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Level:	School Design: Traditional
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Feature Story

The following is an adaptation of a story by Jan Schott, a mathematics teacher at Hampshire Regional High School in Westhampton, Massachusetts. It was originally featured in a professional development package that ENC produced on the topic of Teacher Change. The full story, in its original form, can be found on ENC Online at:

http://www.enc.org/focus/change/document.shtm?input=CDS-000381-381

(Editor's note: Jan Schott has participated in numerous SummerMath for Teachers Institutes and graduate courses at Mount Holyoke College. She also took part in the Southeastern Consortium for Minorities in Engineering at the University of Florida.)

For several years I have been working on making my classroom more interactive as the students construct their own learning and ideas. My previous definition of teaching can be simply summed up as follows: Pour information into the students' brains with clear, concise explanations and lectures while simultaneously stopping the leaks. I realized that I needed to do less preaching and teaching. Instead I have been urging my students to explore their ideas and voice their thoughts. I have been working on encouraging them to be mathematicians.

I have found that by working together in small groups and in whole class discussions, students discover mathematics for themselves. One of my goals is to establish a safe environment -- which does not imply that all ideas that are offered are correct or that they are taken to be correct. Contributions are not immediately judged by either the teacher or students but are respectfully discussed and critically analyzed.

I am especially focusing my attention on providing wait time, learning to accept necessary periods of silence, and slowing down the pace of over-anxious students. Sometimes, when the students are engaged in a vigorous discussion, I simply record what I hear without interrupting the flow. When we are working on exercises with specific solutions, I ask for all the different answers the students have developed and record all of them on the board.

Then we have a discussion about how they reached these solutions, how we could determine if the solutions are correct, and if any of the solutions are, in fact, equivalent. Lively discussions result as students argue their points of view and listen to others defend theirs. Looking at the board, students struggle to make connections between the ideas of one student and those of another.

In one of my classes, an advanced algebra class of eighth- and ninth-graders, a recent assignment involved reading a short lesson from the text -- Elementary Algebra by Harold Jacobs -- and doing a set of exercises on direct variation. This book is not a typical algebra text; there are not a lot of drill problems with the same set of directions. Rather, it offers short, very readable lessons on the topic and then reinforces and further develops these concepts in the problem set.

The concept of direct variation is introduced using a chart that compares the height of an object to the length of its shadow at a given time of the day. The standard formula for direct variation is y = ax, in which a is the constant of variation. If the height of the object is doubled, the length of the shadow is also doubled. These are the main points of the short lesson.

After a very lively discussion of the homework problems on direct variation, I posed the question, "What is direct variation? How would you describe direct variation?" The silence that followed was unusual, and when they finally spoke, the students could only voice their confusion.

I was surprised at this turn of events. The students did not have a handle on direct variation and all the subtle and not-so-subtle interconnected thoughts. The excellent book presentation had not been any better at teaching direct variation to these students than my lecturing had taught it to classes in the past. Yet these students had managed the exercises very well, and they had had a very lively discussion of the problem set. I realized the class needed a second chance to create their own meaning. They knew more than they realized but just hadn't put the pieces together.

Many associates at my school and in the mathematics education courses I had taken had talked about using writing in math classes. Some assigned journals in which the students wrote about what they were learning, what was causing difficulty, and what they did not yet understand. Teachers would then respond to the student journals with thoughts, questions, and encouragement. Several had mentioned use of essays on math tests to determine whether students had a firm understanding of a particular concept.

For homework that night, I asked my students to write a paragraph describing direct variation. The next day we had a fruitful discussion in which we determined together that the slope is defined as the change in the y coordinate over the change in the x coordinate, or the rise over the run.

Later that day, I had a chance to read the students' paragraphs. It was obvious that the exercise of writing had helped them organize their thoughts and at least begin to make connections. The writing had fueled the fire of our discussion. However, their paragraphs revealed that they had not made all the connections. In fact, many were not clear and/or complete in their definition and also, evidently, in their understanding. It was the combination of their paragraphs and the interactive classroom that created the opportunity to discover as much as they did in their discussion. Great inquiring minds met in that classroom with kernels of ideas that germinated and sprouted in the discussion.

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