The ethos of a classroom can support mathematical thinking or it can block mathematical thinking. Approaches which expect pupils to follow the teacher’s examples may not give the pupils enough time to question and explore what they understand. In this way development of mathematical thinking may be inhibited.
Approaches which encourage and conjecture are more likely to help pupils fully understand the concepts they are being taught.
The way that teachers phrase questions and respond to pupils answers and comments can have a powerful effect on the learning environment and it is helpful to develop a vocabulary which supports a conjecturing atmosphere.
If all interactions in the classroom are based on the principal that everything anyone says is an idea that may be adjusted or changed, then learners can participate in an atmosphere of growth, adjustment and modification, and ideas can be strengthened and deepened to improve understanding.
The way in which we respond when pupils disagree, for example can either support a conjecturing atmosphere or not. For example the phrase “that’s interesting, how else can we think about this” might be more productive to mathematical thinking than “no that’s wrong”. In this way pupils are encouraged to build and develop ideas rather than knock down ideas or overcome hurdles. In a conjecturing atmosphere everything said and done is said and done with the expectation of trying it out and adjusting and extending it as and when necessary.
In a conjecturing atmosphere those who are sure hold back or support others who are less sure by offering examples or potential counterexamples; those who are less sure take opportunities to try to express themselves. Making these things explicit in the classroom by encouraging pupils to speak when they are not sure can help.
A classroom in which people are willing to say something because they know it will be respected, listened to, probably challenged, but in which their efforts will not be demeaned, participants become much more willing to discuss ideas and put forward suggestions. In this classroom pupils are likely to learn mathematics more deeply.