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Allowing pupils to record their mathematical ideas as they wish can
open up some interesting thoughts in a teacher's mind.
I was doing an activity rather like
but on this occasion we were making twenty. One of the
adults in the room made some unencouraging comments about this
piece of work that they saw a 7-year old had done:
I was intrigued so I asked the young
girl to tell me about what she had done. She spoke, as if disturbed
that I should need to ask, and said, "Well, YOU said make twenty in
any way we like. So here are ..." (she started to count the number
of numbers that she had written down) "... fourteen numbers and ...
there's the six that I did not write, that makes twenty." There was
a pause and then she added, "I've written them this way so that
when you put a mirror next to them you can see what the numbers
It's so easy and tempting to draw
conclusions about what we see, viewing it as a piece of maths that
an older and more experienced child might present, therefore
judging it to be very poor. It could have been assessed as a piece
of work illustrating that the pupil has no idea about writing
numbers or of performing a calculation that would give $20$ as an
So, I'm pleading for caution on making
assumptions, when looking at a child's representation of the
mathematics they have done. We need to consider the child's ability
to communicate (and the opportunities we provide for
communication), the child's understanding of his or her own
mathematics and to bear in mind that this may not necessarily, at
this stage, be totally in line with 'school maths'. Children's Mathematics; Making Marks,
Making Meaning by Worthington and Carruthers (2006) can help
us to broaden our understanding of these issues. (This can be
purchased from Abebooks from
I believe there are quite a few good
reasons for pupils to record their work when using and applying
mathematics, and these are not just about satisfying adults that
the work has been carried out! These are (in no particular
- If pupils leave their work - to go to
toilet, lunch, sharpen a pencil etc. they'll know where they've got
to on return.
- To help the child express his or her
- To be able to see if - when
applicable - he or she is doing unnecessary repeats.
- To give pupils opportunities to see
any patterns that may be evolving.
- To get some extra idea of further
work they could do - ones in a sequence that they have missed out -
this might be applicable, for example, in
Red Express Train and
- To help them get into a system for
- To aid working with a friend, sharing
aspects of the work and discussing the progress so far.
- To guide conversation and explanation
with classroom adults.
- To lead, later, to explaining to a
group or the whole class what they have been doing.
- To lead, much later on, to being able
to present some work to an adult, leave it with them and the adult
fully understand what has been carried out - even in a test or
When the pupils are presenting their
ideas on paper then there is a need for respect rather than to be
in a hurry to get 'it' to conform to 'school maths'.
I was visiting a classroom and went to
a 9-year old boy and this is what I saw:
He was just about to start the next
one, from the work card from which he was copying his work, when I
asked him to tell me how he was doing the calculations. Very
confidently he explained the last one saying, "Seven and eight are
fifteen so I put down the five and carry the one", (showing me
where the little one was). "Four, five, one and five make fifteen."
(He pointed to the four and five under the tens, the one he carried
and the five that was in his answer for the units.) Now I realised
that, considering what he said in isolation, it was quite
I have re-lived this with a number of
teachers and they have suggested that the boy had all kinds of
problems - lack of understanding of place value, a real need for
some counters, not ready for these kinds of sums yet etc.
I had decided to find out more about
his thinking so I pointed with my wide finger to the T U and asked
him what they said. "Tens and Units," he said in a voice showing
that he was anxious to get on.
"What about these?" pointing to the
"That's a five and an eight!"
So I decided this was what I wanted
to pursue. I wondered if he could say something different if they
were written elsewhere so I wrote on a nearby paper, $58$.
"That's another five and eight," he
said rather impatiently - he was picking up his pencil to carry
So I tried to think of another
situation where he might naturally come across the $58$.
I thought of buses (since we were in
a big town) so I started asking, "Suppose a bus went by and it
...". He interrupted me by saying, "It'd be a fifty-eight."
"Could we pretend that this
(pointing to his sum) is a fifty-eight as well?"
"Yes, if you like," he said in a
very nonchalant voice.
"What would this be?" pointing to
"It'll be forty-seven. Do you want
me to add it?" he asked.
Then, as quickly as it can be said,
he replied, "Forty and fifty is ninety, seven and eight is fifteen.
The answer's one hundred and five."
"Yes!" I said, "Well done!"
What a surprise when, what seemed
like an inability to deal with these additions, turns out to be
evidence of a teacher not establishing a relationship between his
mental maths ability and the formal calculations that were
Can we therefore help the children
in our care to feel confident about their own maths and help them
gradually to make the links to what is seen as 'school maths'
written work? Their thoughts and ideas need to be valued and we
should allow them to have ownership of their mathematical ideas and