Professor Martin Anthony talks about the different types of maths problems you may encounter in assignments and how to approach them.
Even within the quantitative subjects, you may be faced with a variety of different types of problems to solve, ranging from routine 'methods' problems (whose ultimate purpose is to train you to correctly implement a particular procedure or technique) through to more abstract problems such as modelling (applying the theory you have learned to translate a description of a problem - which may not be mathematical in nature - into a mathematical one for analysis) and even proofs.
Because of this, there is no single way to solve problems. However, we shall make note of a well regarded approach to problem solving put forward by the mathematician George Polya in his book How To Solve It. Polya devoted a great deal of his career to thinking and writing about methods of solving problems in both teaching and research, and his books are still in active use (despite the fact that "How To Solve It" was first published well over 50 years ago).
Polya's approach consists of four steps:
1. Understand the problem
A simple enough step, but one that is not always followed by students, some of whom seem to think they can dive into a problem immediately and hope that any uncertainties they have will work themselves out as they progress through the problem.
2. Devise a plan
If the problem is a 'methods' problem, then the plan may present itself to you immediately. However, for more abstract problems such as proofs, this may not be the case. There is no rule for immediately identifying a correct approach when faced with such a problem - this will come with time and practice. So therefore, try something. If it does not work, go back and try something else. In this way you will start to learn where some approaches will work and where they are unsuitable. There is no fast way to learn this - it comes with continued practice.
In that sense, the best advice to follow is "Do not be afraid to fail".
3. Carry out the plan
While the hard part might be to devise the plan, be sure that you follow it carefully and accurately (for example, if your plan calls upon a particular technique, make sure that you are performing that correctly). Do not discard a plan if it becomes difficult (that can often happen - not all problems you will encounter are easy to solve) but do not be afraid to discard a plan and return to the second step if your original one continues to fail.
4. Review
Once you have executed your plan, you should first check that you have indeed completed the task required of you (sadly, it is not uncommon to see students answer a different question to the one that was set, even in exams!). But review should not be limited to just ticking off a task on a to-do list.
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Reflect on what you have done and how you accomplished it. Try to understand what worked and what didn't, and why.
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Could you have completed the task in another way?
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Can you use this approach for other problems?
In this way, you will learn from your experiences and hone your skills as you attempt more and more of the work.
Of course, the principles above - particularly the last step of reviewing and reflecting upon your work - could also be implied to your general approach to learning, and not just problem solving.
For more on reflecting and reviewing your work, see Review and reflect on your learning.