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# Exercises versus Problems

Learning mathematics is closely associated with doing exercises. Most of us will have experienced mathematical learning as: receive some information on a new (to us) area of mathematics and then do some exercises to increase and test that learning.

In this post I want to make a distinction between between exercises and problems. For me exercises are routine while problems involve deeper thinking.

Thus, an exercise may be a straightforward calculation, or involve checking that something satisfies the conditions of a definition, or require the application of a theorem and so on. It may be something for which a worked example is in the notes. A problem, on the other hand, requires combining a number of ideas or seeing some concept in a different way to produce the solution. It would not be possible to give a worked example for a problem.

The distinction between the two can be in the eye of the beholder though. For stronger students a problem may be an exercise and for weaker students an exercise may be a very difficult problem. However, I wish to distinguish the two for later while accepting the dividing line between them is not strict.

The practical advantage of the distinction is that it becomes easier to set exercise/problem sheets for students: Start with exercises and increase the difficulty level to problems.