From the editor
This magazine is just one of the many resources on the portal where we may find materials and ideas that we have not met before. What actually happens when we try out these ideas with our students depends to a great extent on the learning atmosphere that we have created in our classrooms.
In everything that we and our students do, we are sending each other subtle messages about what we believe. We are indicating how we believe people learn, and what we believe doing mathematics involves. And these, usually unintentional, messages create the learning atmosphere.
If we think sometimes about what we may be unintentionally ‘telling’ students, we may find that by making small changes in our behaviour we can make big changes in the atmosphere; that we can make our classroom a better place in which to learn.
For example, when a student asks ‘What do I do now?’, is your automatic instinctive response to tell them? If you believe that we always learn by being shown what to do, that is what you will believe a teacher should do – and the student wouldn’t have asked the question unless she believed that she was expected to learn mathematics by doing what she was told to do. If, sometimes, instead of telling the student what to do, you prompt them to think about something that will help them work out what to do for themselves, you challenge that student’s belief – you begin to change the classroom atmosphere.
Are students in your classroom happy to reveal that they are struggling to ‘get their heads round’ mathematical ideas? Do they feel that that their efforts to explain things about which they are not yet at all confident will be respected and valued by everyone, at least as much as confident statements from students who know that they are right?
I first encountered the expression ‘conjecturing atmosphere’ in ‘Expressing Generality’, a booklet in the Open University Project Mathematics Update series published in 1988. In the introduction, John Mason writes: “A conjecturing atmosphere is fostered by simply noticing the manner and content of contributions and responses to others, and modifying that behaviour when appropriate. It is based on the explicit premise that you learn much more from trying to express ideas that are still fuzzy and half-formed, than you do from telling someone things about which you are confident… even though, perhaps because, you are uncertain, others can also learn from your struggle. The essence of working in a conjecturing atmosphere is therefore listening to and accepting what others say as a conjecture which is intended to be modified. Consequently, it is well worth noticing how you go about developing and using a vocabulary which fosters conjecturing (‘I suggest that…’, ‘perhaps…’ rather than ‘No!’ or ‘That’s right!’), and listening to others and being listened to.”
Asking myself, over and over again, on occasion after occasion, what kind of learning atmosphere I was encouraging, has helped me gradually begin to create better learning environments in which to try out new ideas. Hopefully, there will be some ideas in this issue that are new to you and that you can try out in your learning atmosphere.