As teachers, it is important that we vary the mathematical activities we give our students. The learning that takes place in the classroom is, in one way or another, affected by the kind of tasks that we give them. These tasks may be classified into three: exercises, problems, and investigations.
A math exercise is a task where students know what is asked AND know a direct way of doing it. Task 1 is an example of a math exercise. In this task, students are asked for the number of squares that make up the fourth figure. This can easily do this by looking at the pattern or by counting.
Math exercises are usually given after examples were demonstrated. They are commonly used to enhance the basic computational skills of students.
A math problem is a task where students know what is asked, BUT do not know a direct way of doing it. Task 2 is an example of a math problem. The task is for the students to find the number of squares in Figure 100. In this case, there is no direct way of solving it. Although they can draw the 100th figure by continuing the pattern and then counting the number of squares in that figure, it would take them a long time to do it.
Math problems should be given at the start of the lesson as context to develop a concept or at the end of the lesson to assess understanding. These tasks intend to elicit thinking skills such as recognizing and generalizing patterns, making conjectures, proving claims, giving counterexamples, to name a few.
A math investigation is a task where students do not know what is asked AND do not know a direct way of solving it. Task 3 an example of a math investigation. In the example below, students are just asked to investigate. It is for them to choose a problem that they would like to pursue. They can investigate the number of squares in the nth figure, the perimeter of the figures in the nth figure, the number of horizontal segments, the number of vertical segments, etc. (Can you think of others?).
In math investigations, students create their own problems and find ways to solve them. Math investigations are usually done for a longer period of time. At the end of an investigation, students may be asked to report their findings written or orally.
The different tasks discussed above are all important in their own right. Exercises are used to reinforce mathematical computation skills, problems are used to improve higher order thinking skills, and math investigations are used to train students in problem posing and self-directed exploration. However, if we want our students to be critical thinkers, we should do our best to scaffold them so that they would be able to do problem solving and investigation. In addition, the tasks above also show that the same problem can be used to elicit different levels of thinking. It only depends on the questions we ask.