We are required to teach formal proof in GCSE and A-level maths, for example “Prove C2 is irrational”. This is an area I find difficult to teach, for surely in exams we are not just testing the recall of someone else’s work?
When I was at school we had to learn geometric proof at an early age.
I can remember our teacher telling us she expected these proofs to be written in a particular order and by the time O-levels arrived, this was imprinted on my brain. We learned as many proofs as we could, rather than trying to understand them. This more than likely hindered progress in my own mathematics later, while also taking the fun out of deduction.
There is a generally held belief, as you suggest, that formal proof activities are not really that profitable. Informal arguments are more successful, though standard proofs have their place within the curriculum - there is a certain beauty to the way they are constructed. But we should not pretend that by presenting pupils with these pre-determined proofs that we are teaching them about it. Nor should we think that by requesting that they recite certain facts that they understand proof because, as you say, it is easy to recall the information without understanding it.
David Tall, professor in mathematical thinking at Warwick University, says on his website (www.davidtall.com): “Fundamentally, I don’t believe that mathematicians understand what proof is unless they are talking to other mathematicians and ‘do’ it rather than ‘explain’ it. I have written about this too, at various levels from school to university. I have incorporated this into what I term ‘three worlds of mathematics’: one is what we perceive and conceive in our mind, doing thought experiments; one is what we symbolise in arithmetic and algebra to prove by calculation and manipulation of algebraic symbols; and the third is the formal proof of mathematicians.
“This last world turns everything we do on its head. Instead of seeing things and getting a sense of them and describing them in words, we start with the words and define them. This confuses ordinary folk who believe that the things we talk about already exist and naturally have certain properties. In formal maths we state a few properties verbally as a definition and then deduce everything from this definition. I believe here that the problem is that, in order to handle proof properly, you need to have a sound conceptual structure about what it is you are trying to prove and the context in which you are trying to prove it. Mathematicians have their structure of formal proof (which they can’t explain to non-mathematicians). Therefore, the pressure to give children experience of proof in schools is flawed. Some kids understand what is going on, but most, embroiled in everyday meanings, do not. Most teachers do not.”
Kona Macphee of the online magazine Plus has written about the origins of proof (www.plus.maths.orgissue7featuresproof1).
Barbara Ball, professional officer of the Association of Teachers of Mathematics, offers course about proof (www.atm.org.uk).
Much research has been carried out looking at the way pupils reason in mathematics, including the Longitudinal Proof Project (www.ioe.ac.ukproof) at the Institute of Education, University of London.
The Institute is developing in-service courses based on its research which it will launch next year. Contact the School of Mathematics, Science and Technology, Institute of Education, University of London, 20 Bedford Way, London WC1H OAL.
Wendy Fortescue-Hubbard is a teacher and game inventor. She has been awarded a three-year fellowship by the National Endowment for Science, Technology and the Arts (NESTA) www.nesta.org.uk to spread maths to the masses. Email your questions to Mathagony Aunt at teacher@tes.co.uk Or write to TES Teacher, Admiral House, 66-68 East Smithfield, London E1W 1BX Who said we couldn’t make words into maths? I bet primary pupils could come up with some great maths poems using a similar structure:
Inductive definition: Un+1 = Un + 2 words
If ‘Un+1 = Un + 2 words’, U0 = 0 and U4 = ‘Please substitute
in here for the next term’, find U1 U2 and U3
Un+1 = Un + 2 words
U1 = U0 + 2 words
U1 = Please substitute
U2 = U1 + 2 words
U2 = Please substitute + in here
U3 = U2 + 2 words
U3 = Please substitute in here + for the
U4 = U3 + 2 words + next term
U4 = Please substitute in here for the next term
