reproductions supplied by edrs are the best that can be ... · ed 443 694. author title institution...

ED 443 694

AUTHORTITLEINSTITUTIONPUB DATENOTEAVAILABLE FROM

PUB TYPEJOURNAL CITEDRS PRICEDESCRIPTORS

ABSTRACT

DOCUMENT RESUME

SE 063 846

Langbort, Carol, Ed.; Curtis, Deborah, Ed.Mathematics Education.San Francisco State Univ., CA. School of Education.2000-00-00108p.; Published annually.San Francisco State University, College of Education, 1600Holloway Avenue, San Francisco, CA 94132.Collected Works - Serials (022)College of Education Review; v11 spec iss Spr 2000MF01/PC05 Plus Postage.Elementary Education; *Evaluation; Higher Education;*Mathematics Curriculum; *Mathematics Instruction; MiddleSchools; *Problem Solving; Teacher Attitudes

The focus of this special issue is mathematics education.All articles were written by graduates of the new masters Degree program inwhich students earn a Master of Arts degree in Education with a concentrationin Mathematics Education at San Francisco State University. Articles include:(1) "Developing Teacher-Leaders in a Masters Degree Program in MathematicsEducation" (Carol Langbort); (2) "The Impact of a Mathematics Summer CampProgram on Girls' Spatial Visualization Skills and Career Aspirations inMathematics and Engineering" (Lorraine L. Hayes and Deborah A. Curtis); (3)

"The Effects of a Student-Generated Rubric on Third Graders' Math PortfolioSelection Criteria" (Thais Akilah Kagiso); (4) "Observation and Analysis ofthe Mathematical Problem Solving Behaviors of Six African American Fourth andFifth Graders" (Norman S. Mattox IV); (5) "Teaching Students To Write aboutSolving Non-Routine Mathematical Problems" (Delia Levine and Ann Gordon); (6)

"The Use of Rubrics by Middle School Students To Score Open-Ended MathematicsProblems" (Ford Long, Jr.); (7) "A Survey of Teacher Beliefs aboutPre-Requisite Experiences for Student Success in Algebra" (Audrey Adams); (8)

"Fifth Grade Teachers' Attitudes towards Implementing a New MathematicsCurriculum" (Lorene B. Holmes and Deborah A. Curtis); and (9) "Ethnicity as aFactor in Teachers' Perceptions of the Mathematical Competence of ElementarySchool Students" (Kathlan Latimer). (ASK)

Reproductions supplied by EDRS are the best that can be madefrom the original document.

1

College Of Education

TO THE EDUCATIONAL RESOURCESINFORMATION CENTER (ERIC)

PERMISSION TO REPRODUCE ANDDISSEMINATE THIS MATERIAL HAS

BEEN GRANTED BY Review EDUCATIONAL RESOURCES INFORMATIONCENTER (ERIC)

Office of Educational Research and ImprovementU.S. DEPARTMENT OF EDUCATION

This document has been reproduced aseceived from the person or organization

originating it.

Minor changes have been made toimprove reproduction quality.

Volume 11Points of view or opinions stated in thisdocument do not necessarily representofficial OERI position or policy.

BEST COPY AVAILABLE

San Francisco State University Spring 2000

2

College Of Education

ReviewVolume 11

0-

j10.".....S.,...

San Francisco State University Spring 2000

3

San Francisco State UniversityCollege of Education Review

EditorJack R. Fraenkel

Research & Development Center

Guest Editors for this IssueCarol LangbortDeborah Curtis

AdministrationJacob Perea, Dean

Vera Lane, Associate DeanDavid Hemphill, Associate Dean

Alicia Jalipa, Administrative Assistant

College of Education Review 1

IN THIS ISSUE . . .

3 Editor's Corner

4 Introduction

FEATURES

5 Developing Teacher-Leaders in a Masters Degree Program in MathematicsEducation

Carol Langbort

10 The Impact of a Mathematics Summer Camp Program on Girls' SpatialVisualization Skills and Career Aspirations in Mathematics and Engineering

Lorraine L. Hayes and Deborah A. Curtis

24 The Effects of a Student-Generated Rubric on Third Graders' Math PortfolioSelection Criteria

Thais Akilah Kagiso

37 Observation and Analysis of the Mathematical Problem Solving Behaviors of SixAfrican American Fourth and Fifth Graders

Norman S. Mattox IV

50 Teaching Students to Write about Solving Non-Routine Mathematical ProblemsDelia Levine and Ann Gordon

64 The Use of Rubrics by Middle School Students to Score Open-EndedMathematics Problems

Ford Long Jr.

A Survey of Teacher Beliefs about Pre-Requisite Experiences for StudentSuccess in Algebra

Audrey Adams

84 Fifth Grade Teachers' Attitudes towards Implementing a New MathematicsCurriculum

Lorene B. Holmes and Deborah A. Curtis

95 Ethnicity as a Factor in Teachers' Perceptions of the Mathematical Competenceof Elementary School Students

Kathlan Latimer

105 Information for Authors

2 College of Education Review

We are pleased to bring you thiseleventh issue of the College of Edu-cation Review, the official journal ofthe College of Education at SanFrancisco State University. As is ourcustom, the articles lit this issue ofthe Review present a variety of ideasand viewpoints on a topic of interestto the educational community today.Every so often we produce a specialissue of the Review. This is one ofthose special issues. A few years ago,for example, we organized an issuedealing with the topic of scienceeducation. In that issue, we dis-cussed a' number of state-of-art top-ics having to do with the teaching ofscience in elementary and secondaryschools. The issue was well-receivedby members of both the educationaland scientific communities, and wereceived many complements on it.

In this issue, we present a num-ber of articles that deal with the topicof mathematics education. Guest edi-tors Carol Langbort and DeborahCurtis, both professors of educationhere at San Francisco State Univer-sity, have compiled a variety of arti-cles in which the authors discussnew and innovative ways to teach

EDITOR'S CORNER

mathematics. All of the articles inthis issue were written by graduatesof San Francisco State's new Masterof Art's Degree in Education, with aspecial concentration in MathematicsEducation. This program, under theleadership and direction of ProfessorLangbort, has been funded since1993 by a grant from the EisenhowerMathematics and Science EducationState Grant Program. All of the arti-cles are based on work completed bythe first cohort of graduates from theprogram. We think you will find in-teresting what they have to say, and(we hope) you will gain several newideas about how to teach mathemat-ics.

We hope that you will be pleasedwith this issue, and we welcome anyreactions you may have to the ideasexpressed herein. We will publishyour responses in a future issue ofthe journal.

Lastly, a word of special thanksto my assistant, Michele Larkey, forher very good work and calmnessunder pressure.

Jack R. FraenkelSpring, 2000


Introduction

We are extremely pleased to bringyou this special issue of the College ofEducation Review, the official journal ofthe College of Education at San Fran-cisco State University. The focus of thisspecial issue is Mathematics Education,and all the articles were written bygraduates of our new Masters Degreeprogram, in which students earn a Mas-ter of Arts degree in Education with aConcentration in Mathematics Educa-tion.

The introductory article presents de-tails about the development of this pro-gram, which has been funded by theDwight D. Eisenhower Mathematicsand Science Education State Grant Pro-gram since 1993. The remaining articlesare based on the theses and field studiescompleted by the first cohort of mastersstudents, who received their degrees inMay 1997. There are a variety of topicsaddressed in these papers, examiningboth teachers and students in elemen-tary and middle grades. Several articlesfocus on teachers: teacher beliefs aboutprerequisites needed for learning alge-bra, teacher attitudes about imple-menting new curriculum materials, andteacher perceptions of mathematicalcompetence. Other articles focus on stu-dent assessment: third graders develop-ing portfolios, and middle school stu-dents using rubrics to score open -endedmathematics problems.

Two papers focus on mathematicalproblem-solving: one looks at teachingstudents to write about their mathe-matical thinking, while the other de-scribes the observation and analysis ofAfrican American children's problem-solving behaviors. Lastly, one looks atgender issues, and evaluates the impactof a summer camp math program forgirls.

We wish to give special thanks toLinda-Barton White, manager of Aca-demic Programs of the Eisenhower StateGrant Program, who had the foresightto fund projects such as this. We alsowish to thank the project co-director,professor emeritus Jose Gutierrez, andthe other faculty who worked with thesestudents during this time: Drs. BarbaraFord, Cecelia Wambach, Anne Gordonand Pamela Tyson, San Francisco StateUniversity, and Lise Dworkin andTheresa Hernandez-Heinz from the SanFrancisco Unified School District. All ofthese faculty members had enormousimpact on the development of the cul-minating projects of these graduate stu-dentstheir masters thesis or fieldstudy.

We hope that you will be pleasedwith this special issue. We welcome anyreactions you may have to the ideas ex-pressed. This issue is very special to usand we hope you enjoy it as much as weenjoyed putting it together.

Carol LangbortDeborah Curtis

April 2000

4 College of Education Review7

Developing Teacher-Leaders in aMasters Degree Program in

Mathematics EducationiCarol Langbort

This article describes the development and im-plementation of this interdepartmental mastersdegree program in mathematics education for K-8teachers, beginning with the award of a planninggrant in the year 1993. Twenty students gradu-ated in May, 1997, and 19 will graduate in May,2000. The emphases of the program are describedin some detail: increasing the mathematicalknowledge of the teachers, discussions of the cur-rent issues in mathematics education and the de-velopment of teacher-leaders. The development ofnew courses and descriptions of final projects arealso included.

COE Review, pp. 5-9

I am a mathematics educator in theDepartment of Elementary Education atSan Francisco State University. Since the1970s I have been personal friends andcolleagues with mathematics educatorsin the Mathematics Department. Overthe years we have worked together in avariety of ways induding professionaldevelopment grants for elementary andmiddle school teachers and serving inprofessional organizations. We havebeen an important means of collegialsupport for each other. It was no accidentthen, in 1993, when we received a Re-

quest for Proposals (RFP) from theDwight D. Eisenhower Mathematics andScience Education State Grant Programto develop a master's degree program inmathematics education for minorityteachers, that we were ready and willingto take on this task.

Jose Gutierrez, a professor in theMathematics Department, and I wel-comed this opportunity to write a plan-ning grant for this purpose. Jose hasspent most of his career teaching mathe-matics education courses for under-graduates planning to become elemen-tary teachers. For several years, we hadbeen discussing the possibilities for ajoint masters degree program, but it wasthe RFP that gave us the impetus tomove forward.

We had been working with K-8 class-room teachers through the San FranciscoMath Leadership Project (one of the Cali-fornia Mathematics Projects) since 1984and knew of the plight of our teachers.

Carol Langbort is a Professor of Elemen-tary Education at San Francisco State Uni-versity, and Co-Director of the MathematicsEducation Masters Degree Project.

'This article was originally published in the Fall, 1998 Newsletter of the Mathematicians andEducation Reform Forum. Reprinted with permission.


They had become interested in mathe-matics education; they realized that theyreally could learn and understandmathematics, and indeed, problem solve.Overall, they had a renewed interest andenthusiasm about mathematics learningand teaching. Unfortunately, there wasno place for them to go to continuestudying in this area. These teachersneed to be in an environment where theycan work with others, feel free to makemistakes and learn from them; and theyneed challenging content, which, at thesame time, is related to school mathemat-ics. Typically, mathematics departmentsdo not offer the kinds of courses thatwould be appropriate for these teachers.Another problem is that few courses areoffered at hours that classroom teacherscan fit into their busy schedules.

Planning and Beginning the DegreeProgram

We wanted to design a mathematicseducation degree program would vali-date the efforts of teachers to improvethe mathematics programs in their class-rooms and in their schools, and wouldalso give the teachers official recognitionof their work in this field. It would en-courage them to continue in their studiesand add a depth to their knowledge inboth the content and the pedagogical is-sues of the current mathematics reformmovement. And so this degree was de-veloped between the Department ofElementary Education and the Mathe-matics Department with courses fromboth departments. We utilized some ex-isting courses and designed a few newcourses, in particular: Analyzing Cases ofMath Teaching (Ed), Developing Leadershipin Mathematics Education (Ed), Assessing

Mathematical Thinking (Ed), and Mathe-matical Investigations: Dissection and Inte-gration of Topics (Math). Requirements forthe degree would include four coursesfrom the Mathematics Department, fourElementary Education courses, a coursein Research Methods in Education, andwriting a thesis or conducting a fieldstudy.

Following the planning grant, we re-ceived the full grant to implement thistwo and one-half year program. As thegrant included stipends for teachers, weexpected that the recruitment processwould be successful. However, we werenot prepared for the many, many teach-ers who attended our information meet-ing about the program, and had toquickly change the room to an audito-rium to hold the more than 160 teacherswho attended. We received 80 completedapplications for our first cohort, and se-lected an ethnically diverse group of 30students. Going through the sequence ofcourses as a learning community, theteachers provided an enormous amountof support for one another. The first co-hort graduated in May, 1997; the secondcohort of 19 students will graduate inMay, 2000.

Mathematics, Current EducationalIssues, and Developing Teacher-Leaders

One focus of this program is to in-crease the mathematical knowledge ofthe teachers. There are four courses inthe Mathematics Department: Geometry,Measurement and Probability; Curriculumand Instruction in Mathematics; Computersand Elementary Mathematics, and Mathe-matical Investigations. Completing thesecourses gives the teachers the opportu-


nity to apply for a supplementary au-thorization in mathematics for middleschool teaching.

Another focus of the program is cur-rent issues in mathematics education.With the many changes occurring inmathematics education, these teachersneed to be on the cutting edge of the cur-rent reform movement.

The third focus of the program, de-velopment of teacher-leaders, is the mostunusual aspect of the program. We needmore teacher-leaders, and we need moreteacher-leaders of color. We've learnedthat leadership can take a great variety offorms. Presenting workshops at theschool, district, state, and national levelis one type of leadership. Writing articlesand developing curriculum materials isanother type of leadership. Planning con-ferences for teachers and teachers-to-be,taking leadership positions in math or-ganizations, state councils, and statewidecommittees, and working with pre-service teachers are additional examplesof roles that our graduates have taken.Through this program, teachers have theopportunity to strengthen their leader-ship skills in these areas and are encour-aged and supported in developingstrengths in new areas.

Leadership Development inMathematics Education

Although the entire program contrib-utes to developing leadership, the courseLeadership Development in MathematicsEducation was designed specifically forthe program to focus on leadership is-sues. It is offered early in the program sothat from the beginning the teachers seedeveloping leadership skills as an expec-tation of the program. The course focuses

on three topics: examination of major is-sues in mathematics education; reviewand analysis of major documents of thereform movement in mathematics educa-tion; and examination of leadership anddissemination practices.

We first focused on issues in mathe-matics education. Class discussions werebuilt around articles from NationalCouncil of Teachers of Mathematics(NCTM) Yearbooks on Multicultural andGender Equity in the Mathematics Class-room (1997) and Teaching and LearningMathematics in the 90s (1990). The teach-ers then selected issues to pursue ingreater depth and were expected to pre-sent different perspectives of the issue tothe class. The teachers also wrote indi-vidual papers on a second issue of theirchoice.

Although the teachers were familiarwith some of the major documents of thereform movement, and even had copiesof some documents, they had not reallylooked at them in depth. Some time wasspent examining more closely the 1992California Mathematics Framework, NCTMsCurriculum and Evaluation Standards forSchool Mathematics (1989), NCTM's Pro-fessional Standards for Teaching Mathemat-ics (1991), and Everybody Counts - A Re-port to the Nation on the Future of Mathe-matics Education (National ResearchCouncil, 1989).

We also spent considerable time prac-ticing leadership. Selected articles fromthe NCTM 1994 Yearbook on ProfessionalDevelopment for Teachers of Mathematicswere assigned as useful backgroundreading: "Changing Mathematics Teach-ing Means Changing Ourselves: Implica-tions for Professional Development," byJulian Weissglass and "Ten Key Princi-


1 0

pies from Research for the ProfessionalDevelopment of Mathematics Teachers,"by Doug Clarke. One of the class activi-ties was to develop and get feedback on aworkshop to be presented at the teach-ers' schools. Some students had never be-fore presented a workshop and the sup-port of the group was invaluable. In ad-dition, the grant provided funds for allthe teachers to attend the CaliforniaMath Council's Annual Conference aswell as the annual conference of NCTM.We hoped they could envision them-selves presenting at these conferences inthe future.

Real Life Math Education Activities andFinal Project

There was a great deal of mathemat-ics education activity taking place inCalifornia during the Fall of 1997, thevery semester that the leadership coursewas taught. The students had an oppor-tunity to hear William Schmidt, Execu-tive Director of the Third InternationalMath and Science Study (TIMSS), talkabout the TIMSS results. They also had achance to attend hearings on the newmath standards being developed by theState of California. In fact, a few teachersactually testified at these hearings. Forone class assignment the teachers were tosubmit responses to the draft of the newMath Framework for California Schools,which was being developed during thesame semester. These real life opportuni-ties for leadership contributed greatly tothe teachers' development as teacher-leaders.

The final project, either a field studyor thesis, plays an important role in thedevelopment of teacher leadership. Sev-eral teachers from the first cohort have

had opportunities to present their workat local and national conferences. The is-sues chosen for study by the teachersrepresent real world questions that haveemerged for them in the course of theprogram, but also reflect their concernsas full-time teachers. A sampling of titlesillustrates the variety of topics: A SecondGrade Spanish Curriculum Unit for Devel-oping Mathematical Language; Ethnicity as aFactor in Assigning Mathematical Compe-tence to Elementary Level Students; Teach-ing Students to Write About MathematicalProblems; Observation and Analysis of 4thGrade Latino Students' Problem Solving Be-haviors Using a videotape Protocol; The De-velopment, Implementation and Evaluationof Family Volunteer Workshops that PromoteMath Literacy Young Children; Using Hy-perstudio as an Assessment Tool; A Surveyof Teachers' Beliefs About Pre-Requisite Ex-periences For Student Success in Algebra.

Benefits for Faculty

Having a grant allowed us to recruitfaculty from outside the university, aswell as enlist faculty from several de-partments. There are a total of nine in-structors, including five university pro-fessors and four faculty from outside theuniversity, two of whom are from theSan Francisco Unified School District. Afaculty member teaches courses, workswith the teachers on their final projects,or both. The nine faculty members serveon the project Advisory Board as well asattend regular meetings to discuss theprogram and the progress of the stu-dents.

Two weekend retreats helped de-velop the collegiality among faculty andteachers. The first retreat, held at the startof the program, is an opportunity (1) for


participants to get to know each other,(2) to prepare the teachers for the pro-gram, and (3) to give introductory ses-sions to the courses for the following se-mester. At the second retreat, held at thebeginning of the intense work period forthe final project, teachers write, meetwith faculty, and have an opportunity todevelop and expand their ideas with avariety of people.

During the Fall, 1998 semester, wetried something new--three all-day Sat-urday meetings for the teachers and fac-ulty as a kick-off to the research methodscourse to be taught in the spring. Teach-ers presented their beginning ideas fortheir final projects and the faculty madesuggestions and elaborated on the teach-

ers' ideas. The faculty benefited greatlyfrom this exchange.

One of these Saturday meetings wasdevoted to a discussion of qualitative re-search methods. The discussion, basedon the transcript of an earlier teacher dis-cussion about a case study, allowed theteachers to study themselves as subjects.The interest level was extremely highand both teachers and faculty learned agreat deal about the challenges of doingcareful qualitative research.

Any undertaking in designing andrunning a new program yields unex-pected outcomes. The benefits to the fac-ulty in this project is one of those pleas-ant surprises.

College of Education Review

12

The Impact of a Mathematics SummerCamp Program on Girls' Spatial

Visualization Skills and CareerAspirations in Mathematics and

EngineeringLorraine L. HayesDeborah A. Curtis

The purpose of this study was to design, imple-ment, and evaluate a summer program for sixthand seventh grade girls designed to increasetheir spatial visualization skills and their inter-est in engineering careers. A significant changefrom pretest to posttest was found for one of thetwo measures of spatial visualization. Therewas a significant change in the career interestsof the girls from pretest to posttest. In particu-lar, at the end of summer camp, nearly half ofthe girls indicated that they would be interestedin engineering as a career. Student evaluationsof the effectiveness of the summer program wereoverwhelmingly favorable. Results are discussedin terms of the value of intensive summer pro-grams for middle school girls.

COE Review, pp. 10-23

By the year 2005, the engineeringfield will increase by about 200,000 jobs,with civil engineering jobs increasing by25,000 positions and electrical engineer-ing positions increasing by about 60,000jobs (Krantz, 1994; U.S. Department ofLabor, 1994). Current information indi-cates that only 17 percent of engineeringgraduates are women (Basta, 1989). Re-searchers have found that there are sev-eral reasons why girls are not enteringinto the engineering field. These include

(a) lack of spatial visualization skills,and (b) lack of exposure to appropriatefemale role models. Spatial visualizationskills are used when solving three-dimensional problems. Mastery of theseskills is needed to be successful inhigher level math and science classesleading to engineering degrees (Misty,Colton, Rossi & Berger, 1994).

One type of spatial visualization skillis the ability to visualize three-dimen-sional shapes in various positions by ro-tating the shape or flipping the shapeover to see the reflection of that shape asit would be seen in a mirror. An electri-

Lorraine Lanam Hayes currently teachesfirst grade at Marshall Elementary School inFremont, California. She also is a mathemat-ics coach for the Fremont Summer SchoolIntervention program, and provides supportand training for teachers during summerschool for children who are at risk for reten-tion in Fremont Schools. Ms. Hayes is alsoan adjunct faculty member at National Uni-versity. Deborah A. Curtis is an AssociateProfessor of Interdisciplinary Studies inEducation at San Francisco State Univer-sity.


cal engineer uses this skill to write com-mands to a robot so it moves its arm toremove a bomb safely from the ground.Another spatial skill is the ability to seeembedded figures in two- dimensionaldrawings. A civil engineer uses this skillwhen visualizing the hidden shapes in-side a tunnel to decide which directionthey should dig to be able to getthrough a mountain. Studies have foundthat females do not perform as well asmales on these tasks (Gallagher & De-Lisi, 1994; Hyde, Fennema & Lamon,1990).

Some researchers consider spatialvisualization skills innate. There areother researchers, however, who havefound that training can improve the waystudents are able to perform spatialvisualization tasks. For example, in twostudies which required students to buildmodels with blocks and to practice rota-tions of shapes through varying de-grees, students improved their ability toperform spatial tasks (Ferrini-Mundy,1987; Vandenburg, 1975).

Koballa (1988) and Smith and Erb(1986) have also shown the positive im-pact that female adult role models canmake on math and science course selec-tions of teens. Studies have shown thatexposure to female role models in sci-ence careers can encourage girls to entermale-dominated careers (see, for exam-ple, McNamara & Scherrei, 1982; Swin-dell & Phelps, 1991). However, there islittle evidence that a spatial visualiza-tion training program combined withexposure to female role models in mathand science will inspire more girls toconsider higher level math and scienceprograms and future careers in thesefields.

The purpose of this study is to evalu-ate a summer program for sixth and sev-enth grade girls which was designed toincrease (a) their spatial visualizationskills and (b) their interest in math andengineering careers. The first componentof the summer program consisted of acollection of activities that taught spatialvisualization skills. These activities pro-vided opportunities for the girls to learnhow to rotate and reflect shapes by usingblocks and computers. There were alsogames and tasks that, helped the studentsto see hidden geometric figures in agiven shape.

The second component consisted ofbringing guest speakers to class. Thesespeakers were women, of various eth-nicities, who have been working in a va-riety of engineering fields including civilengineering and electrical engineering.

REVIEW OF THE LITERATURE

The Standards of the National Coun-cil of Teachers of Mathematics (NCTM)emphasize the importance of providingequal opportunities for all students todevelop math power. This emphasis issummarized in the policy statement thatsays "Every student can and shouldlearn to reason and solve problems. Byevery child we mean specifically stu-dents who are female as well as thosewho are male" (NCTM, 1991, pp. 21-22).However, there are a number of studiesthat have shown that these NCTM stan-dards are often not met for female stu-dents. In particular, research has indi-cated that girls are not given the sameopportunities as boys to develop prob-lem solving skills, specifically those skillsneeded to perform spatial visualizationtasks (Bohlin, 1994; Hyde, Fennema, &


1411

Lamon, 1990).There are a number of issues that

need to be addressed in discussing thedevelopment of spatial visualizationskills. Although some studies haveshown that there are differences in theability of boys and girls to perform spa-tial visualization tasks (e.g., Johnson andMeade, 1987; Linn & Petersen, 1985), re-searchers disagree on the age at whichthese gender differences appear and thereasons for these differences (Johnson &Meade, 1987; Sherman, 1980). Some re-searchers have argued that the differ-ences are biologically based (see, for ex-ample, McGuiness, 1976; Petersen, 1976;Sanders & Soares, 1986). Other research-ers have contended that the differencesare primarily based on life experiences(Boekaerts, Seegers & Vermeer, 1995;Bohlin, 1994; Hyde, Fennema & Lamon,1990; Jakobsdottir, Krey and Sales, 1994;Kaiser-Messmer, 1993; Kies ler, Sproull &Eccles, 1983; Serbin, Conner, Burchardt &Citron, 1979; Skolnick, Langbort & Day,1982).

Having underdeveloped visual spa-tial skills has been shown to have a nega-tive effect for girls' overall performanceon college entrance exams. These examshave historically had spatial skills ques-tions weighted heavily in the math com-ponent (Gallagher & DeLisi, 1994; Hyde,Fennema & Lamon, 1990; Mistry, Colton,Rossi, & Berger, 1994). There is likelihoodthat lower scores of girls on these testscould be keeping girls from enteringmath /science - related- majors. Theselower scores could also be limiting thegirls' choice of colleges.

Some studies have suggested that ifboys and girls are given the same oppor-tunities to learn spatial tasks, the skills

can be equally developed (Ferrini-Mundy, 1987; Flores, 1990; Vanderburg,1975). Research has also indicated thatexposure to successful female role mod-els can encourage girls to enroll in male-dominated courses which would thenprovide opportunities to develop spatialskills (Koballa, 1988; McNamara &Scherrei, 1982; Smith & Erb, 1986; Swin-dell & Phelps, 1991).

Developmental Differences BetweenBoys and Girls' Spatial Skills

Using tasks that measured spatialskills, researchers have found that girlsand boys develop these spatial abilitiesdifferently. For example, Linn and Peter-sen (1985) reported that spatial percep-tion tasks (horizontality) were more dif-ficult for females than for males. Fur-thermore, they found that spatial rota-tion, which is the task of abstractly doingseveral degrees of shape rotation, wasmore difficult for females.

Spatial visualization tasks, such aspaper folding and finding geometricshapes in a unidimensional representa-tion, were found to be equally difficultfor boys and girls. However, researchershave found that as children get older,males perform consistently better thanfemales on spatial visualization tasks(Johnson & Meade, 1987). It has beenargued that hormones can explain thesedevelopmental differences between thesexes. In particular, McGuiness (1976)argued that spatial skill development ishormonally based, with a positive rela-tionship between spatial visualizationskills and puberty or the presence ofhormones, specifically, testosterone.

Learning styles have also been dis-cussed as another possible innate factor


that may affect spatial visualizationskills. It has been reported that malesand females come to the classroom withdifferent learning styles, which, in turn,can affect their ability to approach theproblem-solving strategies needed inspatial visualization skills. Bohlin (1994),for example, studied differences instyles of learning. In her research, Boh-lin classified learners into two types,namely serialistic learners and holisticlearners, and found serialistic learnerstended to be female, and holistic learn-ers tended to be male. The serialisticlearners were found to have "ruleswithout reason," and to prefer problem-solving strategies that used a step-by-step approach. Serialistic learners werefound to be uncomfortable with explor-ative-type problem-solving strategies,which are the type of strategies that areused in spatial skill development. Holis-tic learners, by contrast, were found toneed to see the whole picture before fill-ing in the details. They preferred to ex-plore solutions to problems, such as spa-tial visualization problems, and did notstart with algorithms to try and find theanswer. Bohlin stated that this holisticlearning style, which was more commonfor boys, leads to the most success insolving unfamiliar problems.

Effects of Lack of Training and Experi-ence on Girls' Spatial VisualizationSkills

Some researchers have argued thatthe lower level of spatial skills in girls isdue to a lack of experience and training.Recreational activities have been shownto limit female exposure to spatial tasks.Serbin, Conner, Burchardt and Citron(1979) found that by age 3, children al-

ready influence each other to play withsex-appropriate toys. In addition, parentstended to buy gender-oriented toys.These toys, along with other life experi-ences at this age, may explain unequalspatial development. The game of pool,for example, provides an excellent op-portunity to develop angle and speedconcepts, and yet the pool hall is consid-ered "male territory." The present-daypool hall is video arcades. Kies ler,Sproull, and Eccles (1983) did an exten-sive study on video games. They foundthat most video games showed a definitegender bias in terms of their content,which, in turn, could adversely effectfemales who might be interested in try-ing the games. In particular, the re-searchers found that the themes of mostvideo games are war, violence, and malesports like baseball, basketball and foot-ball. According to these researchers,these games provide excellent opportuni-ties to develop spatial skills as asteroidsand battle cruisers, for example, need tobe shot down or speed and ball direction,as a second example, need to be assessedto win the games. However, the re-searchers noted that girls were not seenplaying these games, and concluded thatthis was because they did not includetopics of interest to most females.

When children enter school, the edu-cational environment has an impact onspatial skill experiences. Researchershave reported that teachers do not takeinto account learning styles when pro-viding experiences for developing spatialskills. Bohlin (1994), for example, arguedthat classroom teachers tend to use thedesigned lesson, rather than the explor-ative lesson meaning that teachers tendto teach the idea and the algorithm, then

1 GCollege of Education Review 13

test what was taught. While this methodallows serialistic, often female, problemsolvers to perform well on a class mathtest, it does not provide this same groupwith the higher level cognitive process-ing activities needed to develop prob-lem-solving skills used to perform spatialvisualization tasks.

This lack of training continued asgirls make subject choices in high school.Studies have indicated that more boystake higher level math and sciencecourses. Sherman (1980) did a studywhich covered grades 8 to 11. She foundthat even though the math backgroundsand math performances of boys and girlswere not different in 8th grade, there wasa significant decline in female math per-formance and female math participationin the 11th grade. The two areas cited bySherman as the reasons for the changewere girls' attitudes and their lower spa-tial visualization performance.

Hyde, Fennema and Lamon (1990)reported on a study of California highschools which found that girls only madeup 38 percent of the physics classes, 34percent of the advanced physics classesand 42 percent of the chemistry classes.These higher level math and scienceclasses provide the advanced problem-solving and spatial visualization skillsneeded to take college level math andengineering classes, yet these courses aretaken by only a minority of female stu-dents.

Effect of Spatial Visualization Trainingon Future Educational Goals

Studies have shown that the lack ofexposure to problem-solving strategiesand spatial visualization activities couldbe affecting the choices that girls have in

their future educational goals. Problemsolving and spatial skills are the skillsneeded to score well on the SAT-MathCollege entrance examinations (Mistry,Colton, Rossi, & Berger, 1994). The scoreson these tests could affect a student's ma-jor and college choices.

Gallagher and DeLisi (1994) foundthat overall, girls scored about 50 pointslower than boys on the SAT-Mathematics. Girls, however, performedbetter in the areas of computational prob-lems and algebra. The researchers arguedthat this was due to the girls' ability toperform better at tasks that use conven-tional strategies or tasks that can be an-swered through memorization. How-ever, girls scored significantly lower inword problems and geometry tasks thatrequired problem solving and spatialskills. As Hyde, Fennema and Lamon(1990) reported, these are the skills thathave been taught to a minority of girlswho enrolled in the higher-level mathclasses. Hyde, Fennema and Lamon alsoargued that because the SAT-Math test isused as a criteria for college admissionand scholarship grants, girls' lowerscores influence their college and careerchoices as these tests become the filter formath-related fields like engineering andphysics.

Effect of Spatial Visualization Trainingon Spatial Skills

Studies have shown that even if stu-dents have had fewer spatial task experi-ences, they can be trained to developthese spatial skills. Vandenburg (1975)found that sixth grade girls improvedtheir mental rotation skills after receivingspatial training on building models withblocks. Ferrini-Mundy (1987) studied the


successful training of high school stu-dents. By practicing spatial tasks, thesewomen were able to visualize the rota-tion of solid objects while doing revolu-tion-type problems, a skill seen as essen-tial to the study of engineering.

Effect of Female Role Models on Engi-neering Career Choices

Researchers have found that girlshave limited their own exposure tohigher level math skills, including spatialvisualization skills, by not entering thehigher level math courses that providedopportunities to develop these skills(Hyde, Fennema, & Lamon, 1990). Kai-ser-Messmer studied the reasons thatfemale students do not take these classes.One reason given by the girls in thisstudy was the feeling that there was noneed for these courses in their futuregoals. Other researchers have found thatgirls recognize the value of these courseswhen female role models are introducedwho showed the girls how they success-fully .used the skills taught in the higherlevel math classes (McNamara &Scherrei, 1982; Smith & Erb, 1986; Swin-dell & Phelps, 1991).

As women enter into male-dominatedcareers, researchers have found thatsame-sex role models continued to influ-ence career choices. 'In a study of 30 geo-graphically and ethnically diversewomen pursuing careers in science,mathematics, and engineering (McNa-mara & Scherrei, 1982), women were in-terviewed to determine the influentialfactors that effected their career choices.The study found that as children andteenagers, the women recalled looking toteachers and older female students forencouragement. As they got older, they

said they looked for visible female pro-fessionals in their field who they coulduse as models for their careers and per-sonal lives.

The Current Study

For the current study, a summer mathcamp was developed for sixth and sev-enth grade girls. The program was basedon spatial visualization activities requir-ing problem-solving strategies. The girlswere also given the opportunity to meetand talk to female engineers. It was hy-pothesized that the spatial visualizationtraining and an exposure to female engi-neers would lead to increased spatialvisualization skills and increased inter-ests in engineering careers .

METHODS

Participants

The study took place in an urbanmiddle school in the San Francisco BayArea. All the participants were enrolledin the middle school's Chapter One pro-gram, which funded the summer camp.To select the participants, six middle-school teachers were asked to identifytwo groups of girls as candidates for thecamp: (a) girls who were considered tobe high performing math students, and(b) those ranked as low performing mathstudents. The teachers were instructed toexclude students receiving special ser-vices in the school resource programs(i.e. students with IEPs), students whowere identified as limited English profi-cient, and students who were receivingspecial services in the gifted and talentedprogram. Based on this procedure, 21girls were admitted to the program, and15 of these girls completed the program.

In terms of demographics, the girls'

18College of Education Review 15

ages ranged from 10 to 12. In terms ofethnicity, over half the group was Latino,and the remainder were from EuropeanAmerican, African American, and EastIndian backgrounds.

The Summer Math Camp Program

The summer math camp was held atthe same middle school that the studentsattended during the regular school year.The middle school principal and parentsof the students gave permission to un-dertake the study. Testing was con-ducted in the classroom the first and lastdays of the camp. The camp was held forone week.

A program was developed thatwould expose adolescent females to thetwo areas of concern for this study (a)spatial visualization tasks and (b) engi-neering career opportunities. Several or-ganizations were contacted in order tofind female engineers who would speakto the group. Activities were then devel-oped which could then be incorporatedwith the speakers in order to develop thespatial visualization skills.

The program was a five-day programmeeting from 8:30 a.m. until 12:30 p.m.Each day had a theme. Monday was Ex-ploration Day. In terms of spatial visuali-zation training, students were given twodifferent activities to do: (a) pentominopuzzles and (b) a pentomino computergame. Students were then allowed tofreely explore any of the other gamesthat were provided. In terms of career in-terests, the students revealed their cur-rent career aspirations on the pretestquestionnaires. They also completed aquestionnaire of skills needed to be anengineer using Equal's Spaces unit (Fra-ser, 1982).

Tuesday was Civil Engineering day.A female guest speaker brought slidesand discussed her job as a county civilengineer. In terms of spatial visualizationtraining, the students did civil engineer-ing activities by trying to create a cube,first from 3-foot dowels and rubber-bands. The students then constructed thecubes from skewers and rubber bands.They ended the day by making accuraterepresentations of their constructions onisometric paper.

Wednesday was Aeronautical Engi-neering day. The guest speaker was afemale electrical engineer who was in-volved in designing the newest Concordewith NASA. She brought in a samplepiece of the wind tunnel and explainedhow airplanes are able to fly. The stu-dents were then given two paper-foldingactivities: the first was on following di-rections to create origami structures, andthe second one was following computerinstructions for creating paper airplanes.The students then had a contest to seehow far their planes could fly.

Thursday was Writing DirectionsDay. The guest speaker was a femaleelectrical engineer who designed robotswith Lawrence Livermore Laboratories.She explained that designing robots con-sisted of writing directions so that a ro-bot could follow them. She brought a ro-bot for the students to use. The robotmoved its arm when students gave it di-rectional cues with a remote control. Thestudents were asked to pour water fromone cup to _another using directionalcues. Spatial visualization activities forthe day included working with and play-ing games with the Tangram puzzle.

Friday was Overview Day. The guestspeaker was a female quality control en-


19

gineer from the NUMMI automotiveplant. She explained how she found de-fects in car production and how effectivean assembly line could be in production.For further spatial visualization practice,students were asked to make a blockbuilding from a picture and then drawthe entire structure on isometric paper.The students were also given posttestsand questionnaires that were used toevaluate changes in spatial visualizationskills, knowledge of engineering as a ca-reer and new career interests.

Instruments

Students were given several pretestand posttest assessments. Two differenttypes of spatial visualization tests, whichwill be described in the next section,were given. The pretest and posttest re-sults were used to evaluate spatial visu-alization skills. A Career Interest Ques-tionnaire was also given before and afterthe camp to assess changes in the girls'career choices.

Spatial Visualization Tests

Developmental Test of Visual Motor In-tegrations (VIVA). The students were giventwo spatial visualization tests on a pre-test-posttest basis. The VMI is the Devel-opmental Test of Visual Motor Integra-tions (Beery and Buktenica, 1989) forages 3 through 18. This test measuresperceptual and motor skills. More spe-cifically, it evaluates .the students' abilityto look at a picture and then use smallmotor skills to reproduce the picture ac-curately. In this test, students are askedto copy 24 geometric shapes into a testbooklet. The designs are arranged in or-der of increasing difficulty. The score isdetermined by a point system that evalu-

ates the accuracy of the student's repro-duction of the design. The number ofpoints attained is then converted usingan age and sex norm table to determinethe final score.

Spatial Ability Test. The Spatial AbilityTest (Barrett & Williams, 1992) is typi-cally given by career counselors to helpdetermine spatial visualization percep-tion skills. This particular test is onecomponent of a battery of tests used toevaluate aptitudes that might indicatesuccess in different careers. The SpatialAbility Test determines the test-taker'sability to visualize a solid three-dimensional object when only given lim-ited two-dimensional information. Theseventy-five-point test has nineteen de-signs, which are analyzed in four ways.Tet-takers are asked to examine theoriginal design and then pick out, from aseries of four choices, the resultingshapes derived from the original design.

The test consists of two differenttypes of tasks, and each task measures aparticular kind of spatial visualizationperception skill. In the first type of task,students are asked to evaluate whichthree-dimensional shape is a result offolding a two-dimensional design. In thesecond type of task, students are giventwo shapes and are asked to determinethe resulting shape if the smaller piece isremoved from the larger shape. Bothtypes of skills are important in applica-tions such as understanding technicaldrawings and the relationships betweenobjects in space.

Career Interest Questionnaire

For this study, a Career InterestQuestionnaire was developed and pi-loted on 36 males and 52 female

rS4,0College of Education Review 17

TABLE ONE

Pretest and Posttest Spatial Visualization Test Results (n=15)

Pretest Post test Gain

Test Mean SD Mean SD Mean SD TSpatial Ability 41.33 5.39 49.67 6.16 8.33 5.02 6.42*

VMI 38.33 17.67 45.27 19.67 6.93 13.81 1.94

*Statistically Significant (p<. 05)

students to evaluate its reliability andface validity. After minor revisions, thisquestionnaire was then given on a pre-test and posttest basis. The results werethen analyzed in two ways: (a) questionsregarding the girls' career interests werecompared from pretest to posttest to as-sess changes in the girls' career aspira-tions; and (b) open-ended questions re-garding knowledge of engineering ca-reers were compared from pretest toposttest.

RESULTS

Spatial Visualization Skills

The spatial visualization tests wereanalyzed for possible growth in spatialvisualization skills. In particular, pretestand posttest means on the two testswere compared using matched pair t-tests at a .05 significant level. As shownon Table One, a quantitative analysis ofthe two spatial visualization tests gaveinconsistent results. The difference be-tween the VMI pretest and posttest be-tween the Spatial Ability Test pretestand post test means was statisticallysignificant (t (14) := 6.42, p < .05), indicat-ing a significant improvement in spatialvisualization perception skills on thistest. Students gained an average of 8.33points on this latter test.

Career Interest Questionnaire

Students were asked the open-endedquestion, "When you are 30 years old,what type of job do you think that youwill have?" on a pretest and post test ba-sis. They were given space to write twopossible career choices. The responseswere tallied for specific occupationalchoices. In Table Two, the results aredisplayed for the percentage of re-sponses falling into "technical and pro-fessional" occupation. There was a sub-stantial increase in students choosingengineering careers (+42.9 percent) witha marked decline in students choosingteaching careers (-14.3 percent).

Knowledge about Engineering

At the end of the summer camp, stu-dents' were asked, "Did you learn any-thing about engineering from this pro-gram?" Of the fourteen students whoresponded, all indicated that they hadlearned something. The students werethen asked "What did you learn?" Aninductive qualitative analysis was un-dertaken on students' written responsesto this question. The themes which arosewith regard to this question are pre-sented in Table Three. (Note that stu-dents' original spelling was unchangedfor this table). The answers fell into


2 1

TABLE TWO

Pretest and Posttest Responses to the Question:"When you are 30 years old, what type of job do you think you will have?"

(n=30)

Career Choice Pretest Posttest ChangeEngineer 0.0% 42.9% +42.9%

ComputerProgrammer

0.0% 3.6% +3.6%

Doctor 14.3% 14.3% 0.0%

Lawyer 14.3% 3.6% -10.7%

Teacher 17.9 3.6 -14.3

TABLE THREE

Summary of Responses to the Open-Ended Question:"What did you learn about engineering from the program?"

(n=14)

Theme 1: Engineering as it Relates to Mathematics

I learned that you should know your math and you will use it in life and it will be easyif you know it.Tlearned that girls can do most anything and that math is important.That they you need math for all the jobes.Engineering is cool math.

Theme 2: Women in Engineering

I learned that very fuew wemon ar engineersI learned that a lot of women are not in engineering.That there are very few women in engineering.

Theme 3: Feelings about Engineering as a Career

I learned that being an engineer is hard work. But all I need is practice.I learned that having that job could be fun.I learned that being an engineer is a lot of fun.I learned that women have 2 work very hard to be a good engineer because somepig-headed people think that it's a "man's" job. They have 2 work extra harder to beaccepted in that type of workforce.


22

three thematic areas. In terms of the firsttheme, several girls indicated that theylearned about the connection betweenengineering and math (N=4 students).An example of this is the response froma student who said "Engineering is coolmath." In terms of the second theme,several girls said that they learned thatthere are very few women in engineer-ing jobs (N = 3 students). For example,one girl said "I learned that very fuewwemon ar (sic) engineers." In terms ofthe third theme, several girls indicatedthat they came to recognize the field ofengineering as a career choice (N = 4students). Here, one of the girls said "Ilearned that having that job could befun."

DISCUSSION

Overall, the summer camp programwas a success both in terms of changes invisual perception skills and changes ingirls' career goals. In particular, a statis-tically significant increase was found interms of girls' spatial perception skills.Moreover, the female speakers appearedto have a positive influence on the girls,as reflected in their changes in careergoals.

Spatial Perception Skills

By providing a variety of activitiesfor all the students, the girls were able tobenefit and develop the repertoire ofstrategies that Linn and Petersen (1985)suggested students need in order to do-well on these spatial visualization tasks.The activities were structured in a man-ner where learning about spatial taskswas acquired through exploration, notjust "rules without reason" as described

by Bohlin (1994). These experiential ac-tivities allowed the students to developthe problem-solving strategies necessaryfor the spatial visualization test.

Visual Motor Skills

There was no significant change ingirls' visual motor skills, as measuredby the VMI. This may be because theVMI measures skills that were not em-phasized in the summer camp. In par-ticular, because these skills are meas-ured by recreating pictures with a pen,and because no erasing is permitted onthis test, an additional factor of fine mo-tor drawing proficiency might have en-tered into this task. In addition, oneweek may not have been a sufficienttraining period for this type of skill. Anadditional factor that could explain thenonsignificant results could be that forsome students, drawing is associatedwith artistic ability. Thus, if a studenthas little confidence in her drawing abil-ity, her lack of confidence could nega-tively effect her performance.

Career Choices

There was a notable change in careerchoices after the program. The change ofcareer interest to engineering was mostlikely based on two factors. The first fac-tor was an initial lack of exposure to en-gineering careers. In the pre-camp ques-tionnaire, students were asked the ques-tion "Do you know anyone who has oneof these jobs?" Two of the choices werecivil engineer and electrical engineer.Only one student responded that sheknew a civil engineer (her father). Thesecond factor, which may have affectedthe career choice changes, was the expo-sure to female role models in these posi-

20 College of Education Review2 3

tions. This exposure prompted studentsto not only see how women are success-ful in these careers, but the studentswere also able to hear how women havedeveloped their math skills in maledominated math/science classes.

Limitations of the Current Study

The current research has severallimitations. The findings, therefore,need to be viewed as tentative. First, interms of internal validity, there was nocontrol group. Therefore, the changesfound from pretest to posttest couldhave been the result of confoundingvariables such as maturation or testing.Second, in terms of external validity, theresearch is based upon a small samplesize, and the study took place in oneparticular locale in California. Thus,generalizability is limited.

Conclusion

This study suggests that spatial vis-ualization skills can be developed in asummer math camp setting. These skills,in turn, might then help girls enterhigher-level math programs. By provid-ing opportunities to develop these skills,we gave girls one more tool that theycan use in their future careers. "HowSchools Shortchange Girls," a report bythe AAUW (1992), recommended that"girls must be educated and encouragedto understand that mathematics and thesciences are important and relevant totheir lives." This program showed thevalue of girls learning about and speak-ing with successful female role modelsin non-traditional female careers likeengineering or science. It helped them tosee that these careers could be possiblefor girls to attain.


24

REFERENCES

AAUW Report (1992). How Schools Shortchange Girls. New York, NY: Marlowe &Company.

Barrett, J. & Williams, G. (1992). Test Your Own Job Aptitude. Middlesex, England:Penguin Books.

Basta, N. (1996). Opportunities in Engineering Careers. Lincolnwood, IL: VGM CareerHorizons.

Beery, K. & Buktenica, N. (1989). Developmental Test of Visual-Motor Integration.Cleveland, OH: Modern Curriculum Press.

Boekaerts, M., Seegers, G. & Vermeer, H. (1995). Solving math problems: Where andwhy does the solution process go astray? Educational Studies in Mathematics, 28(3),241-262.

Bohlin, C.F. (1994). Style factors and mathematics performance: Sex-related differ-ences. International Journal of Educational Research, 21(4), 387-398.

Ferrini-Mundy, J. (1987). Spatial training for calculus students: Sex differences inachievement and in visualization ability. Journal for Research in Mathematics Edu-cation, 18(2), 126-140.

Flores, P. (1990). How Dick and Jane perform differently in geometry: Test results onreasoning, visualization, transformation, applications and coordinates. Paper pre-sented at American Educational Research Association, Boston, MA.

Fraser, S. (1982). Spaces. Palo Alto, CA: Dale Seymour Publications.Gallagher, A. M., & DeLisi, R. (1994). Gender differences in Scholastic Aptitude Test:

Mathematics problem solving among high-ability. Journal of Educational Psychol-ogy, 86(2), 204-211.

Hyde, J., Fennema, E. & Lamon, S.J. (1990). Gender differences in mathematics per-formance: A meta-analysis. Psychological Bulletin . 107(2), 139-155.

Jakobsdottir, S., Krey, C. & Sales, G. C. (1994). Computer graphics: Preferences bygender in Grades 2, 4, 6. Journal of Educational Research, 88(2), 91-100.

Johnson, E. S. & Meade, A. C. (1987). Developmental patterns of spatial ability: Anearly sex difference. Child Development, 58, 725-740.

Kaiser-Messmer, G. (1993). Results of an empirical study into gender differences inattitudes towards mathematics. Educational Studies in Mathematics, 25(3), 209-233.

Kiesler, S., Sproull, L., & Eccles, J.S. (1983). Second Class Citizens! Psychology Today,17(3), 41-48.

Koballa, T. (1988). Persuading girls to take elective physical science courses in highschool: Who are the credible communicators? Journal of Research in Science Teach-ing, 25(6), 465-478.

Krantz, L. (1994). The Jobs Rated Almanac. New York: World Almanac.Linn, M.C., & Petersen, A.C. (1985). Emergence and characterization of sex differ-

ences in spatial ability: a meta-analysis. Child Development, 56, 1479-1498.


McGuiness, D. (1976). Sex differences in the organization of perception and cogni-tion. In B. Lloyd & J. Archer (Eds.), Exploring Sex Differences (pp. 123-156). Lon-don: Academic Press.

McNamara, P. & Scherrei, R. (1982). College Women Pursuing Careers in Science,Mathematics, and Engineering in the 1970's. Los Angeles, CA: University of Cali-fornia, Los Angeles, Higher Education Research Institute.

Mistry, M., Colton, M., Rossi, P., & Berger, L. (1994). Up Your Score. A Guide to theNew SAT and PSAT. New York, NY: Workman Publishing.

National Council of Teachers of Mathematics (1991). Professional Standards for Teach-ing Mathematics. Reston, VA.

Serbin, L., Conner, J. M., Burchardt, C. J., & Citron, C.C. (1979). Effects of peer pres-ence on sex-typing of children's play behavior. Journal of Experimental Child Psy-chology, 23, 303-309.

Sherman, J. (1980). Mathematics, spatial visualization, and related factors: Changesin girls and boys, grades 8-11. Journal of Educational Psychology, 7(4), 476-482.

Skolnick, J., Langbort, C. & Day, L. (1982). How to Encourage Girls in Math and Science.Palo Alto, CA: Dale Seymour Publications.

Smith, W. & Erb, T. (1986). Effect of women science career role models on early ado-lescents' attitudes toward scientists and women in science. Journal of Research inScience Teaching, 23 (8), 667-676.

Swindell, R. & Phelps, M. (1991). Designing and Implementing Science Enrichment Pro-grams for Rural Females. Cookeville,TN: Tennessee Technology University,

U.S. Department of Labor (1994). Occupational Outlook Handbook. Washington D.C.:U.S. Bureau of Labor Statistics.

Vandenberg, S. G. (1975). Sources of variance in performance of spatial tests. In J.Eliott & N.J. Salkind (Eds.), Children's Spatial Development. Springfield, IL: CharlesC. Thomas.


2v

The Effects of a Student-GeneratedRubric on Third Graders' Math Portfolio

Selection CriteriaThais Akilah Kagiso

The purpose of this study was to test the hy-pothesis that a student-generated mathematicsrubric would help third grade students to applymeaningful, objective criteria when selectingpieces of problem solving work for inclusion intheir mathematics portfolios. This 12-week studywas conducted in a self-contained classroom.Students were required to provide written justi-fication for their portfolio selections. Studentjustifications and selections were analyzed be-fore and after the development of the student-generated rubric. The results were generallyconsistent with the hypothesis. Although thestudents did not totally abandon their own crite-ria for evaluating their work, a significant num-ber of the students were able to integrate thestandards of the rubric into their selection crite-ria.


The aim of this study was to test thehypothesis that a student generated ru-bric would help students to apply mean-ingful, objective criteria when selectingpieces of problem solving work for in-clusion in their math portfolios. This 12-week study was conducted in a self-contained, third grade classroom. Stu-dents were required to provide writtenjustification for their portfolio selections.Student justifications and selectionswere analyzed before and after the de-velopment of the rubric. The results pre-sented here show support for the hy-pothesis. Although they did not totallyabandon their own criteria, a large

number of students were able to inte-grate the standards of the rubric intotheir repertoire of selection criteria.

One of the objectives of the recentmathematics reform movement is to ac-tively involve students in the assess-ment of their own learning (NationalCouncil of Teachers of Mathematics[NC'TM], 1995). By involving students inthe assessment process, teachers can as-sist students in coming to an under-standing of the goals of instruction andcan learn about the extent to which stu-dents are meeting those goals. Someteachers seeking to enter balanced as-sessment partnerships with their stu-dents have involved those students inportfolio assessment systems (Stenmark,1991). In such partnerships, teachers andstudents have come together to compilecollections of work that provide evi-dence of students' abilities to think andcommunicate mathematically. Researchindicates that when teachers and stu-dents work together in this way, theycan both benefit (Kenney & Silver, 1993).The potential benefit to teachers is hav-ing an assessment instrument thatevaluates performance over time. Thepotential benefit to students is becoming

Thais Akilah Kagiso has been teaching for14 years. She currently teaches 3rd grade atBeverly Hills Elementary School in Vallejo,California


2 7

informed evaluators of their learning.Evidence is mounting in support of

the value of student self-assessmentpractices in upper elementary or highergrades (Kenney & Silver, 1993; Warloe,1993). Research has yet to provide aclear answer as to whether or not suchpractices can be successfully imple-mented in primary elementary grades.Questions remain as to whether or notprimary grade students are develop-mentally capable of evaluating theirown work. There are indications that theproblem may not be in the developmen-tal level of the students, but in the de-sign of the assessment systems (Schunk,1994; Stipek, 1981). Therefore, the chal-lenge for primary grade classroomteachers is to find developmentally ap-propriate ways to support their studentsas they engage in authentic self-assessment activities that yield mean-ingful results.

The purpose this study was to exam-ine whether modification of the studentself-assessment component of a thirdgrade mathematics portfolio system hadan effect on the quality of student par-ticipation in that system. On the basis ofresearch in support of self-regulatedlearning, as well as research whichraised concerns regarding self-assessment by primary grade students, Ihypothesized that in order for thirdgrade students to fully participate in se-lecting pieces of work for their mathportfolios, they would need support inunderstanding appropriate criteria forsuch selections. I set out to evaluatewhether participation in the develop-ment of a scoring rubric, which estab-lished standards of performance, wouldprovide the support these third graders

needed. If, as indicated by research onself-assessment, students in the thirdgrade are capable of objective self-assessment, I hypothesized that the useof a rubric they helped to developwould provide the students in the studywith enough support to make perform-ance standards-based selections for theirmathematics portfolios. To that end, thefollowing question was investigated inthis study: What effect does the use of astudent-generated rubric have on thecriteria third grade students use to selectpieces of work for their math portfolios?The following sub-questions were con-sidered:

1. In what ways do student portfolioselection comments reflect the standardsof the rubric and how do those com-ments change after introducing the ru-bric?

2. In what ways do student portfolioselections reflect the standards of therubric?

3. In what ways are the commentsthat students make on their work sup-ported by the work itself?

4. In what ways do student com-ments regarding the purpose of a port-folio change after introducing the ru-bric?

METHODS

The Sample

This 12-week project was conductedin a self-contained, third grade class-room, at a K-6 year-round elementaryschool, in a North San Francisco BayArea school district. The student popu-lation of this class was diverse: 47 per-cent of the students were of Europeandescent, 22 percent were of African de-scent, 16 percent were of Filipino de-


scent, 9 percent were of Chinese de-scent, 3 percent were of Middle Easterndescent, and 3 percent were Biracial.

The Design.

Information was collected on onecomponent of the class' mathematicsprogram, namely problem-solving ses-sions. The design of the study called forstudents to engage in problem solvingsessions once a week. Every third week,students chose one of the three problemsolving work products for inclusion intheir mathematics portfolio. The result-ing four cycles of data collection werereferred to as the first, second, third,and fourth data collection periods.

The Rubric

The student-generated rubric wasdeveloped during discussion sessionsthat occurred during the period betweenthe first portfolio selection and secondportfolio selections. During these ses-sions, students participated in discus-sions which analyzed solutions to prob-lems they had solved during the firstthree weekly problem solving sessions.During rubric discussion sessions, thestudents identified elements of studentwork which provided evidence of suc-cessful problem solving. For example,students identified the use of a table asan effective way to organize informa-tion. I used the student comments towrite a rubric with descriptors of fourperformance levels. The language of therubric was designed to be developmen-tallytally appropriate.

Instrumentation

Information was collected from stu-dents using three instruments, namely(a) a math questionnaire, (b) problem

solving worksheets, and (c) portfolio se-lection comment sheets. The math ques-tionnaire was given to students oneweek before they engaged in their firstproblem solving session. The problemsolving worksheets were completed bystudents during each of the 12 weeklyproblem-solving sessions. (See Appen-dix 1). Students completed the portfolioselection comment sheets every thirdweek of the study, during portfolio se-lection sessions. (See Appendix 2). Stu-dents completed a post-study mathquestionnaire, identical to the initialmath questionnaire, after the final port-folio selection session.

Portfolio selection comment sheets. Thestudent comments on the portfolio se-lection comment sheets were analyzedfor evidence of the language and stan-dards of the rubric. Student commentswere coded in one of three categories:(a) Rubric Only, (b) Non-Rubric Only,and (c) Combination of Rubric and Non-Rubric. Responses in which all com-ments reflected the language and stan-dards of the rubric were placed in theRubric Only category. Responses inwhich all comments had no relation tothe language and standards of the rubricwere placed in the Non-Rubric Onlycategory. Responses that contained bothrubric and non-rubric comments wereplaced in the Combination category.

Problem solving worksheets. The prob-lem solving worksheets selected by thestudents for inclusion in their portfolioswere analyzed, section by section, forevidence of the standards of the rubric.Part 2 (Descriptions), in which studentsstated what they thought they needed todo to solve the problem, was analyzedfor three rubric standards: (a) Clarity,


29

(b) Completeness, and (c) Accuracy. Aclear description was one in which thestudent used language understandableto the reader. A complete descriptionwas one that included the major ele-ments of the problem. An accurate de-scription was one that recorded the stu-dent's correct interpretation of the prob-lem.

Part 3 (Solutions), in which studentsrecorded their solutions to the problem,was analyzed for four rubric standards:(a) Clarity, (b) Correctness, (c) Re-cordings, and (d) Multiple Representa-tions. Clarity in this section referred toreadability, as well as identification ofthe solution. Correctness referred to theaccuracy of the solutions or partial solu-tions. Recordings referred to methodsused to record the solutions such asequations, charts, diagrams, and tables.Multiple representations referred to anyattempt to solve a problem in more thanone way. This standard did not apply tothe many combinations found whensolving multiple solution problems.

Part 4 (Justifications), in which stu-dents justified their solutions in relationto the elements of the problem, was ana-lyzed for three standards: (a) Clarity, (b)Proof, and (c) Prior Learning. Clarity inthis section was demonstrated throughthe student's use of lucid language toexplain his or her thinking. Proof wasdemonstrated by the use of supportivereferences to elements of the student'ssolution. Prior learning was demon-strated by any references to prior prob-lems or mathematical experiences thataided the student in obtaining his or hersolution.

Student portfolio selections andportfolio selection comments were ana-

lyzed for evidence of consistency be-tween the comments students madeabout the quality of their portfolio selec-tions and the actual quality of the se-lected work. This analysis consisted oftwo categories: (a) Accurately Sup-ported and (b) Inaccurately Supported.

Math questionnaires. The pretest andposttest math questionnaires were ana-lyzed for evidence of change in the un-derstanding of the purposes of mathportfolios. This analysis was limited tothe responses to two of the four ques-tionnaire items: (a) question 3 (If youhad a math portfolio, how would youdecide what to put in it?) and (b) ques-tion 4 (What could people learn abouthow you do Math from looking at yourMath Portfolio?). Student responses toQuestions 3 and 4 were placed in fivecategories: (a) Evaluative, (b) Process, (c)General Mathematics, (d) Off Topic, and(e) No response/Incomplete.

RESULTS

Before examining the specific resultsin detail, it is important to address theimpact of the vacation break on thework completed during the fourth col-lection period. Results indicated thatthere was a significant increase in stu-dent selection criteria after the introduc-tion of the rubric (i.e. from the first tothe second data collection period). Thisincrease was maintained from the sec-ond to the third data collection period.But from the third to the fourth data col-lection period, there was a clear decline.The time delay between these two peri-ods appeared to have impacted the stu-dents' abilities to produce and evaluatethe quality of their work. Therefore, thedata from the fourth collection period is


not included here.

Portfolio Selection Comments

First Selection CommentsDescriptive statistics for the portfolio

selection comments are presented in Ta-ble One. Students made their first port-folio selections before developing therubric. None of the students made Ru-bric-Only comments during this selec-tion session. Non-Rubric Only com-ments were made by 85 percent of thestudents. The remaining 15 percent ofthe students made Combination com-ments, some of which reflected stan-dards that would appear later in the ru-bric.

Second Selection CommentsStudents made their second portfolio

selections the week after developing therubric. At total of 56 percent of the re-sponses included comments reflectingthe standards of that rubric. Rubric-Only comments were made by 30 per-cent. of the students, and Combinationcomments were made by 26 percent.Less than half the students, 44 percent,responded with Non-rubric only com-ments.

Third Selection CommentsThe third portfolio selection was

made three weeks after the second selec-tion. Rubric-Only comments peakedduring this selection period, with 41percent of the student responses fittinginto this category. A total of 28 percentof the students made Combinationcomments.

Change in Portfolio Selection CommentsThe data indicate that the develop-

ment and availability of the rubric dideffect the comments that students made

when justifying their portfolio selec-tions. The strongest effect was observedduring the third selection period, when69 percent of the students included thestandards of the rubric in their selectioncomments. It is significant to note thatthe majority of the students did notcompletely abandon their own selectioncriteria. On average, approximately 38percent of the students integrated therubric standards with their existing se-lection criteria, whereas approximately31 percent of the students replaced theirown selection criteria with the standardsof the rubric. (This 31 percent is equal tothe average number of students who didnot use the rubric standards in additionto or in place or their own criteria.)

Portfolio Selections

Each student selection was examinedfor evidence of rubric standards in thework. The purpose of this analysis wasto determine whether or not there wasany change in the quality of studentwork produced after the introduction ofthe rubric. To support the validity of theanalysis, a qualified second reader ana-lyzed about 25 percent of the selections.Inter-rater agreement was 91 percent.

The student selections did not show asignificant amount of evidence of thework that met the Solutions/MultipleRepresentations or Explanations/PriorLearning standards. Therefore, theanalysis of these two standards is notpresented here.

First-Portfolio SelectionsDescriptive statistics for the portfolio

selections are presented in Table Two.The pieces of work selected during thisfirst pre-rubric collection period had the


TABLE ONEPercentage of Students Making Each

(Based Upon Student Responses to the PortfolioType of Selection Comment

Selection Comment Sheets)

Data Collection PeriodType of Selection Comment First Second Third

Rubric 0% 30% 41%

Non-Rubric 85% 44% 31%

Combination Rubric/non Rubric 15% 26% 28%

RubricTABLE TWO

Standards Evident in Student Selection Pieces

Data Collection PeriodFirst Second Third

Description Standards(Students state what they think they need to do to solve problem)

Clarity 52% 70% 67%Completeness 4% 40% 18%

Accuracy 36% 74% 67%Solution Standards (Students record their solutions to the problems)

Clarity 72% 96% 81%Completeness -56% 89% 52%

Accuracy 76% 100% 96%Explanation Standards(Students justify their solutions in relation to the elements of the problem)

Clarity 68% 74% 63%Proof 72% 56% 33%

lowest correlation of student work andrubric standards. The standard mostrepresented in the work was the Solu-tions/Recorded standard. Over 75 per-cent of these selections met this stan-dard.

Second Portfolio SelectionThe highest correlation between stu-

dent work and rubric standards wasevident in the pieces of work selectedduring the second data collection pe-

riod. Description standards were met inan average of 61 percent of this work.Evidence in support of the Solutionstandards was found in an average of 95percent of this work. The Explanationstandards were met in an average of 65percent of the work.

Third Portfolio SelectionThere was a decrease in the average

number of rubric standards from thesecond data collection period to the

32.College of Education Review 29

TABLEStudent Portfolio Selection

in the Selected

THREEComments Supported by Evidence

Pieces of Work

Data Collection PeriodFirst Second Third

Rubric-related comments 12% 64% 70%

a. Accurately supported 12% 51% 54%b. Inaccurately supported 0% 13% 16%

Non-rubric related comments 88% 36% 30%Total number of comments 32 55 57

third data collection period. This may bebecause the rubric development discus-sion sessions occurred during the timestudents were solving problems whichwould be considered for the secondportfolio selection. The rubric was avail-able to the student when they workedon the remaining six problems, but itwas not discussed after the develop-ment period ended.

Student Comments Supported in theStudent Work

All rubric-related comments wereanalyzed for evidence of support in thestudent work. For example, if a studentstated that one of the reasons he or sheselected a particular piece of work wasthe use of a chart, that work was exam-ined for the presence of a chart. Descrip-tive statistics for this analysis are pre-sented in Table Three.

First Portfolio Selection CommentsThe students made a total of 32 selec-

tion comments to justify their first port-folio se ection pieces. Of the our rub-ric-related comments made by studentsduring the first selection session, allwere supported by evidence in the stu-dent work. These 4 comments consti-tuted 12 percent of all selection corn-

ments made during the first session.

Second Portfolio Selection CommentsEvidence of support was found for

28 of the 35 rubric-related commentsmade during the second portfolio selec-tion session. This meant that 51 percentof the selection comments made duringthis session were accurately supportedby the students in relation to the ap-propriate rubric standard. This was anincrease of 31 percent over the first port-folio selection comments. Students at-tempted to apply the standards of therubric in an additional 20 percent of thecomments made during this selectionperiod. Those comments, however, werenot supported in the student work.

Third Portfolio Selection CommentsStudents made the most rubric-

related comments when justifying theirwork choices during the third selectionsession. Fifty-four percent of the com-ments made in this session accuratelyidentified standards of the rubric pre-

-sent in =the= work. Only 16 percent ofthese comments referred to rubric stan-dards that were inaccurately attributedto the work.


Changes in the Math QuestionnaireResponses

The changes in the responses to themath questionnaires provided clear evi-dence of the benefit of involving stu-dents in an on-going assessment system.By the end of the study, more than 50percent of the students were able toidentify portfolios as tools for evalua-tion. Another 20 percent or more knewthat portfolios involved some sort of se-lection, which in this case, was related tomath performance.

DISCUSSION ANDRECOMMENDATIONS

The results of this study were en-couraging. Many students selected qual-ity work for inclusion in their portfolios.However, the students did not consis-tently apply the appropriate criteria ofthe rubric to their portfolio selections. Inother words, they recognized goodwork when they saw it, but did notidentify what made that work good.

The questions answered and, moreimportantly, left unanswered by thisstudy point the way to areas of future

research. Central to these questions areissues of the language. By language, Imean both the wording of the rubric,and the mode of student expression. It isimportant to determine what effect thelanguage of the rubric has on the stu-dents' willingness and/or ability to in-corporate the standards and language ofthe rubric into their repertoire of selec-tion criteria. It is also important to de-velop effective ways to support the abil-ity of intermediate grade students to ex-press themselves clearly when writing.A study comparing the oral and writtenselection comments of students couldhelp determine if it is reasonable to ex-pect third graders to accurately expresstheir selection criteria in written form.

I encourage teachers and researchersto continue to explore ways of makingassessment activities meaningful toprimary and intermediate grade stu-dents. The results of this study indicatethat self-assessment, when appropri-ately supported, is within the cognitiveand developmental reach of third gradestudents.

,College of Education Review 31

REFERENCES

Kenney, P. A., & Silver, E. A. (1993). Student self-assessment in mathematics. InAssessment in the Mathematics Classroom, 1991 Yearbook of the National Council ofTeachers of Mathematics, edited by Norman L. Webb, 229-238.

National Council of Teachers of Mathematics (1995). Assessment Standards forSchool Mathematics. Reston, VA: National Council of Teachers of Mathematics.

Schunk, D. H. (1994). Goal and self-evaluative influences during children'smathematical skill acquisition. Paper presented at the annual meeting of theAmerican Educational Research Association, New Orleans, April, 1994.

Stenmark, J. K. (1991). Mathematics Assessment: Myths, Models, Good Questions, andPractical Suggestions. Reston, VA: National Council of Teachers of Mathemat-ics.

Stipek, D. J. (1981). Children's perceptions of their own and their classmates' abil-ity. Journal of Educational Psychology, 73, 404-410.

Warloe, K. A. (1993). Assessment as a dialogue: A means of interacting withmiddle school students. In Assessment in the Mathematics Classroom, 1991 Year-book of the National Council of Teachers of Mathematics, edited by Norman L.Webb, 152-158.


APPENDIXES

APPENDIX #1: Problem Solving Worksheet

Problem Solving#:Name: Date:Partners:

1. Read the problem.

2. Write what you think you have to do or find out.

3. Solve the problem.

4. Write how you know your answer fits the problem.

APPENDIX #2: Math Portfolio Selection Comment Sheet

Today's Date

Dear Teacher,

I have decided to put the following piece of work in my math portfolio.

Date:

Number:

Title

I chose this piece because:

Signed,

College of Education Review 33. .

3

APPENDIX #3: Scoring RubricLevel 1A. You are not sure if the student knew anything about the problem because his

or her description in #2:was not understandable, ordid not make sense, orgave information that had nothing to do with the problem, ordid not provide any information.

B. You are not sure how the student figured out the problem because in #3 he orshe did not show much, or any, of his or her work. The student did not cor-rectly use any of the following:

tables,diagrams,lists,equations,charts,pictures,

The student also:did not obtain a solution or obtained an incorrect solution,did not identify his or her solution

C. The student's explanation in #4 was:incomplete or totally missing,said that the student didn't know the solution or how to find it,had nothing to do with the problem and/or the student's solution.

Level 2A. You are not sure if the student knew something about the problem because his

or her description in #2:was confusing, ordid not make sense, orgave some incorrect, ordid not provide enough information.

B. You are not sure how the student figured out the problem because in #3 he orshe did not show much of his or her work. The student did not correctly useone of the following:

tables,diagrams,lists,equations,charts,pictures


37

The student also:obtained an incorrect solution,did not clearly identify his or her solution

C. You are not sure if the student knew his or her solution was incorrect be-cause in #4 the student's explanation:

was confusing,gave reasons for proof that did not fit the problem and /or the solution.

Level 3A. You can figure out how the student solved the problem because in #3 he or

she showed his or her work. The student may have used one of these ways:tables,diagrams,lists,equations,charts,pictures

The student also:obtained a correct solution,identified his or her solution

B. You believe that the student knew the solution was correct because in #4 thestudent's explanation:

was fairly clear,gave some proof from the problem and the solution.

Level 4A. You can tell that the student knew what the problem was because the descrip-

tion in #2:was clear,made sense,gave the correct information,gave all of the information.

B. You can clearly tell out the student solved the problem because in #3 he or sheshowed all of his or her work. The student may have used one or more ofthese ways:


The student also:obtained at least one correct solution,

3: a College of Education Review 35

identified his or her solution,showed his or her solution in more than one way.

C. You know that the student knew why the solution was correct because in #4the student's explanation:

was very clear,gave proof from the problem and the solution,wrote about other problems like this one.

D. You can figure out how the student solved the problem because in #3 he orshe showed his or her work. The student may have used one of these ways:


The student also:obtained a correct solution,identified his or her solution,

E. You believe that the student knew the solution was correct because in #4 thestudent's explanation:

was fairly clear,gave some proof from the problem and the solution.


39

Observation and Analysis of theMathematical Solving Behaviors

of Six African-AmericanFourth and Fifth Graders

Norman S. Mattox IV

The purpose of this research was to observe andanalyze selected mathematical problem-solvingbehaviors of six 4th and 5th grade AfricanAmerican students. The behaviors that were ob-served consisted of mathematical fluency, origi-nality, elegance, communication, collaboration,and persistence. Pairs of African American stu-dents were videotaped during two mathematicalproblem-solving sessions. The performance of thethree pairs of students, two high-performing, twoaverage- performing, and two low- performing,was observed and scored by a panel of research-ers. A 4-point observational rubric (1 = begin-ning, 2 = developing, 3 = maturing, and 4 = ac-complished) was used to score the pairs of stu-dents. The pair of high performing students re-ceived scores in the 'accomplished' level of therubric. The average-performing pair of studentsreceived scores in the 'maturing' level of the ru-bric, while the low performing pair achievedscores in the 'developing' level of the observa-tional rubric.


Traditionally, African American stu-dents have not been the focus of mathe-matics education research. There is verylittle research that has been done to ex-amine the mathematical problem solvingabilities of African American students atthe elementary school level. In the past,most of the emphasis in elementaryschools was on mastering basic compu-tational skills. However, in order forstudents to be successful with the

mathematical content of the Curriculumand Evaluation Standards for SchoolMathematics, the National Council ofTeachers of Mathematics (NCTM) statedthat "problem solving should be the cen-tral focus of the mathematics curricu-lum" (1989, p. 23). Problem solving hasbecome the network and vehicle thathelps the learner navigate the variouspaths to mathematical understanding.

Current research (Kneidek, 1995;Matthews, 1984; Rech, 1994; Tate, 1995)has indicated that many factors have tobe considered in order for AfricanAmerican students to be successful inthe field of mathematics. The culturalheritage and social structure of the Afri-can American child and their familymust be taken into consideration. Themathematics teacher must present situa-tions that are relevant to the child's ex-perience in order for the children to bemotivated to participate in their ownlearning. Also, the mathematics teacher

Norman S. Mattox is currently Vice Prin-cipal at Cesar Chtivez Elementary School inthe San Francisco. Unified School District.He is interested in aligning and integratingmathematical problem solving strategieswith the 'coming to understand' phases ofother curricular content areas.


should pose problems that motivate theAfrican American child to develop anduse computational skills in authenticsituations (Bell, 1994; Delpit, 1995; Fos-ter, 1994; Hart, 1984; Kunjufu, 1984;Kuykendall, 1989; Shade, 1994; Stiff &Harvey, 1988).

Most mathematics curricula in ourpublic schools has been developed byand for the analytical, logical-mathematical thinker, who is of Euro-pean-American descent, and who speaksstandard English. School mathematicshas been described as the expression ofhow European cultures view the world(Stiff & Harvey, 1988). The learning stylepracticed by the European and Euro-pean-American cultures values analyti-cal thinking and systematic approachesto problem solving. These learners tendto be more interested in the details andthe understanding of abstract ideas forthe sake of having that information ac-cessible to them (Kuykendall, 1989; Stiff& Harvey, 1988).

African-American children bringtheir own unique and distinctive per-spective that is in contrast to the logical-mathematical approach validated intextbooks and most classrooms. Unfor-tunately, one of the barriers to makingmathematical problem solving accessibleto the African-American learner is thereliance on textbook word problemswith contrived situations that are notrelevant to the student's experience (Ga-raway, 1994; Kunjufu, 1984; Shade,1994). When- the context of the problemsituation is familiar to the student, it be-comes easier for the student to recognizewhat the problem is, and identify thenecessary mathematical skills that are inuse in every day life (Garaway, 1994;

Hale-Benson, 1986; Ladson-Billings,1994).

In addition to the lack of attentionpaid to cultural issues, efforts in curricu-lum development have failed to ac-knowledge the special linguistic differ-ences between standard and nonstan-dard English. Caraway (1994), for exam-ple, considered the issue of language asan influence on mathematical learning.According to Caraway, many AfricanAmerican students need to recognize thedifferences between standard Englishand nonstandard English usage in theacademic environment.

Garaway further argued that AM-can- American students who are of ele-mentary age also have to translate themathematical concepts learned in theclassroom into terms that are familiar intheir experience. He argues that re-searchers have supported the idea of us-ing the student's own language and thepractical problems that come from thestudent's own society as a foundationfor the teaching of culturally relevantmathematics. Because the African-American student is more likely to man-age concept development differentlythan other children, often it is the Afri-can-American child's cultural learningstyle that is being assessed and not his orher mathematical abilities (Bell, 1994;Kuykendall, 1989; Shade, 1994; Stiff &Harvey, 1988).

For the current study, this researchersought to gain a better understanding ofthe mathematical abilities of AfricanAmerican students by observing andanalyzing the mathematical problemsolving behaviors of six African Ameri-can students enrolled in an urban publicelementary school. Specifically, this re-

38 College of Education Review 41

search attempted to describe the charac-teristics of fluency, originality, elegance,communication, collaboration, and per-sistence that are observable during amathematical problem solving session.

METHODSThe purpose of this research was to

observe and analyze selected problemsolving behaviors of two high, two aver-age and two low performing fourth andfifth grade African American students.The students worked with a partner atthe same ability level. Each pair of stu-dents was given two problems to solve.Each problem was to be solved within aone-hour time period. Fifteen minutes ofeach hour-long videotape were ana-lyzed.

A Description of the Problem SolvingSessions

For each session, the process con-sisted of administering a warm-up prob-lem, followed by the problem solvingsession that was videotaped and moni-tored for the targeted problem solvingcharacteristics, such as fluency, original-ity, elegance, communication, collabora-tion and persistence, for each pair ofstudents. The content of these video-taped sessions provided the materialthat was analyzed using a 4-point obser-vational rubric. In addition to studyingthese videotapes, each student's teacherwas interviewed to understand the stu-dent's disposition in his or her regularlearning environment.

Analyzing the Problem SolvingSessions

The videotaped problem-solving ses-sions were observed and analyzed withspecial attention given to the problem-

solving characteristics. The researcherselected a fifteen-minute portion of eachvideotaped problem that indicated thestudents' understanding of the mathe-matical problem and their engagementin the solving of that problem. Theseportions of the videotaped problem solv-ing sessions were observed and scoredby a panel of researchers involved instudying the mathematical problemsolving processes of culturally diversestudents.

This panel of researchers observedand discussed the level of proficiency ofthe targeted problem solving characteris-tics. In particular, the students' workfrom the problem solving sessions wasreviewed and rated by the panel. Basedon these ratings, a consensus score wasrecorded for each characteristic. Thiswas done, not only to neutralize the sub-jective bias of the researcher, but moreimportantly, to validate the score thatthe individual student received for thetargeted problem-solving characteristic.One of the problem-solving characteris-tics, persistence, was rated exclusivelyby the researcher because of his partici-pation during the entire videotapedproblem solving session and access torepeated observations of the videotape.This helped the researcher to identify thenonverbal communications between thepartners, non-verbal communicationsbetween the individual student and theresearcher, and the nonverbal communi-cation with the problem-solving task.

The Participants

The participants were 4th and 5thgrade students from a small, ethnicallydiverse public school in San Francisco.African American and Spanish-speaking

4 2 College of Education Review 39

students make up the majority of theschool population. There were severalcriteria for student selection: theteacher's assessment of the student'sabilities during mathematics activitiesand instruction; the teacher's experiencewith the student during mathematics in-struction; and the student's practice ofmathematical problem solving. For thepurposes of this research, mathematicalproblem solving was defined as the useof mathematical thinking to understanda problem, select an appropriate strat-egy, apply the strategy to achieve a solu-tion and to reflect on the solution proc-ess. Students were rank-ordered accord-ing to their problem solving abilities.Specifically, students who exhibitedproblem-solving abilities in their class-room practice were ranked high on thelist. Those students who did not performwell during mathematics instructionwhile exhibiting problem solving abili-ties were ranked lower on the list. Thosestudents who did not perform well dur-ing mathematics instruction and showedlittle mathematical problem solving abil-ity were ranked lowest the list. From thislist of the fourth and fifth grade AfricanAmerican students, the participantswere selected by alternate ranking.

Mathematical Problems to Solve

The two problems that were selectedfor the students to solve both involvedthe use of money. These problems werechosen under the assumption that fourthand fifth grade students have a practical

.familiarity with mathematical conceptsusing money.

Problem 1. The first problem pre-sented to the students was as follows:

Slim has $4.00 to spend on

hamburgers and colas. Ham-burgers cost 80 cents (80c)apiece. Colas cost 40 cents(40C) apiece. List all the possi-ble ways Slim could spend hismoney on hamburgers andcolas.

The key mathematical concept to rec-ognize in solving this problem is the two-for-one relationship between the price ofa hamburger (80 cents) and the price of acola (40 cents). In one approach to listingthe possible ways that Slim could spendhis money, if one hamburger were elimi-nated, two colas could be substituted inits place. This approach to solving theproblem would result in six possibleways for Slim to spend his money onhamburgers and colas. Another approachto listing the possible ways that Slimcould spend his money takes into consid-eration the possibility that Slim did nothave to spend all of his money on a com-bination of hamburgers and colas. If Slimonly purchased one hamburger withoutbuying anything else, then his changewould have to account for the rest of thefour dollars. This approach to solving thisproblem results in thirty-five possibleways for Slim to spend his four dollars onhamburgers and/or colas.

This first problem that the pair ofstudents were asked to solve involved thesearch for the possible combinations thatwould add up to a total amount of fourdollars. Of the two problems, this prob-lem was the more traditional moneyproblem which required facility withcounting in multiples of four (or forty)and eight (or eighty).

Problem 2. The second problem givento the students consisted of presentingeach pair with a pile of coins, and then


43

asking:

Can you get exactly one dol-lar's worth of change from thepile by starting at any coinalong the edge, moving fromcoin to touching coin, andending on another edge coin?No coin may be crossed overmore than once.

This second problem challenged thestudents to follow a path of coins toachieve a total of one dollar within spe-cific guidelines. This problem requiredthe students to stay within these guide-lines while keeping track of an accumu-lating total of coin values.

Pilot Study

In order to develop the parametersfor the observation, researcher-studentinteraction, and videotaping protocols, apilot problem solving session was con-ducted. A money problem served as awarm-up problem to the 'actual' prob-lem solving session. During this warm-up, the researcher modeled thinkingaloud and sharing one's problem solvingprocesses with a partner. The researcherparticipated as a partner with each of thethree pairs of students during this initialstage of project development. Not onlydid this give the students a sense ofwhat was expected of them during theproblem solving sessions, it also estab-lished a level of comfort between the re-searcher and the students.

After the warm-up problem, anothermoney problem was administered. Thissecond problem served as the 'actual'problem solving session that was videotaped. The pair of students was situatedat a desk so that they were facing the

camera as they worked on the problems.The pair was provided with the prob-lem, written out on one sheet of paper,so they would be encouraged to worktogether. Both student/participants weregiven their own pencil to provide equalaccess to recording during the problemsolving session. The students were givenmaterials that would help them solve theproblem. These materials consisted ofcoins, scrap paper, and calculators. Inthe case of the second problem, differ-ent-colored markers were provided todifferentiate 'solution' paths. With theaid of these resources, the student pairswere expected to share their strategiesand ideas, both verbally and mathemati-cally. These pilot sessions were instru-mental in developing the parameters forresearcher-student interactions. The pilotsessions also provided information onwhat types of communication dynamicscould be expected between the pair ofstudents.

The Researcher's Role During theProblem Solving Sessions

The teaching actions for problemsolving (Charles, Lester, & O'Daffer,1987) were used as a guide for interac-tions between the researcher and thestudents. Before proceeding with thevideotaped session, the researcher andthe students read the problem aloud anddiscussed it to clarify any necessarywords or phrases. Once the problemsituation was understood, the studentswere encouraged to proceed on theirown, and the videotaped observationbegan.

The researcher intervened during theproblem solving session only when ei-ther of the students had a question that


would assist them both in understand-ing the problem situation more clearly. Ifthe students proceeded in their problemsolving paths for a period of time with-out speaking with one another, the re-searcher reminded the students to "talkabout what you are doing now." If oneof the students was proceeding along asolution path without acknowledginghis or her partner, the researcher re-minded the students to, "solve it to-gether." If the researcher saw that therewere repeated solutions, then the stu-dents were advised of the repetitions. Ifboth of the partners concluded that theyhad found all of the possibilities for aparticular solution, and there were, infact, more possibilities, then the re-searcher would say, "there are more."These interactions occurred during theproblem solving session.

When the students believed that theyhad encountered all the possible solu-tions and would not proceed any fur-ther, the video camera was turned offand the 'after' problem solving interac-tions started. This was a period when theresearcher became the mathematicalproblem-solving instructor. The studentswere encouraged to keep track of theirwork. The student pairs received directinstruction in how to organize theirwork so that they might be able to keepbetter track of their work, check overtheir work and to recognize patterns thatwould help lead to a solution.

The Observational Rubric

The content of the observational ru-bric evolved from a synthesis of descrip-tions found in several sources thatevaluated mathematical problem solvinghabits and mathematical communication

skills. The language used to describe theproblem solving behaviors and thecommunication skills was developedfrom (a) a problem solving assessmentthat was prepared by Sawada (1996), (b)a rubric that is part of a mathematics as-sessment used by the California De-partment of Education, and (c) amathematics problem solving scoringguide used by the Oregon Departmentof Education.

A description of the targeted prob-lem solving characteristics was devel-oped from these various sources andfrom discussions with other colleaguesinterested in mathematical problem solv-ing. Consider these characteristics inmore detail. Fluency is concerned withthe level of understanding of mathe-matical concepts and applying that un-derstanding to the context of themathematical problem. It also involvesthe ability to internalize this understand-ing, and then engage in productive prob-lem solving. Originality is the develop-ment of a unique idea, or, making an in-sightful observation about the mathe-matical task at hand. Elegance is con-cerned with the expression of the indi-vidual's thinking in mathematical terms.Elegance is also concerned with the con-verse of this relationship. Expressingmathematical concepts as metaphors re-lated to the individual's experience isalso considered elegance. Communicationis the individual's ability to conveymathematical thinking and reasoningthrough written and spoken words, andactions. Collaboration is the willingness ofthe individual to share and accept ideasand materials with a partner. Persistenceis the ability to continue to focus on aproblem in order to find a solution and


the willingness to pursue other solutionpaths, even when the process becomesstalled.

Each of the targeted problem solvingcharacteristics, namely fluency, original-ity, elegance, communication, collabora-tion and persistence, were rated on a 4-point scale, as follows: (1) Beginning; (2)Developing; (3) Maturing; (4) Accom-plished.

The researcher selected a 15-minuteportion of the videotaped problem solv-ing session to be viewed by the observ-ers that indicated the students' under-standing and engagement of the prob-lem. During this period, the observersmonitored and scored the pair of stu-dents. The rationale for each observer'sscoring was discussed based on the ob-servational rubric. The scores that theobservers gave to each pair of studentswere recorded.

After the observation period wasover, the observers discussed how theyscored each of the characteristics. Thisdiscussion helped the researcher to cal-culate the consensus score that would berecorded. This consensus scoring was anaverage of the scores that were discussedamong the observers.

RESULTS AND DISCUSSION

The results of the analysis of thevideotape using the observational rubricwere summarized for each of the prob-lem solving behaviors for each of thetwo problems. These data, presented inTable One, indicate the consensus scorefor each of the students. Recall that thepairs of students were designated as ahigh-performing pair, average perform-

ing, and low performing.During the search for solutions in

both problem situations, the pair of highperforming students Key and Joanexhibited an 'accomplished' level of pro-ficiency in the communication of theirmathematical thinking and in their col-laboration with each other. There wasfrequent interaction between the stu-dents at this level. They spoke with eachother in unfinished phrases thatsounded like the audible portions of amental dialogue between the two.

Because of Joan's persistence, the pairof high performing students were able tofind all the possible solutions to the firstproblem. Key was unsure if there wereanymore possibilities when they foundthe penultimate solution to the firstproblem. In the second problem, whenthere was a growing sense of frustrationin both students, Joan seemed to be thecheerleader for both partners. She re-minded Key of what they both answeredin their student attitude surveys aboutnot 'giving up.' "Try until we die,'member?" was her motivating state-ment to Key.

The pair of average performing stu-dentsLenard and Raywere proficientin that they both received consensusscores in the 'developing' and 'maturing'levels of the observational rubric. Duringthe first problem solving session bothpartners chose to use coins to help themlist the possible ways of spending thefour dollars. It was Lenard's decision tolist ways that included getting change.back, instead of spending the whole fourdollars. This approach was not organ-


46

StudentTABLE ONE

Performance on the Observational Rubricfor Problems 1 and 2

Low Performing Average Performing High PerformingEl I Barry Lenard I Ray Key I Joan

Scores for Problem #1Fluency 3 3 3 3 4 4Originality 2 2 3 2 3 3Elegance 2 2 3 2 4 4Communication 3 2 2 3 4 4Collaboration 3 2 2 3 4 4Persistence 3 2 3 2 3 4Scores for Problem #2Fluency 2 3 3 3 4 3Originality 2 3 3 2 3 4Elegance 2 3 3 3 3 4Communication 2 3 3 2 3 4Collaboration 2 3 3 3 4 4Persistence 2 3 3 3 3 4

ized, so it did not facilitate finding pat-terns. While Lenard did not communi-cate his random selection of possible so-lutions, Ray spent most of the time look-ing over Lenard's shoulder and movingcoins that he assumed Lenard wascounting. This caused confusion be-tween the two. A lot of time was spentcomparing the quantity of the other'spile of coins and it seemed to put a drainon their motivation. While they wereable to communicate their mathematicalthinking to the researcher, they did notshare their insights with each other, un-less they were prompted.

The second problem was one thatchallenged the spatial abilities of bothpartners. Though Ray received a consen-sus score of two for his "developing'level of originality, he made a remarkthat was not witnessed by the observers.Ray noted that in "...other problems wecould keep track... chart it out and wecould come up with the (solution)."

When he was asked to clarify what hemeant, Ray continued, "This one has theanswer in it. There's a dollar inside thesecoins. It's like an adventure. We gottafind clues so we don't lose track." Theseremarks reflected Ray's persistence andhis effort to make a connection to otherexperiences outside of mathematics.

The low performing pair-- Barry andElreceived consensus scores of two,almost exclusively. This indicated thattheir mathematical problem solving be-haviors were in the 'developing' level ofproficiency. Overall, both El and Barrywere limited in their understanding ofthe mathematical concepts involvedwith solving these problems. They alsodemonstrated limitations in their facilitywith some of the basic mathematicalskills that would help them to progressthrough the problem solving process.Most of the calculations were donecounting by ones and on fingers, theonly strategy these two students used.


47

Even when they requested the use ofcoins to help them solve the moneyproblems, both El and Barry counted byones and on their fingers to keep track ofthe coin values. At times, if El took arisk, and totaled the addition of one cointo a quantity that was already added up,he would call out the amount, and referto the researcher for a signal as towhether he was correct or not. Othertimes, if Barry was voicing his rationalefor an answer, he looked to the re-searcher to validate his response. Be-cause of their general insecurity with themathematics concepts and applications,both Barry and El were very dependenton the approval of the researcher forhow they were proceeding during theproblem solving sessions.

Though Barry and El may have beenlacking in their experiences withmathematical problem solving strate-gies, they seemed to recognize eachother's limitations. They were suppor-tive of each other's efforts and tried toencourage one another when frustrationstarted to show. This mutual encour-agement helped motivate them throughsome unsuccessful attempts and pro-pelled them to continue the problemsolving process. Towards the end of thesecond problem solving session, and be-yond the short portion of the videotapeobserved by the panel of mathematicscolleagues, both Barry and El started tobuild in their confidence that they weregetting close to a solution. When theirtotals were near one dollar, Barry stated,"I think I'm gonna get this. I'm fittin' tofind this!" And as El started to feelBarry's confidence growing, he added,"I'm starting to think it can be done be-cause we're getting so close." They did

make more unsuccessful probes towardsfinding a solution path and they werewilling to continue the problem solvingprocess until the end of the allotted time.

The success the high performing pairof students shared was primarily due totheir competence and facility with thebasic mathematical operations. This fa-cility gave them the confidence to ma-nipulate the numbers through the vari-ous problem-solving strategies they at-tempted. This confidence in their num-ber sense was a reminder that they couldfind all of the possible answers, and in-spired the high performing pair of stu-dents to persist until all the solutionswere found.

The results for the average perform-ing pairs of students seemed to be in-dicative of their inexperience with work-ing in collaboration with a partner. Bothstudents were competent with their ba-sic mathematical skills. But their reluc-tance to communicate or plan a mathe-matical problem solving strategy to-gether led to incomplete solutions to thegiven problems. Though the studentspersisted in their attempts to find a solu-tion during the allotted time, their indi-vidual efforts were mostly unsuccessful.

The low performing pair of studentswere not competent in their basicmathematical operations. Even thoughthe two students communicated openlyduring the problem solving session andneeded little prompting to work togethertowards solving the given problems,their apparent lack of fluency with num-bers diminished their confidence and in-hibited any risk taking during the prob-lem solving process. The low performingpair of students had a difficult time de-fining their problem solving approach


and they spent most of their time usingthe 'Guess and Check' strategy.

Limitations of the Study

There were two main limitations tothe study. First, there was a problemwith the fact that the panelists onlyviewed a portion of the problem solvingsessions. Specifically, it had been de-cided that the panelists' observationswould begin once the students under-stood the problem and had articulatedthe problem in their own words. Eventhough a consensus was reached in howto score the problem solving behaviorsand what to observe during the problemsolving sessions, there were actions andcomments that were made beyond theportion that was viewed that could havepromoted changes in the individualscoring. The researcher was present dur-ing the entire problem solving sessionand had access to the videotape of thosesessions. Comments made during dia-logue between the problem solvingpartners and other subtleties of non-verbal communication were not avail-able to the other members of the scoringpanel.

The second limitation concerns thecultural backgrounds of the panelistsand students. In spite of any backgroundinformation the researcher may havegiven to that panel of observers, all theminutiae of communication that occursbetween members of the same culturalgroup can be difficult to articulate. Thismay have created assumptions that bi-ased the results of the scoring based onsuch a limited demonstration of problemsolving behaviors. By being part of samecultural group as the students, the re-searcher was aware of the familiarity

with cultural communication cues thatcreated an understanding which wasdistinct from the other members of thepanel.

During the discussions that followedthe observations of the videotaped prob-lem solving sessions, this familiaritymay not have been conveyed so that theother observers could get a sense ofwhat was happening during the 'realtime' of the problem solving session.This might account for the "developing"scores that El was given for the portionof the videotape that was viewed by theobservers. In terms of persistence, El wasnot engaged by this problem initially.Ultimately, he and his problem-solvingpartner, Barry, worked through to theend of the videotaping period. El re-marked that he was starting to believethat there was a solution to the problembecause they were getting closer andcloser to the goal of a one-dollar path. Aresolution to this issue might be for thepanel of observers to view the entirevideotaped problem solving session.This could give the observers a bettersense about the problem-solving envi-ronment created by the students and theresearcher.

Future Considerations

If a teacher presents a problem-solving task to a selected a heterogene-ous group of students in his or her classand observed their performance, theseobservations would bear fruit for theteacher as well as for the students. Ini-tially, the students' performance wouldgive the teacher a sense of how the restof the class would fare under similar cir-cumstances. With the aid of the 'problemsolving behaviors' rubric, the teacher


would be able to orient his or her focusto evaluate what and how students un-derstand a given mathematical conceptand its application. The teacher couldalso share the problem solving behaviorsrubric with the students so that the ex-pected level of competence, the mind-set, and the attitude towards mathemati-cal problem solving would be more ex-plicit. Once the students are aware of theteacher's expectations, they can becomemore critical of their own performanceduring the mathematical problem solv-ing periods in their class.

It is extremely important for studentsto realize that mathematics is muchmore than just knowing facts and arith-metic operations. Students can havemore success in mathematics if they arecognizant of the variety of problem solv-ing strategies, are flexible in applyingthose problem solving strategies, andpersist beyond the first unsuccessful at-tempt at solving a particular mathemati-cal problem. If teachers are willing to fa-cilitate, students can present an explana-tion of their solutions to an audience oftheir peers. Then, students will be able todemonstrate their mathematical reason-ing and hone their communication skills.This activity is one that would give theteacher an opportunity to take advan-tage of the different student abilitieswithin the classroom community.

An area that needs to be exploredfurther is the discrepancy that exists be-tween the teacher's perception of thestudent's demeanor within a whole classpopulation compared with the student'sself-perception as a student within thewhole class population. Whereas the

teacher might perceive the student asunmotivated and disinterested in learn-ing, the student might be under the in-fluence of other distresses, i.e. separationfrom family members, transient livingarrangements, or inappropriate respon-sibilities for elementary aged children,that distract them from performing ade-quately in their academics. This elementof resilience, where the student performssuccessfully in spite of negative condi-tions, needs to be explored further.Those individuals need to be validatedand appreciated for overcoming theirdifficulties outside of school and doingwell in their academic pursuits. Bydocumenting the routes these willfuland successful students take to accom-plish their tasks, the teachers can becomeinformed of the students' problem solv-ing abilities in the classroom commu-nity.

Another consideration for future re-search might be to look at mathematicalproblem solving behaviors when theproblem situations are set in the contextof the student's life. A mathematicalproblem situation can be tailored to re-flect any cultural experience. As we be-gin to recognize the variety of ways inwhich children demonstrate their under-standing of a problem situation and thesolution strategies that satisfy the pa-rameters of the problem, it becomes im-perative that the instructor becomeknowledgeable, if not 'expert,' in thecurricular goals and objectives of themathematics content area. This will en-able the instructor to take advantage ofthe mathematical connections that existin the students' every day life.


REFERENCES

Bell, Y. R. (1994). A culturally sensitive analysis of black learning style. Journal of BlackPsychology, 20(1), 47-61.

Charles, R., Lester, F. & O'Daffer, P. (1987). How to Evaluate Progress in Problem Solv-ing. Reston, VA: National Council of Teachers of Mathematics.

Delpit, L. (1995). Other People's Children: Cultural Conflict in the Classroom. New York,NY: The New Press.

Foster, M. (1994). Effective black teachers: A literature review. In E. R. Hollins, J. E.King & W. C. Hayman (Eds.), Teaching Diverse Populations, Formulating a Knowl-edge Base, 225-241.

Caraway, G. B., (1994). Language, culture, and attitude in mathematics and sciencelearning: A review of the literature. The Journal of Research and Development in Edu-cation, 27(2), 102-111.

Gay, G. (1975, October). Cultural differences important in education of black chil-dren. Momentum, 30-33.

Hart, L. C. (1984). Some Factors That Impede or Enhance Performance in MathematicalProblem Solving. Doctoral Dissertation, Georgia State University, Columbus, GA.

Kneidek, T. (Winter 1995). Two Worlds in One Classroom. NorthWest Education.Kunjufu, J. (1984). Developing Positive Self-Images and Discipline in Black Children. Chi-

cago, IL: African-American Images.Kuykendall, C. (1989). Improving Black Student Achievement By Enhancing Students'

Self-Image. The Mid-Atlantic Equity Center, The American University, Washing-ton, DC.

Ladson-Billings, G. (1994). The Dreamkeeper: Successful Teachers of African AmericanChildren. San Francisco, CA: Jossey-Bass Publishers.

Matthews, W. (1984). Influences on the learning and participation of minorities inmathematics, Journal for Research in Mathematics Education, 15(2), 84-95.

National Council of Teachers of Mathematics: Commission on Standards for SchoolMathematics (1989). Curriculum and Evaluation Standards for School Mathematics.Reston, VA: NCTM.

Rech, J. (1994). A comparison of the mathematics attitudes of black students accord-ing to grade level, gender, and academic achievement. Journal of Negro Education,63(2), 212-220.

Shade, B. (1994). Understanding the African American Learner, In E. R. Hollins, J. E.King & W. C. Hayman (Eds.), Teaching Diverse Populations, Formulating a Knowl-edge Base, 175-189.

Shenk, M. (May, 1985). Loose Change: A Pocketful of Coin Puzzles. Games Magazine,44 45.

Stiff, L. V., & Harvey, W. B. (1988). On the education of black children in mathemat-ics. Journal of Black Studies, 19, 190-203.

Tate, W. F. (Summer, 1995). Returning to the root: A culturally relevant approach tomathematics pedagogy. Theory into Practice, 34(3), 166-173.


APPENDIX ONE

Can you get exactly one dollar's worth of change from the pile of coins bystarting at any coin along the edge, moving from coin to touching coin, and endingon another edge coin? No coin may be crossed over more than once.

BEST COPY AVAILABLE


52

Teaching Students to Write about SolvingNon-Routine Mathematical Problems

Delia LevineAnn Gordon

A two-week writing intervention was developedand used with sixth grade students. Pre-testsand post-tests were administered to measurechanges in use of problem solving strategies,mathematical understanding, persistence anduse of appropriate representations. Results indi-cate that mathematical proficiency improved andwritten explanations became longer, displayingincreased understanding of non-routine mathe-matics problems. Teaching students to writeabout their mathematical thinking impactedtheir understanding of mathematical concepts asdemonstrated in students' written explanations.


Researchers and educators in thedomain of language arts have long ar-gued that having an opportunity towrite deepens students' understandingand helps them grasp abstract conceptsmore completely, because the act ofwriting requires students to re-organize,and therefore reprocess, the ideas theyare writing about (Archambeault, 1991;Fortescue, 1994; Miller, 1991). A numberof individuals have extended these find-ings into the field of mathematics(Burns, 1995; Countryman, 1992; Miller,1991; NCTM, 1989, 1991; Steward &Chance, 1995). In fact, writing has cometo be seen as an important part of prob-lem-solving instruction in many class-rooms. For example, Szetela and Nicol(1992) claim that although it is often dif-ficult for students to communicate their

thinking to others, it is through com-munication that we can truly assess ourstudents' problem-solving abilities. Belland Bell (1985) also found writing to bean effective tool for teaching math prob-lem-solving.

However, many teachers find that itis not easy to get students to write abouttheir mathematical thinking. Studentsoften have had little practice doing thiskind of writing and they often don'tknow how to proceed or what, exactly,is being asked of them. Teachers havefew strategies available to them forhelping students overcome these diffi-culties. In fact, little attention has beenpaid to developing this kind of student

Delia Levine has been an educator for 30years. She is currently teaching sixth gradeat A.P. Giannini Middle School in SanFrancisco. Ann Gordon is an educationalpsychologist who has spent the past 26 yearsteaching children, collaborating with class-room teachers, and supervising studentteachers and new teachers. For more than 12years, Dr. Gordon has directed school-basedresearch on innovative educational and pro-fessional development programs. This workincludes a six-year study of the impact ofMath Case Discussions (Math Cases,WestEd) on teachers, classrooms and stu-dents.


writing, or how to develop an exposi-tory writing program in a math class. Inan effort to address this problem, thecurrent study was conducted. The pur-pose of the study was to evaluate the ef-fectiveness of an instructional model in-spired by the Bay Area Writing Projectto teach sixth grade students how towrite about their mathematical problemsolving.

METHODS

Thirty-five percent of the students inthe school where this study was carriedout scored below the 40th percentile ineither reading or mathematics on a statestandardized test. Following an intro-duction and a week of pretests, the pro-ject began with a two week writing in-tervention and continued for 14 weeks,during which time students solved non-routine problems in collaborativegroups, were instructed in how to writeexplanations, and wrote explanationsindividually. Pre-tests and post-testswere administered to measure changesin the use of problem-solving strategies,mathematical understanding, persis-tence and the use of appropriate repre-sentations over time. In addition tothese measures, data consisted of stu-dent computational, pictorial and writ-ten work.

Four problem types were used inthis study: (a) two-constraint problems,(b) pattern problems, (c) grouping prob-lems, and (d) number-at-the-beginningproblems. These types were chosen be-cause they were believed rich enough toprovide students with opportunities touse multiple solution strategies and ex-panded explanations of their thinking.The first three of these problem types

were used throughout the study and thelast was used as a measure of transfer.Each student had a chance to work ontwo examples of each type by the end ofthe study, while the transfer type onlyappeared on the pre- and post-tests.

In the two-constraint problem, twoconstraints are given and the student'ssolution to the problem must satisfyboth constraints. For example:

Ted is at the race track. Thereare lots of horses and jockeysrunning around the track.There are 82 feet and 26 heads.How many horses are there?How many jockeys?

A pattern problem asks a question inwhich a particular pattern or sequenceneeds to be discovered in order to suc-cessfully solve the problem.

Jeremy keeps pennies in a jar.He adds 1 cent for the firstweek, 2 cents the second week,4 cents the third week, 8 centsthe fourth week, and so on.How many pennies will he addin the twelfth week?

Grouping problems require that a mul-tiple set of criteria be met in order tosuccessfully solve the problem. A solu-tion needs to be found that satisfies allthe clues given in the problem.

Marvin is counting his marblecollection. He counts more than40, but less than 70. When heputs the marbles in groups of 5,he has 1 left over. When he putsthem in groups of 4, he has 1left over. When he puts them ingroups of 3, he has 1 left over.


How many marbles doesMarvin have?

And finally, number-at-the-beginningproblems include information about thesituation being described and the endstate, but do not give information aboutthe initial state. In order to solve theproblem the initial state needs to be fig-ured out, using the information given.

Dad is doing his weekly gro-cery shopping. The first storehe enters is the meat market. Atthe meat market, he spends halfhis money and then spends $10more. The second store he en-ters is the bakery. Here hespends half of his remainingmoney and then spends $10more. He now has no moneyleft. He is totally broke. Howmuch money did he have whenhe started at the meat market?

During the two week training pe-riod, following a structure similar tothat used in the Bay Area Writers Pro-ject, students were taught how to writea) a problem statement, or explanation ofthe problem in their own words that in-cludes both the facts and the question,b) a process paragraph in which they wereexpected to describe the solution or so-lutions obtained, as well as their think-ing as they did the problem, c) a solutionparagraph, in which they were expectedto give a short explanation proving theirsolution(s), and d) an evaluation para-graph, in which they were expected togive their opinion about the relative dif-ficulty of the problem. A class discus-sion followed each problem solving ses-sion, in which strategies were shared.

Written explanations were first done asa whole class activity, modeled by theteacher, then as a group activity amongthe students, and finally, as an individ-ual independent activity.

The primary question of concern waswhether the model used would result inbetter student-generated explanations ofindividual problem-solving activity. Inorder to answer this question, studentwork collected over the course of thisstudy was evaluated along four dimen-sions: Did the student use an appropri-ate strategy or strategies for solving theproblem? Did the mathematics and/ orwritten work evidence understanding ofthe problem? Did the student do morethan was necessary to solve the prob-lem, or did he or she continue to explorefeatures of the problem after a solutionhad already been reached? And finally,did the student make appropriate use ofany drawings, charts, graphs, diagrams,or tables needed to solve the problem?The overall quality of a particular pieceof work was then rated using criteria toidentify high, medium, and low per-formance. The criteria used for ratingeach task were as follows:

High Performance RatingReached a correct solution.All mathematics work is clearly

shown.Written explanation is clear and

explains all the mathematics.If appropriate, mathematics and

written explanation shows whatwas done before a solution' wasreached.

Other strategies or other solutionsmight be mentioned.


55

Medium Performance Rating'Reached a correct solution.Might have reached an incorrect so-

lution but is on the right track(might have made a carelessarithmetic error).

'Mathematics is clear but writtenexplanation may be confusing orvisa versa.

'Might only explain the solution andnot the other mathematics orwhat was done before a solutionwas reached.

'Written explanation is short -things are left out.

'Shows understanding of the prob-lem either through the mathemat-ics or the written explanation butnot necessarily both.

Low Performance RatingSolution is unrelated to the prob-

lem (unrealistic solution) or nosolution found.

'Mathematics work doesn't connectto the clues of the problem.

.No written explanation.'Might have reached a correct solu-

tion but no support for it (nomathematics or written explana-tion).

DATA ANALYSIS

Results indicate that mathematicalproficiency improved, students' writtenexplanations became longer, and expla-nations displayed increased under-standing of non-routine mathematicsproblems that the students were askedto solve. Consider the quantitative datasummarized in Table One. Each of the29 students in the study were given a setof four pretest problems, four posttestproblems, and four one-month delayed

post-test problems. For each time pe-riod; one problem was a two-constraintproblem, one was a patterns problem,one was a grouping problem, and onewas a transfer problem. Students' writ-ten responses to each of these twelveproblems was rated as either "high,""medium," or "low," using the ratingsystem described previously. Table Onesummarizes the changes in student per-formance that were found over thecourse of this study. These results wereobtained by analyzing individual stu-dent work quantitatively and what fol-lows is an analysis of what thesechanges looked like in more qualitativeterms, using the two-constraint problemtype as a case in point.

Two-Constraint Problem

As previously indicated, in a two-constraint problem two pieces of infor-mation are given. The student's solutionto the problem must satisfy both ofthese constraints. The two-constraintproblem type was probably the mostfamiliar of the four problem types. Twoof the three practice problems, used thefirst three days of school, were two-constraint-type problems. This type ofproblem was also used for three of thefour problems used for the intervention.For example, the following problem wasgiven as a pre-test item during Week 2of the study:

Ted is at the racetrack. There arelots of horses and jockeys run-ning around the track. There are82 feet and 26 heads. How manyhorses are there? How manyjockeys are there?


5

TABLE ONEProblem Solving Performance across Time (n=29)

Problem TypePerf or-manceRating

Pretest Posttest

OneMonthDelayedy

- ePosttest

ChangePre -to

ttPosttest

ChangePost-toDelayedPosttest

1.Two Constraints H

M

L

0%

38%

62%

28%

38%

35%

38%

28%

35%

+28%

0%

-27%

+10%

-10%

0%

2 Patterns H

M

L

3%

45%

52%

24%

48%

28%

38%

52%

10%

+21%+3%

-24%

+14%

+4%

-18%

3. Groupings H

M

L

3%

45%

52%

21%

66%

14%

14%

66%

21%

+18%

+21%

-38%

-7%

0%

+7%

4.Number at be-ginning (TransferProblem)

H

M

L

0%

34%

66%

7%

31%

62%

31%

41%

28%

+7%

-3%

-4%

+24%

+10%

-34%

Since the problem says that therewere 26 heads, the number of horsesplus the number of jockeys needed to be26. In order to answer the question suc-cessfully, furthermore, students need tomultiply the number of horses times thenumber of feet on each horse and addthat to the number of jockeys times thenumber of feet on each jockey, to get atotal of 82 feet on the horses and thejockeys together.

As-reported in Table One, 62 percentof the responses to this question on thepre-test only met the criteria for a lowrating, while none of the responses onthis question met the criteria for a high

rating. Many of the responses received alow rating because they gave an answerthat did not fit either constraint of theproblem. One student said, for example,that there were "19 horses and 18 rid-ers" [S-I, pre] which would mean thatthere was a total of 37 heads and 112feet. Another responded that "Theamount of horses were 15 and 22 peo-ple" [S-N, pre] which would mean thatthere were 37 heads and 104 feet. Somestudents who received, a low rating onthis problem did so because they onlydealt with one of the two constraints.One student said there were "13 jockeysand 13 horses" [S-A, pre] which was the


57

correct amount of heads, but wouldyield 78 feet; another said there were "4men and 22 horses" [S-M, pre; S-R, pre],which would also result in the correctnumber of heads (26), but would yield atotal of 96 feet, which is more than al-lowed by the problem.

A low rating was also given if the re-sponse could not be related to the prob-lem being asked. Students wrote thingslike, "42 people" [S-E, pre], which wasaccompanied by a picture that did notappear related to the current problem,or, "I counted them up and my ansercame up to be 41 feet and heads . . ." [S-D, pre], which was not the answer to thequestion that was asked.

However, by the time of the posttest,only 35 percent of the responses to asimilar problem were like those de-scribed above, while 28 percent nowmet the criteria for a high rating. Fur-thermore, by the time of the delayedpost-test, which took place a month af-ter the end of the study, 38 percent ofthe responses met the criteria for a highrating. The following are the two-constraint problems used on these tests:

Post-test problemTrisha is visiting her grandpa's

farm. She notices that he raises onlyhens and hogs. She counts 38 headsand 100 feet in the barnyard. Howmany hens and how many hogs doesher grandpa have in the barnyard?

Delayed post-test problemThe Creepy Critters Scary Tunnelhas two types of cars for the chil-dren to ride. One type seats 4 andone type seats 6. There are 24 carsthat seat a total of 114 children.How many cars seat 6 children

and how many cars seat 4 chil-dren?

The following patterns emerged in thehigh responses: (a) most of the re-sponses included lots of written expla-nation, (b) they included a processparagraph which explained the step-by-step procedures that they used to solvethe problem, (c) a problem statementwhich retold what the problem wasabout, and (d) a solution paragraphwhich gave proof according to the cluesthat the solution was correct. For exam-ple:

First I made two charts. Onechart is for the hens and one is forthe hogs. For the hen's chart. Onthe right side I put the mutiplesof 2 for the hens leg because henshave 2 legs. And on the right sideI put the numbers of heads by themutiples of 1. because each henhas 1 head. For the hogs I put thesame thing except that I put themultiples of four for the legs be-cause hogs have 4 legs. For thehen's I stoped at 37 heads and 74legs because the problem saidthat there are 38 heads all to-gether and that there are hensand hogs. For the hogs. I stoppedat 25 heads and 98 legs becausethe problem said there are 100legs and if I go farther there willbe 102 legs. Then I matched theheads from the hogs to the headsof the hens to see if it is equal to38 because the problem said thereare 38 heads. And if it is equal to38. I matched the hen's legs to thehog's legs and if it is equal to 100legs that means the answer is


5a

right because the problem saidthere are .100 legs all together.And I have only one answer andit is 26 hens and 12 hogs [S-L, postProcess paragraph].

I know my answer of 26 hens and12 hogs is right. because there are38 heads and 100 legs all to-gether. And there are hens andhogs [S-L, post Solution paragraph].

Some of the responses in the highgroup indicated that the student hadfound his/her solution by using thestrategy of making a systematic list andfrom that list was able to figure out thatonly one correct solution was possible.All of the strategies described in thehigh responses led to an efficient solu-tion of the problem. If guess and checkwas used, the guesses were not random.Since there were 24 cars, all the guessesof six- seated and four-seated carsequaled a total of 24 cars, which wasone of the constraints of the problem.One such guess was eight six-seatedcars and 16 four-seated cars. When theyfound out the number of children thatwas seated, for example 112, they statedthat they knew that they were close tothe answer so the next guess was ninesix-seated cards and 15 four-seated cars,which ended up being the correct solu-tion.

Another efficient strategy used bythe students who were given a high rat-ing was making a list of the multiples offour and six and then finding a numberfrom each list that would equal 114children. They also had to make surethat their two numbers kept the con-straint of 24 cars. Yet another efficientstrategy was using up all of the children

from one constraint and then alteringthose amounts to fit the other constraint.For example, making 24 circles to repre-sent the cars and first putting four chil-dren in each car allows room for 96children. They then realized that theyneeded to make room for 18 more chil-dren. If they added two more childrento nine cars, they would end up with 15four-seated cars and nine six-seated carsthat can seat 114 children.

Problem-Solving Strategies

e On the pre-test, many children wereunable to solve the problem correctly;although most gave an answer of somekind, they did not indicate that they didnot know what to do. It was difficult todescribe the strategies used by thesestudents, because many of them carriedout activities that appeared entirely un-related to the problem. For example:

I found the anser buy Drawing 82lines the I pretended were feetand I counted them up and my ansercame up to be 41 feet and headsIt was quite simple and you don'tneed a calculator to figer it out

6 0. Piclist IcAc4 Idi el

.11 11 q 11, /./ I/ /./ 0/t /1.1/

XN..1)....ti 1.1_11.11.. t1 11

t..k it t i lie/ IL 11 .

.

[S-D, pre] .

I put them into groups of one and4 beans under each chip. Then Icounted that there were 26 chipsand 82 beans. I found out that


59

there were 19 horses and 18 rid-ers [S-I, pre].

By the post-test, however, many stu-dents were using strategies and explain-ing their choices. For. example:

For my work to find out my an-swer I used all guess and check.For my 1st guess I guessed 15 - 6seats and 9 - 4 seats. That waswrong because when you times 6seats to 15 because my guess was15 you get 90. Then when youtimes 9 to 4 you get 36. 90 + 36 =126. I know that is wrong becausethe total of children in a seat was114 just like the problem says.Next I guessed 12 - 6 seats and 12- 4 seats. That was wrong becausewhen you times 12 to six you get72 and when you times 12 to 4you get 48. When you add thosetogether you get 120. I know thatis also wrong because the prob-lem said that 114 children is thetotal. All the guessed numberhave to equal 24 because that isthe number of cars so 15 - 6 seatsand 9 - 4 seats is the number ofcars and seats. I also guessedsome more numbers till I got myanswer /S-X, delayed Process para-graph].

I know my answer which is 6seats in 9 cars and 4 seats in 15cars is correct because when youadd 6 seats in 9 cars you get 54.When you add 4 seats 15 timesyou get 60 - 60 + 54 = 114. justlike the problem says thereshould be 114 children [S -X, de-layed Solution paragraph].

First I drew 24 circles because inthe problem it says that there are24 cars. Then I drew 6 dots ineach circle for 12 circles and 4dots in each circle for 12 circlesbecause in the problem they saidthat there are 2 types of seats oneis 4 and one is 6 children. the cir-cles represent the 24 cars and thedots are the children. Then I x 6 x12 because there are 6 children inone seat for 12 cars to find outhow many children are in thos 12cars and I x 4 x 2 to find out howmany children in the cars. Thereare 72 children in the 6 childrenseats and there are 48 children inthe 4 children seats. I add those 2numbers (48 + 72) because I wantto see if it equaled to 114 becausein the problem it said that thereare a total of 114 children [S-AA,delayed Process paragraph].

Now they have a language to talk aboutwhat strategies they are choosing to use;they name their strategies, explainwhich ones didn't work, and say whatthey are going to do next.

Mathematics

On the pretest, most papers didn'tshow any mathematical work at all.Students either just gave a solution withno work shown and no explanationgiven, or they explained the mathemat-ics in the written explanation, but therewas no mathematics visible. In othercases, the mathematical explanationswere confusing. For example:

There are 20 horses and there are27 Jockeys. The way I got this an-swer is as soon as I knew that


60

some of the feet could be human

feet I knew the answer. See 20 x 4= 80 and there are two remainingfeet. Then I added the 1 feet tothe 26 heads and I got 27 [S-Y,pre].

First I get 82 feet and 26 heads.Then I divided them into 4 and 3.After that I found that there are 4heads and 22 feets. Then I get 4heads and 22 feets. Then I seper-ate the 4 heads and I divided the22 feets into 5 & 6. After I did Ifound there are 2 horses and 2jockeys. This is how I found outby acting it out, makes a modeland guess and check IS- P, pre].

First I took 82 chips out. Then Itook 3 chips and put them in onegroup. When I was finish I addedall the heads up then I added allthe feets up. After that I countedall of the groups and I turned outhaving 50 bodys. So then "P' usedthe 50 and took away 26 andhave 24. So al together there ar 24people [S-Q, pre].

By the time of the posttest, many ofthe students wrote out the mathematicsthey were using and clearly explainedthe mathematics that was shown. In ad-dition, the mathematics work alsoshowed what was tried prior to reach-ing a correct solution. For example:

For _this POW I use Guess andCheck. I thought of 19 hens and19 hogs which make 38 heads. Sofirst I multipled 19 by 2 (becausehen have 2 feet ) and I got 38 legs.

Next I multiplied 19 by 4 (be-cause hogs have 4 legs) and I got76 legs. Then I added 76 with 38,but I got 104 legs. the problemsaid that there are 100 legs, thatmean 19 hens and 19 hogs is in-correct. Next I tried 18 hens and20 hogs, it adds up to 38 heads.18 hens and 20 hogs is incorrectbecause when I added up the to-tal legs of the hens and hogs to-gether, I had 116 legs instead of100 legs. I decided to draw a pic-ture next. I drew 100 circles forlegs and divided all of them intogroups of 4. Out of the 100 legs,there are 25 groups of 4. So if Isubtracted 38 by 25, there wouldbe 13 left. So I put 1 line in eachgroups 13 times to make somegroups of 2. I had an answer of 12hogs and 26 hens. I did somemultiple mathworks to makesure it's right and it is. I did somemore multiple mathworks to findanother anwer, but I couldn'tfind another answer.


61

looccim0000loo

oco-p0 o

aolooe,

JWclocoti,c0000

26 1- 12 = 38 headsx2.5 100 195

2 8 I- 11 = 39 kackas

56 -I- 11445. -= IOC) IeaS

30 + ID = Li0 keactx 2 6.160 Li0 ^ 1DD legs

29-1- 1 3 = 3 7 hp_octsx2-

403'k 52= IUo teas

[S-C, post Process paragraph]

First I drew 24 circles. The 24 cir-cles represent the cars. There aretotal of 114 children. So in each 24circles I put 4 children becausethe problem says how many carsseat 4 children. After that I did 24x 4 to see how many childrenthere are total. When I multiplythe answer is 96 children which iswrong because the problem saysthere are 114 children, so 114 - 96

to see how much more children Ineed. The answer is I need 18more. So I add 2 more children ineach 9 circles because 9 x 2 = 18.After that I found that it is rightbecause 9 x 6 = 54 which meansthe place were I add 2 children ineach of the 9 cars. The answer is54. 15 x 4 represent that there are15 cars which have 4 children.The answer is 60 so 54 + 60 = 114.Which is the correct answer. Be-cause the answer is 114 childrenjust the problem says. So thereare 9 cars seat with 6 childrenand 15 cars seat with 4 children.

tor "ksrcars

= 214 I

q (0 children I

ll

moSmiD§64.6)

&3)S-

(oo

I lq

[S-P, delayed Process paragraph].


Persistence

On the pre-test, most of the re-sponses were short or incomplete. Stu-dents either just gave solutions, somecorrect, some incorrect, or wrote a veryshort written explanation. For example:

Therer are 13 jockeys and 13horses Frist I used chip anddidn't Work so I deived and itworked [S-A, pre].42 people [S-E, pre].

There is 52 horses & 2 Jockeys. Iuse some red circles & Beans tosub. it for getting how manyhorses. And I use some coloredchips to Sub. for getting the an-swers [S-K, pre].

On the other hand, by the end of thestudy, responses were longer in lengthand usually had the mathematics andwritten explanation on two separatepages. The student work also showsmathematics work attempted after acorrect solution was reached:

First I made 24 cars because thepaper said that there were 24 cars. Ifilled 19 cars with 6 because 19 times6 make 114 children. Next I startedto filled the 5 blank cars, and to dothat I must take two children fromeach cars until I fill the left over cars.I took out 2 children from 10 differ-ent cars because 10 divided by 2equal 5. Then I count the cars thathave 4 children and-those that have6. There ar 15 cars that have 4 and 9that have 6, then I add 15 and 9 (tomake sure) and I got 24 cars. I alsoadd up the total children in all of the

cars of 4 and the total in all of thecars of 6 and I got 114 children. Tofind another answer, I took 2 cars of6 which make 3 cars of 4. The tookchildren is till 114, but the are 25 carsand there is 24 cars in the problem. IfI keep taking 2 cars of 6 to make 3cars of 4, the cars will add up. Next Itook 3 cars of 4 to make 2 cars of 6,the amount of children is the same,but the number of cars is 23, not 24.If keep taking 3 cars of 4 to make 2cars of 6, the amount of cars willgrow less.

15 Gars beal _

co.rs secti.

2 4(5 t car::


6 3,

[S-C, delayed Process paragraph].

Another indication of persistencewas demonstrated by the work of thosestudents who took the time to constructsystematic lists of all the possibilities fora given problem. In those cases, therewas no need to continue looking forother possible solutions.

Representations

Many of the responses on the pre-test either did not show any representa-tions or used representations that wereinappropriate to the problem. These re-sponses sometime included tally marksor circles to represent the 82 feet or 26heads, but the representations didn'tmatch the clues.

40o00AilkA4 nfi,Htn00

111.1.\0 \\\Ns4

\ \ \ \\.\\

e t 0

[S-E, pre].I put them in to groups of oneand 4 beans under each chip.Then I counted that there were 26chips and 82 beans. I found outthat there were 19 horses and 18riders

O. _o oo o 00 04( 4/ v 4/ 41 4/

0 '0 0 0 0 0/ (/ (./ zi V

[S-I, pre].

On the post-test however, the repre-sentations were much more sophisti-cated. Many students made two charts,one for hogs and one for hens. Theycontinued their list until they reachedthe limit of either constraint, either the38 heads or 100 feet. They then lookedfor amounts from each chart that whenadded together would give both con-straints asked for in the problem.


CONCLUSION

Similar analyses as the one just de-scribed were carried out for each of thefour problem types and the results ofthese analyses, as already suggested,strongly support the conclusion thatstudents who participated in this projectdid, in fact, improve in their mathemati-cal proficiency and became better able toexplain their mathematical thinking. Us-ing a model inspired by the Bay AreaWriting Project to teach students towrite about their mathematical prob-lem-solving resulted in students' beingable to write clearly about their mathe-matical thinking. When students were

given examples of how to explain theirmathematical thinking they had a betteridea of what was expected of them.However, not all students benefitedequally from this intervention, suggest-ing that case studies of individual chil-dren with different learning profilesmight be useful. Most importantly,faced with difficulties getting studentsto write about their mathematical think-ing, this approach might be relativelyeasy for teachers to implement, particu-larly if they have already had experiencewith using similar approaches in theirlanguage arts programs.


65

REFERENCES

Arachambeault, B. (1991). Writing across the curriculum: Mathematics. Paper presented atthe Annual Meeting of the National Council of Teachers of English, Seattle, WA.

Bay Area Writing Project/California Writing Project/National Writing Project: An Overview.(1979). (Report No. CS-205-424). Berkeley, CA: California University, Berkeley.School of Education. (ERIC Document Reproduction Service No. ED 184 123).

Bell, E., & Bell, R. (1985). Writing and mathematical problem solving: Arguments in fa-vor of synthesis. School Science and Mathematics, 85(3), 210-221.

Burns, M. (1995). Writing in math class? Absolutely! How to enhance students' mathe-matical understanding while reinforcing their writing skills. Instructor, 104 (7),40 - 42.

Countryman, J. (1992). Writing to learn mathematics--Strategies that work K - 12. Ports-mouth, NH: Heinemann.

Fortescue, C. (1994). Using oral and written language to increase understanding of mathconcepts. Language Arts, 71, 576-580.

Miller, D. (1991). Writing to learn mathematics. Mathematics Teacher, 84(7), 516 521.National Council of Teachers of Mathematics (1991). Professional standards for teaching

mathematics. Reston, VA: National Council of Teachers of Mathematics.Szetela, W., & Nicol, C. (1991). Evaluating problem solving in mathematics. Educational

Leadership, 49(8), 42


The Use of Rubrics by Middle SchoolStudents to Score Open-Ended

Mathematics ProblemsFord Long Jr.

Student work was analyzed to investigatewhether students who became involved in theassessment process would improve their per-formance on open-ended mathematics questions.Twelve seventh grade students were selectedfrom 150 students to participate in this study.The pretest and posttest consisted of open-endedquestions adapted from those used by the Bal-anced Assessment and New Standards Projects.Students solved a Problem of the Week for sixweeks and then scored them using a teacher-designed rubric based on Polya's model. Resultsindicate that the lowest students benefited themost by being involved in the assessment proc-ess in this way.


Historically, students in traditionalmath classes have been taught specificprocedures for solving math problemsand the primary focus has been on learn-ing computational skills (National Coun-cil of Teachers of Mathematics (NCTM),1991). Since the purpose of testing hasbeen simply to evaluate students' compu-tational skills, standardized tests consist-ing of multiple-choice items have histori-cally been used. These tests have beenpopular because they are easy to score,but it has been argued that they do notmeasure students' mathematical thinking.New assessment techniques haveemerged, including ways of assessingstudent work on open-ended tasks, ob-serving students at work, and keeping

student portfolios (Branca, 1994; Califor-nia Mathematics Framework, 1992). Overthe years the word 'testing' has been re-placed by the term 'assessment' to em-phasize the broader scope that has beenencompassed.

Recent educational research findingsindicate that learning occurs when stu-dents actively assimilate new informationand experiences and construct their ownmeanings (NCTM, 1991, p.2). This re-search has generated a move away fromthe traditional classroom practice of rotememorization of facts and has led to newteaching practices as well as to new con-tent that includes more than arithmetic.New standards and frameworks for teach-ing mathematics have been developed(NCTM, 1989; California MathematicsFramework, 1992). These new guidelinesfor curriculum materials have created aneed for new assessment tools (Resnick,1994; National Research Council, 1989).

Ford Long has been a seventh and eighthgrade middle school mathematics teacher inthe Laguna Salada School District in Pacifica,CA for the past ten years. He has had a varietyof leadership roles, as a cluster leader in theMiddle School Mathematics Renaissance Pro-ject and as a mentor teacher in his school dis-trict. Currently he works with K-8 teachers asa mathematics resource teacher and as amathematics coach.


New assessment programs havebeen funded to monitor the effective-ness of instructional strategies that em-phasize thinking and reasoning inmathematics (Katims, Nash, & Tocci,1993; Lane, 1993; Resnick, 1994). A keycomponent of these programs is open-ended questions that provide opportu-nities for students to demonstrate theirmathematical understanding as op-posed to multiple choice questionswhich were designed to measure com-putational skills.

The purpose of this study is to exam-ine students' work to determine if theirperformance on open-ended questionswould improve after the students areinvolved in the assessment process. Inthis study, students revised a scoringrubric based on George Polya's four-step problem solving model and usedthis rubric to assess a series of Problemsof the Week (POWs) for six weeks. Amodified New Standards Project (NSP)rubric was also used by the students toscore a pretest and posttest of open-ended questions adapted from the NewStandards Project and the Balanced As-sessment Project (BAP). These tests wereused to evaluate improvement.

It was hypothesized that students'performance on open-ended problemswould improve after they were involvedin the assessment process. One expectedoutcome was that, after being taught as-sessment techniques, students would bebetter prepared to explain and justifytheir mathematical thinking by usingmore precise mathematics vocabularyand multiple representations to illus-trate their solutions. In addition to thepretests and posttests, the effectivenessof involving the students in the assess-

ment process was evaluated by compar-ing the students' assessments to the tea-cher's assessments. The research ques-tions for this study are:

1. Will using Polya's model as a ru-bric help students to set standards forquality work and improve their per-formance on open-ended mathematicsproblems?

2. How do student scores compare toteacher scores on open-ended mathe-matics problems?


For the past century traditional class-room pedagogy and assessment systemshave been based on the behavioraltheory of learning which stresses that"content can be broken down into smallsegments to be mastered by the learnerin a linear sequential fashion" (Rom-berg, 1995, p.5). Historically, thesetraditional teaching practices were ac-companied by standardized tests. Morerecently, researchers have found thatstandardized tests can not measuremathematical thinking or mathematicalliteracy and encouraged development ofalternative assessment systems (Ku lm,1994; Romberg, 1995; NCTM, 1995).

Constructivist research indicates thatlearning occurs when students use priorknowledge to actively assimilate newinformation and experiences and con-struct their own meanings (Resnick,1987). In other words, learning is a proc-ess where the student gathers, discov-ers, or creates knowledge in the courseof performing a purposeful activity.Katims, et al. (1993) cautioned thatwhen the students are first introducedto complex nonroutine problem solving,their initial reaction is frustration and


68

they are at a complete loss as to how tobegin. As the students become familiarwith problem solving activities, the pe-riod of confusion and frustration at thestart of each activity shortens considera-bly and their planning time increases.She explains that performance assess-ment can not be successful withoutsome retraining of the students.

Some studies support the need forinvolving students in the assessmentprocess (Stiggins, 1994; Higgins, Harris,& Kuehn, 1994), and some researchershave developed instructional programsto assist students in learning how toevaluate their own responses and to re-vise and improve their own work(Katims, et al. 1993; Lane, 1993; Lappan& Ferrini-Mundy, 1993; Resnick, 1994;Baker, O'Neil, & Linn, 1994). A majorcomponent of each of these programshas been open-ended questions that areevaluated by using alternative assess-ments.

Several studies have found that peerinteraction enhanced the problem solv-ing process and helped students per-form better on open-ended questions(Hart, 1993; Larson, 1996). Schoenfeld(1989) also mentioned how Mason(1982) used the ideas generated by hisstudents to help them develop stan-dards by which their arguments wouldbe considered. In this study, the stu-dents worked with their peers to revisea teacher-designed rubric, which theythen used to set standards for qualitywork. They also worked with their peersto score their work.

Scoring rubrics provides a structureto assist students in their problem solv-ing efforts. Pate, Homestead, andMcGinnis (1993) described a rubric as a

scaled set of criteria that defines for thestudent and teacher what a range of ac-ceptable and unacceptable performancelooks like. They stated that rubrics couldbe used to evaluate process as well ascontent, and can guide students in self-assessment. They believed that the "ru-brics should have such detail that noone (students, teachers, parents) hasquestions as to how well the activitywas done" (p. 27). Rubrics have beenused to provide descriptions of eachlevel of performance in terms of whatstudents are able to do and assignsvalues to these levels (Herman, Asch-bacher, & Winters, 1992). The rubrics aremeant to assist the students in develop-ing the self-reflection or metacognitiveskills used by expert problem solvers.

According to Wilson, Fernandez,and Hadaway (1985), most formulationsof a problem solving framework in U. S.textbooks attribute some relationship toPolya's stages; however, the use of lin-ear models used in these textbooks doesnot promote the spirit of Polya's stagesand his goal of teaching students tothink. They described the following de-fects in the traditional models.

They depict problem solving as alinear process.

They present problem solving as aseries of steps.

'They imply that solving mathemat-ics problems is a procedure to be memo-rized, practiced, and habituated.

'They lead to an emphasis on an-swer getting.

These linear formations are not consis-tent with the genuine problem solvingfocus which should continually cycle be-tween the four phases: understand,plan, try, check.


69

Szetela and Nicol (1992) stated that"effective assessment of problem solv-ing in math requires more than a look atthe answers students give. Teachersneed to analyze their processes and getstudents to communicate their thinking"(p.42). Charles, Lester, and O'Daffer(1987) described scales that focus moreattention on solution procedures. TheCalifornia Assessment Program (Pandy,1991) included comprehensive descrip-tions of various levels of performancefor specific problems. This was appro-priate for large-scale assessment pro-grams. However, the classroom teacherhas little time to construct scales for in-dividual problems. Teachers need as-sessment procedures and scales thatthey can modify or use intact for a widerange of problems (Szetela & Nicol,1992). Because of this shift in focus,teachers must be provided with tools toevaluate their students' problem solvingskills. This study attempts to developways of using alternative assessments ina classroom setting.

The uniqueness of this study is thatthe students designed their rubric basedon the questions Polya considers usefulto the problem solver who works byhimself (Polya, 1957). They also reviseda modified New Standards Project ru-bric that was used to score their testsand solutions on previously scored an-chor papers, which serve as examples ofparticular scores.

METHODS

The students in the study attended amiddle school located about 15 milessouth of San Francisco. The studentpopulation is approximately 63 percentCaucasian and 27 percent other ethnic

groups. Students are heterogeneouslygrouped and randomly assigned to theirclasses. All students in the five 7th grademath classes were given a pretest, aProblem of the Week (POW) each weekfor six weeks, and a posttest.

The instruments used for the as-sessments were open-ended questionsdesigned for middle school studentsfrom the New Standards Project 1994Reference Exam and the Balanced As-sessment Project. The pretest consistedof one short-item from NSP and onelong-item from BAP. The posttest con-sisted of a short-item and a modifiedlong-item both from the NSP. One of theshort items asked the students to deter-mine the better airline deal based on thespecials that they were offering. One ofthe long items involved a situationwhere three schools were planning ascience fair. There were four parts tothis problem in which the students wereasked to allot space and dollars to eachschool based on its population.

The students scored their pretest andposttest using a modified rubric, whichthey designed, based on the criteria es-tablished by the NSP. The students alsowrote justifications for their score. Thesescores and justifications were the self-assessments that were later compared tothe teacher assessments. As part of thetraining in scoring, all the studentsscored anchor papers with the officialscores removed. They did this twotimes, once at the beginning and once atthe end of the study. The anchor paperspresented two solutions for each short-item question from the tests. After scor-ing the papers, the students thenworked in pairs to revise and improvethe solutions.

7QCollege of Education Review 67

A subset of twelve seventh-gradestudents were selected from the 150students in the five classes using a strati-fied sample. Two female and two malestudents, each from the top, middle, andbottom third of these lists, were selectedusing a table of random numbers.Eighty-five additional students' scoresof the anchor solutions were also com-pared.

The purpose of this study was to in-vestigate whether students who becameinvolved in the assessment processwould be better able to explain and jus-tify their mathematical thinking. The re-searcher developed a rubric based onthe four-part model suggested by Polya:understand, plan, try, check (See Ap-pendix 1). Each part had a 4-point scalethat the students used to score theirPOW's. Each week the emphasis wasput on one part of this model and thestudents, with guidance from theteacher, refined the language of the ru-bric. The students then used the revisedrubrics to score the following week'sPOW. This self-assessment process con-tinued for six weeks. It was hypothe-sized that using the rubric to assess theirown and other students' work wouldbetter prepare them to establish criteriafor quality work.

The students also used a modifiedNSP rubric in the beginning and at theend of the study to score their tests andto score a range of anchor solutions withthe scores removed. The teacher and thestudents discus_sed the meaning of thescoring criteria for each score point andrevised the language of the rubric. Thestudents used this revised rubric toscore anchor solutions and their ownposttests. The process of revising both

rubrics was designed to help the stu-dents identify criteria for quality workand help them to better express theirmathematical thinking. The responseson the open-ended questions from NSPand BAP were analyzed to determineimprovement.

RESULTS

To determine if using a scoring ru-bric based on Polya's model would helpstudents set standards for quality workand improve their performance onopen-ended questions, scores of 12 stu-dents on a two-item pretest and posttestwere compared. The teacher scores forthe twelve students are reported in Ta-ble One. On the long item, five studentsimproved on the posttest; seven stu-dents' scores remained the same. Fur-thermore, four of the five increasedscores were for students who scored a"1" on the pretest. This indicates thatlow scoring students might have bene-fited more from this process. These stu-dents had partial success in answeringthe question and were able to demon-strate some understanding of themathematics required. The scores of allthe students either stayed the same orimproved. However, on the short-itemthe posttest scores of four students haddecreased and the scores of three stu-dents had increased. One possible ex-planation for these lower posttest scoresis that the single question gave the stu-dents less opportunity to demonstratetheir understanding of mathematics.

A comparison of student assess-ments to teacher assessments is reportedin Tables Two and Three. In cases wherethe student scores matched the teacherscores the difference was zero. If a stu-


71

TABLE ONETeacher Scores on Long and Short Items

from Pretests and Posttests

Student ID Long Term Student ID Short TermPretest Posttest Pretest Posttest

1132 3 4 1132 3 2

1137 3 3 1137 2 2

1139 2 2 1139 4 2

1153 3 3 1153 4 41229 2 2 1229 2 3

1238 1 1 1238 2 1

1241 1 2 1241 2 2

1253 3 3 1253 3 21307 1 2 1307 1 1

1308 1 2 1308 1 1

1321 3 3 1321 1 3

1355 1 2 1355 3 4

dent score is higher than the teacherscore, the score was positive. None ofthe students scored their tests lowerthan the teacher on the pretest or post-test on either item. One explanation forthe students' high scores could be thatmost of the students assumed their solu-tions and justifications were correct,therefore they scored their work a 3 or 4.

Table Two compares the student andteacher scores on the long-items. TableThree compares the student and teacherscores on the short items. Four student'sscores matched the teacher scores on thelong-item from the pretest; only onematched on the posttest. However, 8 outof 12 students (scores highlighted in ta-bles) scored their work within one pointof the teacher scores on the pretests.This indicates a high level of agreementon the pretests between the student andteacher on the scoring of their work.There was less agreement on the post-tests. On the long-item, seven students'

scores were within one point of theteacher's scores. On the short-item therewere five. One reason for less agreementon the posttests was that almost all thestudents scored their work a 4. Thismakes the comparison of the scores onthe posttests less meaningful.

The teacher and student scores werealso compared by having all seventhgrade students score short item anchorpapers at the beginning and end of thestudy. Table Four shows the scores ofthe 85 students who scored both sets ofpapers, which included both a strongand a weak solution for each item. Ineach case the anchor scores were un-known to the students. For the weakstudents seemed to score the work a 3 or4 because the answer was correct, evenif the mathematics to support the an-swer was incorrect or if the explanationor justification was inaccurate or un-clear. For the strong anchor papers,most of the students concur with the


TABLE TWOStudent and Teacher Scores on Long and Short Items

from Pretest and Posttest

Pretest PosttestStudent

IDStudentScore

TeacherScore

Difference StudentID

StudentScore

TeacherScore

Difference

1132 3 3 0 1132 4 4 01137 3 3 0 1137 4 3 +11139 4 2 +2 1139 4 2 +21153 '3 3 0 1153 4 3 +11229 4 2 +2 1229 4 2 +21238 2 1 1238 2 1 +11241 3 1 +2 1241 4 2 +21253 4 3 - +1 1253 4 3 +11307 3 1 +2 1307 4 2 +21308 2 1 +1 1308 4 2 +21321 4 3 +1 1321 4 3 +11355 1 1 0 1355 3 2 +1

TABLE THREEStudent and Teacher Scores on Long and Short Items

from Pretest and Posttest

Pretest PosttestStudent

IDStudentScore

TeacherScore Difference Student

IDStudentScore

TeacherScore

Difference

1132 3 3 0 1132 4 2 +21137 4 2 +2 1137 4 2 +21139 4 4 0 1139 4 2 +21153 4 4 0 1153 4 4 01229 3 2 +1 1229 4 3 +11238 3/2 2 +1 1238 4 1 +31241 3 2 +1 1241 4 2 +21253 4 3 +1 1253 3 2 +11307- 1 +3 1307 3 1

1308 3 1 1308 4 1 +31321 3 1 +2 1321 4 3 +11355 3 3 0 1355 4 4


73

TABLE FOURStudents' Scores Compared to Anchored Scores on Short Item

Beginning of Study End of Study

Scoregiven bystudents Anchor Score 1 Anchor Score 4 Anchor Score 1 Anchor Score 3

4 4 79 11 24

3 46 5 41 56

2 .31 1 31 5

1 4 0 2 0

n 85 85 85 85

score, either the 3 or the 4. This agree-ment may occur because the studentsseem to be generous in their scoringoverall but the less generous scoring onthe weak anchor papers indicates thatthe students were able to distinguishthat the work was not quite right. Moststudents stated they scored the paper a 3instead of 4 because the individual didnot show their work for the calculations.

DISCUSSION

In class discussions students wereconsistently prompted to represent themathematics in different forms usingcharts, diagrams, or different numericforms. It was hoped that this would leadthem to check their work and help toensure a mathematically correct solu-tion. The intensity of the focus on theprocess seems to have directed studentsaway from checking their answer. Thestudents acted like the college studentsin Schoenfeld's study who continueddown a dead end path and neverstopped to check their first results. This

differs from the mathematician whowould check to see if his solution madesense, make adjustments in his thinking,and then would proceed down a newpath (Schoenfeld, 1989).

The assignment of a score from 1 to 4based on the rubric also took on a life ofits own for the students. The majority ofstudents assigned themselves a 4 on theposttests because they believed thattheir work met the criteria of a 4 paper:

mathematically accurate and ex-plained well,

each step of the solution is shown,all numbers are labeled with words

or symbols,you showed and explained how

you checked your work,if appropriate, a diagram or

illustration is included.If a student showed all their work

and explained their solution, they feltcomfortable assigning their work a 4 be-cause they were unable to considerwhether or not their mathematical rea-soning was correct. The students were


7 4

not able to stop and check to see if theirsolution was appropriate for the ques-tion being asked.

Most students represented their so-lution in more than one way and gave awritten justification for their solution.However, their multiple representationswere not sufficiently independent toprovide a reasonable check for the an-swer; they reported the same incorrectsolution in a different form. Althoughmost students were able to identify thecarefully structured criteria of a qualitypaper, they were not able to identify anincorrect solution or justification.

The six weeks of the study did notprovide enough opportunity for thestudents to make significant changes.Future researchers should provide moretime and training for students to focusmore on checking the correctness oftheir answers. It is also necessary tohave a variety of rich tasks where sev-eral different approaches to solve theproblem are readily demonstrated. Theyneed to open up the problem solvingprocess of students so that a range ofstrategies are available and even re-quired for rich and engrossing prob-lems.

From classroom experience, the re-searcher noticed that many studentswho are not successful at mathematicstend to shut down and give up on prob-lem-solving. It was hoped that thismodel would provide these studentswith some of the problem solvingstrategies used by students who are suc-cessful problem solvers. One weaknesswith the model presented was that stu-dents used the questions in it as a check-list to simply answer yes or no. For ex-ample, one of the questions on the 'De-

vise A Plan part of the model was, "Canyou solve a related or simpler prob-lem?" Many students wrote, "Yes, I cansolve a simpler or related problem."They didn't actually look for a sub-problem or simpler problem that couldbe solved as a first step. Many studentswere not able to do this on their own.

The students suggested rewriting thequestions in the first model presented asstatements. The intention of these ques-tions, later statements, was that theycould be used as a guide for the stu-dents who became bogged down in theproblem solving process. The modelwas meant to illustrate the cyclical na-ture of Polya's problem solving phasesand also to provide the students with alinear structure or outline for the finalwrite-up and presentation of their solu-tion. Many of the low achieving stu-dents continued to use the model as achecklist even after the questions werechanged to statements.

Inexperienced problem solvers needmore instruction on how to use Polya'smodel. The most important phase of thismodel is to understand the problem.Students need more training in how tomove back and forth between thephases as they gain more information inorder to check the reasonableness oftheir solution path. They can then moveon to the 'carry out your plan' phase.With this approach they can determineif their strategies make sense and im-pxove their overall understanding of theproblem.

The students' revisions of the word-ing of each phase each week so that themodel made more sense to them did notappear to help them identify when theywere heading down an inappropriate


solution path. Also, asking the samequestion in different phases did not ap-pear to help the students check the ap-propriateness of their solution path. Forexample the question, "Did you use allthe data?" appears twice, in the 'devisea plan' and 'look back' phase. The stu-dents did not see the need to answer itagain and left out an important stimulusfor the cyclic process. More examplesneed to be provided where the studentsgain new information and then go backto the original question to determine ifthey are using all the available data intheir solution. Students also need to bepresented with more problems that havethe correct answers, but incorrectmathematics to support their answers orunclear explanations. It might be morebeneficial to solve several problems as aclass completing every phase of thewrite-up together instead of breakingthe Polya model into separate phasesthat are introduced one per week. Con-solidating the process in this way mighthelp the students to constantly moveback and forth between the understand,plan, try, and check phases.

CONCLUSION

Ku 1m (1994, p. '25) stated that, "In re-cent work on mathematical thinking, at-tention has been given to the notion thatpeople have strategies that guide their

choice of what skills to use or whatknowledge to draw upon during thecourse of problem solving, investiga-tion, or verification of a discovery." It isclear that middle school students needto be taught these strategies. This studywas intended to help students furtherdevelop their problem-solving strate-gies. The process of scoring studentwork each week related to Polya'smodel was utilized so that the studentswould be involved in the assessmentprocess and they would have examplesof student work to revise and improve.It was hoped that by working with theirpeers to improve student work, theywould be better able to apply the strate-gies they learned to improve their ownsolutions to open-ended questions. Theresults indicate that perhaps because ofthe design of the study, which devel-oped Polya's steps, one at a time, the fi-nal step, check, did not get internalized.

Most middle school students havenot developed strategies to help themplan and carry out the steps in solvingproblems. Ku 1m (1994, p. 26) stated thatthis strategic knowledge must be explic-itly taught, modeled, and practiced. Thisstudy reinforces and further supportsthis conclusion. Students need assis-tance in developing lines of reasoningand decision making strategies used bysuccessful problem solvers.

76- College of Education Review 73

REFERENCES

Baker, E. L., O'Neil, Jr., H. F. & Linn, R. L. (1994). Policy and validity prospects for per-formance-based assessment. Journal for the Education of the Gifted, 17(4), 332-353.

Balanced Assessment. (1995). Available from Balanced Assessment, Graduate School ofEducation, University of California, Berkeley, Ca. 94720-1670.

Branca,N. (1994). California Mathematics Project. (Available from CMP, 6475 AlvaradoRoad, Suite #206, San Diego; CA 92120-5006)

California Department of Education (1992). Mathematics framework for California publicschools: Kindergarten through grade twelve. Sacramento, CA: Author

Charles, R., Lester, F. & O'Daffer, P. (1987). How to evaluate progress in problem solving.Reston, VA: National Council of Teachers of Mathematics

Hart, L. C. (1993). Some factors that impede or enhance performance in mathematicalproblem solving. Journal for Research in Mathematics Education, 24(2), 167-171.

Herman, J. L., Aschbacher, P. R. & Winters,L. (1992). A practical guide to alternative assess-ment. Alexandria VA: Association for Supervision and Curriculum Development.

Higgins,K.M., Harris, N. A. & Kuehn, L. L. (1994). Placing assessment into the hands ofyoung children: A study of student-generated criteria and self-assessment. Educa-tional Assessment, 2(4), 309-324.

Katims,N., Nash,P. & Tocci, C. M. (1993). Linking instruction and assessment in a middleschool classroom. Middle School Journal, 25(2), 28-35.

Kulm, G. (1994). Mathematics assessment: What works in the classroom. San Francisco, CA:Jossey-Bass.

Lane, S. (1993). The conceptual framework for the development of a mathematics per-formance assessment instrument. Educational Measurement: Issues and Practice, Sum-mer, 16-22.

Lappan, G. & Ferrini-Mundy, J. (1993). Knowing and doing mathematics: A new visionfor middle grades students. The Elementary School Journal, 93(5), 625-641.

Larson, R. (1996). Teacher perceptions of effective mathematics practices associated with highperformance on the open-ended mathematics assessment. Report prepared by NorthwestRegional Education Laboratory, Portland, OR.

Mason, J. L. & Stacey, K. (1982). Thinking mathematically. London: Addison Wesley Pub-lishing Co. Inc.

National Council of Teachers of Mathematics (1989). Curriculum and evaluation standardsfor school mathematics. Reston, VA: Author.

National Council of Teachers of Mathematics (1991). Professional standards for teachingmathematics. Reston, VA: Author.

National Council of Teachers of Mathematics (1995). Assessment standards for schoolmathematics.. Reston, VA: Author

National Research Council (1989). Everybody counts. Washington, DC: National AcademyPress.


7 7

New Standards Project. (1994). Available from New Standards, 4th floor, 300Lakeside Drive, Oakland, CA 94612-3350.

Pandey, T. (1991). A sampler of mathematics assessments. Sacramento, CA: CaliforniaDepartment of Education.

Pate, E. P., Homestead, E. & McGinnis, K. (1993). Designing rubrics for authenticassessment. Middle School Journal, 25(2) 25-27.

Polya, G, (1957). How to solve it (2nd ed.). Princeton, NJ: Princeton UniversityPress.

Resnick, L. B. (1987). Education and learning to think. Washifigton, DC: NationalAcademy Press.

Resnick, L. B. (1994). Standards, assessment, and educational quality. Journal forthe Education of the Gifted, 17(4), 409-420.

Romberg,T. A. (1995). Reform in school mathematics and authentic assessment. Al-bany, NY: SUNY Press.

Schoenfeld, A. H., (1989) Mathematical problem solving. (In Toward a thinking cur-riculum: Current cognitive research, Association for Supervision and Curricu-lum Development, ASCD yearbook.

Stiggins, R. (1994). Assessment literacy for the 21st century: A progress report. Unpub-lished report adapted from Student-centered classroom assessment. New York:Macmillan.

Stiggins, R. (1994). Student-centered classroom assessment. New York: Macmillan.Szetela, W. & Nicol, C. (1992). Evaluating problem solving in mathematics. Educa-

tional Leadership, May, 42-45.Wilson, Fernandez, & Hadaway, (1985). Mathematical problem solving. InSchoenfeld, A. H., Mathematical Problem Solving (pp. 57-77). Orlando, FL: Aca-demic Press.


78

APPENDIX ONERubric Used by the Students to Score Their 'POWS' Each Week

NAME DATE PERIOD

PROBLEM OF THE WEEK = POWerful MathematicsGEORGE POLYA SAYS: Understand Plan. Try and CheckAs you solve the problem check a bracket for each line you completed. If you checkedfour brackets give yourself a '4'; three brackets a '3', two brackets a '2', one bracket a '1.'

I. UNDERSTAND THE PROBLEM III. CARRY OUT THE PLAN( ) Rewrite the main question in your own ( ) Show all your work.words. ( ) Use math vocabulary and/or math( ) Underline the questions, circle the word symbols to explain your answer.that will describe the answwer (cows, $,feet, etc.).

( ) Label all numbers with words orsymbols.

( ) List the data. What can you conclude ( ) Prove each step is correct by showingfrom the data? your solution more than one way.( ) Make a predictiOn or estimate theanswer.Circle one of the following: 4 3 2 1 0 Circle one of the following: 4 3 2 1 0

II. DEVISE A PLAN IV. LOOK BACK( ) List three problem-solving strategies to ( ) Explain and show how you checked thesolve the problem (Guess & Check; Work result.Backwards; Lists; Equations: etc.). () Explain and show how you checked all( ) Explain how you are going to use at of the relevant data.least one strategy. ( ) Show how you used your answer in a( ) Make a sketch, illustration, or chart of different way. (Graph, equation, Usedthe problem. implied information).() Explain how you will use all of the ( ) Make up a similar problem of yourrelevant data. own.

Circle one of the following: 4 3 2 1 0 Circle one of the following: 4 3 2 1 0Overall rating from above:

14-16 pts = 4 12-13Circle your overall score below:

Span4 Scholar3 Practitioner

.2 Apprentice1 Novice

pts = 3 8-11 pts = 2

Grade Scale

1-7 pts =1

18-2015-176-141-- 5

Well Done/Complete WorkAcceptable/ Minor RevisionRevision NeededRestart / Incomplete

Limit your write up to one page. ANSWER IN OUTLINE FORM. Work 15minutes each night. Write at least 2 sentences that explain what you did to solvethe problem. Attach all scratch work.


7 9

A Survey of Teacher Beliefsabout Pre-Requisite Experiencesfor Student Success in Algebra

Audrey Adams

In this study, algebra teachers responded to sur-vey questions about their beliefs about whatmathematical skills or experiences students needto be successful in their algebra classes andabout how frequently they used different teach-ing strategies. Teachers were grouped accordingto their self-identification of their teachingstrategies as traditional, non - traditional; or amix of both. ,Results showed the three groups tobe similar in terms of their ratings of necessaryskills and experiences, but dissimilar in types ofteaching strategies used in their classroom

COE Review, pp 77 -83.-

For those students who either do nottake algebra, or who take it and fail theclass, algebra serves as a gatekeeperwhich bars them from further educa-tional and, often, economic opportunity(NCTM, 1994). The fact that more thanhalf of the students who fail algebranever take another math course furtheremphasizes the importance of algebra tostudents' mathematics futiire (Mullins,Dossey, Owens, & Phillips, 1991). Stu-dents must be successful in algebra tocontinue in more advanced mathematicscourses, which are generally a prerequi-site for many professional career paths.

The responsibility for preparing stu-dents in algebra cannot be left to algebrateachers alone. The entire kindergartenthrough eighth grade mathematics cur-riculum needs to prepare students to

develop the abstract concepts necessaryfor algebra. All elementary school andmiddle school teachers have a role toplay in this preparation, but because al-gebra is typically offered beginning ineighth and ninth grades, sixth and sev-enth grade mathematics teachers play aparticularly critical role.

The way in which an algebra class istaught can affect the algebra teacher'sexpectations of the prerequisite skillsand experiences needed for student suc-cess. Because of the mathematics reformefforts of the past decade, algebra isnow taught in a variety of ways. Non-traditional algebra teachers teach from aproblem-solving perspective and em-phasize conceptual understanding andreal-world applications, whereas moretraditional teachers tend to emphasizememorization and the manipulation ofsymbolic representations.

It is clear that success in algebra isneeded for higher-level mathematics.Therefore, all teachers, but particularly

Audrey Adams has been a middle schoolclassroom teacher for eleven years in SanFrancisco, California and a mentor teacherin mathematics for six years. Although ateacher of all subjects, a special passion formathematics spurred her to complete theMasters Degree in Mathematics Educationthat resulted in this paper.


8 U

middle school mathematics teachers,need to know what preparation willhelp ensure that success. The purpose ofthis study is to investigate the beliefs ofalgebra teachers regarding prerequisiteskills and knowledge necessary for stu-dent success in algebra classes. In par-ticular, the major research questions forthis study are as follows: (a) What priorexperiences do algebra teachers thinkare necessary for student success in al-gebra?, and (b) Do traditional and non-traditional teachers differ in their viewsabout what those experiences shouldbe?


The mathematics achievement ofstudents in the United States does notcompare favorably with students frommany other countries. The Third Inter-national Mathematics and Science Study(TIMSS) compared the mathematicsachievement of eighth grade students in41 countries and found that the UnitedStates ranked in 28th place (TIMSS,1996). Another troubling statistic is thatonly half of U.S. students take morethan two years of mathematics in highschool (Everybody Counts, 1989). The lowmathematics achievement for eighthgrade students in the United States, ac-cording to the conclusions of TIMSS, isdue, at least in part, to the lack of focusin the U.S. mathematics curriculum.Teachers are expected to cover too manytopics, resulting in a fragmented, un-connected curriculum that does not al-low students enough time to study anyone topic in depth.

How students are taught is anothercause for concern. TIMMS researcherscompared lessons taped in real class-

rooms in the United States and abroad.They found that in comparison to othercountries, U.S. students spend more oftheir time practicing computationalskills rather than analyzing relation-ships, solving problems, and developingdeeper conceptual understanding.

The mathematics reform movementin the United States is largely basedupon the idea that children learn bestwhen they construct their own meaning(Resnick, 1987). Many of the problemschildren experience in learning mathe-matics are thought to result from tryingto use formal procedures and algo-rithms without having conceptual un-derstanding. Kamii and Lewis (1991),for example, found that second gradestudents who were taught in a construc-tivist program demonstrated greaterability to use higher-order thinkingskills than similar students taught in amore traditional program which em-phasized memorization of algorithms.

The research about how childrenlearn mathematics have led somemathematics educators to reconsiderwhat constitutes meaningful mathemat-ics content and effective instructionalstrategies for algebra classes. The NCTMCurriculum and Evaluation Standards(1989) reflect this new understanding. Inparticular, the NCTM standards de-scribe effective algebra instruction aspresenting many varied experiences ininquiry and application that actively in-volve students in those processes andinclude opportunities to reflect uponand express mathematical ideas.

Since algebra classes are changing, itis necessary to delineate the basic ideasand concepts which should be taughtprior to enrollment in a formal algebra


81

class, especially if algebra is to be madeaccessible for all students (NCTM, 1994;Moses, Kamii, McAllister, Swap, &Howard, 1989). In preparation for alge-bra, students need to learn computationin a problem-solving setting, under-stand the vocabulary of mathematics inorder to understand the symbols, beable to recognize patterns, have lots ofexperiences collecting, graphing andanalyzing data, and develop numbersense (Howden, 1990). They must per-sonally construct symbolic representa-tions in order to create mathematicswith context and meaning (Moses et. al.,1989). In terms of specific arithmeticconcepts and specific skills essential forsuccess in algebra, children need be ableto solve equations such as the meaningof the equals sign and the order of op-erations (Herscovics & Linchevski,1994). In addition, it has been arguedthat the understanding of proportional-ity is a bridge between arithmetic andabstract algebra (Post, Behr, & Lesh,1988).

The view of what algebra classesshould be like is changing because ofthe implications of research as well asthe introduction of technology. Thismeans that the prerequisite skills andknowledge necessary for success in al-gebra are changing and, therefore, thepreparation of students for success inalgebra must also change.

METHODS

Based on the review of the literature,a three-part questionnaire was designedto elicit information from algebra teach-ers. The first part contained questionsabout the teacher's view of the studentskills necessary for success in his or her

classroom. The second part containedquestions about (a) whether the teachercharacterized himself or herself as tradi-tional, non-traditional or a mix of both,and (b) the teaching strategies and/ ormanipulatives, textbooks and curriculaused by the teacher. The third part in-cluded demographic questions aboutgender, age, ethnicity, teaching experi-ence and educational background.

The questionnaire was sent to allteachers of eighth and ninth grade alge-bra in the San Francisco Unified SchoolDistrict. Of the 102 teachers to whomsurveys were sent, 51 (50%) teachers re-sponded.

RESULTS

The survey respondents consisted of31 males and 25 female algebra teachers.European-Americans comprised thelargest ethnic group (N = 29), followedby Chinese-Americans (N = 7), Latinos(N = 4), African-Americans (N = 3), Fili-pinos (N = 2) and all other (N = 6). Theages were fairly evenly distributed, withslightly more teachers in the 51 to 60year-old category. The number of yearsof teaching experience ranged from sixmonths to 36 years.

Teachers were asked to identify theirclassroom teaching strategies as tradi-tional, non-traditional or a mixture ofboth. Two-thirds of the 56 teachers iden-tified themselves as a mixture of bothtraditional and non-traditional (N = 37teachers). Of the remaining third, 11identified themselves as traditional and8 identified themselves as non-traditional.

A comparison was made betweenthe three groups of teacherstraditional,mixed, and non - traditional - -in terms of


82

their number of years of teaching ex-perience. It was hypothesized thatnewer teachers would tend to identifythemselves as non-traditional. Considerthe teachers who had been teaching be-tween two and five years, namely thenewer teachers. Of these newer teachers,only 18 percent (2 of 11) consider them-selves traditional and 50percent (4 of 8)non-traditional. At the same time, whenconsidering the teachers who had beenteaching 20 years or more, 55 percent (6of 11) considered themselves traditionaland only 25 percent (2 of 8) non-traditional.

The teachers were asked how fre-quently they used certain instructionalstrategies during a typical week. Thenine strategies were separated by thisresearcher into three groups of threestrategies each: traditional (multiple al-gebraic exercises, memorizing proce-dures, and whole class instruction), non-traditional (the use of manipulatives,use of cooperative groups, and writingabout mathematics) and those whichdid not fit easily into either of the firsttwo categories (pencil and paper graphs,real-world problems, and graphing cal-culators).

A scale was then created to comparethe self-identification of each teacherwith the strategies used in his or herclassroom. The strategies in each cate-gory listed above were assigned a valuethat was then multiplied by the numberof times per week the teacher used thatstrategy. A score was then calculated foreach teacher. A low score meant that theteacher frequently used non-traditionalstrategies; conversely, a high scoremeant that a teacher frequently usedmore traditional classroom strategies.

The teachers who identified themselvesas traditional all scored in a range from35 to 43. Those teachers labeling them-selves as non-traditional had scoresranging from 19 to 34. The remaininggroup of mixed designation teachershad scores ranging from a low of 22 to ahigh of 40. Thus, the range for teachersin the mixed group was greater than foreither of the other groups and wasslightly higher than for the lowest scoreof non-traditional teachers and slightlylower than for the most traditionalteacher. The designations chosen byteachers and their use of particularclassroom practices seem to be consis-tent with what the research says are tra-ditional and non-traditional teachingmethods.

Teachers were also asked to respondto a series of eleven 5-point Likert scalequestions scale that asked them toevaluate specific mathematics skills andmathematical experiences in terms oftheir importance as pre-requisites forsuccess in algebra. A rating of "1" indi-cated that a skill or experience was con-sidered to be highly important and a rat-ing of "5" indicated that a skill or experi-ence was considered least important.The skills in Table One are ordered frommost traditional to least traditional.

A comparison of the mean ratings ofeach of the three teacher groups (tradi-tional, non-traditional, and mixed) indi-cated agreement, for the most part,about the relative importance of theseskills. But there were some notable ex-ceptions when comparing the threegroups. First, on average, the skill of


83

TABLE ONETeacher Ratings of the Relative Importance

of Specific Skills and Experiences as Prerequisites to Algebra

Ability to: Traditional MixedNon-

TraditionalMean SD Mean SD Mean SD

1. Master operations withwhole numbers 1.55 0.69 1.62 0.68 2.00 0.932. Master operations withfractions 2.09 1.3 1.95 0.97 2.50 1.073. Master operations withintegers 1.55 0.69 1.76 0.83 2.38 1.194. Memorize and useformulas 2.73 1.19 3.30 1.05 4.25 0.875. Understand order ofoperations 1.46 0.69 1.84 0.83 2.25 1.046. Solve equations with anunknown 1.36 0.51 2.16 1.14 3.25 1.397.Conceptually understandproportional reasoning 2.10 0.88 1.97 0.93 2.13 0.84

8. Recognize patterns andmake generalizations 2.36 1.5 1.95 0.88 1.75 0.46

9. Problem-solve 1.55 0.69 1.62 0.76 1.63 0.7410. Communicate math ideasin writing 3.18 1.25 2.11 1.09 1.86 0.3511. Work cooperatively ingroups 3.73 1.01 2.35 1.06 2.25 0.71

memorizing and using formulas (Skill 4)was rated as more important by the tra-ditional and mixed teachers than thenontraditional teachers. (Overall, acrossthe groups, the skill of memorizing andusing formulas was rated as less impor-tant, on average, than the other skills).Second, the skills of writing aboutmathematics (Skill 10) and working co-operatively in groups (Skill 11) wererated as more important by the nontra-

ditional and mixed teachers than thetraditional teachers.

DISCUSSION

The number of teachers who partici-pated in this study constituted morethan half of the algebra teachers in afairly large urban school district. Two-thirds of the teachers identified theirteaching strategies as a mixture of bothtraditional and non-traditional. Most


84

teachers in the survey indicated thatthey sought a balance between the prac-tice of routine skills and the develop-ment of conceptual understandings, us-ing what they believed was valuable tocreate an effective algebra program.

When comparing the teachers' self-identification of strategies used andtheir years of teaching experience, thegeneral presumption that newer teach-ers tend to be more non-traditional andolder teachers tend to be more tradi-tional was supported in this case. Whenthe self-designations of teachers werecompared with their actual classroompractices, the traditional and non-traditional teachers identified them-selves in ways that are consistent withthe literature, thereby validating theseself-reported data.

All three of the teacher groups be-lieved that students need to have astrong foundation in the traditional, ba-sic operations using whole numbers, in-tegers and fractions. There was alsoagreement among the three groupsabout the importance of certain skills orexperiences typically identified as non-traditional. In particular, there wassome agreement among the threeteacher groups about the importance ofthe skills of proportional reasoning, rec-ognizing patterns and making generali-zations, and problem-solving. The re-sponses of the teachers in this studyclearly indicate that there is moreagreement than disagreement aboutwhat skills and experiences are neces-sary for student success in algebra.


85

REFERENCES

Everybody Counts: A Report to the Nation on the Future of Mathematics Education.(1989). Washington, DC: National Academy Press.

Herscovics, N. & Linchevski, L. (1994). A cognitive gap between arithmetic and al-gebra. Educational Studies in Mathematics, 27, 59-78.

Howden, H. (1990). Prior experiences. Algebra for Everyone. National Council ofTeachers of Mathematics.

Kamii, C. & Lewis, B. A. (1991). Achievement tests in primary mathematics: Per-petuating lower-order thinking. Arithmetic Teacher, 38, 4-9.

Moses, R. P., Kamii, M., McAllister Swap, S. & Howard, J. (1989). The Algebra Pro-ject: Organizing in the spirit of Ella. Harvard Educational Review, 59(4), 27-47.

Mullins, I. V., Dossey, J. A., Owen, E. H., & Phillips, G. W. (1991). The State ofMathematics Achievement: NAEP's 1990 Assessment of the Nation and the Trial Assessmentof the States. Washington, DC. National Center for Education Statistics.

National Council of Teachers of Mathematics (1994). A Framework for Constructing aVision of Algebra. Draft. National Council of Teachers of Mathematics. Reston, VA.

National Council of Teachers of Mathematics, Commission on Standards forSchool Mathematics. (1989). Curriculum and Evaluation Standards for School Mathematics.Reston, VA.

Post, T., Behr, M. & Lesh, R. (1988). Proportionality and the development of preal-gebra understandings. The Ideas of Algebra, K-12. National Council of Teachers ofMathematics. Reston, VA.

Resnick, L. B. (1987). Education and Learning to Think. Commission on Behavioral andSocial Science and Education. National Research Council, Washington, DC. NationalAcademy Press.

Third International Mathematics and Science Study (1996). National Science Founda-tion. Washington, DC. National Center for Education Statistics.


a6.

Fifth Grade Teachers' Attitudestowards Implementing a New

Mathematics CurriculumLorene B. HolmesDeborah A. Curtis

Math Land is a relatively new mathematics cur-riculum program that has been adopted by anumber of school districts in the San FranciscoBay Area. This curriculum is a product of themathematics reform movement, and requires amajor shift in mathematical content and peda-gogy. The purpose of this study was to surveyfifth grade teachers in one particular school dis-trict that was in its second year of implementingMathLand. In particular, the goal of this studywas to assess fifth-grade teachers' opinionsabout the MathLand curriculum as well as theiropinions about the current mathematics reformmovement in general. Although the teacherstended to agree with many of the currentmathematics reform principles, they were gener-ally dissatisfied with the MathLand curriculum.A number of teachers cited the lack of practicewith basic skills as a limitation of the new cur-riculum. Results are discussed in terms ofimplications for classroom practice.

COE Review, pp 84-94

A major concern of educators inmany California school districts is thatthe mathematics education reformmovement has entered into the class-rooms by way of new mathematics text-book series adoptions. In California, the1994 adoption of instructional materialsin mathematics was a milestone in thestate's ten-year effort to advancemathematics education reform (Califor-nia Basic Instructional Materials in

Mathematics Adoption Recommenda-tions, 1994). This adoption of mathemat-ics instructional materials reflected theinfluence of the 1992 California Mathe-matics Framework, which called for in-struction to develop "mathematicallypowerful" students, including both in-struction in basic computational skills aswell as the opportunity to develop theseskills through real-world problem solv-ing and investigations.

In the San Francisco Bay Area, theadoption of the MathLand curriculum in1995-96 by one of the large, urbanschool districts in the area was a majorshift toward math reform for that dis-trict. This adoption was more than amovement from an old mathematicstextbook series to a new one. The Math-Land curriculum includes a majorchange in pedagogy and mathematicscontent. Consequently, in-service train-ing for teachers was an essential part ofthe district's implementation process.During the first year of implementation(1995-96), all the kindergarten through

Lorene B. Holmes is a mathematics andscience middle school teacher in the SanFrancisco Unified School District. DeborahA. Curtis is Associate Professor of Interdis-ciplinary Studies in Education at San Fran-cisco State University.

84 College of Education Review8r7

fifth grade teachers in the school districtwere required to attend three profes-sional development days on the Math-Land curriculum. This in-service coveredthree Math Land units, namely numberstrategies, geometry, and logic, whichwere to be taught by all teachers duringthe 1995-96 school year.

Teachers' attitudes and beliefs aboutthe mathematics education reformmovement will influence the way inwhich a new mathematics curriculum isimplemented in their classrooms. Con-sequently, a number of questions aroserelated to the adoption of this curricu-lum. Does the adoption of Math Landpromote movement toward math re-form in the district? How do teachersview the new curriculum? How does itaffect their teaching and beliefs aboutmathematics teaching and learning?What support do teachers need to fullyimplement new curriculum materials?

The purpose of this study is to exam-ine fifth -grade teachers' views and atti-tudes toward the Math Land programand the influence of the new curriculumon their teaching of mathematics; theirinvolvement in math education reform;and their fifth-grade teaching experi-ence with the new curriculum. Fifthgrade teachers were chosen becausefifth grade is perceived of as being criti-cal to the transition from elementaryschool to middle school.


The Math Land series contains manyof the current mathematics reformmovement's goals and defining charac-teristics. The current call for reform inmathematics education (National Coun-cil of Teachers of Mathematics, 1989,

1991; California Mathematics Framework,1985, 1992) has been shaped by a con-structivist approach to teaching andlearning mathematics. How does theimplementation of new curriculumsupport the mathematics reform proc-ess? A review of the literature suggeststhat the answer to this question is verycomplex.

Mathematics education in the UnitedStates has been in a state of reform formore than 30 years. Ever since the Rus-sians launched Sputnik into space in1957, Americans have worried that ourchildren are not being taught enoughmathematics and science to competewith other nations.

During the 1950s through the 1970s,the mathematics reform movementsprang from many roots and took onmany different (and sometimes oppos-ing) forms. Hence, the curricular re-forms undertaken during those yearsunder the slogans of "modern mathe-matics" or "new math" left a mixed leg-acy to American mathematics education(Everybody Counts, 1989). Jack Price(1995), president of the NCTM, wrotethat the "new math" reform of the 1950sto 1970s failed for several reasons, in-cluding: (a) top-down attempts at de-velopment and implementation; (b) alack of consensus within the mathemat-ics community as to it value; and (c) alack of understanding of the goals onthe part of teachers, administrators,boards of education, and other policymakers.

The "new math" era was followedby a "back to basics" emphasizingmathematics as a core of rules, proce-dures and facts during the mid-1970s.Then, in 1980, the National Council of


Teachers of Mathematics (NCTM) pub-lished An Agenda for Action: Recommen-dations for School Mathematics of the1980s, which called for the implementa-tion of a 10-year reform program. Onegoal of this agenda was to move the fo-cus of school mathematics curriculumbeyond basic skills objectives to a moreproblem-solving conception of mathe-matics content and instruction (NCTM,1989).

The NCTM Standards, unlike previ-ous mathematics reform attempts, havebeen supported by many professionalorganizations, colleges and universities,large segments of the private sector, andthe general public (Hatfield & Price,1992; Price, 1995). For instance, in 1988,the National Council of Supervisors ofMathematics (NCSM, 1988) updated its1977 basic skills position statement todescribe twelve essential mathematicalcompetencies that citizens would needto begin adulthood in the next millen-nium.

Textbooks drive what is taught inschools (Flanders, 1987). In fact, 70 to 90percent of classroom curricular deci-sions made by teachers are based on thetextbooks adopted by their schools ordistricts. Chandler and Brosnan (1994)found that mathematics textbooks pub-lished after 1989 tended to be longer andthat there was a shift in emphasisamong the five content areas of arithme-tic, measurement, geometry, data analy-sis and algebra. These researchers alsofound that there were changes in theamount of content development, drill,word problems and problem solving,and there were increases in the use ofestimation and calculators. According tothe Third International Mathematics

and Science Study (TIMMS Report),(Schmidt, McKnight, & Raizen, 1996)most American mathematics textbooksare unfocused in a number of ways, in-cluding the tendency to cover too manytopics with a "breadth rather than depthapproach."

The MathLand program has no text-books for the students, but does providea teacher's Guidebook for each gradelevel that covers an entire year of cur-riculum. Each Guidebook consists of tenunits, each taking from two to fiveweeks to teach. It is the lack of text-books, along with the activity-based les-sons and the use of manipulatives inthis program that has brought a flood ofcriticism from educators, parents, andthe public. Debra Saunders (1996), oneof the most vocal critics of the mathe-matics reform curricula (which she callsthe "New-New Math"), wrote that pro-grams like MathLand are causing testscores to drop because these programslack basic computational skills, use con-fusing time-intensive "fuzzy math" ac-tivities and do not prepare students forstandardized tests.

However, the MathLand authors(Randolph & Charles, 1996) argued intheir position paper that in order forstudents to have real math power, theymust memorize basic arithmetic facts.Thus, the formal memorization of factsbegins in second grade in the MathLandprogram. They also argued that Math-Land provides several different vehiclesfor students to practice basic skills in-cluding games in the Guidebook, andpractice in the Daily Tune-Ups andArithmetwist booklets. The MathLand au-thors strongly believe their program willdevelop students into mathematical


thinkers. The authors' goal is that stu-dents using Math Land will become con-fident problem solvers.

Does the adoption of a new mathe-matics curriculum promote movementtoward mathematics education reform?According to some researchers (e.g.Cohen &, Ball 1990; Cronin-Jones, 1991)the answer is no. However, other stud-ies (e.g. Yore, 1991) have shown thattextbooks are the main influence inteaching mathematics, since most teach-ers adhere closely to the strategies andexamples used in textbooks for planningtheir lessons. In general, teachers do notalways implement a curriculum in theirclassrooms in same way the curriculumwas designed to be implemented(Cohen & Ball, 1990; D'Ambrosio, 1991;Cronin-Jones, 1991).

Researchers have begun to atknowl-edge the powerful influence teachershave on the curriculum implementationprocess (Cohen & Ball, 1990; Good,Grouws, & Mason, 1990; Cronin-Jones,1991; Brosnan, 1994). Even if teachersare provided with all the tools (all thematerials, in-service training and sup-port), the mathematics reform move-ment might still be hindered by teach-ers' beliefs, if these beliefs are not sup-portive of the newly recommended styleof instruction. Consequently, it is the at-titudes of the teachers and their beliefsabout mathematics education reform is-sues that will influence what is imple-mented.

METHODS AND RESULTS

The purpose of this research was tostudy fifth grade teachers' attitudes to-ward the new mathematics curriculumprogram, and their opinions about

mathematics education in general. Asix-page questionnaire was the primarytool used to assess their beliefs and atti-tudes about the new mathematics cur-riculum. In particular, this questionnaireaddressed: (a) teachers' attitudes towardthe new curriculum; (b) the implementa-tion of the new curriculum; (c) the sup-port teachers' believed they needed toadequately implement the new curricu-lum; and (d) their opinions about themathematics reform methods. Therewere two open-ended items that askedfor the main strength of MathLand andthe main weakness of MathLand. Theteachers were chosen by cluster randomsampling, where a subset of 30 elemen-tary schools in the district were ran-domly selected, and then all of the fifthgrade teachers in the selected schoolswere surveyed. Seventy-three fifth-grade teachers were sent a surveypacket. Thirty-eight teachers (52 per-cent) returned the questionnaires. Asummary of their opinions is presentedin Table One.

The teachers' responses were for themost part consistent with ideas of theNCTM Standards and the CaliforniaMathematics Framework. For example,teachers tended to agree with the state-ment "In teaching mathematics, one ofmy primary goals is to help studentsdevelop their abilities to solve problemsand think mathematically." Also teach-ers tended to agree that students learnmathematics best when they constructtheir understanding from a variety ofexperiences.

One of the goals of mathematics re-form is to make mathematics learningmore meaningful to students. Teachers


9U

TABLE ONETeachers' Opinions about Teaching Mathematics in General

Mean SD nStudents learn mathematics best in groups with students ofsimilar abilities. 2.64 1.02 36Some students are inherently more capable in mathematicsthan others. 3.28 1.21 36Students learn mathematics best when they construct theirunderstanding from a variety of experiences. 4.26 0.92 35Students should learn computational skills within the con-text of problem solving 3.43 1.17 35Students need to master basic computational facts andskills before they can engage effectively in mathematicalproblem solving. 3.33 1.43 36In teaching mathematics, one of my primary goals is tohelp students develop their abilities to solve problemsand think mathematically 4.39 1.08 36In teaching mathematics, one of my primary goals is to helpstudents master basic computational skills 3.94 1.24 36Note: Opinion Scale: strongly disagree=1 to strongly agree=5

were asked to rate the importance ofcertain teaching methods for effectiveinstruction in mathematics. A summaryof their responses is presented in TableTwo.

Again, teachers' responses were forthe most part consistent with ideas ofthe mathematics reform movement. Forexample, most teachers (72 percent)thought it was "very important" to in-troduce applications of mathematics indaily life. But less than half the teachersthought it was "very important" to use avariety of assessment strategies.

The teachers were asked to reportonhow much time was spent on certaininstructional strategies since the adop-tion of Math Land. These data are pre-sented in Table Three.

More than half the teachers reportedthat they were spending more time inhaving the students discuss differentways to solve problems and having thestudents use manipulative materials ordrawings to solve problems. None of theteachers reported spending more timehaving the students work alone on as-signments.

The teachers were asked to indicatewhich Math Land units they taught dur-ing the first year of implementation ofthe new curriculum (Year 1) as well aswhich units they planned to teach dur-ing the second year (Year 2). A sum-mary of these data are presented in Ta-ble Four.


91

TABLE TWOTeachers' Ratings of the Importance of Certain Teaching Methods ,

for Effective Mathematics Instruction (n = 36)

PercentagesNot

Important

2.8

SomewhatImportant

2.8

FairlyImportant

22.2

VeryImportant

72.2a. Introducing applications ofmathematics in daily life.b. Using a variety of assessmentstrategies. 5.6 11.1 36.1 47.2c. Engaging students in open-ended investigations. 5.6 8.3 30.6 55.6d. Making connections betweenmathematics and other disciplines. 2.8 5.6 38.9 52.8

e. Having students explain theirthinking with other students. 5.6 5.6 33.3 55.6

TABLE THREETeachers' Responses to the Statement

"Since the adoption of MathL and, the amount of class timespent on the following is:"

Percentage of Math TimeMoreTime

LessTime

SameAmount

a. Students practice multiplication facts. 8.3 36.1 55.6 36

b. Students solve story problems or other problemsthat don't have obvious solutions.

143 40.0 35

c. Students discuss different ways that they solveparticular problems.

58.8 11.8 29.4 34

d. Students discuss mathematical ideas, as a class orin small groups.

42.9 11.4 45.7 35

e. Students practice or drill on computational skills. 5.7 48.6 45.7 35f. Students use manipulative materials or drawingsto solve problems.

8.5 34.3 35

g. Students work alone on assignments. 0.0 51.4 48.6 35h. Students work on self-assessments. 24.2 12.2 63.6 33I. Students work in cooperative groups. 42.9 5.7 51.4 35j. Students work on long term projects. 37.1 17.1 45.7 35k. Students reflect on their own math learning. 38.9 8.3 52.8 361. You explain concepts or computational procedures. 22.2 30.6 47.2 36


92

,

TABLE FOURPercent of Teachers Who Taught Specific Math Land Units during Year 1

and Their Plans for Year 2

PercentagesYear 1 Year 2

a. Unit 1: All About UsCollecting, Communicating, and Interpreting Data 63.9 77.8b. Unit 2: Using PredictionsExploring Patterns as a Problem-Solving Tool 55.6 64.9c. Unit 3: StrategiesBuilding a Network of Number Relationships 41.7 55.6d. Unit 4: OrganizationsCreating Attribute Sets and Exploring Logical Operators 52.8 44.4e. Unit 5: MillionsExplorations with Everyday Objects, Models, and Equations 16.7 25.0f. Unit 6: Inside/OutsideExplorations in Volume and Capacity 16.7 61.1g. Unit 7: Equivalence ThinkingExploring the Relationships between Whole and Fractional Numbers 36.1 69.4h. Unit 8: The Triangle AngleExploring Angles and Triangles 38.9 63.9i. Unit 9: System ThinkingAnalyzing and Interpreting Number Systems 16.7 19.4j. Unit 10: ConfidenceDeveloping Probability Sense through Sampling and PossibleOutcomes 8.3 27.8

In the first year, the district providedinservice training on three Math Landunits, namely Unit 3, Unit 4, and Unit 8,and teachers were expected to have im-plemented these units in their classesduring Year 1. Roughly 40 to 50 percentof the teachers reported using thesethree units in Year 1 of implementation.Generally, the teachers reported thatthey planned to use more of = the Math-Land units during the second year.

Finally, teachers were asked to ratetheir overall satisfaction with Math Land.Results are presented in Table Five.

The teachers tended to disagree withthe statement "I am very satisfied withMath Land" (Item g). Furthermore, theytended to disagree that Math Land hadeverything they needed to teach theirstudents (Item b). Additionally, they didnot think it prepared their students formiddle school mathematics (Item f).However, they tended to agree with thestatement "Math Land's explanations ofthe Key Mathematical Ideas are helpful"(Item d).

Twenty-nine of the 36 teachers wrotecomments for the main strength of the


TABLE FIVETeachers' Responses to Satisfaction with MathL and

Mean SD na. Math Land is a well-balanced mathematicscurriculum. 2.61 1.18 36b. Math Land has everything I need to teach mystudents fifth grade mathematics. 2.12 1.27 35c. Math Land's layout is easy to follow. 2.85 1.33 34d. Math Land's explanations of the KeyMathematical Ideas are helpful. 3.18 1.03e. The assessment piece in Math Land is aneffective means to look at growth and under-standing of math concepts. 2.74 1.21 34f. Math Land prepares my students for middleschool mathematics. 2.50 1.14 30g. I am very satisfied with Math Land. 2.65 1.32 34Note. Satisfaction Scale: strongly disagree=1 to strongly agree=5

Math Land program. Four different the-mes arose from the written comments:(a) the usefulness of using manipula-tives, (b) the strength of using coopera-tive groups, (c) the worthwhile prob-lem-solving and critical thinking activi-ties, and (d) the benefits of writing.Thirty -two teachers wrote comments onthe main weakness of the Math Landprogram. These comments were dividedinto four themes: (a) lack of sufficientpractice in computational skills; (b) lackof addressing items on the CI 8S test; (c)too much time needed for preparationand for teaching topics; and (d) the needto supplement Math Land with additionalmaterials.

DISCUSSION

The results indicated that these fifth-grade teachers' opinions about teachingmathematics and using certain teachingmethods for effective mathematics in-struction were compatible with the un-

derlying philosophy of the mathematicsreform movement. However, the resultsof this study also revealed that the atti-tudes that these teachers have towardMathLand itself were mixed, with anumber of teachers indicating theiroverall dissatisfaction with the curricu-lum.

Consider first teachers' beliefs aboutmathematics instruction in general. Theteachers agreed that students learnmathematics best when they constructtheir understanding from a variety ofexperiences. Also, the teachers tended toagree that it was very important to in-troduce their students to applications ofmathematics in daily life. Moreover, theteachers believed that one of their pri-mary goals is to help their students de-velop their abilities to solve problemsand think mathematically. These threeresults reflect some of the recommendedpractices in the 1992 California Frame-work in order to empower students


9 4

mathematically.These teachers' opinions were com-

patible with the 1992 California Frame-work and the NCTM Standards, whichwould suggest that they were ready tomake changes in their teaching, pro-vided that the opinions they reportedwere actually internalized into their be-lief systems. These teachers were awarethat mathematics can be taught in waysother than simply memorizing facts,rules, and procedures, and were awarethat mathematics can be taught in waysthat will empower students mathemati-cally. Thus, the teachers should havebeen in a good position to implementthe new Math Land curriculum.

Now consider the results regardingthe Math Land curriculum itself. The re-sults also indicated that the teachersmight be going through some of theconflicts and dilemmas similar to otherteachers described in the research litera-ture (Putnam, Heaton, Prawat & Remil-lard, 1992). In terms of the quantitativedata, roughly half of the teachers didnot implement the three units they wereexpected to implement in Year 1. In ad-dition, their overall satisfaction withMath Land was mixed, and the teacherstended to disagree that Math Land is awell-balanced curriculum. These con-flicts with the new curriculum are fur-ther highlighted in the teachers' re-sponses to the open-ended' questionsabout the main strength and mainweakness of the Math Land Program. Al-though teachers cited the problem-solving activities, cooperative learning,and the manipulatives as the mainstrengths of MathLand, one of most fre-quent main weaknesses of MathLand re-

ported by the teachers was lack of com-putation practice of basic skills. Thismay be an indication that these teachersstill believe that mathematics is a subjectof rules and procedures, to be learnedthrough practice in mastering the basicskills.

According to Wood, Cobb, andYackel (1991), struggling with conflictsand dilemmas are important in thechange process for teachers. Teachersmust experience some elements of dis-equilibrium so that they can questiontheir own understandings about mathe-matics education reform and their ownteaching practices (Ball, 1996). The con-flicts these teachers have with the newcurriculum and math reform could pro-vide an opportunity for them to reflect,debate and challenge their understand-ings about mathematics reform educa-tion. This would be a good opportunityto shift staff development toward a for-mat where critical discussions couldtake place about the rationale and theo-retical perspectives underlying themathematics reform movement. Thenideally, teachers can make informed de-cisions about mathematical content andinstructional strategies when imple-menting a new curriculum.

This study looked only at one gradelevel of teachers during the implementa-tion of a new mathematics program. Theconclusions drawn from the results arelimited to this particular group, andtherefore may not be applicable to thebeliefs and attitudes of the teachers ofother grade levels in the district. Furtherresearch is needed with teachers of dif-ferent grade levels.


REFERENCES

Ball, D. L. (1996, March). Teacher learning and the mathematics reform. What we thinkwe know and what we need to learn. Phi Delta Kappan, 500-508.

Brosnan, P. (1994). An Exploration of Change in Teachers' Beliefs and Practices during Im-plementation of Mathematics Standards. Columbus: Ohio State University (EricDocument Reproduction Service No. ED 372 949).

California State Department of Education (1992). Mathematics Framework for CaliforniaPublic Schools, Kindergarten through Grade Twelve. Sacramento, CA: Author.

California State Department of Education (1994). California Basic Instructional Materialsin Mathematics Adoption Recommendations of the Curriculum Development and Sup-plemental Materials Commission to the State Board of Education, Sacramento, CA:Author

Chandler, D. G., & Brosnan P. A. (1994, Fall). Mathematics textbook changes from be-fore to after 1989. Focus on Learning Problems in Mathematics, 16(4), 1-9

Cohen, D. K., & Ball, D. L. (1990, Fall). Relations between policy and practice: A com-mentary. Educational Evaluation and Policy Analysis, 12(3), 331-338.

Cronin-Jones, L. L. (1991). Science teacher beliefs and their influence on curriculum im-plementation: Two case studies. Journal of Research in Science Teaching, 28(3), 235-250.

D'Ambrosio, B. S. (1991, February). The modern mathematics reform movement in Bra-zil and its consequences for Brazilian mathematics education. Educational Studiesin Mathematics, 22(1), 69-85.

Flanders, J. R. (1987, September). How much of the content in mathematics textbooks isnew. Arithmetic Teacher, 3(1), 18-23.

Good, T. L., Grouws, D. A., & Mason, D. A., (1990). Teachers' beliefs about small-groupinstruction in elementary school mathematics. Journal for Research in MathematicsEducation, 21(1), 2-15.

Hatfield, M. M., & Price, J. (1992, January). Promoting Local Change: Models for Im-plementing NCTM"S Curriculum and Evaluation Standards. Arithmetic Teacher,34-37

National Council of Supervisors of Mathematics. (1988). Essential Mathematics for the 21stCentury (Position Paper). Minneapolis, MN: Author

National Council of Teachers of Mathematics. (1989). Curriculum and Evaluation Stan-dards for School Mathematics. Reston, VA: Author.

National Research Council. (1989). Everybody Counts: A Report to the Nation on the Futureof Mathematics Education. Washington, DC: National Academy Press.

Price, J. (1995). President's report: Selling and buying reformIf we build it , will theycome?. Journal for Research in Mathematics Education, 26(5), 487-490.

Putnam, R. T., Heaton, A. M., Prawat, R. S., & Remillard, J. (1992, November). Teachingmathematics for understanding: Discussing case studies of four fifth-gradeteachers. The Elementary School Journal, 93(2), 213-227.


9G

Randolph, M. & Charles, L. (1996). Adopting New Programs: What about Arithmetic?.http://www.mathland.com (3 pages). City: Publisher.

Saunders, D. J. (1996, November 3). New-New Math Alert. San Francisco Chronicle, p. 11.Schmidt, W. H., McKnight, C. C., & Raizen S. A., (1996) A Splintered Vision: An Investiga-

tion of U.S. Science and Mathematics Education Executive Summary. TIMMS Report.Michigan State University.

Wood, T., Cobb, P. & Yackel, E. (1991, Fall). Change in teaching mathematics: A casestudy. American Educational Research Journal, 28(2), 587-616.

Yore, L. D. (1991). Secondary science teachers' attitudes toward and beliefs about Sci-ence reading and science textbooks. Journal of Research in Science Teaching, 28(1),55-72.


9

Ethnicity as a Factor in Teachers'Perceptions of the Mathematical

Competence of Elementary SchoolStudents

Kathlan LatimerThe purpose of this study was to evaluate the ex-tent to which teachers use ethnicity in attribut-ing mathematical competence to their students.A group of elementary teachers who were teach-ing an ethnically diverse group of students wereasked to describe a "mathematically competent"student as well as to describe the behaviors andcharacteristics of their most and least competentstudents. Although some differences were foundin teachers' perceptions of student competencebased on student ethnicity, greater differenceswere found based on teachers' perception of thelevel of student competence. The most commondescriptors of students judged as most compe-tent were (a) having a positive attitude and dis-position and (b) being studious; the most com-mon descriptors of students judged as least com-petent students were (a) having poor work hab-its and (b) exhibiting negative behavior.


Effective teaching and learning arefacilitated by the teacher's awareness ofa student's knowledge and capabilities.Although a teacher may have a generalimpression of a student's abilities andhave access to the student's standard-ized test results, it is difficult at best forthe teacher to determine exactly what astudent does know and how well he orshe can perform. One important sourceof information about student knowledgeand performance is teacher judgment.

This study looks at the factors teach-

ers use to make judgments about themathematical competence of students.The way in which teachers define-mathematical competence as well astheir expectations for mathematicsachievement are also examined.

BACKGROUND

Currently, people of color compriseapproximately 25 percent of the totalU.S. population. In California, people ofcolor represent 44 percent of the popula-tion (U.S. Dept. of Commerce, 1993),whereas 87 percent of the teaching forceis European American (National Centerfor Education Statistics, 1995b). Califor-nia's student enrollment is 5,341,025-59percent are students of color. By theyear 2000, it is anticipated that the pro-portion of students of color enrolled inCalifornia public schools will grow to 65percent (EdSource, 1995). Who will thesechildren? If current trends continue,these students will continue to be taughtby teachers whose ethnicity and/or cul-ture is different from their own.

In this study, I attempt to examine

Kathlan Latimer has more than twentyyears of experience as an educator. Cur-rently, she is a third grade teacher in Fair-field, California.


the impact, if any, of the cultural andethnic mismatches between teachers andstudents. As stated by Viadero (1996),schooling does not occur in a culturalvacuum; school is a cultural enterprise.Culture, according to Anderson (1988),is built upon a philosophical worldviewor conceptual system of patterns of be-liefs and values that are transmitted tomembers of a cultural group throughsocialization. It has been suggested thatthese values and worldviews may be ac-companied by differing cognitive stylesand patterns of communication. There-fore, differences in culture or ethnicitymay call for modifications of the peda-gogical strategies utilized by teachersand a re-examination of teachers' per-ceptions of student ability. This studybegins work in this area by examiningteachers and their cultural lenses.

Teacher judgment of a student's ca-pacity to learn, i.e. teachers' perceptionsof student competence, is the focus ofthis study. This study examines judg-ments of mathematical competence thatteachers make based on teachers' per-ceptions of their students. The use ofrace and ethnicity as a basis for teachers'differential expectations of students formathematical achievement is consid-ered. Specifically, the purpose of thisstudy is to investigate teachers' use ofethnicity or racial identification as a fac-tor in assigning mathematical compe-tence to elementary level students. Ofparticular interest is the situation inwhich the ethnicity of the student doesnot match that of the teacher.

The research questions to be investi-gated are:

1. What factors do teachers ascribe to amathematically competent student?

2. Do the factors considered vary withthe ethnicity of the student?.3. Do these factors vary with the eth-nicity of the teacher?

In this study, these factors included, butwere not limited to, social class, gender,reputation, ethnicity/culture, physicalattractiveness, verbal skills, attentive-ness, maturity, and family characteris-tics.

METHODS

Sampling. Fourteen elementaryschool teachers of ethnically diversestudents participated in this study. A to-tal of 519 students were taught by theseteachers. Most grade levels as well asmulti-age classes were represented.

Instrumentation. A questionnaire wasdeveloped which was designed to elicitinformation on teachers' conceptions ofmathematical competence. The ques-tionnaire consisted of twelve items. Tenof these items called for open-ended re-sponses; the remaining two items askedfor demographic information. Specifi-cally, each teacher who participated inthis study was asked to select, in termsof mathematical competence, the topthree and bottom three students in hisor her class. The teachers were asked todescribe the attributes and behaviors ofthese selected students.

Data analysis. Teacher responses tothis questionnaire were analyzed to un-derstand the bases upon which theteachers made assignments of compe-tence. Responses to the open-endedquestions were categorized according tosimilarity of response. These categoriesgenerated themes, which were then la-beled for further analysis.


99

TABLE ONECharacteristics of Students Selected by Teachers (n = 84 Students)

Ethnicity Most Competent Least Competent TotalEuropean-American 17 14 31

Hispanic 7 12 19

African-American 7 11 18

Asian-American 9 3 12Filipino 3 1 4

Southeast Asian 1 0 1

Eastern Indian 3 0 3

Chinese 1 0 1

American Indian 0 0 0

Other (Biracial) 2 2 4

Bilingual ClassificationEnglish-Only 32 33 65

Fluent EnglishProficiency 7 5 12English LanguageLearner 3 4 7

GenderMale 25 20 45

Female 17 22 39

RESULTS

Demographics

Consider first teacher demographics,which are present in Table One. The 14teachers in this study were predomi-nantly female and, reflecting the overalldistrict ethnicity proportions, EuropeanAmerican. Of the 84 students selectedby the teachers (42 of whom were iden-tified as "most competent" and 42 ofwhom were identified as "least compe-tent"), the largest group was EuropeanAmerican, and most of these studentswere male. Interestingly, althoughteachers selected students from a varietyof ethnic backgrounds, more EuropeanAmericans students were selected as

both "most competent" and "least com-petent" even though the classrooms ofthese teachers had a majority of studentsof color.

Actions and Observable Behaviors ofMost and Least Competent Students

Teachers were asked to make twolists one of the observable actions andbehaviors of the mathematically mostcompetent students in their classes, andone of the observable actions and behav-iors of the least competent students intheir classes. More specifically, for themost competent students, teachers wereasked "What do these students do (ac-tions, observable behaviors; e.g. eagerlyshare ideas in class) that distinguishes


10097

them from the rest of the class?" Teach-ers were again asked this question forthe least competent students.

Teachers' descriptions of the actions andbehaviors of mathematically "most compe-tent" students. An analysis of teachers'responses is presented in Table Two.The descriptions of mathematicallycompetent students fell into six maincategories. A student's positive atti-tude/disposition was the characteristiclisted most often for mathematically"competent" students. In particular,

teachers listed traits such as a student'seagerness to learn, interest in work andpersistence. In order of decreasing fre-quency of response, the other five char-acteristics of mathematically competentstudents identified by the teachers were:communication of mathematical think-ing, facility with procedures and skills,understanding of mathematical con-cepts, reasoning ability and problemsolving ability.

When asked to list observable ac-tions and behaviors of mathematically

TABLE TWOTeachers' Descriptions of the Actions and Observable Behaviors

of Most Competent and Least Competent Studentsby Student Ethnicity (n = 84 Students)

Most Competent StudentsEuropeanAmerican

n=17

African-American

n=7Hispanic

n=7

AsianAmerican

n=9Biracial

n=2Total

42Attitude 17 5 9 10 2 43Communica-tion 6 1 2 4 1 14Skills & Pro-cedures 5 3 1 1 2 12Concepts 5 4 0 1 1 11Reasoning 2 2 2 3 1 10ProblemSolving 4 1 2. 2 9

Least Competent StudentsEuropeanAmerican

n=14

African-American

n=11Hispanic

n=12

AsianAmerican

n=3Biracial

n=2Total

42Work Habits 13 10 5 3 0 31Behavior 7 8 4 0 0 19Attention 9 2 3 2 0 16Attitude 4 5 5 1 1 16Concepts &Skills 2 4 4 0 4 14


1 01

competent students, some variations ofresponses became apparent, but overallfactors used were fairly evenly appliedacross groups. In examining whathighly competent students do, re-sponses relating to attitude appearedmost often across all ethnic groups. Thiswas reported by a large margin for allgroups except for African Americanstudents. For this group; responsesrelating to concepts (e.g., sense ofnumber, grasps new ideas) rivaled thoserelating to attitude. Other categories ofresponse that emerged for this groupwere attitude or disposition,communication of mathematicalthinking, reasoning, conceptualunderstanding, problem solving ability,and knowledge of skills and procedures.

Teachers' descriptions of the actions andbehaviors mathematically "least competent"students. Teachers were also asked to listthe observable actions and behaviors ofmathematically "least competent" stu-dents. When looking at students identi-fied by teachers as least competent, notonly did different categories emerge,but also an emphasis on deficits ratherthan on actual performance was noted.Responses relating more to work habitsand student behavior, particularly nega-tive attention-seeking behavior pre-dominated. Poor work habits were mostoften mentioned, but even more so, interms of frequency, for students of color.Other factors ascribed to the least com-petent students were negative attitude,lack of attention or focus, inappropriatebehavior, and lack of knowledge of con-

cepts and skills.

Characteristics that Set Selected Stu-dents Apart from Other Students

Teachers were asked to make twolists distinguishing characteristics ofthe mathematically most competentstudents, and one of the distinguishingcharacteristics and least competent stu-dents in their classes. More specifically,for the most competent students, teach-ers were asked "Are there characteristics(personality traits, etc.) which set themapart from the other students. " Teach-ers were again asked this question forthe least competent students.

Results are presented in Table Three.In terms of the most competent stu-dents, studiousness emerged as the topcharacteristic for all ethnic groups ofstudents except Hispanic/Latino stu-dents; for this group quietness waslisted more than any other response. Interms of the least competent students,different characteristics emerged; forthese students the most frequent cate-gory listed was inappropriate attitude.However, behavior was identified mostfrequently for the least competent Euro-pean Americans and introversion wasidentified most frequently for Hispanics.It is of further interest that yet again en-tirely different categories emerged forstudents seen as least competent ratherthan the negation of those listed forhighly competent students, thus ques-tioning . if and how the perception ofability impacts the mathematical experi-ences of these students.


1 t =4

99

TABLE THREETeachers' Descriptions of Characteristics that Set Selected Students

apart from Other Students (n= 84 Students)

Most Competent Students

EuropeanAmerican

(n=17)

African-American

(n=7)Hispanic

(n=12)

AsianAmeri-

can(n=9)

Biracial(n=2)

Total42

Studiousness 7 6 2 7 1 23Popularity 6 2 0 2 1 11Confidence 5 1 2 1 0 9Quietness 1 2 4 1 0 8Curiosity 4 1 1 0 1 7Sense ofHumor 2 1 0 2 0 5Leadership 1 1 1 1 0 4

Least Competent StudentsEuropeanAmerican

(n=14)

African-American

(n=11)Hispanic(n=12)

AsianAmeri-

can (n=3)Biracial

(n=2)Total

42Attitude 5 5 4 2 3 19Behavior 7 3 2 2 0 14Work Habits 4 3 3 1 1 12Introversion 0 4 5 0 1 10Attentiveness 3 1 3 0 1. 8Attendance 0 3 2 0 0 5Home Support 0 3 1 0 0 4Maturity 2 0 0 0 1 3

Differences in Perceptions Based UponTeacher Ethnicity

Descriptive statistics for this analy-sis are presented in Table Four. Firstconsider European American teachers'perceptions. In terms of teacher ethnic-ity, European American teachers ap-plied factors evenly except in the case ofpositive disposition or attitude whenjudging the most competent AfricanAmerican students. For African Ameri-can students, positive disposition and

attitude was noted by European Ameri-can teachers equally as often as knowl-edge of mathematical concepts. For allother groups of students, positive dis-position /attitude was clearly listed byEuropean American teachers most often.When considering the least competentstudents, European American teacherslisted factors that were fairly representa-tive across all groups. Work habits werementioned most often by EuropeanAmerican teachers for all student


103

TABLE FOUREthnicity of Students Judged Most Competent and Least Competent

by Ethnicity of the Teacher (n =14 Teachers)

Most Competent StudentsTeacher Ethnicity

Student Ethnicity

EuropeanAmerica)

(n=10)

African-American

(n=2)Hispanic

(n=1)

AsianAmerican

(n=1) Total =42European-Amer 12 3 0 2 17African-Amer 5 1 0 1 7

Hispanic 4 0 3 0 7

Asian-Amer 9 0 0 0 9

Biracial 0 2 0 0 2Least Competent Students

Student Ethnicity

EuropeanAmerica(n=10)

African-American

(n=2)Hispanic

(n=1)

AsianAmerican

(n=1) TotalEuropean-Amer 10 2 0 2 14

African-Amer 9 1 0 1 11

Hispanic 6 3 3 0 12

Asian-Amer 3 0 0 0 3

Biracial 2 0 0 0 2

ethnic groups except Hispanic studentsfor whom attitude was chosen as oftenand for biracial students for whomknowledge of concepts and skills was aclear choice.

There were too few teachers of colorto make general statements about spe-cific teacher ethnic groups. Thereforethe responses of all teachers of colorwere combined for analysis. No teacherof color selected Asian American stu-dents; all teachers of color selectedEuropean American students when suchstudents were a part of the class. Similarto European American teachers, teach-ers of color also selected more studentsof color as least competent than most

competent. Attitude and dispositionfigured prominently in their responsesfor most competent students; factorsapplied to least competent studentswere scattered across categories.

DISCUSSION

Based on the top ranked responses, aprofile of teachers' perceptions of thecharacteristics of a mathematically com-petent student emerged from the data.Such students are curious, excited, andmotivated to learn mathematics. Theirattitude towards mathematics is positiveand they exude confidence. Accordingto these teachers, these students havegood concepts of number and a highly


developed, as developmentally appro-priate, number sense. They are problemsolvers; they are able to reason logicallyand use critical thinking skills. They arepersistent, take risks, and learn fromtheir mistakes. Mathematically compe-tent students understand what mathe-matics is about as a discipline and canmake connections within the .disciplineand to the world around them. For themajority of teachers surveyed these at-tributes set the stage for successfulmathematical achievement.

The attributes of mathematical com-petence described by these teacher par-allel themes detailed in the goals forstudents in the assessment of mathe-matical power in the NCTM Curriculumand Evaluation Standards for SchoolMathematics (1989). In particular, theStandards describe the importance oflearning to value mathematics, becom-ing confident in one's own ability, be-coming a mathematical problem solver,learning to communicate mathemati-cally, and learning to reason mathemati-cally. Additionally, the description ofmathematical competence based onteacher responses is strikingly similar tothe NCTM notion of mathematicalpower:

This term denotes an individual's abili-ties to explore, conjecture, and reasonlogically, as well as the ability to use avariety of mathematical methods effet-tively to solve non-routine problems.This notion is based on the recognitionof mathematics as more than a collec-tion of concepts and skills to be mas-tered; it includes methods of investigat-ing and reasoning, means of communi-cation, and notions of context. In addi-tion, for each individual, mathematical

power involves the development of per-sonal self-confidence. (p. 5)

In examining descriptions of highlycompetent students, factors relating toattitude appeared most often across allethnic groups. Although most factorsthat teachers reported related closely tothose detailed in the NCTM's descrip-tion of mathematical power, a fewteacher comments related to other fac-tors such as attentiveness and studyskills. When looking at students judgedas least competent, by comparison, re-sponses that related more to student be-havior, particularly negative attention-seeking behavior, and work habits pre-dominated. This suggests that the man-agement of student behavior may be-come the focus for those students whoare not seen as competent. This maylead to a downward spiralstudents donot attend to mathematics instruction orexperiences, exhibit acting out or apa-thetic behavior, and miss out on moremathematics. And so the cycle contin-ues, enabling the downward spiraling,in terms of mathematics achievement, ofthese students.

In terms of characteristics of themost competent students, being hard-working or studious were factors usedmost often to describe these studentsacross all groups, whereas negative orinappropriate attitude was seen most of-ten for the least competent students.Thus entirely different categories againemerged for students seen as least com-petent, thus raising the question of howteachers' perceptions of ability impactthe mathematical experiences of thesestudents.

Even in diverse classrooms, Euro-


1 0 5"

pean American students were selectedmore often in both the "most" and"least" competent student categoriesrelative to their proportion of classroomenrollment. Bilingual students werejudged most competent at about thesame rate as they were judged leastcompetent. Yet on the whole, factorswere applied evenly across all ethnicgroups; the major differences in re-sponses were based on perceived levelof competence.

Based on this study, the ethnicity ofthe student does not appear to be a pri-mary factor in assigning competence.Likewise, the ethnicity of the teacherdoes not appear to be a primary factorin assigning competence. Yet teachers'comments on culture and ethnicity werequite interesting. Many teachers em-braced the importance of culture, buteschewed the idea of differential treat-ment of individual students based onculture. However, according to Ander-son (1988):

Whereas it was once fashionable andsometimes academically rewarding todeny the existence of cultural assets andvariation among non-white populations,social scientists and researchers nowrecognize that such traditional ap-proaches have become anachronistic ...

A different set of understandings aboutthe way diverse populations communi-cate, behave, and think needs to be de-veloped by educators. (p. 8)

Emphasis on culture or ethnicity canbe construed negatively, as indicated byKnapp and Turnbull (1990):

First, stereotypic ideas about the capa-bilities of a child who is poor or belongsto an ethnic minority will detract from

an accurate assessment of the child'sreal educational problems and potential.By focusing on family deficiencies, theconventional wisdom misses thestrength of the cultures from whichmany disadvantaged students come.(P. 5)

The influence of culture and ethnic-ity can be positive, but these advantagesare overlooked. This may be due to thehistorically held, but currently unten-able, goal of a colorblind society. Al-though it may be prudent to be objectivein terms of assessing students' ability,overlooking the impact of culture cre-ates a missed opportunity. This missedopportunity results when teachers fail tolook to culture and ethnicity as sourcesof valuable, pertinent information.Given that the culture and ethnicity ofteachers and students will match less of-ten as population demographics change,this is a critical area. As noted by Ander-son (1988):

At the superficial level, cross-racial,cross ethnic teaching, learning and/orservice delivery occurs when the per-sons interacting are of different racial orethnic identities. When one adds to thisequation the differences in degrees ofacculturation and type of cogni-tive/learning style, the examination andexplanation of these differences be-comes more complicated and the ur-gency to identify the critical dimensionsassociated with them more pronounced.(p.8)

Thus the mismatching may be evenbroader than demographics alonewould suggest.

In conclusion, more differences wereseen based on perceived level of abilityrather than on any other factor. Much of


106103

this may be attributable, however, to theinability on the part of teachers to allowfor and/or acknowledge the impact andimportance of culture and its role intheir own assignment of competence.Yet this, the consideration of culture, isthe first step in moving from mere ack-

nowledgement to the affirmation of cul-ture and ethnicity as well as a step to-wards taking advantage of what up tonow has been a missed opportunity, theopportunity to embrace culture as an as-set rather than a liability.

REFERENCES

Anderson, J. A. (1988). Cognitive styles and multicultural populations. Journal of TeacherEducation , 39, 2-9.

Ed Source Report (1995, December). Who are California's students? Menlo Park, CA: Ed-Source, Inc.

Knapp, M. S. & Turnbull, B.J. (1990). Better schooling for the children of poverty: Alternativesto conventional wisdom. Washington, D.C.: U.S. Department of Education.

National Center for Education Statistics (1995). Digest of Education Statistics. Washington,D.C: U.S. Department of Education.

National Council of Teachers of Mathematics (1989). Curriculum and Evaluation Standardsfor School Mathematics. Reston, VA: Author.

U.S. Department of Commerce (1993). Statistical Abstract of the United States. Washington,DC: Author.

Viadero, D. (1996, August). Culture clash. Teacher Magazine, 7, 14-17.


107

San Francisco State University College of Education ReviewManuscript Submission and Review Process

The COE Review is published annually by the Research and Development Center of theCollege of Education and San Francisco State University. It is designed to stimulate thinkingabout a variety of educational issues and to foster the creation and exchange of ideas and researchfindings in a manner that will expand knowledge about the purposes, conditions and effects ofschooling. The focus of the journal is on best practice, supported by research, theory, orexperience. Manuscripts on a variety of tropics are welcome, as long as they are related in someway to educational issues and concerns:

conceptual pieces;empirical and other research investigations;theoretical or philosophical arguments;innovative teaching practices;student work;reviews of literature;book reviews.

Manuscript SubmissionAll manuscripts submitted for publication must conform to the style of the Publication Manual

of the American Psychological Association (3rd ed.). Manuscripts should be typed on 8.5 x 11 inchpaper, upper and lowercase, double-spaced, with 1.5 inch margins on all sides. Subheads shouldbe at reasonable intervals to break the monotony of lengthy texts.

Four copies of each manuscript (one original and three copies) must be submitted. Allmanuscripts are sent (names removed) to three outside reviewers. As such, the complete title ofthe manuscript and author(s) name(s) should be listed on a separate sheet of paper to assureanonymity in the review process, and the first text of the article should have the complete title ofthe manuscript, but no list of author(s). In addition, please send a 3.5" disk containing yourmanuscript (including any tables and graphics); formatted in Microsoft Word for the Macintosh.Once reviewed, manuscripts are accepted, accepted pending revisions, or rejected.

All materials should be addressed to:

Dr. Jack Fraenkel, EditorCollege of Education ReviewResearch & Development CenterSan Francisco State UniversitySan Francisco, CA 94132

Authors are responsible for the quality of presentation on all aspects of their manuscripts.This includes correct spelling and punctuation, accuracy of all quotations, complete and accuratereference information, and legible appearance. Authors are also responsible for preparing theirmanuscripts so as to conceal their identities during the review process. Manuscripts that do notconform to the guidelines specified here will be returned unread.


i 0' 8

105

U.S. Department of EducationOffice of Educational Research and Improvement (OER1)

National Library of Education (NLE)Educational Resources Information Center (ERIC)

REPRODUCTION RELEASE(Specific Document)

I. DOCUMENT IDENTIFICATION:

s9J&Y14-Co

ERIC

Title:

College of Education Review, Vol II

Author(s): Jack Fraenkel, Editor; Carol Langbort & Deborah Curtis, Guest Editors

Corporate Source:San Franicsco State University

Publication Date:Spring 2000

II. REPRODUCTION RELEASE:

In order to disseminate as widely as possible timely and significant materials of interest to the educational community, documents announcedin the monthly abstract journal of the ERIC system, Resources in Education (RIE), are usually made available to users in microfiche, reproducedpaper copy, and electronic media, and sold through the ERIC Document Reproduction Service (EDRS). Credit is given to the source of eachdocument, and, if reproduction release is granted, one of the following notices is affixed to the document.

If permission is granted to reproduce and disseminate the identified document, please CHECK ONE of the following three options and signat the bottom of the page.

The sample sticker shown below will beaffixed to all Level 1 documents

1

PERMISSION TO REPRODUCE ANDDISSEMINATE THIS MATERIAL HAS

BEEN GRANTED BY


Level 1

Check here for Level 1 release, permittingreproduction and dissemination in microfiche or otherERIC archival media (e.g., electronic) and paper copy.

Signhere;please

The sample sticker shown below will beaffixed to all Level 2A documents

PERMISSION TO REPRODUCE ANDDISSEMINATE THIS MATERIAL IN

MICROFICHE, AND IN ELECTRONIC MEDIAFOR ERIC COLLECTION SUBSCRIBERS ONLY,

HAS BEEN GRANTED BY

2A


Level 2A

ElCheck here for Level 2A release, permitting reproductionand dissemination In microfiche and in electronic media

for ERIC archival collection subscribers only

The sample sticker shown below will beaffixed to all Level 2B documents

PERMISSION TO REPRODUCE ANDDISSEMINATE THIS MATERIAL IN

MICROFICHE ONLY HAS BEEN GRANTED BY

2B


Level 2B

ElCheck here for Level 2B release, permitting

reproduction and dissemination in microfiche only

Documents will be processed as indicated provided reproduction quality permits.If permission to reproduce is granted, but no box Is checked, documents will be processed at Level 1.

I hereby grant to the Educational Resources Information Center (ERIC) nonexclusive permission to reproduce and disseminate thisdocument as indicated above. Reproduction from the ERIC microfiche or electronic media by persons other than ERIC employees andits system contractors requires permission from the copyright holder. Exception is made for non-profit reproduction by libraries andother service agencies to satisfy information needs of educators in response to discrete inquiries.

Signature:

1114041Ocrialorrildress:

ge of Education/1600 Holloway AveS an rancisco, CA 9413

Printed Name/Position/Title:

7.6. C.14 FP' 4 4, IN 1..elTel9phqge: - FAX:

41°5-21R-2299 4125-33R-0672

t-maa ACIateSS: uate:

III. DOCUMENT AVAILABILITY INFORMATION (FROM NON-ERIC SOURCE):

If permission to reproduce is not granted to ERIC, or, if you wish ERIC to cite the availability of the document from another source, pleaseprovide the following information regarding the availability of the document. (ERIC will not announce a document unless it is publicly available,and a dependable source can be specified. Contributors should also be aware that ERIC selection criteria are significantly more stringent fordocuments that cannot be made available through EDRS.)

Publisher/Distributor:

N/A

Address:

Price:

IV. REFERRAL OF ERIC TO COPYRIGHT/REPRODUCTION RIGHTS HOLDER:

If the right to grant this reproduction release is held by someone other than the addressee, please provide the appropriate name and address:

Name:N/A

Address:

V. WHERE TO SEND THIS FORM:

Send this form to the following ERIC Clearinghouse:

N/A

However, if solicited by the ERIC Facility, or if making an unsolicited contribution to ERIC, return this form (and the document being contributed)to:

ERIC Processing and Reference Facility4483-A Forbes BoulevardLanham, Maryland 20706

Telephone: 301-552-4200Toll Free: 800-799-3742

FAX: 301-552-4700e-mail: [email protected]

WWW: http://ericfac.piccard.csc.comEFF-088 (Rev. 2/2000)