 | ON-Math Fall 2002 | Volume 1, Number 1 | |
How many times have you found activities, whether introduced in your textbook or presented at a conference or an in-service workshop, that seem to deal with patterning and sequential thinking but in which the formalized mathematics behind the activity is missing from the materials?
I have encountered elementary and middle school materials that contain engaging activities described as encouraging students' cooperative learning. These activities are often suggested for use at the beginning of the school year or as new groups are formed in the classroom, when in fact they are excellent opportunities for teaching algebraic thinking.
The Traffic Jam lesson was one of those activities for me. I first encountered it as an icebreaker during the 1996 Math Forum Summer Institute. As the institute began and after a few brief introductions to the subsequent week's activities, the participants went out onto a porch, where the floor conveniently consisted of a grid of square tiles. The institute participants and staff divided into groups of six each to experiment with the activity.
I remember the problem as being one that I did not understand but that was a fun way to get to know people. After the institute, when I opened the textbook to start preparing for the opening of the school year, I was surprised to find the same activity in the book.
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The problem involves seven stepping stones and six people. On the three left-hand stones, facing the center, stand three of the people. The other three people stand on the three right-hand stones, also facing the center. The center stone is not occupied. Everyone must move so that the people originally standing on the right-hand stepping stones are on the left-hand stones, and so that those originally standing on the left-hand stepping stones are on the right-hand stones, with the center stone again unoccupied.
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Interestingly, the textbook version of the problem was also presented as an icebreaker activity. No expectation was conveyed that the students would learn any mathematics behind the activity.
The first year that I used Traffic Jam, I introduced it as I had first learned it and as it was written in the textbook—a full-body, kinesthetic activity with six students in each group. The result was chaos. The students had no idea how to cooperate or get a handle on the problem, let alone solve it mathematically. Mike Morton, a Swarthmore student also at the 1996 Summer Institute, had written a Traffic Jam applet. I tried that version with the students, but in so doing, I still wasn't asking students to think of the mathematics behind the activity.
The following school year, however, as I reflected on appropriate uses of technology and how I could combine those uses with activities and get to a point of formalizing the mathematics, the Traffic Jam Activity started unfolding as a perfect example of a mathematically rich experience.
I found that combining the use of a manipulative activity with a technological activity both extended the time spent on the problem and gave students additional opportunities for understanding. Classroom discussion throughout the process was important, as were formalizing the mathematics and encouraging a connection between the real-world experience and the symbolic representation.
In the following pages I have tried to capture the experience that resulted in my students’ getting to a point of full understanding of the kinesthetic activity and the algebra behind it.
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