Glenn StevensAl Cuoco Wayne HarveyRyota Matsuura Steve RosenbergSarah Sword The Impact of Immersion in Mathematics on Teachers Focus on Mathematics

Abstract Over 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? Focus on Mathematics

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? Focus on Mathematics

Mathematics Knowledge Mathematics Knowledge for Teaching Mathematical Thinking Focus on Mathematics

“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 Focus on Mathematics

Elementary teachers should know basic principles and concepts in: number sense and numeration patterns and functions geometry and measurement data analysis Focus on Mathematics

Middle and H.S. teachers should know: algebra geometry trigonometry discrete math intro to calculus history of math use of technology Focus on Mathematics

H.S. teachers should additionally know: abstract algebra number theory calculus probability & statistics trans. Geometry applied math Focus on Mathematics

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 Focus on Mathematics

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 Focus on 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 Focus on Mathematics

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 Focus on Mathematics

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 Focus on Mathematics

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. Focus on Mathematics

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. Focus on Mathematics

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. Focus on Mathematics

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. Focus on Mathematics

Lawrence Public Schools Focus on Mathematics

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 Focus on Mathematics

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?…)‏ Focus on Mathematics

Student Research Projects Total Number of School, Projects, and Students Participating in Math Fairs, ; early 2007 results. Focus on Mathematics

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 Focus on Mathematics

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 Focus on Mathematics

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. Focus on Mathematics

Specific Examples From PCMI 2004 From a Study Group in Chelsea, MA Another example Focus on Mathematics

Institute for Advanced Study Park 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 Focus on Mathematics

Today’s Example PCMI a combinatorics course Overarching problem: “simplex lock” A few guidelines for working on the problem Focus on Mathematics

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

What participants did Focus on Mathematics

What participants did Focus on Mathematics

“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. Focus on Mathematics

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? Focus on Mathematics

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. Focus on Mathematics

This is Bowen and Ben! Focus on Mathematics

Bowen and Ben take the Problems to PCMI Focus on Mathematics

This is Al! Focus on Mathematics

What Al Might Say It'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. Focus on Mathematics

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.) Focus on Mathematics

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? Focus on Mathematics

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