the class was fully engaged
and participating in the work. The same was true with the boys, who were all sewing
cross-stitch patterns and having a lot of fun. There were still differences by gender at
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the school – the girls wanted to sew, and the boys wanted to saw – but Janne said that
separating the students allowed everyone to participate and made it possible for
everyone to participate equally. It was a strange example of separate but more equal.
This expectation of participation also extended to physical education time, as
well as recess time, which was interesting for me as someone coming from an
American background. I remember students opting out of gym class and other
activities because they had a note from home, but in Finland, everyone participated
except when sick or injured. Recess was probably the most noticeable example, as it
happened numerous times each day. Students would sometimes ask to stay inside
during recess, and Janne and the other teachers consistently said no. The rule at
Kalevala was that students would not be required to go outside if the temperature
dropped below -20 degrees Celsius, though a couple of times the temperature did drop
that low, and the students still went outside. Rules at other schools were similar,
though the exact temperature requirement differed (Abrams, 2011). Adults were
present during recess time, though they were present more in case they were needed
than there to supervise. The students also did a fair amount of cross country skiing
and ice skating during the winter months, and in the summer months they occasionally
went out to play Finnish baseball (called Pesäpallo, it is the national sport of Finland,
though in my experience ice hockey was more of the national obsession). I enjoyed
this game in particular because it seemed as crazy to me as it did to Clara, who not
only had never played Finnish baseball, but also had never played American baseball
either. Every student played, and even the reluctant ones could be seen standing in the
outfield hoping that a ball did not come their way.
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2. The Role of the Teacher
As the above stories and explanations have already shown, teachers in Finland
are expected to be multi-talented in terms of teaching ability (e.g. being able to teach
multiple subjects in high school), mentors or helpers to other teachers, and
knowledgeable about their students and their needs. Elementary school teachers are
also expected to be able to develop deep relationships with their students and problem-
solve relationship issues, even when those issues are between themselves and a
student. Working extra hours is normal, although not as expected as one would see in
Japan, and more in line with one might see in an American school.
One additional note, related to the above, is the role of the teacher in terms of
working with parents. Here again the Finnish teachers’ roles fall somewhere in
between what one might see in Japan and the United States. Japanese teachers have a
strong and defined set of requirements in terms of meeting and working with parents,
while American teachers have perhaps one or two official open houses or parent nights
per year and then work with parents on a case-by-case basis. While Finnish teachers
do not have such formally defined requirements and deal with parents on a case-by-
case basis as in the United States, the long-term nature of the relationship with the
students directly affects the parental relations. Parents in Finland would often cross
the line between what an American teacher might see as the division between school
and home. Behavior issues at home were a topic to be discussed with the teacher. In
two cases with the students I worked with, the parents would talk with Janne about a
past divorce or a divorce in progress that was affecting the student. In essence, the
role of the teacher in Finland included a slightly higher degree of parenting.
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3. Teachers are Qualified
The statistics for desire to become a teacher in Finland, and how hard it can be
to get accepted into a program, are well known and easy to find. In an often-cited
national poll, over 26% of upper secondary school graduates rated teaching as the
most desirable profession, despite the fact that the salary for teachers was just barely
over the national average (Sahlberg, 2007). As a result, admission into teacher
education programs in Finland is highly competitive, in some cases with only 10% of
applicants accepted (Westbury, Hansen, Kansanen, & Björkvist, 2005) and an overall
acceptance rate into all programs of about 25% (Sahlberg, 2007). Of course, when I
was in Finland, I was able to experience this reality rather than just read the statistics.
In terms of the prospective student teachers I met while in Finland (which
unfortunately was limited to the tail end of my time there while collecting research
data), I found them to be highly engaged, talented, and intelligent people. Several of
the students I interviewed revealed to me that they were not at all interested in my
work, but they valued the opportunity to practice their English and speak intelligently
about mathematics education. These students were aware of current educational
research and how theory and practice could be linked to improve classroom
instruction. When I told one of them that I was from Stanford, she asked, “Do you
know Hilda Borko? She’s my favorite researcher.” The expectation of quality, at
least in terms of recruiting prospective teachers, is at the very least a proven fact
through the rigorous application process and demand far exceeding supply.
In terms of the practicing teachers I met, I also found this expectation of
quality to continue. Teachers spoke well of their colleagues, both the known ones and
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the theoretical ones at other schools across the country. That is not to say that
everyone always spoke well of each other; there were complaints about style and
personality, as there would be between any two people or any two professionals. But
teachers consistently believed that their colleagues were quality teachers. Students
always had good things to say about Janne (aside from the playful teasing that was
commonplace), and the parents that I met during the performance of Hamlet also had
wonderful things to say about him. My experience with the later was limited,
however, as most of the parents spoke to Janne exclusively in Finnish. Others came
over to introduce themselves to me, as I had been working with their children for some
time, and they wanted to meet me in person. The expectation that teachers were doing
a quality job was pervasive, and the assumption naturally seemed to be quality rather
than an expectation of lack of quality. It was a feeling that I personally rarely had in
the United States, but Janne said that, even when he was feeling down or
overwhelmed by teaching or frustrated when dealing with parents, he always felt
valued.
CONCLUSION
I left Finland on May 30, 2010, having said goodbye to Janne and my students
on May 26 as they left for a class trip to Slovenia. I left with mixed feelings, knowing
that there were many things that I could have done better to maximize my time and my
learning. I thought back to my time in Japan and how it took me three years to learn
what I had, and how I felt that three years completely immersed in teaching was not
enough time to learn everything I wanted. One year only partially immersed in
teaching left many unanswered questions.
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Still, I left with more knowledge than I had when I arrived, and though I no
longer had a class of high school students where I could easily apply my learning, I
would have the chance later on to teach mathematics at a local community college.
There, to a smaller extent, I was able to apply my learning, such as continuing to get to
know my students on a personal level and help them with issues beyond what they
were learning in my class. I also adjusted how students participated in my classes, and
for those students who were not inclined to raise their hands (whether from shyness or
lack of English language skills or anything else), I made sure that they had
opportunities for group conversations or even opportunities for me to check in with
them individually. I truly believe that this change made a difference, as faculty who
had taught my students in previous quarters would comment that “that student never
participated in my class.”
My experiences with the prospective mathematics teachers also add additional
sources of knowledge and data that I will explore in more detail in the following
section. Thus, in some ways this section will follow along in parallel with the
preceding sections detailing Japan and the United States, but for this reason and
others, the structure will diverge.
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CHAPTER 9: Finland (teacher training)
There is a mystique that surrounds education in Finland. Its reputation in math
education is a positive one, and it is worth contributing knowledge concerning the role
that teacher education plays in the process. Research data for Finland come from two
distinct sources, described below. As both of these data collections involve interviews
with professors, interviews with students, and different universities, the pseudonyms
involved can become a little hard to follow. To make reading easier, throughout this
chapter I will occasionally reinforce the connection, and at the end of this section I
have included a brief list of the people and places involved.
Catherine University was selected by the CATE project as one of their two
study sites in Finland and the only one where the research was conducted in Finnish
(the other university was located in the Swedish-speaking part of Finland). Of all the
data in this study, these are the only data that were not collected by me personally.
Like both Yamato University and Foster University, Catherine University is
internationally known for many programs, including its teacher training program, and
is located in a major metropolitan center. The teacher training program at Catherine
University, like all such programs in Finland, is a Masters degree program. It is
important to note, however, that the Masters program in Finland is not a separate
program as it is in the United States; students enroll in the Masters program from the
onset of what would be considered their undergraduate studies and typically graduate
somewhere around 5-6 years later. The teaching component of the program occurs
toward the end of the program at what are called normal schools, or sometimes
practice schools, which are high schools that have a direct connection with the
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university but are otherwise typical public schools. Coursework takes place at the
same time as the teaching component and includes pedagogy classes as well as content
classes and research classes, as all prospective teachers are required to conduct
research to obtain their degree. Also, it is common for students to obtain licenses in
multiple disciplines at the same time, so a student might be graduating with licenses in
both mathematics and physics, for example.
Data for the CATE program were collected by a Finnish graduate student in
October of 2012. The relevant pedagogy course (which the Finns refer to as
Didactics) takes place two times a week from 12:15 to 1:45pm, and observations were
conducted twice in the first week of October, once in the second week, and once in the
third week. There are also notes from a class on October 16 for a class from 10:15 to
11:45 (the same day as one of the Didactics course observations), but it is unclear
what class this is. There are also transcripts from a teacher interview and a student
interview.
These data do provide some interesting and valuable insights into mathematics
teacher training, but the data are somewhat limited in use. The teacher interview is
perhaps the strongest source of information as the notes are detailed and mostly follow
the CATE interview protocol. The class observation notes are at times incomplete or
inscrutable, as they are often a few words or a phrase describing something that
happened in a way that might be meaningful to the observer/writer but are mysterious
to an outside reader such as myself. The notes for the first observations are the most
detailed, and the final observation notes are the least detailed. The student interview,
which is the only data source provided in English (all the others were originally
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written in Finnish), is almost completely without value, and it is also unclear whether
the interview is with one student or two, as the protocols required. In addition, the
interview almost completely ignores the CATE interview protocol. The following is a
sample of data directly from the notes that is typical of this interview:
Interviewer: Do you have any courses about learning difficulties?
Student: There were a lot of special education courses.
Interviewer: Were they useful?
Student: Yes.
Some basic information can be gleaned from the interview, but little of substance.
Fortunately I have access to another data set, which was collected by me
during my time on the Fulbright program. The interview protocols for this project are
included in Appendix D and Appendix E. While the purpose of these interview
questions is to understand more about Finnish prospective mathematics teachers’
experiences with, and opinions, regarding heterogeneous and homogenous groupings,
there are some questions about their program in general and their experiences in the
program. These data of course are not parallel to the CATE data and have limitations,
but in conjunction, they help to form a more complete picture.
My Fulbright project takes place at Lapinkaari University, which is also a well-
known university in a major metropolitan area in Finland. Its reputation
internationally is not as established as Catherine University, but within Finland it is
highly regarded. Near to campus is what is called a Normal school, which is a public
school that is connected to the university and allows researchers and student teachers
access to classes and students. Interviewees for the project are selected from the 2009-
2010 cohort of students taking mathematics Didactics classes based on their interest in
being interviewed in English. A total of 16 volunteer after a presentation in their class
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(there are two classes, one taught by Hanna, the other by Pauli), and 13 eventually
agree to, and sit for, an interview. These interviews are conducted entirely in English,
and the IRB protocol for this project allowed for audio recordings of the interviews
followed by transcription. Data are collected over a one month period beginning in
March 2010.
Project Pseudonym Context
CATE Catherine University CATE data collection university
CATE Saara Professor of Didactics classes at Catherine
University
Fulbright Lapinkaari University Fulbright data collection university
Fulbright Hanna Professor of Didactics at Lapinkaari
University
Fulbright Pauli Professor of Didactics at Lapinkaari
University
Fulbright Students 1-13 Prospective mathematics teachers at
Lapinkaari University TABLE 1: THE CATE TEACHER AND STUDENTS
The professor interviewed for the CATE project data collection At Catherine
University is named Saara. Although I did not collect this information personally, in
an interesting twist of fate, I happened to have met Saara while she was working at
Lapinkaari University and even attended one of the classes she taught at the normal
school. She taught the Didactics course from 2002 to 2009 and was not someone I
interviewed as part of my study. Saara was also working on her PhD at Lapinkaari
University at the time, and she graduated in 2010. The interview notes say that her
dissertation is about mathematics teachers and experiential teaching in mathematics.
Saara also spent three and a half years as a full time teacher at the normal school
teaching mathematics, computer science, physics, and chemistry, and she also worked
as a textbook author, writing eight short math books and teacher guides.
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Saara’s Mathematics Didactics class should have met twice a week, though the
dates on the research notes appear to be inconsistent (one week was Monday and
Tuesday, another week Wednesday only, for example). Saara states in her interview
that the planning of teaching was perhaps the most emphasized aspect of the course,
with 15-20% of the course focusing on theory and the remainder focusing on practice.
The field notes indicate the topics of the class and give general notes, and though it is
impossible to say accurately how detailed the lessons are, there are many notes that
support the idea that practice is highly emphasized. In the October 1 class, for
example, the topic is listed as “mathematics learning difficulties.” Class begins with a
group discussion of what types of learning difficulties the prospective teachers have
encountered and how those difficulties could have been prevented or supported. A list
of student answers under the heading of “why students do not like math” includes
weak self-motivation, emotional problems, inefficient study habits, lack of support at
home, and neurocognitive problems. About 30 minutes into the class, the topic shifts
slightly to how a teacher can support students with learning difficulties, with being
encouraging, providing scrap paper, encouraging the development of learning
techniques, deconstructing tasks, and differentiation listed as examples of supports.
Much of the remainder of class is spend discussing how these factors play out in the
classroom, with what appears to be discussion about parental support (or lack thereof)
and parental lack of interest or ability in mathematics.
In contrast, the Didactics class the following day (October 2) begins with a
mathematical warm-up asking students to incorporate a mathematical phrase of
formula into a cartoon. Examples of the phrases include the sum of the angles in a
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triangle, surface area, and the volume of a cone, to name a few. After 15 minutes, the
class begins to discuss Hanna’s dissertation, which dealt with the teaching of discourse
and involved mathematical cartoons to some extent. I imagine that Saara and Hanna
interacted quite a bit before and during my time at Lapinkaari University, so it is only
somewhat surprising to see Hanna’s name explicitly referenced in the field notes. The
details in the field notes are sparse (the description at 20 minutes into the class reads
“the teaching of discourse,” and 25 minutes in is “learning discourse” and “will also
do this by example” to cover the entire 10 minute block of time). What is relevant
here, however, is that the October 1 class clearly deals heavily with the practice of
teaching, and the October 2 class deals with mathematical tasks and connecting recent
research and theory to the practice of teaching. As the class continues, the students
begin to look at mathematical tasks (such as the volume of a gas tank, or something
with the Leaning Tower of Pisa) and discuss the nature of discourse in these activities.
Other classes deal with Finnish curriculum and the educational system or with
mathematical ideas such as percentage or probability.
THE FULBRIGHT TEACHERS AND STUDENTS
The complementary data set for Finland comes from a total of 15 interviews (2
with professors, 13 with prospective mathematics teachers) from Lapinkaari
University. The semi-structured interview protocol (see Appendix D and Appendix E)
is designed around questions concerning mixed ability grouping, but it also includes
questions that provide supporting information about the teacher training program.
There are two professors of mathematics pedagogy classes (which will
hereafter be referred to as Didactics, as that is the English word that the interview
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subjects use): Hanna and Pauli. Hanna, whose dissertation was referenced above in
the CATE data, has been teaching at Lapinkaari University for 9 years at the time of
the interview. She is also working on her dissertation at the time, which she completes
the following year. Hanna has both a mathematics and physics background and taught
for 7 years at the high school level and 2 years at the primary level. She also teaches
Didactics classes for some of the science classes at the university. Pauli has been
working in the Department of Teacher Education at Lapinkaari University for just a
little over 4 years. He actually worked for one year in the Mathematics Department
prior to changing departments, and before that he was a secondary school teacher for
“20 or 25 years.” He also has a background in both philosophy and computer science,
both of which he taught at the secondary level.
When I ask Hanna and Pauli “in your opinion, what are some of the most
important things that the students here learn about teaching,” Hanna replies
“cooperation” without hesitation. The fact that she answers without hesitation is
important because Hanna’s English level was not so high, and she stumbles in
answering every other question in the interview. When asked why cooperation was
her answer, she says:
They have to take cooperation with each … teachers. They don’t … they can’t
work at school alone. They have colleagues, they have the other teachers
around them, and we have to make together work, because we have the same
pupils. And then, there is cooperation with teacher and students. So,
cooperation is the most important word.
Cooperation is something that Hanna teaches directly in her courses, and she
deliberately organizes group work as much as possible to foster it. Group work is also
something that she highly values in mathematics classrooms, so she models what she
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values. Pauli, on the other hand, offers a more philosophical answer (perhaps not
surprising, given his philosophy background): “Teaching is not at all a single thing.”
Part of what prospective mathematics teachers need to know is content, another part is
working together, and another part is pedagogy. Though Pauli in some ways
effectively dodges the question, his answers are consistent with what I later learn from
students are essential parts of what they are learning as teachers.
The prospective teachers have a similar question in their interview: “What do
you think are some of the most important things you’ve learned so far about
teaching?” The interview also ends with the question, “Finally, what would you say
are your goals as a teacher? In other words, at the end of the school year, what would
you want your students to say about their experience learning mathematics?” which
also helps in a small way to understand what the students are learning about teaching
and how this information affects how they want to be perceived, though for the most
part answers are more personal than programmatic (e.g. wanting their students to like
math, just like they do). Some of the students have already spent a fair amount of time
working and teaching in the normal school, while others have not yet spent any time
student teaching. Their background experiences are also quite diverse, with some
students having volunteered or worked in schools, and others who have not and for
whom teaching is not necessarily their end goal (teaching certification in Finland is in
and of itself a valuable credential to have, much like in other countries, as it shows a
diversity of skills). The students are also candid in their interviews, which is a fairly
stereotypical Finnish trait. Students are not afraid to say that they are having trouble
making sense of some of the theory they are learning in class, that they support or
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disagree with an idea like mixed performance groups, for example. The majority of
the interviews take place in a coffee shop in the Lapinkaari University student center,
which is an informal and relaxed location. Levels of English vary amongst the
students, and some interviewees remain fairly terse throughout the interview, while
others sometimes say what amounts to almost a full page of single-spaced transcribed
notes for a single question. Overall the students provide a wealth of interesting
information that helps me begin to understand their views on teaching and some of
their conflicts over the value of mixed performance groups.
A PLACE FOR THEORY BUT MOSTLY PRACTICE
While there is ample evidence of theory being taught in classes, coming from
all three professor interviews and from the interviews of the 13 prospective teachers at
Lapinkaari University, there is a clear focus on practice. Saara, from Catherine
University, states directly that she deliberately aims to have her Didactics classes
focus only 15-20% on theory, with the rest of the time spent on “practical work.”
References to research are apparent in the field notes from her classes. As previously
mentioned, the dissertation written by Hanna (from Lapinkaari University) is a topic
of discussion in the October 2 class, and there are references to research articles and
researchers in other classes as well. For example, in the October 1 class, there is the
note “Dyscalculia manifestations ICD October 2009.” A Google search reveals that
the most likely reference is the International Classification of Diseases, published by
the World Health Organization. More mathematics-specific references can be found
for example in the October 9 class, which references “Lahdes 1997 page 39” and
“Millar 2001.” A search through Google Scholar shows a Finnish book by Lahdes,
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which translates as “New Didactics for the Comprehensive School,” and further search
reveals Lahdes to be an author of curriculum work and research into Didactics classes
(Lahdes, 1997). A discussion of this work in another source (Simola, 2014) quotes
Lahdes as writing that there are three “determinants” in teacher work. The first
determinant, the pupil, consists of “singular, unique, and individual pupils” who are
the objects of education but who also interact with the teacher and the other students in
the class (p. 17). The second determinant is “the branches of knowledge,” which is
“the knowledge and skill to be transmitted through education to the next generation,”
and the third determinant is “society,” as a “personified actor” who “pays the costs of
education, answers for the administration of the schools and proposes general goals for
the comprehensive school in particular.” Millar appears to be a researcher in science
education, particularly in physics and chemistry, who writes about such things as the
nine “ideas-about-science” that should be taken into account (Osborne, Collins,
Ratcliffe, Millar & Duschl, 2003). In the October 9 class, these research-based
references appear within the first 30 minutes of class and are referenced all the way
through the end of class. This situation is unique amongst the field notes, however, as
the majority of classes have only a single reference to research. The October 16 class,
for example, deals almost exclusively with mathematics topics and the teaching of
them, and no references to theory appear at all.
At Lapinkaari University, Pauli says little about theory in classes, saying only
that “teaching is not at all a single thing.” Hanna is a little more direct and states that
every class needs to follow “clear theory” every time, and that she makes sure that
“Each of the Didactics seminars is interspersed with three days of familiarization in
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the training school.” Once again, the focus is on practice rather than theory, though
theory does make an appearance and is considered essential by the professors. The
students at Lapinkaari University are only asked about theory as it relates to mixed
performance group classes, and responses are thus somewhat limited. Many of the
students are not aware of same performance group classes (or tracked classes, as in the
United States), and the idea of learning theory for classes that do not exist is confusing
to them. The prospective teachers do, however, talk about instances where they have
or have not learned about strategies for teaching classes where there is a range of
student ability, though they are likely to talk more about what they have learned in
observations of classes or from their cooperating teachers (if they are engaged in
practice teaching). Given the strong connection between the Normal school and the
university and the emphasis on practice, this result is not surprising. Specific to the
different ability groups and the question of how the teacher training program at
Lapinkaari University prepares them, the students frequently cite differentiation (or,
lacking the precise English word, describe the process of differentiation). Group work
is mentioned as a strategy for all students, even when the prospective teachers are
talking about high performing students or low performing students. Advanced work
and extra tasks are common strategies to assist the faster students, while more
remedial tasks, repetition of tasks, and emphasis on the basics are cited as practical
strategies for low performing students.
A FOCUS ON AUTHENTICITY
The role of authenticity in the classes at both Catherine University and
Lapinkaari University directly relates back to the strong role of practice in the courses
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taught by Saara, Hanna, and Pauli. Authenticity here refers to dealing with the
realities of teaching and preparation for the actual practice of teaching. For example,
in the October 1 class at Catherine University, Saara talks about calculation
difficulties, attention deficit disorder, and reading and writing difficulties. She then
spends the next 25 minutes talking about how the teacher can support students with
these challenges, giving strategies such as being encouraging and deconstructing tasks,
and noting that factors such as earlier grades or scores are not as important as one
might think. These lessons are combined with the realities of teaching rather than just
discussion; prospective teachers in Finland spend a great deal of time in actual
classrooms, either as a student teacher (similar to the Japanese or American models) or
simply as an observer. Saara notes that her students are in the Normal school for
about 3 days per week for about 10 hours per week, and that every week there is a
different theme for the prospective teachers, such as monitoring teacher methods or
ethics. Each prospective teacher is then expected to write a 2-3 page report on what
they learned regarding that week’s topic. Saara also emphasizes two key elements of
her classes: planning teaching and reflection. Saara says, in paraphrase, that she is
always asking the question of “how does this relate to mathematics teaching?” for
every lesson and every part of her lessons. Given that she cites planning of teaching
as the most emphasized part of her course, this connection makes sense. Reflection, in
addition, is a heavily emphasized tool. Each teaching session is designed to give
positive feedback (Saara specifically uses the word “positive” here, revealing a critical
detail of her focus) to the prospective teachers, and she clarifies this choice by saying
that giving positive feedback is part of a teacher’s daily role. In the field notes for
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Saara’s classes, there are daily references to some sort of feedback mechanism
involving yellow strips of paper. For example, in the October 2 class during the first
15 minutes of class time, the notes “positive feedback” and “give yellow strips on
which to write only good things for two people” appear. While the specific details are
missing from the notes, it is clear that Saara does indeed incorporate positive feedback
into her classes, and it is precisely noted as such by the note-taker.
Pauli is a fairly reticent interviewee, but his focus tends to be on the content
knowledge that teachers would require in the field. He says:
Perhaps still the most important thing is this quite traditional thing. You have
to do a lot of work in the subject area. It is very difficult to become a teacher if
you do not have skills in your subject, enough. In mathematics you must
always train and think about those things like a pianist or someone else.
Yet even with a strong mathematical rather than pedagogical focus, Pauli does
also show that teaching and learning to teach is more than just mathematics. He later
says “Perhaps the other thing is this idea that you must always remember that in every
human being there is really a human being, and you must respect also those people,
those young people who perhaps just now are your good pupils.” The human aspect
of teaching, of being true to your students, is important to him.
Hanna confirms that content is important, citing that the students who are
accepted to the Lapinkaari University program are those who come in already having
strong content knowledge. As previously mentioned, both Pauli and Hanna are not
exceptionally strong English speakers, and Hanna was very reserved at first in her
interview. In the beginning, her answers are often only a few words or a request to
come back to a question. She also sometimes needs time to think about the questions
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to form her answer. Later in the interview, however, when I ask the question about the
most important things students are learning about teaching (see Appendix E, question
5), Hanna replies without hesitation “cooperation.” The full quote is included on page
149. Hanna’s impression of the most important part of teaching is about what happens
in schools and what her students will need to know to be strong, successful teachers.
Not only that, she indirectly supports what he colleague Pauli has to say by thinking
about both the students and the teachers (both current and prospective) as people.
Neither Pauli nor Hanna talk about assessment, though to be fair that will take place in
another course (Hanna notes in her interview that it will be a subsequent course, not a
previous or concurrent course).
The students at Lapinkaari University make mention of some facets of teaching
also relevant here, though mostly in reference to what they observe in their high
school classes. Student 9 does say of her coursework, though, that “I’ve also learned
that, like, you don’t have to be in one model, like everybody doesn’t have to be the
same, and then [teaching] works. So there can be some differences and different
people are doing [teaching] in different way.” Of note is that Student 9 is talking
about what actually worked rather than simply the theory of things working, and she
finds that this idea from her courses plays out in the real classroom. In talking about
the most valuable learning from her coursework, Student 13 says, “I just think, every
time I have to go in front of the classroom, and I plan the class, and I do it, and
afterwards they give you feedback about things you are doing good and things you are
doing bad. That’s the stuff that really develops your talents. Sort of, doing it, and
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then learning from your mistakes.” These are authentic experiences that are a
deliberate part of the program.
THE GOALS OF MATHEMATICS TEACHING
Mathematics education has years of experience and history so that the first 15
minutes of tasks are done directly by the teacher. The new model is really
flexible. … The students at first do not have the experience that mathematics
can be learned in other ways, and my biggest challenge is to try to break this
pattern. (Saara, Catherine University)
The above quote comes directly from the notes taken by the Finnish graduate
student who collected data for the CATE program. While I cannot guarantee that it is
a direct quote, the language is direct and fascinating. Saara also says that
“mathematics can be student-centered education, and it can involve investigation. It
can be done in groups.” An important point to remember when studying Finnish
education is that the much-vaunted innovations in education are not very new.
Student 5, from Lapinkaari University, states that at 42 years of age he is the oldest
student in the program and perhaps the only one there, besides the teachers
themselves, who were taught in the “old” style of ability-grouped classrooms. Saara
would also fall into that same age category as Student 5, meaning that she was a
learner in an old system yet teaching future teachers to thrive in the new one, which in
their case would be the only system they had known or experienced. Jukka Sarjala, a
member of the Finnish Ministry of Education from 1970 – 1995, said, “It took several
years, in some schools until the older teachers retired, for these reforms to be
accepted” (OECD, 2011a). Saara’s perspective very clearly represents the modern
view on Finnish mathematics education.
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Pauli and Hanna are directly asked about the most important things that
students are learning, as discussed previously. Pauli feels that knowing mathematics
is important, and notes that practicing for teachers is similar to what a trained pianist
might do. The first thing he says, as mentioned previously, is that “teaching is not at
all a single thing.” This statement in and of itself is perhaps not noteworthy, but
Hanna also gave a first answer that has no clear tie to mathematics teaching versus
teaching in general. For both of these professors, the mathematics is a supporting
detail to the larger topic of teaching.
To be fair, the professors are not directly asked about mathematics, but rather
the goals of teaching, so perhaps a more general answer is not necessarily indicative of
their view of mathematics in particular. The students interviewed at Lapinkaari
University, however, are directly asked as the last question in the interview what they
would want their students to say about the experience of learning mathematics (see
Appendix D, question 12). Given the number of responses to the question, I was able
to organize them into four categories (not mutually exclusive). The most popular
answer, given by 10 of the 13 interviewees, is that they wanted their students to see
how mathematics connects to the real world, that it is applicable, or that it relates to
something that will provide future benefit in some way. Student 10, for example,
says, “… so they really see the connections between the stuff you teach in the
classroom and what happens in real life. And that’s probably the goal. If they go
outside and see something happening there, they would see the connection.”
Similarly, 9 of the 13 more generally say that they hope that students have at least
learned something. Many of the students directly say something of that form; for
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example, Student 1 says, “They have actually learned something,” 7 of the 13 said that
they hoped students would find math fun or that they would leave the class liking
math. Student 2 says that she would like her students to be “happy for understanding
maths,” and Student 8 says while laughing, “And math isn’t that bad! Math can be
fun.” Interestingly, 5 of the 13 also say that they hope students will walk away
thinking that math was not as hard as they thought, or that math is not bad, which is
different from liking math. Student 4 says, “Maybe that it’s not as boring as they
thought it would be,” and Student 9 says, “So that nobody would hate mathematics.”
A FOCUSING CULTURAL ISSUE: EQUITY VERSUS ELITISM
Because my Fulbright research in Finland deals with ability grouping, which is
a common practice in the United States but now almost unheard of in Finnish public
schools, I had many conversations with students, teachers, professors, and non-
education people about these systems. Finland is sometimes referred to as a welfare
state, which is an overly simplistic way of saying that Finland has both laws and
strong cultural values that prohibit social stratification and large disparities in wealth.
According to many measures, Finland ranks near the top of all OECD countries in
terms of strong level of income equality (OECD, 2011b). Still, the Finns would often
see such famous American figures as Bill Gates and wonder if their system prevented
them from producing outliers. At the time I was in Finland, there were concerns about
the future health of Nokia, which accounts for an incredibly high percentage of the
technical jobs in Finland. One person directly said to me that he worried that Finland
was not producing “geniuses,” which could lead to economic problems. Thus, the
discussion of mixed performance classes (or non-tracked classes) versus same
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performance classes (or tracked classes) led to informative conflict of equity versus
elitism.
One of the defining features of the prospective mathematics teachers’
interviews, reflected in their valued statements, is that of sense-making, or the verbal
process of talking through the potential advantages and disadvantages of mixed and
same performance groups. For nine of the interviewees, the concept of same
performance groups is a completely theoretical concept, and for the remaining four,
the stereotypical American idea of separation is only partially understood at best.
When asked about how same performance groups would potentially work or not work,
Student 4 says, “Well, I’ve never really thoughts about high performance or low
performance groups, so I don’t know how it would be.” For convenience and
simplicity, interviewees are told that American schools in general tend to separate
students into groups of high, middle, and low performance, but they are not told
exactly how this process is accomplished or what instruction or learning would look
like in this context. As such, responses by the interviewees represent individual
constructions of how same performance education is structured and enacted.
As described in the Methods section, the interview transcripts for the 13
Lapinkaari University students were first open coded, then coded for statements that
were positive or negative towards same performance groups and mixed performance
groups, then coded once again for high performing students and low performing
students. The codes of same/mixed, positive/negative, and high-performing/low-
performing/general create 12 distinct categories for analysis.
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A view of performance groups as positive for high performing students is the
only category amongst the 12 from the second round of coding for which every
interviewee makes at least one comment. In other words, every interviewee makes at
least one statement concerning the perceived benefit of same performance groups for
high performing students. Most of the prospective teachers self-identify as high-
performers themselves, and as such easily put themselves in the theoretical position of
high-performing students. Given that teaching in Finland is considered one of the best
possible professions and that acceptance rates into teaching programs stand at
approximately 10%, this is not surprising (Westbury, Hansen, Kansanen, & Björkvist,
2005). As such, the prospective teachers are able to tell stories of their experiences as
high performing students in mixed classrooms and contrast their experiences with the
hypothetical shift to a same performance classroom. Some people, such as Student 13,
report frustration when, as a student, she understood the topic being taught but felt
slowed down by the rest of the class. In contrast, Student 7 reports being “[thrown]
out of class with a book on matrix calculus” so that he could continue learning more
advanced topics.
Although only one student knew the precise English word, all 13 interviewees
distinctly mention classroom differentiation in some way, and many mention it almost
as a side remark, as if such things are commonplace in classes across the country.
Eight interviewees report boredom for high performing students in mixed performance
classrooms as a potential problem, and four interviewees state that high performing
students might be slowed down or that their time might not be advantageously
utilized. In thinking about ways that teachers work with high performing students,
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nine interviewees report that a common solution is to provide high performing
students with extra work, and some mention giving students their homework early so
that they can finish before class ends. One interviewee states that providing for high
level students is easier, as they often are not high performing in other areas, and extra
time can be devoted to additional practice in these topic areas.
Based on these concerns about mixed performance classrooms, descriptions of
what a high performance classroom would look like are not surprising. Common
themes are that a high performance class might move more quickly through material
and go deeper into topics. As Student 4 says, “With the higher performance groups
you could go over the basic stuff more quickly because they understand it the first
time. And get to the more interesting stuff.” Reduction of content is never mentioned,
but several interviewees talk about progressing through basic material as quickly as
possible. Interviewees also believe that high performance classrooms might result in
higher learning for these students, including those who mention earlier in their
interviews that mixing high and low performing students in groups is beneficial for
everyone. Criticism of high performance classes tends to focus on social relationships
and equity issues rather than academic issues. Since social relations are an integral
part of Finnish education, however, this is not a trivial statement (Kupiainen,
Hautamäki, & Karjalainen, 2009). Student 8, for example, reports that same
performance classes during her study abroad time in the United States were really not
very heterogeneous (in terms of ability to learn), so she switched from an AP class to a
lower level class to be with her friends.
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Whereas interviewee perceptions of high performance classrooms are
generally favorable, perceptions of low performance classrooms are conversely quite
negative. 11 of the 13 interviewees make at least one statement concerning how
performance groups would not be good for low performing students. Conversely,
three interviewees make one comment each concerning how low performance classes
might be beneficial for low performing students, showing a strong denial of value for
low performance groups. In mixed performance classes, low performing students are
often described as noisy or disruptive, though almost never as dumb or stupid, the
exceptions often being for emphasis rather than as a pejorative. Differentiation is
mentioned substantially in this context, predominantly in the form of providing extra
help for low performing students either during class or after school. The idea of
making groups where high and low performing students work together was not as
commonly or explicitly mentioned as might be expected, though the idea of making
groups in mixed performance classrooms is mentioned by all but three of the
interviewees.
Similarly, the perceived structure of a low performance classroom is the
opposite of the perceived structure for a high performance classroom. Interviewees
state that low performance classes would move more slowly, focus on the basics or on
practical/real-world problems, and involve less content. These statement lead to many
contradictions within interviews, many of which the interviewees themselves notice as
they try to make sense of performance groups. One of the most direct cases is Student
9, who says, “With the mixed group, I think the weakest students, they don’t learn so
well. They have just moments that it’s too difficult to them.” Then, she describes her
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ideas of a low performance group by saying, “Like, thinking for example of the low
performance group, it shouldn’t be so much with the theory, or some things. Like
really the practical things, like maybe just having some cooking recipe and maybe just
dividing it for two or dividing by two or just calculating some area of the room or
something like that.” Student 9 concludes, however, with the statement, “But also I
think that low performance group students wouldn’t learn so much – than now in
mixed groups. Of course, I don’t have any scientific background for this, it’s just
intuition.”
This particular contradiction exemplifies the struggle the interviewees face
when talking about low performance groups. Interviewees are able to make sense of
high performing students and the perceived benefits of high performance classrooms
quickly, but when faced with low-performing classrooms, they are generally unsure of
the potential benefit to the students. Given the Finnish educational commitment to
high achievement for all combined with the fear of not producing talented students
(OECD, 2011a), the seeming contradiction in statements makes sense. In numerous
cases, the interviewees are able to explicitly say that a separation of low performing
students would be detrimental to learning:
So … leaving talented people alone in the room doesn’t hurt so much as
leaving the less talented people alone in the room. [Student 5]
Because if I had … weaks and good ones in separate groups, then weaks
haven’t got done anything. [Student 1]
Also, I think that it’s important for the weaker kids that they are not put into a
different place and told from the very early age on that you are weak and you
can’t do this. [Student 6]
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Yet this reluctance to separate students into low performance groups does not mitigate
the preference for high performance groups. As Student 9 states, “I think that our
system now isn’t making any really good students, like some of the really highest top
is missing. But I think also the lowest ones, they are put somehow equally, just
compressed into average.” This view is particularly interesting in light of the opposite
finding through the PISA data (OECD, 2010), which was never mentioned by any of
the interviewees. The potential benefits of ability-grouping for high performing
students are alluring, yet the interviewees feel the need to qualify the potential
detriment to low performing students -- a feature that is by deliberate design not a part
of the Finnish educational system.
Another excellent example of one of the prospective teachers attempting to
articulate her difficulty in making sense of low performance classrooms comes from
Student 11:
It’s hard to analyze it because I don’t understand the whole system. Is it some
kind of punishment when you are put in the lower performance group, or what
is the motive of keeping them separate? What is it?
Like most of the other interviewees with no prior experience with same performance
groups, Student 11 engages in sense-making throughout the interview to try to reason
through questions of how student learning might potentially differ. She is one of the
only interviewees to ask questions of me during the interview in this effort. Student
11 is thus able to make an illuminating statement concerning her perception of same
performance classes: “Actually I don’t have a problem with that high performance
group. I would have a normal performance group and a high performance group.”
Student 6, who had experience as a student in one of the few schools in Finland
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comprised solely of high ability students, says much the same with a clarifying
analogy:
I think it’s better for the good students to have their own groups, but it would
be bad for the weaker students to be just … very limited. So the most practical
model would be some kind of mixed groups, and then the top ones could be in
their own group. Like you’re playing chess or baseball or something, it’s no
use training with average people after some point. But then again, if you are
not good at baseball, I think you enjoy it more when you play it with average
people than when you play it with quite, quite … people who are not.
The tension between elitism (i.e. the statements of students who believe that
high performance groups would be beneficial for the “elite” students) and equity (i.e.
that low performance groups would hinder students and possibly prevent access to
opportunity) is very apparent in these interviews. In some ways we can see the
philosophy of Foster University here with its lens of equity, but writ large as a more
general cultural statement. As with any cultural generalization, it is impossible to say
that a country having a belief is indicative of each and every person in the country.
Finland believes in equity in education much the same way that the United States
views concepts such as freedom, with all the complexities and contradictions and
interpretations inherent in that analogy. The Finns have a strong believe in these
mixed performance groups, yet it is perhaps natural for them to wonder what
advantages a different system offers. Yet in the end, the majority of those interviewed
favored equity over elitism, consistent with the general philosophy of the country.
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CONCLUSION
And some told that it’s so that we have … we have best teacher education
institution, but I don’t know it’s much (laughs). Too much saying that. But
maybe it’s told that we have good teachers, we have teacher expertise in
Finland in higher level. We have good teachers, we have teachers who want to
teach, who want to take care their students. And our students have
pedagogical content knowledge enough. Maybe. I don’t know. [Hanna,
Lapinkaari University Didactics professor]
For a group of teachers and educators internationally lauded for their incredible
teaching ability, the Finns were remarkably humble and introspective. Teasing out
what they think are their strengths is difficult; learning about their challenges and
concerns is surprisingly easy. Many possible explanations for the very strong math
performance of Finnish students have been suggested by both domestic and foreign
researchers (Hautamäki et al, 2008; Darling-Hammond & McCloskey, 2008), though
it is also important to note that some Finnish scholars have challenged the idea that
test scores, such as those on PISA, accurately reflect the learning in Finnish schools
(Astala, Kivelä, Koskela, Martio, Näätänen & Tarvainen, 2005; Tarvainen & Kivelä,
2005). So too, it seems, do its teachers and teacher training professionals. Pauli, who
was very reticent on most questions, said the following regarding Finland’s PISA
scores:
The limitations are clear that, as I just said, PISA questions, they concern the
so-called everyday skills and so on. And so it is quite natural to have such a
consequence that more theoretical aspects, they are not so good. So something
like that you can hear from the university teachers who are quite worried about
the skills of the young people after the secondary school when they are coming
to the university. And especially I have heard complaints concerning the
algebraic skills, for example, you don’t … not so good … manipulating
rational expressions, and so on. And also I have heard complaints concerning
even the basic logical level, so to say.
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Perhaps this is one important lesson to learn from Finland: they have little desire to
“show off” to the world. Along with the strong idea of equality comes the companion
feeling of not acting better than your neighbor, and while the Finns may have
experienced some glowing successes, they are more concerned with providing a
quality education than basking in glory.
Still, it is important to recognize that there is value in examining an
educational system that produces top marks in student performance, even with
reservations concerning the assessment mechanism. What is it that Finland does well?
One clear result of these data is the strong emphasis on practice, which mimics what
we have seen in both Japan and the United States. Prospective teachers mix a high
level of content knowledge with training in methodologies to meet the needs of
diverse learners with diverse needs (Centre for Educational Assessment, 2008).
Future mathematics teachers – from elementary teachers to high school teachers –
train in and are examined in both pedagogy and mathematical knowledge. Prospective
teachers also spend a great deal of time learning about what to do in schools, and they
spend time in schools teaching, observing, and reflecting. These lessons of practice
include not only mathematical knowledge and pedagogy, but also thinking about who
students are and what students need. Saara said, “And because in school, in
mathematics lessons there are so many other … point. Not only mathematics. We
have to learn social skills. We have to take care each others.” Perhaps this is a
cultural point, or perhaps it is in some ways a result of mixing students of different
backgrounds and potential ability levels together in one class that must work together.
Regardless, it happens. The attitudes that teachers bring to the classroom concerning
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what their students can do form a large part of how these students progress and
ultimately how well they succeed – or do not succeed – in mathematics (Dweck, 2006;
Pajares, 1992).
In addition, Finnish society in general places strong emphasis on the concept of
equality, which is made visible in national policies in education such as the Basic
School Law (which abolished tracked classes – see introduction for more details)
(Laukkanen, 2007). Finland has also been described as a “welfare state” due to
popular concern for what society considers the basic rights of citizens, which include
health care, food, shelter, and education. Yet detractors of such a system still look to
Finland for educational ideas and guidance. The Finnish prospective teachers clearly
have conflicting ideas about the needs of the “gifted” versus the needs of the “weak,”
yet they are not willing to sacrifice those who struggle for the sake of those who excel.
Student 9 says, when comparing the strengths and weaknesses of mixed performance
groups, “I think that our system now isn’t making any really good students, like some
of the really highest top is missing. But I think also the lowest ones, they are put
somehow equally, just compressed into average.” This statement well sums up the
general feelings of the Finnish prospective teachers interviewed. While they worry
about the results of the best students, they also worry about the results of the
struggling students. In short, they worry about all students, and in that attitude is
strength.
Throughout the last six chapters we have looked at three distinct cultures and
countries, and three exemplary teacher preparations programs. We have seen teachers
who care about their students, various combinations of theory and practice, and
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examples of how one’s own cultural lens affects the questions asked and the
interpretation of teaching and learning. So what does this all mean? In the concluding
chapter I look across all three countries to discuss some of the differences, but I also
emphasize the surprising similarities and what is suggested about mathematics teacher
preparation from these cases.
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Chapter 10: Conclusion
Ben: In August, I’ll be back at my own university, and I will be working with
American teacher trainees again. What would you want me to tell them about
becoming a math teacher in Finland?
Hanna: [laughs] Oh! Ah! To train a teacher is not like making cake. There is no only
recipe what is the most, what is the best way to train teachers. Teacher training
is a process, and you need good trainers, good students.
As I sit in my home writing this conclusion, I think how lucky I am to have
experienced such wonderful teaching in three very distinct countries and contexts.
From a rural school in Japan to a suburban school in the United States to an urban
school in Finland, the experiences have been both memorable and life-changing. My
one regret is that I am not currently in the classroom, though once the graduate school
chapter of my life comes to a conclusion as well, perhaps I can find ways to continue
to teach and to learn. And who knows? Perhaps there is even a fourth country in my
future.
My views on education continue to grow and expand, and I am constantly
amazed at how much there is to learn about teaching. Even as I collected these data,
sitting in a room full of prospective teachers while I had over 10 years of teaching
experience, I felt that there was so much I had forgotten and so much that I still
needed to think about or to learn. Thinking about all the good practices that I saw
espoused and practiced by these wonderful and thoughtful teacher educators made me
think of how few hours there are in a school year, and how hard it is to help even one
student in that amount of time, let alone over a hundred. Teaching is a complex
profession, and it is perhaps the profession of teaching people how to be effective
teachers that ranks as the most complex.
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It is fitting that this journey through multiple cultures and teaching contexts is
a chronological journey with ideas and conclusions that build. If you had asked me
after my time in Japan if themes like “authenticity” or “theory and practice” would
characterize what I learned in Japan, I can honestly say that the answer is no. These
themes were only apparent to me after contrast and reflection, and some only apparent
after encountering yet another context and supplementing it with research skills
learned in graduate school. I have also learned that my experiences are but a small
sample of the possible experiences in these cultures, which is a focusing yet limiting
factor. My high school experience in the United States is vastly different from the
high school teaching experiences of many of my friends, and my teaching experiences
in Japan were different from others even in similar contexts at the same time in the
same general geographic region. I cannot claim that my experiences speak to the
“true” nature of these countries, whatever that means. I can only claim that I have
seen and experienced excellent teaching and excellent teacher preparation in these
cultures, and I have returned to my own country a better person for it, though there is
still much I wish to know.
MY NEW AMERICAN QUESTION
The choice to focus on exemplary schools of education as part of this project
was deliberate, and not only because learning from the best is a wise choice. I also
come from a country where the teaching profession is under attack now perhaps more
than any time in its past, and I am extremely protective of teachers. Looking at
“average” schools or anything other than the best is to diminish the role teachers play
in our society.
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However, in looking at my own country, it would be foolish never to look to
improve ourselves or our way of educating students, and finding systematic ways to
do so could be of tremendous value. What can we learn from these excellent
examples, including (deliberately) an example from within this country? This new
American question is quite large and complicated, and I characterize it as a new
“American question” as it does include a bit of naiveté, as did the previous American
questions. There is a great deal to be learned from international comparisons and
studies (Darling-Hammond, 2009), but Laukkanen also writes that:
Individual countries can use [international comparisons] as mirrors in which to
reflect their own performance and policies. Although it is not wise to import
policies from other countries as such, countries can benchmark their own
products with products from elsewhere. Countries can also learn from each
others' good practices (2007, p. 319).
My emphasis is on the last sentence – to learn from each others’ good practices. As
Hanna said in the quote at the beginning of this section, training teachers (and
education in general, one might argue) is not a fixed recipe. There is not one way to
do it, and even if we were able to determine precisely what works best for any one
particular student or to train one particular teacher, that process might not be ideal for
another student. This is also so concerning the immense range of differences from
classroom to classroom, school to school, and country to country. But perhaps there is
something much like a recipe here, just not in the overly prescriptive sense. A good
batch of chocolate chip cookies requires flour and sugar. It also requires chocolate
chips (tautologically), but there are numerous brands and levels of sweetness even in
this necessary ingredient. Option ingredients abound, and the debate as to whether
walnuts should be included could go on forever.
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Though this analogy is a little strained, hopefully the point is clear. No study
will ever give the definitive answers to the question of what education needs, and as
such even the answer to the question of what we can learn from these case studies is a
matter of perspective and interpretation. As someone who has taught in all three of
these countries and conducted research on mathematics teacher education in all three,
here are some of my conclusions.
RELATIONSHIPS WITH TEACHERS MATTER
One of the key features described in the previous chapters is relationships, and
perhaps the strongest component of these relationships is the teacher. Jackson
describes the role of the American teacher quite clearly when he defines their role as a
combination of, “traffic cop, judge, supply sergeant, and time-keeper” (1990, p. 13).
Interestingly, Jackson also immediately writes that “such functions must be
performed” (italics added). The function of the classroom is intellectual development,
and the role of the teacher is to maintain the atmosphere and authority necessary for
such a function to occur. The key to authority is often characterized by the simple
concept of respect, and to establish respect the teacher must maintain a sense of
“other,” of not being in the same category as the students (Shimahara & Sakai, 1995).
Beginning teachers are admonished not to treat their students as friends, to maintain an
emotional distance. In fact, it might even be referred to as a professional distance, so
important is this concept in the culture of American teaching. One of the American
teachers studied by Shimahara and Sakai said,
I feel it is important to maintain formal relationships with kids. I need to be in
power. I do not have to be liked, I do not need to be in a popularity contest.
The rapport I would have with the kids must be professional and appropriate. I
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would like to be in a warm and open environment, but I do not have to be
liked. (p. 82)
While this quote definitely defined my own teaching training, it stands in stark
contrast to the data in these three cases, including the teacher educators from Foster
University in the United States. These relationships do matter.
While the Japanese educators had the most formal titles with their students, the
relationships were characterized by getting to know their students and caring about
them as people, not just as students of mathematics. Whereas Jackson writes that the
American teacher is, “chiefly concerned with only a narrow aspect of a youngster’s
school experience,” Japanese teachers are concerned with zenjin kyoiku, or whole-
child education (Jackson, 1990, p. 3). Teacher Kenji Furukawa said,
It is important to understand children as human beings whose characteristics
are expressed in their activities. It is my belief that all children can do their
best and concentrate on work. But it depends on a teacher’s approach and
desire. I am not concerned with how to teach children; rather, I try to
understand them first, by developing personal relations” (Shimahara & Sakai,
1995, p. 169).
The “personal relations” that Japanese teachers share with their students extends even
beyond the classroom into what would be seen in America as a dangerous domain: the
home. Japanese teachers are actually required to visit students’ homes once per
semester and talk with the students and their parents, in large part to establish a sense
of community but also to learn as much about the student as possible. Teachers will
eat lunch with students, socialize with them in the teacher’s room (which is open to
students), and ask them to write personal journals that they (the teacher) will later
read. The wall that American education has built between teachers and students
simply does not exist in Japan; once again the idea of going beyond the curriculum to
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help students is an American concept, whereas doing so is part and parcel in the
Japanese system.
This wall similarly does not exist in Finland in the same way that it does in the
United States; once again, the relationship between teachers and students is closer,
more personal, and more involved than with their American counterparts. Even
compared with their European counterparts, Finnish teachers measure low on
“depersonalization” with their jobs (Rasku & Kinnunen, 2003). Teachers in Finland
are still sources of authority, and Janne still at times had to act as the “traffic cop” as
part of his duties, but he would never define that as his role. Janne also valued the
idea of whole child education, though he never said those words in particular. His job
was to create learners and future adults.
Note as well, however, that the Foster University teacher educators also spent a
great deal of time talking about getting to know students. True, much of it was
contextualized with mathematics, as that was the nature of the class they were
teaching, but knowing how students think, what they value, and how to connect with
them were at the core of the lessons and explicitly mentioned as a tenet of good
teaching. In fact, all three teacher education programs focused on this idea. The
Japanese program emphasized learning about how students think; the United States
program focused on what students value and related to; and the Finland program
talked about understanding learning difficulties and attitudes toward mathematics and
the challenges those students may face. Perhaps it might be a little too much to say
that these programs teach the prospective teachers to care about their students, but I
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believe that this idea is on the right track. Teachers learn to care about their students’
learning, but in turn, they learn to care about the students as people as well.
The difference points not at the attitude or qualification of teachers, but more
broadly to the expected role of teachers. We have already seen differences in this
regard: teacher education programs in Japan and Finland are highly selective, and
teachers are viewed with respect in these countries as much is expected of them. I
would argue that the societal view of teachers precipitates the desire to join the
profession rather than the opposite, though I have only common sense to back up that
argument. Still, I find it hard to believe that people would flock to the teaching
profession, and then society’s impression would change toward the positive. This
causal argument is important because, as a society, the United States does not trust its
teachers and increasingly looks for ways to rate teachers, and the dialogue is about
identifying “bad” teachers rather than rewarding “good” teachers. As previously
mentioned, Finnish teachers are expected to help each other, and the collaborative
nature of Japanese teachers (e.g. sharing office and work space, having regular
meetings with grade level teachers) leads to great support and little to no need to
evaluate teachers so visibly or harshly. I also never heard much in either of these
countries about unions or lawyers, as perhaps teachers in these countries were not in
need of protection as much as American teachers.
Societal support for teachers to fulfill these higher expected roles comes from
multiple places, but the ones that struck me the most as different from the United
States are administration and parents. In many ways, I saw many aspects that were
similar to negative stories from the United States. I met and heard about unsupportive
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principals in Japan and in Finland, and I had the opportunity to talk with some of these
people in person. Similarly I heard about and met challenging parents who were
overly involved in schooling and questioned the choices the teacher was making. I
want to be clear that I cannot characterize either of these countries as “always” one
way or another. Still, the preponderance of experiences and stories were positive. My
teacher colleagues in Japan rarely if ever complained about parents, and Janne only
had one or two who were frequent bothers. The administrators that I worked with in
Japan were, for the most part, good people who supported the teachers. Dislikes were
personal, not professional, and as such never seemed to interfere with the work of
education.
All of these factors point to the general theme that relationships with teachers
matter. From the expectations of teachers to the eventual role of teachers, the way that
teachers interact with students influences education, potentially for the better. These
are lessons that I have both consciously and unconsciously taken away from these
countries. As I was writing this section, I received a phone call from a former student
inviting me to his graduation party (for his Masters degree – I left this school in the
spring of 2008). Even seven years later, the relationships that I formed with my
students bear fruit. I learned this lesson unconsciously in Japan, and now, years after
being this student’s teacher, I still see this relationship of teacher to student being
important and formative.
PRACTICE, SUPPORTED BY THEORY, IS KEY
Looking in particular at the teacher training programs in all three countries,
another common theme is a heavy emphasis on practice, with the corresponding
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theory to support it. Reading through the transcripts, I was often reminded of one of
my former martial arts instructors, who said that the idea of “practice makes perfect”
is potentially quite destructive. If you practice the same wrong thing over and over
again, a habit is formed, and the practice makes you worse. In his opinion, “perfect
practice makes perfect” is the more appropriate and correct aphorism. Indeed, these
programs do emphasize practice, but there are opportunities for observation, small
lessons, feedback, reflection, and discussion. The simple act of practice without all
these supports might seem like a productive way of learning to teach, but it is these
supports that keep the practice as perfect as possible.
At first glance, it may not seem like the Yamato University classes support the
assertion of an emphasis on practice. Yoshida-sensei very directly said that he did not
emphasize mock teaching lessons, but he said in his interview that the goal of the
program is to prepare teachers “who can teach lessons in which importance is placed
on the process and the way of thinking – we want them to become such teachers. We
have an image of lessons based on problem solving.” Yoshida-sensei’s classes were
heavily about problems and problem-solving, which in his mind is the ultimate goal of
practice. By probing student thinking, having students share answers, and then having
the same students dissect their own thinking to better understand their students and
how to develop their problem-solving skills, Yoshida-sensei is engaging in a fairly
authentic form of practice. This idea of practice is supported in multiple ways, one of
which is the direct connection to Takeyama-sensei’s class, which focuses more
heavily on watching videos of teaching, attending classes at the practice school, and
debriefing with the whole class or in small groups. Students are exposed to practice in
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multiple ways in this manner, all of which will support their eventual time as student
teachers, which would begin the following September (recall that the interviews took
place near the end of the spring semester in April).
The emphasis on practice and on being present in authentic classrooms is
abundantly evident at Foster University. When the program director was asked “What
opportunities does your program provide to help candidates practice teaching?” (see
Appendix C, question 10), she laughed and said “I don’t know if there is a more
extensive program. There are 19 days in the summer that they are in a school setting
and 180 days in the academic year in schools.” In total, she estimates that the
prospective teachers at Foster University spend over 700 hours in schools over the
course of the year. Approximately 6-8 weeks (at an average of 20 hours per week) are
spent in independent student teaching, and at that time the students “are de facto
responsible for all aspects of teaching.” This emphasis is supported by the interviews
and the observations as well. The prospective students reported in their interview that
the process of creating a real and usable teaching segment was of great value, with one
student stating, “something I learned was how essential lesson planning is for
teaching. How it is not easy, but how essential it is. Without a plan, it’s impossible to
teach.” The students further supported their learning in the program by talking about
how hard self-assessment is and how easy it was to get feedback. “The teachers were
willing to meet a lot and give individualized feedback,” said one, saying that she felt
“really well supported” in her learning. Though the instructors of the course wished
that they had more opportunities for practice teaching, they supported the learning of
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practice with as many valuable, authentic tasks as possible to show how different
theories that were being studied translated directly into useful tools for practice.
All three Finnish teacher educators were also in agreement about the value of
practice, supported by theory. Saara from Catherine University, who was asked these
questions more specifically as part of the CATE interview protocol, states it most
clearly by saying “All of the seminars, the aim is that the theoretical share of class is
about 15-20%. The rest is practical work.” Here again we also see the strong support
of the teacher educators in supporting the prospective teachers by giving practical
advice and feedback about real teaching tasks. Speaking about implementing and
modifying an existing task, Saara says “This is being practiced in lesson plan making.
I give feedback to every person. And I have given them these operational tasks and
concept maps.” Here I would note that concept maps were one of the features of the
learning segment of the Foster University students. Saara finishes her statement by
saying, “Students also practice the administration of feedback … that, too, is part of a
teacher’s work.” Hanna and Pauli were asked more generally about teaching, yet they
too focused on the applications of practice and authentic means of teaching. Pauli
speaks more directly about the need for mathematics in the classroom, yet he also talks
a great deal about how teaching affects students in the real world, that the practice of
teaching is about using these tools to shape the future of children. He says, for
example, when talking about the theories of mixed performance groups, “We have
here in Finland such a starting point, that you have to give everyone right
opportunities into this life. And we don’t want to stop or close the doors too early.”
Hanna also focuses on the practice of teaching through what will be a daily occurrence
182
for Finnish teachers” collaborating with colleagues. In teaching her students about
cooperation, she says, “They have to take cooperation with … teachers. They can’t
work at school alone. They have colleagues, they have the other teachers around
them, and we have to make work together.” Part of Hanna’s responsibility, as she sees
it, is teaching how things are done in the real world of teaching, and her theories on
how to accomplish these tasks play out directly in her classroom.
WHERE PATHS DIVERGE
When comparing any two things, it is facile to say where there is a lack of
similarity. Comparing three things makes such statements more challenging, but still,
it is challenging to avoid the trivial and find the meaningful. I had hoped that the
CATE protocol questions would shed light into differences in national standards and
curricula (see Appendix A question 11 and Appendix B question 8), as these are areas
where significant difference exists. As both Japan and Finland have a national
curriculum, and the Common Core is such a hot topic in the United States, it is
disappointing that I cannot reasonably make any definitive statements. That being
said, there were two areas where there is enough data to show important differences in
these three systems that are related to national standards and issues and that are related
to the idea of national curriculum. Those areas are textbooks and testing.
TEXTBOOKS
It is noteworthy that in all three contexts, textbooks and textbook analysis were
part of the discussion, especially as all three countries have widely divergent policies
regarding textbooks. In Japan, with the national curriculum, there are only a handful
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of companies that are allowed to provide textbooks (Yoshida-sensei mentioned in
class that there are 6 that are authorized to publish junior high school textbooks, and
he mentioned 5 of them by name). Yoshida-sensei spent a great deal of time in the
first observed class directly comparing how different textbooks covered the same
topic. For example, when talking about multiplying two negative numbers, all of them
used a speed-time-distance model for this process through understanding rates and
directions. Yoshida-sensei then pointed out that the examples quickly proceeded to
a purely numerical form, and that the jump was difficult to understand, and that it
would be more valuable “for the students to think about the connection themselves.”
In addition, Yoshida-sensei pointed out at the beginning of the second observed class
that the national Curriculum Guidelines change about once every 10 years, but that
textbooks are revised about every 4 years. He then described at length some of the
differences between a 2008 textbook (under the last set of Curriculum Guidelines) and
a 2012 textbook (under the latest Curriculum Guidelines, released in 2012). One of
the most notable differences was that mathematics classes now meet 4 hours per week
instead of 3, and he noted that the new textbooks are 273 pages instead of 194,
reflecting the increase in content. The point of this description is that, because of the
national curriculum, textbooks are very prescribed, and variations are subtle at best.
This compares directly with Finland, another country with a national
curriculum. The curriculum, however, is not a rigid set of topics and timelines, but
rather a broad set of guidelines to be interpreted by the individual teachers. Included
in that interpretation is the ability to choose a textbook or textbooks, which means that
two teachers of the same class in the same building might be using different textbooks.
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During my time at the Normal school connected to Lapinkaari University, I met a
teacher who did not like any of the published textbooks, so he created his own. I still
have the autographed copies in my office. Although the CATE study observation
notes do not allow me to see what, if anything, happened in the Didactics classes
regarding textbooks, Saara happens to talk about textbooks when answering the
question about planning for teaching (see Appendix A, question 7). Saara says, after
describing how students use a certain program to make their concept maps:
Then, using textbook analysis and task analysis, which is directly related to the
fact that in the future they can think about the types of tasks that can be found
in textbooks and how they are utilized. And what is the role of the textbook
and what is the role of the various tasks in your work. There are always
questions about how this relates to the aspects of mathematics teaching, which
are raised these group sessions.
Here Saara is looking at the textbooks not as limiting factors, but as potential sources
of tasks and ideas for a lesson. Textbooks are more flexible resources that can be
advantageous, but one must examine them to see where value can be derived best.
The United States falls squarely in the middle of these two systems, and as
such, the view on textbooks also falls somewhere in the middle. While the United
States does not have national standards, it does have state standards, and in many
cases it additionally has local standards (such as a county or a school district). As
such there is wide variety in textbooks, and the instructors at Foster University used
this to their advantage. The homework assignment before the first class I observed
involved looking at a topic and then looking at three different textbooks to see how
that topic was taught. Emily said of the task, “we asked you to think about how each
text might be useful to you as the teacher, and also how they might be useful for the
student.” Samantha followed this by saying, “implicit in this task is the question: what
185
is this textbook not useful for? What might it not accomplish?” Here we see the
American prospective teachers looking at textbooks much in the same way as the
Finnish prospective teachers, but more as supplemental materials rather than the
primary materials. For the most part, when these prospective teachers become
teachers, they will not be able to select the textbook they use in their classes; it will be
selected for them. Samantha spent some time during this segment of the lesson talking
about how they could get free copies of textbooks to use as references, from
borrowing them from the library, to writing the publisher for a free copy, to finding
online copies of textbooks. Resources are more widely available than would be the
case in Japan, but the same sort of restrictions on use exist as well.
TESTING
Testing did not come up as a topic as strongly as did textbooks, but there was
an undercurrent of it in several of the interviews. The idea of testing came up most
directly in the Japan interviews, as the prospective teachers talked about juku and its
effects on the teaching profession. Japan has but a few high stakes tests, but they are
of the highest stakes. One exam determines who get accepted into the most elite high
schools and who does not. Similarly, one exam determines admission to university,
and the elite are separated off once again. Juku exists in part to help students gain
advantages in the admissions process, and it makes no claims about critical thinking
skills or conceptual understanding. It teaches for the test.
The United States has many more high stakes tests, and it seems like there are
more every year. Though not as singularly deterministic as the Japanese tests are,
these tests have profound effect on high school tracks, accessibility to remedial
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resources, and, in general, admission to college. Test preparation is a big business in
the United States, and I have personal experience doing part time work for one of
them to support myself through my Masters program. I can say with certainty that
these American-style juku are also all about the test. At Foster University, the
assessment class was concurrent with and deliberately separate from the curriculum
classes I observed. While I have no observation notes, I do have an informative quote
from the program director about assessment: “[the professors] spend all this time
talking about the assessment for learning and of learning, and all those deep
conversations, and then when it comes time to grade, the students are back to 25% for
this and 20% for that.” She laughs. “They want to change the world, but not the
grading system!” Even teachers who are philosophically against standardized testing
get caught up in their own experiences, as they live in a testing culture.
Finland, in direct contrast, is known for not having all the standardized tests
that the United States does. They do have the Matriculation Examination, which
occurs roughly at the end of what would be high school in the United States.
Observers have pointed out (correctly) that this test really does not compare to high
stakes tests in other countries, though I do feel the need often to point out that upper
secondary school in Finland is optional, and that students are able to track themselves
into “higher” or “lower” level classes. The noteworthy point in the Finnish data is that
there is not a single instance of talk about national assessments simply because there
are none. I spent weeks probing the students about what it would be like to teach in
same performance classrooms, and for many of them, they had to imagine what such a
187
concept would even look like. I imagine the same would be true if I had asked them
about their thoughts on yearly assessments for students across the country.
AREAS FOR FUTURE RESEARCH
This study has identified several areas where similarities in high performing
mathematics teacher preparation programs exist. One of the limitations of this study,
as discussed previously, is that only three countries were examined, and only four
universities were researched (two in Finland, one less thoroughly than desired, another
examined in a somewhat different direction). Examining across other countries and
cultures would of course be of interest, though I believe it would be of more benefit to
continue to examine these and other universities in the same country to determine if
these generalizations hold true within countries and across countries.
Looking in particular at the role of teachers and the relationships with students,
focused questions for teachers concerning the nature of their relationships with
students, both in class and out, and with principals to determine the scope of those
relationships as per the vision of the school would help us to understand where these
beliefs come from, how they are enacted, and how they are supported. Teacher
education programs would also provide valuable information in this area as we could
understand if such relationships are implicitly or explicitly supported or opposed, and
what strategies, if any, prospective teachers are learning to help students develop as
“whole children.” Supplementary information from unions or lawyers would also be
informative, if nothing else to raise more “American Questions” regarding why such
questions would even be at issue.
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The area of practice also affords many opportunities for more detailed research
concerning what some of the specific practices might look like. A focus on
authenticity was apparent in all three contexts, and practice supported by theory was
also a clear similarity. But what precisely are the details of these practices? What
specific skills are mathematics teachers learning that might be the same in all three
contexts, or which ones might be specific to the particular culture and irrelevant or of
lesser value for the others? Here also details concerning the nature of the student
teaching practice would be of value. How much learning occurs through the
cooperating teacher? Of that learning, how much of it is in contrast to what is learned
in the teacher education program? How do prospective teachers reconcile any
differences they might encounter? Given that all three of these programs devote an
incredible amount of time to having their prospective teachers placed in schools and
other authentic learning environments, much could be learned from these placements
and supporting teachers.
Also missing from this study is what happens to the prospective teachers after
they leave the program. If I were to run another study, and with more time and
resources, it would be of great value to interview the students multiple times
throughout the program (though choosing corresponding time intervals for the one
year program versus a four or five year program would be challenging) to examine
their views on topics such as the role of the teacher or the important factors of practice
are changing. More importantly, it would be of value to identify these areas and then
track the prospective teachers one year, three years, and five years into their teaching
careers to see what lessons remain part of their practice, and what changes from
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increased experience in the classroom. I would of course like to change the interview
protocols to be more specific to these questions, but also to allow the students and
teachers to share what they think about some of the most salient cultural features of
their teaching. Some of the most valuable things I learned in these countries about
teaching were not from the questions I asked, but more from the lessons that others
were willing to teach me. Setting aside my own cultural views of what teaching is or
should be is challenging, but it is perhaps the best way to learn the answers to
questions you did not know to ask.
190
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Appendix
APPENDIX A: CATE INTERVIEW PROTOCOL, FACULTY
Background
1. How long have you been teaching this course?
2. Can you tell me briefly about your own professional training and experience related
to teaching this course?
a. Have you had experience teaching in K-12/K-13 schools yourself? (How long did
you teach? And how long ago is it that you taught?)
3. Can you tell me a little bit about the main mission or vision of the teacher education
program? By mission or vision, we mean the image of the kind of teacher your
program intends to prepare.
4. How do you describe the goals of this course to your candidates (to the student-
teachers in the course)?
5. Can you elaborate on the ways that your course builds on other courses or others’
teaching in the program?
6. Are there any regular meetings for faculty in your program—when all faculty in the
program are required to meet together? What is the focus of those meetings?
PRACTICE
7. Can you tell me about any of the opportunities candidates have in your class, for
instance, to plan for teaching (to plan lessons or units, to develop materials, or to
design assessment tasks)? How do you do that?
8. What about role play, or enact teaching? How do you do that?
9. What about opportunities to assess pupil learning, using samples or examples of real
student work (like student essays or math problems, or history writing)? How do you
do that?
10. What about opportunities to examine videos of classroom teaching, or to look at
transcripts of pupils’ discussions? How do you do that?
11. What about looking at national curriculum/standards/guidelines? Can you tell me
about how they do that? How do you do that?
12. Do you demonstrate or model the practices that this course emphasizes (for
201
instance modeling different instructional strategies or different ways of engaging
students)? Can you give me an example?
13. Can you describe any opportunities the candidates have to analyze and reflect on
their own field work?
14. Of all those elements—planning, role play, looking at pupil work, examining
video, or national curriculum—could you pick one of them that you use the most and
talk me through a lesson that uses those opportunities?
LINKING THEORY AND PRACTICE
15. Can you tell me about one or two assignments or activities in the class that require
the candidates to make connections between theory and classroom practice?
a. How can you tell that they are learning about the relationship between theory and
practice from doing these assignments? Can you give me an example?
16. We asked you to select an assignment or activity that is most successful in helping
candidates link theory and practice.
a. Can you tell me about the goals of this assignment? How can you tell that it helps
them link theory and practice?
THEORY
17. Finally, I’d like you to select one of the key readings on this reading list—one of
the readings that seems particularly central to your goals in the course—that you have
been reading up to this point in the class. [NOTE: they should be the same readings
they ask the candidates about in the candidate pair interview.] Interviewer should
NAME the reading on the audio-recording so we know what materials they are
discussing.
a. Why is this such a good text for teaching the class?
b. What sorts of activities do you have candidates do in class (or outside of class) to
help them understand the ideas in this reading?
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APPENDIX B: CATE INTERVIEW PROTOCOL, TEACHER CANDIDATES
Background
1. What do you think are the goals of this course?
a. What is your teacher educator trying to help you learn about teaching and learning?
2. Can you elaborate on the ways that this course builds on other courses or others’
teaching in the program?
PRACTICE 3. Can you describe any of the opportunities you have had in this class, for instance, to
plan for teaching (to plan lessons or units, or to develop materials)? (Probe: How?)
4. What about opportunities you have had to role play, or rehearse aspects of real
teaching, to practice what you would say to a student or to practice a mini-lecture or
leading a class discussion? (Probe: How?)
a. What aspects of the teacher role were emphasized in this mini-lecture?
5. What about any opportunities you have had to assess or analyze pupil learning,
using samples or examples of real pupils’ work (like pupils’ essays or math problems,
or history writing)? (Probe: How?)
6. What about any opportunities you have had in this course to examine videos of
classroom teaching, or to look at transcripts of pupils’ discussions, or to read cases of
real classroom teaching? (Probe: How?)
7. Has your teacher educator demonstrated or modeled the practices that this course
emphasizes (for instance modeling different instructional strategies or different ways
of engaging pupils)? (Probe: How?)
8. What about any opportunities you had in this course to look at or review, or critique
national, state or local curriculum? (Probe: How?)
LINKING THEORY AND PRACTICE
9. Can you tell me about one or two assignments in the class that really helped you
make connections between theory (some of the ideas from the readings or the articles,
or the theoretical ideas in the course) and practice? (either what real classroom
teachers do in their practice, or what you yourselves experienced in your field
placements)?
a. What did that help you learn?
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10. We asked you to bring a completed assignment and talk about it with us. Can you
take out that assignment now and tell us a little bit about it.
a. First, can you tell me about what you think your teacher educator wanted you to
learn from doing this?
b. What are some of the things you learned about teaching and learning, from doing
this assignment?
c. What was hard about this assignment? What was easy?
d. What sorts of feedback did you get on this assignment? What do you think the
teacher educator was trying to help you understand?
THEORY
11. Finally, I’d like you to tell me about [reading the faculty selected in the faculty
interview].
Interviewer should NAME the reading on the audio-recording so we know what
materials they are discussing
a. From your perspective, what is the main idea of this reading? Why do you think
your teacher educator wanted you to read this? What did he or she want you to learn?
What do you think you learned from reading it?
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APPENDIX C: CATE INTERVIEW PROTOCOL, PROGRAM DIRECTOR
Program Vision
1. When you imagine the ideal teacher graduating from your program, what kind of
teacher are they?
a. What kind of teacher would you like to see graduating from your program?
b. What kind of teaching would you like them to be doing in the classroom?
c. Are there things you would like them to be doing outside of the classroom--in terms
of service, school reform, community work, or a particular role you would want them
to play in society-- or are you primarily focused upon their work inside the classroom?
2. How does your vision compare, or fit with, the visions of your colleagues in your
program?
a. How do you know that the vision is (similar/different)? Do you work on it together?
3. Does your program have an agreed-upon vision regarding the preparation of
teachers?
a. In my research, I have found that some programs are designed around a particular
vision, for example, to improve the learning and development of children, to address
social inequalities, or to serve a particular community. Would you say that your
program has a particular vision of what the faculty want to accomplish in preparing
teachers?
b. Is the vision of your program different from other programs in your country?
4. Would you say your program has a particular vision of good teaching that your
program is designed to support? How would you describe the kind of good teaching
your program is designed to emphasize?
5. Does your program also have a vision of the role you hope your graduates will play
in society?
6. Does your program also have a vision of the role you hope your teachers will play
in terms of service, for example, serving a particular community, or promoting
national values?
7. Does your program try to recruit teachers that fit that vision?
a. If yes, what qualities or characteristics do you look for in the candidates who apply
to your program?
205
8. Ultimately, of those candidates who come to your program, what kinds of visions
do they have?
a. Rephrase: In my work, I have found that teachers have a variety of reasons to enter
teaching; some want to make a difference in children’s lives; others want to make an
impact upon social inequalities they are concerned about; and some see it as a kind of
service they want to make for their community.
b. Why do you think teachers in your program are choosing to become teachers?
Theory and Practice
9. What are some of the main opportunities that your program provides to help
candidates learn about the relationship between theory and real classroom teaching?
a. Can you give some examples?
b. What about in language arts or mathematics; are there particular assignments in
those classes that are designed to link theory and practice? How are they constructed?
Can you give me some examples?
10. What opportunities does your program provide to help candidates practice
teaching?
a. Can you give me some examples of particular assignments? Or projects?
b. What do you think these assignments help the students learn or understand?
c. How do you support students in doing these assignments? What kind of help do you
provide for them?
Practice
11. Can you tell me about how the fieldwork and student teaching experiences are
organized in your program? [Probe: how many hours of fieldwork; how is the
fieldwork or student-teaching sequenced over time; when do candidates start having
full responsibility for teaching a class of K-13 students (if any)]
12. Are there assignments or experiences in the program that draw upon or require the
candidates to draw from or use those fieldwork or student-teaching in their
coursework in the program? Can you give me an example?
Assessment of Practice
13. How is candidates’ teaching practice assessed in this program? Do you have any
206
shared standards or criteria by which you evaluate candidates’ teaching practice?
[PROBE: For example when a faculty member or supervisor from your program
observes a candidate, are there any common standards or assessments you use to
evaluate their teaching?]
14. Are there any particular assignments or requirements that you use, to help you
determine whether (and when) a candidate is actually ready for full-time classroom
teaching? Will you please describe them?
207
APPENDIX D: FULBRIGHT INTERVIEW PROTOCOL, PROSPECTIVE
TEACHERS
Intended interviewees: Finnish teacher-trainees (mathematics) at the [University]
Today is <DATE>, and this interview is taking place in <LOCATION> with
<NAME>. Hello, <NAME>.
My name is Ben, and I’ve been here in Finland for several months studying
mathematics education. In the United States, most math classes are sorted according
to the performance levels of students. For example, one class may contain students
who have previously performed at a high level, while another may contain students
who have previously performed at a low level. Here in Finland there is no
performance level division in mathematics classes, and I became interested in learning
more.
Today I’d like to ask you some questions about your experiences both as a student and
as a teacher. And I’d like to thank you for taking the time to talk today, and even
more, to talk with me in English.
1. So how did you decide to become a teacher?
2. What subjects are you preparing to teach?
3. How did you choose mathematics? (If this question is partially answered in
question #1, ask for further details)
4. What do you think are some of the most important things you’ve learned so far
about teaching? (If needed, ask why).
5. How has your teacher training program prepared you to teach mixed performance
groups? (Possible add on: what do you expect the range of abilities to be?)
6. How do you think students learn in mixed-performance groups compared to same-
performance groups?
7a. (If interviewee says that all students learn better in mixed-performance) Could
you tell me about an experience you had in school, either as a student or a teacher, that
describes how this happens or happened?
7b. (If interviewee feels that not everyone is learning in mixed groups) How do you
make sure that everyone learns well in these groups?
8. Does your cooperating teacher do specific things to address mixed performance
groups? (Follow-up for details if necessary) (if yes, follow-up for how this has
influenced TT’s teaching)
9. Do you think that students know who the “stronger” and “weaker” students in the
class are? (If yes, how do you see this in the classroom?)
208
10. Imagine that classes were separated – “high performance” students in one class,
and “low performance” students in another. How do you think that would affect
student learning?
11. How do you think such a separation would affect how you teach?
12. Finally, what would you say are your goals as a teacher? In other words, at the
end of the school year, what would you want your students to say about their
experience learning mathematics?
209
APPENDIX E: FULBRIGHT INTERVIEW PROTOCOL, PROFESSORS
Intended interviewees: Finnish teacher-trainees professors (mathematics) at the
[University]
Today is <DATE>, and this interview is taking place in <LOCATION> with
<NAME>. Hello, <NAME>.
So as you know, I’ve been here in Finland for almost 9 months studying mathematics
education. In the United States, most math classes are sorted according to the
performance levels of students. For example, one class may contain students who
have previously performed at a high level, while another may contain students who
have previously performed at a low level. Here in Finland there is no performance
level division in mathematics classes, and I became interested in learning more. I’ve
spent the past several months talking with the teacher trainees, thanks to your help,
and I wanted to finish by asking you some questions from the professor’s point of
view.
1. So just as a background question, how did you become involved in mathematics
education?
2. How did you choose mathematics? (If this question is partially answered in
question #1, ask for further details)
3. Besides your teaching here at the university, what kind of work have you done with
primary or secondary school teachers?
4. What kind of students do you think are attracted to the teaching program here?
5. What do you think are some of the most important things that your students learn
about teaching in the university classes?
6. How much of your time is spent helping these teachers prepare for mixed
performance classes? (follow-up: ask for specific details or strategies or examples)
7. Do you ever talk about the theories behind mixed performance groups and same
performance groups?
8. What experience or knowledge do you have about these different systems?
9. Based on your experience or knowledge, what opinions do you have regarding
mixed performance groups?
10. Based on your experience or knowledge, what opinions do you have regarding
same performance groups?
11. Researchers have been coming to Finland for almost a decade now based on
Finland’s PISA results. In your opinion, what are the reasons for this success?
210
(potential follow-up: what do these results tell us about mathematics education in
Finland? What don’t they tell us?)
12. Finally, an opinion question. In August I’ll be back in California working with
teacher trainees at my own university. What would you want me to tell them about
becoming a math teacher in Finland? (If this question is too tough, ask what advice
they would give to the teacher trainees)
211
APPENDIX F: CATE DATA CODES
Code Example
Authenticity “We have lots of opportunities to learn from actual lessons.” [Yamato
University students]
Theory “And our students have pedagogical content knowledge enough.” [Hanna,
Lapinkaari University professor]
Practice “That’s what the C&Is are all about: the incredible emphasis on clinical
practice.” [Foster University program director]
Textbooks “They can think about the types of tasks that can be found in textbooks and how
they are utilized” [Saara, Catherine University professor]
Curriculum “In today’s lesson too – this is the national curriculum. We show them this.”
[Yamato University professor]
Goals of Math
Teaching
“The candidates should look at things not just in terms of well/not well, but
what happened and why? … The goal was to look for evidence of math
proficiency in students.” [Foster University instructors]
Mock lessons “They use rehearsal, which is done multiple times throughout the 3-course
sequence.” [Foster University instructors]
Juku “The aim of schools and cram schools are different.” [Yamato University
students]
English
Language
Learners
“It might be confusing for English learners that 9, 10, and 11 have no verbs.”
[Foster University observation 1, student]
Testing “This is a personal view, but results of periodic tests in Japan’s schools are
important, and the test results of junior high school students studying to enter
high school are important; almost half of the entrance process for university is
based on high school results.” [Yamato University students]
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APPENDIX G: FULBRIGHT DATA CODES
Code Example
Mixed
Positive
General
“It still may be a good idea to have a mixed groups [sic], because it has some
benefits. I mean pupils come teach each other and of course it’s a more social stuff.”
[Student 5]
Mixed
Positive
Low
“Of course, for less talented students it’s [mixed groups] a benefit.” [Student 5]
Mixed
Positive
High
“So, if you are the stronger student, and you get a chance to teach the weaker one,
then you get better from it.” [Student 7]
Mixed
Negative
General
“And that’s a problem with mixed performance groups.” [Student 8]
Mixed
Negative
Low
“With the mixed group, I think the weakest students, they don’t learn so well. They
have moments that it’s too difficult for them.” [Student 9]
Mixed
Negative
High
“I remember sometimes it was really frustrating because, like sometimes you were
just sitting there, and the teacher is going through the same thing again, and you
already understood a while back.” [Student 13]
Same
Positive
General
“Well, I myself, I think that we really need performance groups.” [Student 6]
Same
Positive
Low
“Um, well, with low performance class you could really, really have the opportunity
to make the subject at least a little bit more interesting than what they think it is.”
[Student 4]
Same
Positive
High
“Well of course, I think high performance group will learn much more, which is
good.” [Student 12]
Same
Negative
General
“I think parents think that they [sic] children have some kind of mark on their
forehead that ‘yeah, now he’s a stupid, and he’s the normal one, and he’s the brilliant
one’.” [Student 12]
Same
Negative
Low
“Well, I get the feeling that people in a low performance group would be even more
dumbed down, you know.” [Student 7]
Same “I think that it’s more problematic for the better students, at least from some point on,
213
Negative
High
like from 7th
to 8th
grade. Because then they get the bottom of the social hierarchy.”
[Student 6]
Mixed = Mixed Performance Groups; Same = Same Performance Groups
Low = Low performing students; High = High performing students; General = no specific
performance group identified
214
APPENDIX H: THE PAINTED CUBES PROBLEM
Painted Cubes
Using 1-centimeter cubes, Ben builds a big cube with side length 3 centimeters.
Ben paints all of the faces of this big cube yellow.
Then he breaks it back down into 1-centimeter cubes.
215
Part One
1. Build a model of Ben’s big cube using the blocks.
2. How many of the 1-centimeter cubes have no faces painted yellow? Which cubes
are these?
3. How many of the 1-centimeter cubes have exactly one face painted yellow? Which
cubes are these?
4. How many of the 1-centimeter cubes have exactly two faces painted yellow?
Which cubes are these?
5. How many of the 1-centimeter cubes have exactly three faces painted yellow?
Which cubes are these?
6. How many of the 1-centimeter cubes have more than three faces painted yellow?
Which cubes are these?
7. Show your conclusions using a picture or a model.
Part Two
Ben decides to build and paint cubes of different sizes using the 1-centimeter cubes.
8. First Ben builds a 444 cube. How many of the 1 centimeter cubes in this big
cube will have
a. No faces painted yellow?
b. One face painted yellow?
c. Two faces painted yellow?
d. Three faces painted yellow?
e. More than three faces painted yellow?
9. What about a 555 cube?
216
10. A 101010 cube?
11. An nnn cube?
12. For what values of n do these patterns hold?
13. Justify your answers to question 11.
217
APPENDIX I: THE TV ANTENNA PROBLEM
TV Antenna Problem
The Situation:
Elder wants a good TV antenna to watch all of FC Barcelona’s games at home, so he
makes a plan to put an antenna on the roof of his house. However, Elder does not
know how much wire he needs to hold the antenna in place from strong San Francisco
winds.
You can help Elder find the length of the wire needed to hold the antenna by creating a
model of his roof and by using your Pythagorean knowledge.
Elder’s Roof:
Elder’s roof is one large, flat rectangle. It is parallel to the ground. His TV antenna is
mounted in the exact center of the roof. Eight wires connect the top of the antenna to
the roof - four to the corners and four to the midpoints of each side. These are the
wires that support the antenna in the wind.
Building a Model of the Situation:
Materials Needed:
Use rectangular cardstock for the roof. Draw the diagonals of the rectangle to find
the center.
Use a straw for the TV antenna. Attach it to the roof using tape or knots.
Use string for the wires.
The Real Life Dimensions:
Elder’s roof is 60 feet long and 32 feet wide. The TV antenna is 30 feet tall.
Question: How much wire does Elder need to hold his TV antenna
in place?
Your Task:
Answer the question above. In other words, tell Elder how much wire he
needs.
Consider the following additional questions:
How many different right triangles did you find?
218
o How many different sized wires does Elder need?
What was special or different about finding the wire length to the four
corners?
Is your model a scale model? Why or why not? If so, what is your scale
factor?
If the dimensions above (roof 60 by 32 ft, antenna 30 ft) were…
o cut in half (divided by two), what would happen to the length of the
wires?
o doubled (times two), what would happen to the length of the wires?
o changed to 72 ft long and 20 ft wide, would it change the amount of
wire?
How long is one of the roof’s diagonals?
If wire costs $3 per foot, how much will Elder need to secure the antenna
on top?
.