glenn stevensal cuoco wayne harveyryota matsuura steve rosenbergsarah sword the impact of immersion...

Glenn Stevens Al Cuoco

Wayne Harvey Ryota Matsuura

Steve Rosenberg Sarah Sword

The Impact of Immersion in Mathematics on

Teachers

Focus on Mathematics

AbstractOver almost 20 years we have developed and refined an “immersion” approach to

professional development through our collective work in programs like PROMYS for Teachers, Focus on Mathematics, and the Park City Mathematics Institute. The approach is based on our belief that deep personal experience of doing mathematics in the spirit of exploration is a necessary prerequisite for developing what the National Academies have called a “mathematical disposition.” Our presentation will describe a few of the experiences we have shared and will attempt to outline the ideas that have grown out of these experiences. The afternoon workshop will offer participants an opportunity to personally experience, albeit briefly, a simple example of a “mathematical immersion.”

Key questions include: • What is the impact on teachers of doing mathematics in “mathematical” ways? • What understanding of mathematics do we hope teachers will gain? • Can/How can our particular approach to immersion be generalized and used

successfully in other settings? • How does this approach affect the involvement of mathematicians in the work of

mathematics education?


The impact of mathematics immersion on secondary teachers

What is our focus and how does this contribute to a meaning for KMT?

What are the desired outcomes?

Where is the impact? What evidence do we have for impact, or lack thereof?

What critical questions are raised?


Mathematics Knowledge

Mathematics Knowledge for Teaching

Mathematical Thinking


“Knowledge of Mathematics for Teaching”

• Knowing (?) mathematics content

• Knowing which concepts are easy or difficult to learn and why

• Knowing ways of representing concepts so that others can understand them

• Knowing how to connect ideas to deepen them

• Recognizing what students might be thinking or understanding

-------------------------------------------------

But the perspective that is too easy to miss, and might be the most critical, is:

– Experience thinking (and struggling) as a mathematician does


Elementary teachers should know basic principles and

concepts in:

• number sense and numeration• patterns and functions• geometry and measurement• data analysis


Middle and H.S. teachers should know:

• algebra• geometry• trigonometry• discrete math• intro to calculus• history of math• use of technology


H.S. teachers should additionally know:

• abstract algebra

• number theory

• calculus

• probability & statistics

• trans. Geometry

• applied math


All mathematics teachers must learn:

• to experiment (wander with ingenuity)

• to use abstraction• to develop and use theories• to solve problems• to struggle and not solve problems• technical fluency


All mathematics teachers must recognize the following qualities

of mathematics:• utility of mathematics• intrinsic beauty• historical value

All mathematics teachers must develop a mathematical disposition. Focus on

Mathematics

A Taxonomy of Mathematics for Teaching

Expert mathematics teachers…Know mathematics as a scholar (the “facts”)

They have a solid grounding in classical mathematics, including:

• Its major results• Its history of ideas• Its connections to pre-

college mathematics


Know mathematics as an educator (the “epistemology”)

They understand the habits of mind that underlie major branches of mathematics and how they develop in learners, including:

• Algebra and arithmetic• Geometry• Analysis


Know mathematics as a mathematician (the “experience”)

They have experienced the doing of mathematics—they know what it’s like to:

• Grapple with problems• Build abstractions• Develop theories• Become completely absorbed

in mathematical activity for a sustained period of time


Know mathematics as a teacher (the “craft”)

They are experts in the uses of mathematics that are specific to the profession of teaching, including:

• The ability “to think deeply about simple things” (Arnold Ross)

• The craft of task design• The ability to see underlying

themes and connections• The “mining” of student ideas


I have a whole new view of mathematics. I always had a philosophy of teaching; now I realize that it is important to have a philosophy of mathematics. Before I had a static view. Now I understand it: it is dynamic; there is an art and science to it; it is more than a body of knowledge.


The first weeks of the program, I could connect to things I knew. Even if I was frustrated one day, the next day I’d have an epiphany – there were lots of ups and downs. Understanding math concepts was not enough. You had to look at things in different ways. It’s not necessarily intuitive. I learned a lot about my own patience. Every time I felt frustrated, I realized something that I wouldn’t have realized without being frustrated.


We would look at a simple question, and start to change the math problem to make it more complex to go more in depth. I’m much more aware of looking at all the patterns and how things are interconnected. The more I work with the mathematician, the more I see that I’ve missed so many patterns in math for many years, and now I’m sharing these with my students, and the patterns pop out at me more and more. This is a direct result of my experience with FoM.


I’ve learned to not underestimate students, expose them to new math ideas even though they may not understand the whole concept, and next time they hear about the math concept or see it they will have a better understanding, making new concepts become familiar.


Lawrence Public Schools District

2004-2005 Race/Ethnicity White 9.1 % African-American 2.4 % Hispanic 85.5 % Asian 3.0 % Native American 0.1 % Selected Population Enrollment Limited English Proficiency 19.7 % Low-income 84.6 % Special Education 16.7 % Total Count 12,272


High Schools

• 54 out of 57 (95%) of high school mathematics teachers across the districts participated in one or more FoM activities

• Average hours per teacher per year: 34

• Within Schools, this average varied from 25 to 46 hours


Middle Schools

• 83% participated in one or more FoM activities

• Average hours per teacher per year: 15

(about half the high school participation)

• Within Schools, this average varied from 4 to 24 hours (hmmm?…)


10th Grade Math MCAS Scores 2002-2006

48%

18%

39%

50%

44%

74%

34%

73%77%

67%

26%

16%

34%

27%23%

0%

25%

50%

75%

100%

1 2 4 5 State 10th

Grade

High School

Percent Scoring Proficient or

Above

2002

2006

Difference


Rates of Teacher Participation in FoM at >

30 Hours/ Year

21.4%

62.5%

33.3%

66.7%

0.0%

10.0%

20.0%

30.0%

40.0%

50.0%

60.0%

70.0%

80.0%

1 2 4 5

High School

Percent Participating


Student Research ProjectsTotal Number of School, Projects, and Students

Participating in Math Fairs, 2004-06; early 2007 results.

Number of School's

Participating

Total Projects

Total Students

2004 6 301 584 2005 13 592 996 2006 16 1252 2180

2007 19 1536+ 2545+


What do you think teachers you’ve worked with have gained from their participation in

FoM? (n=13)

1012Development of math community among teachers

1111Greater comfort/willingness to ask questions and not know the answers

1012Encouraged teachers to explore doing mathematics

0013Increased teachers’ interest in and excitement about mathematics

2011Encouraged teachers to assume more responsibility for their learning

3010Influenced teachers’ work in the classroom

0013Deepened teachers’ understanding of mathematics

Don’t Know

NoYes


If only we had more resources and unlimited access…

• Do mathematics immersion experiences have enduring effects?

• Is there any resulting evidence of impact? What kind? On who? On what systems?

• What are the critical elements in creating the experience needed to be most effective?– How is time a factor?– What are prerequisites?– What conditions are needed for success?Focus on

Mathematics

Teachers and schools are constrained by…

• District curricula they must cover

• Pacing guides

• Short teaching periods

• Disciplinary problems• Students’ English language proficiency• Preparation for State tests• Time available outside the school day• Teacher turnover rates• Student mobility


• It’s the combination of classroom experience and working with math outside of classroom, that’s what makes it powerful.

• The key you are hearing is deepening of knowledge—we go so much deeper—and you can kind of see the structure underneath it all—things show up again and again. It’s all connected—geometry, algebra, number theory—we always come back to it, these same patterns.

• The thing that I liked a lot is, they made us feel like mathematicians. I know before, I felt like just a math

teacher, and now I’m a mathematician.


Specific Examples

• From PCMI 2004

• From a Study Group in Chelsea, MA

• Another example


Institute for Advanced StudyPark City Mathematics Institute

• Problem based courses – Less than 5 minutes of introduction– Discussions were participant-centered

• Diverse group of teachers - grades 5-12 (mathematicians and educators, too)

• Teachers worked for two hours each day for three weeks on mathematics


Today’s Example

• PCMI 2004 - a combinatorics course

• Overarching problem: “simplex lock”

• A few guidelines for working on the problem



This is the first problem that participants saw on the first day of PCMI 2004.

What participants did


“Trains”

At PCMI 2004, participants worked on the simplex lock problem for an hour or so, and then dove into a set of problem sets.


How do these Problem Sets Grow out of FoM?

• A group of FoM teachers, mathematicians, and in-betweens meet six times over a school year

• We choose an overarching problem:– 2004: the Simplex Lock problem– 2006: a “proof” of the Fundamental Theorem of

Algebra – 2007: a “proof” of the Prime Number Theorem

• What mathematics will participants need to solve that problem?


Building “the Soup”

• We discuss mathematics.

• We collect problems.

• We create a rough organization.

• We try them out with a group of teachers at an all-day seminar at EDC.


This is Bowen and Ben!


Bowen and Ben take the Problems to PCMI


This is Al!


What Al Might SayIt's sometimes hard to grasp or even uncomfortable for an expert

in a field to talk with someone who is in the middle of coming to terms with the ideas in that field. Think, for example, of that genre of education research articles that presents verbatim transcripts of interviews with a student learning about some topic. The student says things that are mix of solid insights, half-baked ideas, and misconceptions. There are missing connections and needless assumptions. It may look as if the student has not “understood” what was supposedly taught, but what we see is the very act of coming to understand.

Or think about your own struggles with ideas. I often look back on my own notes from a course and wonder what I could have been thinking.


What Al might say

But it's important to be clear that the sense that participants make out the problems isn't necessarily the same as the sense you make out of it. And that doesn't mean that they haven't been successful. It's a natural part of coming to understand. (At any particular instant in time, any one of us is in the same boat. Lots of things nailed down and lots of things not nailed down, and we sometimes don’t know the difference.)


In Chelsea Middle School (Glenn and Ryota’s Study group)

• Trains, number 10 • People had a “sense” that you could build length 5

trains out of length 3 and length 4 trains - some sense of “Fibonacci-ness” - but there were some unresolved issues:– How do you know you can build every train this way?

– How do you know there aren’t duplicates?


QuickTimeª and aTIFF (Uncompressed) decompressor

are needed to see this picture.


For more details:http://www2.edc.org/cme/showcase.html

www.focusonmath.orghttp://mathforum.org/pcmi/

www.promys.org

[email protected]@edc.org

[email protected]@math.bu.edu [email protected]


http://www2.edc.org/cme/showcase.html

http://www.focusonmath.org/



http://mathforum.org/pcmi/




http://www.promys.org/



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