At Luke's request, we began with one of the BEAM cards. He had
tried a few of these already with his Dad and seemed keen to have a
go at some more (perhaps to get further through the box than his
older sister!). We started with the Counter-Counting Puzzle, which
begins like this:
Even before we had begun the problem itself, Luke demonstrated
his awareness of, and familiarity with, number by counting out the
twelve counters that we needed in twos. At first, we spent a minute
or so just checking we understood what the instructions were asking
of us, for example, "How many counters could we move into this
pile?". It was interesting for me that Luke launched himself
eagerly into the task. He did not appear to worry initially about
getting the "right" answer, he was simply keen to have the same
number of counters in each pile. As he moved counters from one pile
to another, I tried not to interfere with his choices, except to
remind him of the rule. He was triumphant on reaching a solution.
When I enquired how many moves it had taken, he had no idea. I
think this first go for him had been about understanding and
internalising the task rather than worrying about how long it took.
Now with this extra challenge, Luke asked me to count the number of
moves as he manipulated the counters. He appeared a little
concerned that he wouldn't be able to remember how he had done it,
but that didn't hold him back for long.
So, with Luke moving counters and me counting aloud, he managed
to finish with three piles of four counters in three moves:
6, 5, 1
6, 4, 2
6, 2, 4
4, 4, 4
I asked, "Do you think you can do it in fewer moves?". Luke's
response was to try, but again he managed three. "Look in the book,
look in the book!" he suggested. Luke wanted confirmation from the
teacher's book that he had succeeded. This was an interesting
dilemma for me. Which should come first - analysing what we could
do at each stage to convince ourselves what the fewest number of
moves was, or having been told it could indeed be done in a
particular number of moves, trying to find out how? I sensed that
Luke would have been happier with the latter (was that a "cop-out"
on my part?!), so we looked in the book. Of course, we read that in
fact it could be done in only two moves. "That's impossible!" cried
Luke, but in an I-still-want-to-work-out-how way rather than an
I'm-giving-up way. It was then that I felt my role of prompter was
really needed. We took it step by step, looking at the different
possibilities: "What could
we do now?" and in this way, Luke quickly arrived at a solution in
I feel it is worth pointing out that at this stage we hadn't
written anything down, and in fact we didn't at any point as we
worked on this problem. Retrospectively, perhaps I should have
encouraged Luke to address the issue of recording, but it didn't
seem to be hindering him enough to warrant attention at the time.
If we had recorded what Luke had tried, it might have helped him
see the two moves more quickly. It would also have facilitated
other questions, such as "Can we do two moves in another way?";
"Will we always end up with four in each pile?"; "How do you know?"
... Should I have suggested to Luke that he could write things
down? I wonder if he wanted to, but didn't ask?
Having gone through the process of analysis together, Luke
seemed to gain in confidence. The BEAM activity goes on to suggest
starting with the counters in different piles. Luke's own example
was $1$, $1$, $10$ and he was able to cope well with my
questioning: "How do you know you can't do it in fewer moves?". His
justifications centred on looking at the different possibilities at
each stage, as we had done together. He was able to articulate that
he couldn't move any counters into the pile of $10$, so he would
move one from the $10$ into one of the piles of $1$. Again, he
couldn't move any into the resulting pile of $9$, so would move two
out and so on.
Working with Luke in this way highlighted for me the power of
this task in encouraging young children to begin to work in a
systematic way, justifying choices and decisions as they go. Who
says that primary-aged children can't prove results
The second activity that I had chosen, called
Secret Number, was taken from NRICH and was published in July
So, Luke and I played a version of this game. I told Luke that I
was putting a secret number into my calculator and then I was
adding five. I pressed the equals button and showed Luke that the
answer was $8$. Could he work out what my secret number was?
Without hesitation, Luke picked up his own calculator and, after
thinking for a few seconds, began to press keys. He didn't say out
loud what he was doing but after pressing equals, he expressed
delight. He too had $8$ on his display. "What was my secret number,
then?" I asked. The reply was a confident one: "$3$". "How did you
work it out?" I enquired. Luke described how he counted on from $5$
to $8$ in his head, which needed $3$. Using the calculator, he had
performed the same calculation as I had, as a check. I was
delighted. Such a simple question had revealed to me exactly what
Luke understood by addition. It was obvious to him that addition is
commutative - that if $3 + 5 = 8$, then $5 + 3$ is also $8$.
I wondered whether counting on was Luke's only strategy, or
whether he had others to draw on. So, I told him that my new secret
number added to $10$ was $20$, and showed him the calculator
display. This time, Luke knew that $10$ add $10$ is $20$ without
any mental calculation, it was a fact that he was familiar with,
but he did check the sum on his calculator once again.
I invited Luke to think of a secret number and to see whether I
could work out what it was. This seemed to appeal to him, so after
a short time he told me that he'd put his number into the
calculator, then pressed add $100$ (checking with me that he had
entered $100$ correctly), then equals. He showed me the result of
this addition, which was $101$. I knew Luke was very familiar with
numbers up to $100$ - he'd told me that on the way to see me, he
and his Dad had counted all the way from $1$ to $100$ and then in
twos up to $100$ - but he wasn't sure how to read the number on the
display. I read out "one hundred and one", explaining a little by
breaking it up into the $100$ and the $1$, pointing to the display
at the same time. I imagine that adding a large number had appealed
to him, thinking it would be much harder for me to work out his
secret number. It was fascinating to see that as soon as the total
was said aloud, he immediately realised what an "easy" task he had
Building on this, I did the same, adding $100$ to a new secret
number, resulting in $116$ which we read together from my
calculator. Unsurprisingly, Luke needed no prompting and was able
to verify my number immediately. I decided to take this idea in a
slightly different direction, hoping to gain more insight into
Luke's understanding of place value. This time, I added $50$ to a
different secret number and got $56$. This did not phase Luke at
all, as I'd anticipated. So, I thought I would push him a little.
Sticking with adding $50$, I now got a total of $71$ on my display.
This took several stages. Having started with $9$, and so obtaining
$59$, we talked about whether we needed a bigger or smaller number
than $9$ if we were aiming for $71$. So, Luke tried $10$ to get
$60$. This produced an "Aah!" and Luke began with $20$ to get $70$.
He then knew immediately that he needed one more, so told me that
it would be $21$ before he used the calculator as the checking
Would Luke be able to use that calculation in the next? I added
$50$ to another secret number and the total was $81$. Whether Luke
didn't remember what the previous addition had been, or whether it
just didn't occur to him that it might have been useful, I'm not
sure. What he had appeared to take on board was that my number was
going to be larger than $10$. He tried $48$ but realised the total
was too big. So, he then tried $38$, which of course made $88$.
This was the breakthrough moment - he was certain then that my
number was $31$.
Afterwards, I was struck by the fact that playing this game with
Luke, which was based on such a simple idea, had revealed huge
amounts about his understanding of place value and addition. It was
a humbling process for me as the "teacher" in the sense that
without the conversations we had, I would have jumped to inaccurate
conclusions about the level of Luke's understanding. Talking as we
went along meant that Luke's thought processes were gradually
revealed to me and the more I discovered, the better I felt I was
able to encourage him and push him further.
In the last ten or fifteen minutes of our time together, we
looked at a third problem based on
Caterpillars, published on NRICH in September 2007:
Our caterpillars have numbers on each little part - numbers $1$,
$2$, $3$, $4$, ... up to $16$.
You can see their pale blue head, and their body bending at
right angles so that they are lying in a square.
"Can we do diagonals?" he enquired. I thought that we probably
could and in this way he actually made the problem more
challenging. As he began to layout the counters, incorporating a
diagonal arrangement, he realised that he was more restricted. He
talked as he went along, looking ahead to see whether, if he put
that counter there, he was going to be able to place the rest of
the caterpillar. He was able to answer my questions as to why he'd
had to change the position of one segment, for example, in order to
be able to complete the arrangement.
As he had made the different arrangements, he had recorded the
numbers in his own hand-drawn grids. He suddenly decided that he
would join the numbers in numerical order with straight lines. I'm
not sure what prompted him to do this - it wasn't my instigation -
but what was wonderful was that he immediately remarked that the
first two arrangements created the same pattern of lines. He
clearly had not predicted this and found it fascinating. This felt
as if it was his own investigation. He was definitely leading the
way quite happily and in this instance, it led to an interesting
question: why are they the same? I wonder whether this has ever
happened to you as you've worked on some mathematics? Two results
are the same and what a fantastic feeling it is to be able to
explain why, even though you had not anticipated it happening. How
powerful mathematics feels during those moments!
As I intimated at the beginning of this article, I was in a
luxurious situation. How often do we as teachers have the
opportunity to work on a one-to-one basis with any of our pupils?
And yet how valuable those opportunities can be. While working with
Luke, I was particularly struck by the fact that he was prepared to
have a go at each task. Was this the attitude he always adopted?
Was it the fact that there weren't any peers to judge him? Was it
that he regarded our time together as "fun" and was therefore more
relaxed? I don't know. But it made me reflect on where children
pick up a reluctance to "play with the mathematics" from? Is this
something we value and therefore should be actively encouraging
more often? The conversations Luke and I had as we worked together
enabled me to probe quite deeply what Luke was happy with. I was
able to judge his zone of proximal development and move him on
appropriately. Subtleties and nuances in his understanding were
revealed which would almost certainly have passed me by had I been
in a room with twenty nine other children. However, to some extent,
I still felt that I was making decisions on the spur of the moment
which may not have been the best in hindsight, just as we do in the
So, I'm not sure what I conclude. I guess I'm saying that we
should try and grab every opportunity to engage in mathematical
conversations with our pupils, although it's never going to be
easy. We're never going to be able to support every child in a
class as we might if we were working with each one individually.
But perhaps we could just take stock and ask ourselves: are we sure
they have grasped that idea? What questions could we ask to probe a
little more deeply?
Thank you for indulging me.
Thank you also to Luke and his
This article first appeared in Primary
Mathematics, a journal published by The