A conjecture is an assertion or statement of what might be true. To be a conjecture rather than a guess, there has to be some evidence. Whenever you get stuck, try to articulate what you think might be true, and what evidence you have (such as particular examples or cases). Never leave off working on a problem without writing down your current conjectures, so that when (if) you pick it up again, you know what you thought might be the case.

In his film “Let us Teach Guessing”, George Pólya (1965) emphasised the importance of making conjectures, and then disbelieving those conjectures.    If you try to keep things in your head until you are convinced about them, you will have the ‘tumble drier’ experience, where ideas tumble around, changing every so often, without being truly examined and tested. By saying or writing down conjectures, you get them outside of your head so that you can consider them critically. Most teachers know about some students who sit at the back and pretend to themselves that they ‘could have got the answer’ … but don’t commit themselves, so never discover that they were not so clear as they thought. Often these students are surprised at how badly they do on tests. They can be helped by encouragement to commit to a conjecture which is uttered with the intention of modifying it as necessary. You cannot think properly about mathematics, or mathematically, without constantly making and modifying conjectures. The book Proofs & Refutations (Imre Lakatos 1976) displays mathematical thinking as a sequence of conjectures, attempts at proofs and refutations either of the statements or of the reasoning, seeking to be more precise about the conditions under which some statement is actually true.

Establishing an atmosphere or ethos in which mathematics can be done successfully involves a conjecturing atmosphere in which everything said is said as a conjecture to be tested and probably modified or else justified by mathematical reasoning. This who think they ‘know’ and answer are responsible for listening carefully to others while checking their own thinking, and asking questions which provoke further insight (rather than simply stating your own thoughts); those who are unsure take the opportunity to try to say what they think in the full knowledge that they intend to make modifications with the help of others.