Today's session began with a five-question quiz with the words written in Japanese, which none of us knew. Roberta showed us that it is possible to select correct answers based only on familiar number patterns. For example, given the numbers 56 and 7, division was the operation most people chose. The five problems involved these number pairs:
- 56 and 7
- 103 and 98
- 156 and 3
- 4 and 19
- 415 and 5
The majority of the participants scored 80 percent or better on this short quiz. The discussion that followed each problem led participants to generalize and then verbalize the strategies they used to solve these problems. Some people looked for similarities in the words that surrounded the numbers, while others isolated the numbers and looked only at the relation between them. Still others searched for a pattern that would carry through the problems, such as divide, subtract, divide... or divide, subtract, multiply 'because we haven't done that yet...'
The participants expressed a variety of opinions in discussing this nonsense test, but all agreed that such a test is an exercise that tells us little about a student's mathematical thinking or ability.
We then moved on to work in groups to complete an open-ended problem solving activity:
Each group independently defined the problem we were asked to solve. In the follow-up presentations of the groups' solutions it became apparent that not all groups had defined the problem in the same way. For example, there was the question of shipping charges: some calculated only a one-time shipping fee, claiming that they were looking for the cheapest cost and would buy all CD's at one time, while others felt that teenagers would be unlikely to have enough money to do this. The lesson learned was that there can be reasons for multiple solutions to the same real-life problem.
Next we were given sample solutions to the problem written by three different students. After carefully examining the solutions we were asked to assess the work. No rubrics were given and several groups identified the need for one or created their own rubric.
In discussing the way solutions would be assessed, the importance of telling students the criteria by which they will be evaluated became clear. For example, if points will be deducted if a table or graph is not included, then the original problem or directions should state clearly that a table or graph is required. Some participants felt that credit should be given to students who create pie graphs and bar graphs even if these graphs are inappropriately used to represent the given data, because these students have at least learned to make graphs. Others disagreed with this interpretation and felt that credit should only be given if the data were represented in an appropriate way using the appropriate tool.
The discussion would have continued but we were already at the very end of our day. We hope the discussion can continue via email and the geometry-institutes newsgroup, with our on-line participants joining us and sharing their thoughts. Thanks to Roberta Schorr for a thought-provoking activity!
- Judy Ann Brown
22 July 1997
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