I recently posted a reply to Clara Einfeldt's comments on the need to teach college math courses in a manner consistent with the Standards so that teachers become constructivist learners as well as constructivist teachers.
I was very pleased that I received many letters through personal e-mail that indicated that more of a dialogue on this matter would be helpful. Rather than try to answer them all individually, which I truly cannot find the time to do (I am teaching the July summer session courses which are a bit time-intensive), I would like to thank everyone who sent me e-mail and acknowledge the questions in the letters by commenting a bit further.
I think of myself as a teacher, having been in the classroom for more than 30 years--both as an elementary teacher and later as a high school math teacher. My degree is in mathematics, very traditionally taught, and I worked in engineering before becoming a teacher. Now I work with teachers in our graduate program in Math Leadership and visit teachers in their classrooms and also teach Math for Teachers courses at the college. I do not have a doctorate. I still am *just a teacher*. That has been, and still is, my raison d'etre.
The course I talked about is not a course of Math for Teachers or how to teach math to children. My students still have to take that course. Topics in Math is a course for my students to learn math for themselves- as adults. While there is some overlap in the goals of the Math for Teachers and the Topics in Math, the emphasis is quite different. We don't talk about teaching issues. We talk only about math. When I said I emphasized definition, algorithm and all those traditional sounding mathy ideas, I mean that I try to give them the experience of what it is to define a mathematical term with just the right amount of information--not too much, not too little. I try to give them the chance to build an algorithm for themselves so they can see what the role of algorithm is in mathematics. In one of my classes this semester the students discussed the differences between an algorithm and a formula. The discussion came from them. That's important I feel. I didn't have written in my course description two months before I met my students that on day such and such we will discuss the difference between an algorithm and a formula. I feel really strongly that timing is crucial. The time to discuss an idea is when it comes up from the students. I guess this comes from my elementary experience. They also discover patterns and express them as functions--Tower of Hanoi type of experiences.
This course is 3 and 1/2 hours per session twice a week for the month of July. The sessions are mostly cooperative learning. Some students are nervous at first working with any one else. They don't want to appear stupid and are easily discouraged if someone else is perceived as "catching on" faster than they are. So there is a flexibility allowed for going in and out of group work at will--as the need dictates. As trust builds, that seems to change. There is sharing at the end of each session as we celebrate what we have learned and that is different for different people. Everyone doesn't learn the same thing on the same night. But we hear each other and stimulate each other to go on in different directions. One person's findings encourage another person to go on in a similar direction. I guess this makes me nervous too, as I mentioned in my earlier post. The idea of everyone learning the same thing in the same way at the same time is so deeply embedded in me that I somehow have to deal with the guilts of not providing that kind of experience for them---EVEN THOUGH I REALLY BELIEVE that math is not learned in that factory oriented manner. It's hard to shake those demons from the past.
The first night of this course is the hardest for me. I put out about 30 activity cards that are related in some way to triangles. They use materials, problems, geometry, number...Sierpinski is in there too. Geoboards to see what happens to the area when you double the height of a triangle, the base, both. Problems like how can you divide a triangle in half so that each half has the same area. Geo-sticks to investigate the relationships of the lengths of the sides of triangles, piles of triangles to categorize as they see fit, triangles made of foamcore to try to find the point at which they will balance on the point of a pencil and why, distance across the river problems, heights of trees, patterns of triangular numbers, Pascal's triangle. triangles to tessellate, How many degrees in the angles--in the exterior angles...I watch as the students select activities and begin to engage with each other. Later as a group they define a triangle. They bring up many old theorems from hs geometry that they want to check out. From this opening experience I learn alot about each student (they also write a personal math biography) as I watch and interact with them. So far, students have been able to find something to get interested in. It starts the ball rolling. Lots of words come up, lots of questions about their understandings of this common shape...Ican do alot of one on one interactions and teaching.
My goal in this course is to give them a math learning experience that empowers them as adults to learn math, to continue being interested in learning more math and dissipating some of the phobic behavior associated with math, to see the need for math literacy in the healthy functioning of our society. I don't want them to need handholding for learning math in the future.
Lucille L. Peterson Math Leadership Program Bank Street Graduate School of Education Tel: 212-875-4665 610 West 112th Street Fax: 212-875-4753 New York, NY 10025 E-mail: firstname.lastname@example.org