The need for further exploration

Many students will attack a mathematics problem and get an answer. Then they stop right there. Some don't even check to see if their answer is correct from the back of the book. For such students, the end product is the only important outcome (after years of conditioning that this is so.)

No. After finishing the problem, we should consider:

• Is it reasonable (close to the estimation that we made before starting the problem)?
• What have I done and what does it mean? (Where can I use this? How does it relate to what I did last week and what I will be doing next week?)
• Could I have done the problem in another way?
• What will the sketch of my solution tell me? Perhaps I can use a graphics calculator or computer to investigate it further?

And so on.

George Polya, the Hungarian mathematician who was a problem-solving specialist, once said:

Even fairly good students, when they have obtained the solution of the problem and written down neatly the argument, shut their books and look for something else. Doing so, they miss an important and instructive phase of the work.

A good teacher should understand and impress on his students the view that no problem whatever is completely exhausted. One of the first and foremost duties of the teacher is not to give his students the impression that mathematical problems have little connection with each other, and no connection at all with anything else. We have a natural opportunity to investigate the connections of a problem when looking back at its solution.

-- George Polya (1887-1985)

Now, to find the time to do such things...

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