Learners asking ‘Why are we doing this?’ is usually an indication that their beliefs about what mathematics is, and what doing mathematics is, are, at the least, unhelpful for learning. It suggests that, if asked why some mathematical assertion is true, they are likely to say that ‘it just is’, ‘a teacher told me’ or ‘the text book says so’.
In order to develop strategies for changing learners’ beliefs you will need to reflect on your own ideas about what doing mathematics, as opposed to practising examples, is like. During your teaching you can look out for evidence that mathematics is a natural human endeavour which a teacher is able to draw out of individual learners. You are likely to find this evidence when learners are in situations which make it possible for them to build their own mathematics rather than just replicate someone else’s.
If learners are to change their ideas about mathematics, so that they view the structure of mathematics as its own authority, there must be some communication about the ideas. You will need to think about how learners in your classroom come to know what is expected of them in learning mathematics. For example, do you let the learners discuss and decide how they will tackle a problem without first giving them a method?
You will need to develop a classroom culture in which learners gain a sense of what they could be doing. This will involve sustaining dialogue about what is happening in the classroom. You will need to look out for, and comment on, occasions when what learners are doing looks mathematical to you, for example when they are:
- seeing patterns
- looking at simpler cases
- thinking what could be changed and still the method would work
- making conjectures
and so on.
This process of commenting on what learners are doing is often referred to as ‘metacommenting’.
By metacommenting you can draw learners’ attention to ways in which they are using their natural powers, for example when you say:
- ‘Ben’s group had an idea that they tested. But it didn’t work, so they tried another idea. That is what a mathematician would do.’
- ‘Amy has described what she was visualising. Doing mathematics often involves visualising, which is picturing things in your mind.’
- ‘Ellie has tried to convince us that what she found out must be true. That is a very mathematical thing to do.’
Eventually learners will begin to reflect back aspects of doing mathematics on which the teacher has commented. For example you might overhear learners telling each other:
- 'we need to organise this’
- ‘can we think of an example that proves this is wrong’
- ‘let’s try that, and see what happens’
and so on.
If learners have opportunities to ‘play with maths’, to explore results of changes, to make and test conjectures, to generalise, they will gain confidence in their abilities as mathematicians. When they are doing these things they are really doing mathematics, and they are very unlikely to ask ‘Why are we doing this?’.
Ball, B. (2007). Knowing The Answers
. Mathematics Teaching 200 p17-18.
Mason J. Burton L. , & Stacey K. , 1982, Thinking Mathematically
, Addison Wesley, London.
Swan, M. (2006). Collaborative Learning in Mathematics: a challenge to our beliefs and practices
. London: National Institute of Adult Continuing Education.
You might find these useful sources of information to explore this further:
Ball, B. (2007). Knowing The Answers. Mathematics Teaching 200 p17-18.
Mason J. Burton L. & Stacey K. , 1982, Thinking Mathematically, Addison Wesley, London.
Swan, M. (2006). Collaborative Learning in Mathematics: a challenge to our beliefs and practices. London: National Institute of Adult Continuing Education.