Here are some generic techniques that might be useful when approaching problems, that I am going to present to my students tomorrow.
First, consider the question 'as a whole': words, mathematical objects and diagrams. Can you say what 'type of question' it is? Can you articulate what you are going to do? If you can, then do it, remaining conscious of what you are doing throughout, and making sure you check your answer in some way.
If you are not sure what to do, try these 'linguistic' ideas, designed to bring associations and useful actions to the fore:
- Read the whole question aloud, making a mental note of the words/symbols that you stress as you speak. Close your eyes and repeat the question, over-stressing these words/symbols. Why are these words, and the relationships between them, significant?
- Name the mathematical objects that you can see (e.g. quadratic, cubic, ...). Is there a mathematical name for the thing(s) that you are trying to work out? Can you give it a name?
- What assertions being made in the question? Close your eyes and summarise any assertions in your own words. What do these assertions imply (algebraically, graphically, ...)?
- If there is a diagram, look at it carefully. What are your eyes drawn towards? These are probably the key features of the diagram - what do they imply, algebraically?
- What images, associations, techniques come to mind? Are there other things (not mentioned in the question) that are usually associated with these words and objects? Might they be useful?
At some stage during this process, some action(s) may have come to mind. It may be helpful before doing it to articulate what you are going to do (to yourself or someone else). Can you say why this will help you answer the question?
If you are still not sure what to do, some transformation of the problem as it is may be required, such as:
- Can you transform what is given into something more useful or recognisable, perhaps adding something to a diagram, or some algebraic manipulation?
- Would a concrete example be useful, a numerical example or two, a sketch...?
- Would solving a simplified version of the problem be useful in some way?
