# Exercises, Problems, and Math Investigations

November 6, 2009 1 Comment

I love to ask this question to maths teachers: *In your experience, in which part of your lesson do you think your students are really learning mathematics?* Most of them will always share that it is when the students are doing the activities themselves. Teachers will almost always shyly admit that it’s not really during the part where the students are listening to the teacher’s explanation and lectures.

Learning mathematics is not a spectator sport. You got to be in it. Students got to be in it. The quality of mathematics students learn depends on the mathematical tasks or activities we let our students engage in.

Mathematical activities/tasks can be categorized into three types: exercises, problem solving, and, investigation. Samples of these tasks are shown in the picture below:

**Standard exercises** – an activity with clearly defined procedure/strategy and goal.Standard exercises are used for mastery of a newly learned skill – computational, use of an instrument, and even new terms or vocabulary.

**Problem solving activity** – an activity involving a clearly defined goal but the solutions or strategies are not readily apparent. The student makes decision on the latter. If the students already know how to solve the problem then it is no longer a problem. It is an exercise.

**Mathematical investigation** – is an exploration of an open-ended mathematical situation where the student is free to choose what aspects of the situation he or she would like to do and how to do it. The students pose their own problem to solve and extends to a directions he/she wants to pursue. In this activity, students experience how mathematicians work or how to conduct a mathematical research.

**Students need to be exposed to all these type of activities. It is unfortunate that textbooks and many mathematics classes are dominated by exercises rather than problem solving and investigations tasks, creating the misconception that mathematics is about mastering skills and following procedures rather than a way of thinking and communicating.**

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