Mathematical thinking can only take place in what I call a conjecturing atmosphere. This means that everything that is said is treated as a conjecture that may require modifying. If you disagree with someone you “invite them to modify their conjecture” or you ask how their conjecture applies to some example (which perhaps you think might be a counter example).
In a conjecturing atmosphere, when a question is asked, those who are confident listen carefully and quietly check their thinking, while those who are unsure take the opportunity to formulate their own conjecture. If need be, those who are confident that they know might ask questions designed to point in a useful direction.
The aim is to reduce the pressure on ‘being correct’ and rather to emphasise the deeper appreciation and comprehension which can then be the basis for reconstructing the idea in the future when needed. A conjecturing atmosphere requires trust: learners trusting the teacher that they can make progress on the tasks and questions posed to them, because effort is worthwhile; teachers trusting learners to make an effort because they will be rewarded by a more comprehensive appreciation of the topic, and be better equipped in the future than if they simply rehearse procedures over and over. A conjecturing atmosphere based on trust grows over time. It does not just suddenly happen. It depends on senior management trusting that teachers’ ways of working will produce deeper and more effective learning over time, rather than demanding immediate short-term success.
Action: When a learner offers a response to a question, try to catch yourself before you declare whether it is right or wrong; praise it as a conjecture, and invite others to consider whether they agree with it, or whether they would like to suggest a modification or a counter-example. In this way you can be responsible for the process of thinking, while the learners work together to decide correctness.
Eureka Game: One way to support the development of a conjecturing atmosphere
The initiator (the teacher to start with) writes down on a piece of card a rule that three numbers have to satisfy in order to be ‘acceptable’. They then present the participants with one example.
For example, the rule might be “must be an increasing sequence” and the example might be 2, 4, 8. With a smiling face beside it.
Participants then come to the board and offer a triple, and the initiator thinks and then draws a smiley face or a sad face beside it. Each offering can be praised as a conjecture even if it doesn’t satisfy the rule.
No-one ever actually says the rule out loud, or even the rule they think is governing the choices. Every offer is a conjecture. When someone thinks they ‘know the rule’, they say “Eureka” and then offer examples they think will help other people to recognise that rule.
However, it is very easy to be mistaken. When someone thinks they know the rule, they need to test their conjecture by offering some triples that they think do fit the rule, and some that they think will not.
You may want to introduce other sequences as examples, such as three fractions, or a decimal number or a negative number to awaken participants to other possibilities, but only if they seem stuck in a set pattern of thinking or ‘rut’.
For example, the triple 2, 4, 8 usually catches people in a ‘rut’ of thinking of powers, or even numbers. Even when they discover that 1, 2, 3 gets a smiley, they may not think of trying 1, 1, 1 for example.
Advice: Use very simple rules at first, so that learners develop confidence.
The game can be used a number of times, perhaps at the start or end of the week, for a few minutes, in order to raise issues about conjecturing: you would never say “you’re wrong”, but rather “that doesn’t fit with my idea”.
Some teachers like to conduct this game in total silence, offering the pen to people who come to the board, write their triple, and then retreat again. This silence can usefully focus attention on the conjecturing.
If you try Eureka with a class, do send in a brief account of what you noticed. It may take several encounters with the game before learners appreciate the possibilities!
John Mason worked for the Open University Mathematics Department, where he designed two of the mathematics summer schools, contributed to numerous courses, and then helped form and run the Centre for Mathematics Education. He has written numerous books and booklets, as well as research articles and book chapters, the best known being ‘Thinking Mathematically’ (1982, 2010).