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Many people would suggest that proof has no place in primary
mathematics classrooms. It needs to be too formal and rigorous, and
young children find it difficult to follow and even harder to
replicate for themselves. We would disagree and suggest that, on
the contrary, ideas of proof need to be introduced to children when
they are quite young so that they become comfortable with one of
the central ideas in mathematics.
John Mason (1982) suggests that central to ideas of proof are
notions of conviction and he suggests that there are three levels
of this. Firstly we can consider whether we are convinced
ourselves, secondly we can try to convince a friend and, the most
rigorous level, we can try to convince an 'enemy'. This means that
the argument should be strong enough to convince someone who is
sceptical and can dispute any slightly under-justified steps in it.
If these ideas are mentioned to children at quite a young age, we
find that they are responsive to them and start to internalise them
and to use them independently on their own initiative when they are
engaged in reasoning about a mathematical problem.
We have already mentioned several aspects of proof in the previous
paragraph. In their book Primary Questions and Prompts for
Mathematical Thinking, the authors (2004) suggest that
generalising, justifying and convincing are key elements of
mathematical thinking and my own research (Back, 2004) suggests
that classrooms in which teachers encourage pupils to participate
in mathematical reasoning and argument are more likely to be
successful in helping children to be enthusiastic
Many of the problems that we offer here on the NRICH website focus
on looking for general rules by looking at a number of particular
cases. Although this activity does not involve creating proofs as
such, it is an important part of the process of proof. The theme of
the June 2005 website is proof and we are offering several problems
of this type.
Ring a Ring of Numbers , the interactivity allows pupils to
enter their numbers into the boxes and displays the differences
between the two boxes. This means that the focus of the pupils'
activity is on the process of generalising about what is going on,
rather than working out the answers. The generalisations that we
are looking for here are about the sums and differences of odd and
even numbers: an odd plus an odd makes an even, the difference
between two odd numbers is even and so on. We are not asking for a
proof as such but seeking to observe the general rules: the proof
could follow later and might in fact be an appropriate extension to
the question for slightly older children.
problem that links with this is
Make 37 which was published in October 2003. Once again there
is no request for a proof but that is the natural solution of the
problem. We have used this with a number of groups of children, and
of adults, and the initial response is usually the same: people get
stuck in and have a go. We would hate to spoil your enjoyment of
the problem so we won't give the game away - do try it yourself
before we proceed.
It hinges on what happens when you add two odd numbers. The proof
would involve demonstrating that two odd numbers added together
always give an even answer. What would we find acceptable as a
proof at Key Stage 2 level? Here is oneof the
that we put up on the website:
Joshua from Tattingstone School
explained very clearly why the problem was impossible:
I looked at the numbers in the bag
and discovered that they were all odd. I know that it is a
mathematical fact that if you take any two odd numbers and add them
together, you will always get an even number as the answer e.g. 9 +
7 = 16. Therefore any even combination of odd numbers will also
always give an even number as the answer e.g. 7 + 1 + 5 + 9 = 22.
The question asks me to make 37, which is an odd number, out of 10
odd numbers which due to the facts above is impossible.
This is not the kind of proof one would expect from an adult or 'A'
Level student but demonstrates a clearly reasoned and intelligible
argument that convinces us certainly. In a recent article in
'Mathematics Teaching', Alf Coles (2005) talks about proofs that
fail to convince even when one has followed every step of the
logical path. He suggests that we need to offer children
opportunities to gain insights into the mathematical contexts
surrounding the proof before we can expect them to gain a thorough
understanding and ownership of the proof itself. We would like to
suggest that by offering children the opportunity to produce
convincing arguments in writing or speech like the one above we
will familiarise them with the process of creating proof so that
the shifts to more formal expressions of proofs will come more
Several kinds of mathematical proof seem to be appropriate for
introduction to young children. Possibly simplest is proof by
exhaustion - only we have to be careful to choose problems that
won't generate data that proves too exhausting! An example of that
we think works is the problem
from the June 2005 website.
To start with we have an investigation
based on ideas about parallel lines and we move quickly into
collecting data. How do we know that we have found all the
possibilities for each number of sticks? We will need arguments
based on the systems we have adopted for finding them and this will
require data handling skills as well as being organised and
systematic in our approach. The third paragraph moves into higher
numbers and eventually into general number and algebra so should be
a challenge to a lot of children even though the initial setting
will be accessible to most.
Hopefully the experience of dealing with something that is
initially straightforward will lead children into offering us a
proof that makes sense to them and will convince us.
We look forward to receiving your children's proofs on the
Back, J.M. (2004) Mathematical Talk in Primary Classrooms:
Forms of life and language games. PhD Thesis, King's College
Coles, A. (2005). Proof and Insight. Mathematics Teaching 190,
Jeffcoat, M., Jones, M., Mansergh, J.,
Mason, J., Sewell, H. and Watson, A. (2004) Primary Questions and
Prompts for Mathematical Thinking. Derby, ATM.
Mason, J. with Burton, L. and Stacey, K.
(1982) Thinking Mathematically. Wokingham: Addison-Wesley
This article also appears in
Primary Mathematics, a journal published by The
Mathematical Association .