1.1 Definition

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Most important mathematics concepts and procedures can best be taught through problem solving. That is, problems can and should be posed that engage students in thinking about and developing the important mathematics they need to learn.

A problem is defined as any task or activity begins where the student are for which the students have no prescribed or memorized rules or methods, nor is there a perception by students that there is a specific “correct” solution method.

Task selected as problem is effective when it helps students learn the ideas you want them to learn. It must be mathematics in the task that makes it problematic for the students so that it is the mathematical ideas that are primary concern. Where do you look for tasks?

Example of a Problem

The following task might be used in grades 3-6 as part of the development of fraction concepts.

Place an X on the number line about where 11/8 would be. Explain why you put your X where you did. Perhaps you will want to draw and label other points on the line to help explain your answer.

Note that the task includes a suggestion for how to respond but does not specify exactly what must be done. Students are able to use their own level of reasoning and understanding to justfy their answers. In the follow-up discussion, the teacher may well expect to see a variety of justifications from which to help the class refine ideas about fractions that are greater than 1.


  1.2 Value of Problem Solving

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There is no doubt that teaching with problems is difficult. Tasks must be designed or selected each day, taking into consideration the current understanding of the students and the needs of the curriculum. There are good reasons to go to this effort:

Strategy and Process Goals

Places the focus of the students’ attention on ideas and sense making.

Develops “mathematical power”. Students solving problems in class will be engaged in all five of the process: problem solving, reasoning, connections, communication, and representation.

Extend and generalize problems. Students consider results or processes applied in other situations or used to froms rules or general procedures.

Metacognitive Goals
Metacognition refers to conscious monitoring and regulation of your own thought. Good problem solvers monitor their thinking regularly and automatically. Metacognitive goals are

Monitor and regulate actions

Atitudinal Goals
Problem solving develops the belief in students that they are capable of doing mathematics and that mathematics makes sense. Everytime you pose a problem and expect a solution, you say to students, “I believe you can do this.” Atitudinal goals are

Gain confidence and belief in abilities

Be willing to try and persevere

Enjoy doing mathematics