Designing problems to evaluate mathematical thinking

iStock_000018120666XSmallThe National Council of Teachers of Mathematics in its “Curriculum and Evaluation Standards for School Mathematics” calls for a change of direction away from traditional mathematical problem-solving with its emphasis on rules, teacher talk, and passive learning toward active student participation in reasoning and discussion of problem solving.

Walter Szetela, University of British Columbia, and Cynthia Nicol, a mathematics teacher, suggest that to meet these standards, teachers will have to be able to design new types of problems, encourage appropriate interaction in their classrooms, and develop ways to evaluate more complex kinds of problem solving.

Instead of simply scoring answers, teachers will need to evaluate students’ comprehension of problems and concepts. Szetela and Nicol state that even the most capable students are not necessarily inclined or able to communicate their thinking. Therefore, teachers need to devise problem situations and questions that encourage and motivate students to discuss and explain how they solve a problem.

Szetela and Nicol suggest the following:

1. Present an incorrectly “solved” problem and ask students questions which guide them toward discovering the mistake.

2. Describe a problem situation with all the facts and conditions, but have the students design an appropriate question for the problem. Next, have them solve the problem and then write their perceptions about the adequacy of the solution.

3. Have the students complete a partially-solved problem.

4. Present a problem with extraneous information and ask students to delete all but the essential information needed to solve the problem.

5. Ask students to explain, using words only, how they would solve a certain problem and then ask them to construct a similar problem.

6. Ask students to solve a problem and then have them write a new problem with the same structure, but within a different context.

7. Present a problem without numerals. Have students provide appropriate numerals, estimate the answer, and then solve the problem.

“Evaluating Problem Solving in Mathematics” Educational Leadership, May 1992, Volume 49, Number 8, pp. 42-45.

Published in ERN September/October 1992 Volume 5 Number 4

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