[Note to readers on-line: the process of transferring this out of an early version of Word introduced some bizarre bits of quasi-punctuation that I can't seem to eliminate. Sorry!]

Keeping my claim to asynchronicity on firm ground, here's another news hit.It's another preview of an AWM Education Column, with a bit of overlap with the last newsletter, but a fairly small bit, so it rates independent status.

This column rarely features guest writers. This time I am making up for that: I have a whole bunch of them, some anonymous, some pseudonymous, and all under the age of sixteen.

A word of explanation might be in order. A couple of weeks ago, at the recommendation of a colleague, I began reading Jo Boaler¹s Experiencing School Mathematics. I found it both exciting and informative. It is a report on a three-year longitudinal study of students at a pair of middle schools. The middle schools are English ones, but as Alan Schoenfeld points out in his introduction, they could just as well be anywhere in the US. The schools were carefully chosen to match closely in terms of student social and economic status and to diverge maximally in terms of pedagogic stance. At Amber Hill (a pseudonym, of course), conscientious and hard-working teachers adhere to a highly traditional format: students are in eight different tracks, but for each one the class proceeds the same way, with half a period to present a new topic or technique, and half a period for students to practice it. And practice it they do, sitting quietly in straight lines, heads down and pencils moving. At Phoenix Park, on the other hand, all classes are in mixed groupings, and all lessons are project based. Projects last for several days, and the level of freedom and apparent chaos is such that I suspect even my progressively-inclined mind would boggle. There¹s nothing chaotic at the teaching end, though: conscientious and hard-working teachers have spent a lot of time choosing the projects and working on their implementation.

At the end of three years, the students in the study all took a set of very formal tests administered throughout the English system. The Phoenix Park results were appreciably higher than the Amber Hill ones. This alone is impressive but I was even more impressed by the degree to which the Phoenix Park students had developed autonomy, a sense of mathematical responsibility, and a capacity to take their mathematics out of school with them. I was hoping to convey that here with quotations taken from the book, mostly those spoken by students. Unfortunately, most of what is quoted of theirs comes in the midst of research interviews, which makes them slightly cumbersome to use.

The project was just beginning to daunt me when another group of kids crossed my horizon. These were Americans students, and their story is a very different one, but one with some key and very heartening similarities. It seems that a few years ago Linda Foreman, an Oregon teacher, was in the process of developing a middle school curriculum entitled Visual Mathematics, and needed a class to teach it to. She not only got one, she got to keep a number of the students through four years of school. The students developed into a tight and appreciative community, and in 1997, after reading a bit about TIMSS, they decided they would like to tell the National Council of Teachers of Mathematics and the National Council of Supervisors of Mathematics about what they had learned. Many a bake sale and writing session later they had their tickets in hand and their speeches in their suitcases. From those speeches, a booklet entitled ³What¹s the Big Idea?² was made. I had the good fortune to be given a photocopy, and I treasure it. It feels as if these ninth and tenth graders articulated the entire of the philosophy that underlies my own support of the various ³progressive² styles of teaching. What I really want to do is reproduce the entire of the booklet, but lacking that option I¹ll do some extracting of passages that particularly strike me.

Before I turn the column over to the kids, though, I need to point out one more aspect of the overall situation that struck me. At Phoenix Park, the central teaching organizer was the sequence of projects that the teaching staff had developed and honed over a number of years. Students were permitted to confer, and certainly spent plenty of time in conversation, but were basically working individually. In Oregon, the center was the Visual Mathematics curriculum that the teacher was in the process of developing. Students were permitted to go off on their own, and certainly spent plenty of time in individual thought, but were basically working in groups. Despite this difference in approach between the two groups, the outcome in the areas that seem to me essential was identical. Students left with a firm grasp on some chunks of mathematics, and an even firmer confidence that they could figure out whatever part of the rest of mathematics they wanted to. There is no single Right Way to do that. And I might add, thank goodness there isn¹t!

Here are a few of the things that the kids wrote in ³What¹s the Big Idea?² and that I loved reading:

³Every day there are continually more issues that I can find, explore, prove and wrestle with, that¹s what makes the subject of math so exciting to learn aboutŠ.Just when I think I understand one topic completely and can¹t know any more about it, we investigate another one that fits right in. Sometimes the new topic even raises ideas that contradict what I had previously been thinking and help me come to a better understanding.²

³Learning is a journey. Mathematical trust keeps us going and allows us to travel in new directions without worrying about getting lost or taking the routes that others do. Š Our teacher has Œmathematical trust¹ in my classmates and me that is, she believes there is a mathematician within each of us. Therefore, she does not lead, show, or guide us in our journey. Rather she provides an activity that encourages us to try new routes and discover landmarks.²

Another student discusses these activities in more detail:

³In order to determine if an activity is worthwhile, some helpful things to ask are:

… Will it make the students stretch their thinking?

… Will it branch off to other topics?

… Is there more than one way to solve the problem?

… Will it help the students¹ understanding of the idea?

… Will it cause some disequilibrium?²

Other students also comment on disequilibrium:

³Some people think that a student shouldn¹t leave the classroom in disequilibrium. I think that if a student leaves the class with an idea that doesn¹t work it gives her a chance to rethink the problem and maybe get more ideas. Š Disequilibrium is a healthy thing to have, as long as it¹s not out of control.² [I do like that last proviso!]

Another student, discussing the importance and excitement of inventing mathematical ideas, explains ³ "For us, in an ideal class, the teacher gives the class a situation to investigate, and then turns the students loose.²

One of his classmates, on the other hand, recognizes that the teacher¹s role does not stop with the presentation of the situation, but continues with the asking of questions provided the questions are effective ones, which she characterizes as ones which ³allow the student to explain his or her thinking process.² She goes on to give examples of effective questions:

… ³What do you think?²

… ³What if _______?

… ³Is there another way to think about that problem?²

… ³What are some observations you can make?²

… ³Can you explain your thinking?²

… ³Can you predict what might happen next?²

I¹ll finish with a pair of quotations that seem to me a particularly neat illustration of the similarity of the outcomes of the two teaching strategies.

From Oregon: ³To us, real mathematics is not a page filled with numbers and symbols, it is about applying everything you know to solve a puzzle.

From Phoenix Park: ²It¹s [maths is] more the thinking side to sort of look at everything you¹ve got and think about how to solve it.²

References: ³What¹s the Big Idea?², assembled by Linda Cooper Foreman of the Math Learning Center at Portland State University.

Experiencing School Mathematics, Traditional and Reform Approaches to Teaching and their Impact on Student Learning, Revised and Expanded Edition by Jo Boaler [Lawrence Erlbaum Press, 2002]