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Teaching Problem Solving

Learning mathematics involves solving problems. So, is it a good idea to teach students how to solve problems in the sense of an abstract process called “problem solving”. That is, should we provide a course on solving problems where we go through problems – not necessarily of a mathematical nature (for example, the classic you have a lit candle, a box of tacks. How do you attach the candle to the wall?) and then analyse and reflect on the solution?

Polya’s classic book How To Solve It is probably the most famous example of attempting to teach methods for problem solving. His four point plan is as follows:

  1. Understand the problem.
  2. Make a plan.
  3. Carry out the plan.
  4. Look back.

The problem with this is that the second and third parts are basically “solve the problem”. Also, the four point plan perhaps generates unreasonable expectations of how problems get solved. Problem solving is messy with many false starts and dead ends.

Part (i) and (iv) are certainly useful tasks and should be emphasised to students. In his Foreword to Polya, Ian Stewart admits to wanting to cheerfully strangle students who fail to understand a problem by failing to know what a word a means. Reflection — the looking back — is something I find very powerful. Ruminating on a problem afterwards can produce some new insight that produces a better solution or how to make the solution more general.

Leaving aside the slightly dubious plan, there are many ideas within Polya’s book that are worth instilling in students. For example, I often tell students to “Draw a diagram”. This rule forces them to pay attention to the details and to understand the problem.

However, I would never advocate having a course dedicated to solving problems. I don’t think it necessarily hurts to have read Polya but spending a whole course on problem solving techniques is wasteful. Time is better spent doing real problems about real maths instead of problems invented to teach students a lesson about problem-solving. The useful ideas from Polya can be sprinkled through the the course.

There is no substitute to solving lots of problems. The solution to a problem often comes form the solution to a problem already seen and so the more problems you have seen the better. Experience allows you to recognise that a certain technique can be utilised. Just today, my PhD student and I were trying to show that the derivative of a map evaluated at a particular point was the identity. The solution wasn’t obvious and the answer came from considering Taylor series. Why try Taylor series? Well, because we had seen it used in similar questions many times. At no point did we make a plan and carry it out. In hindsight the solution came from starting in the middle, working to the end and then working backwards to the start!

One good reason for not teaching a problem solving course is that I know of no evidence that such a course improves students. I have an explanation for this lack of effectiveness : We’re not very good at transferring skills. In general, we have to consciously transfer them — they don’t transfer themselves. Furthermore, I think that problem solving courses contradict the “Show, don’t tell” dictum. Problem solving classes are the equivalent of telling!

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