# Coming across multiplication and division

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- Coming across multiplication and division

## Coming across multiplication and division

“Mummy! We’re doing multiplication and division now!” This seemingly spontaneous statement is exclaimed by my now six-year-old daughter, as she embarks upon her breakfast before another day at school. I can’t deny, I am caught off-guard. But I look up at her, nod and smile encouragingly, and wonder what little anecdote or piece of information might follow this statement.

But nothing does; the breakfast bowl has distracted her once more.

I ask, “So what is multiplication?” She ponders this, in the very thoughtful way that children do; head on one side, eyes looking up at the ceiling. Finally, she looks at me, and says, “It’s this” (holding up her hands and making a cross with her two index fingers). I pause, a little surprised. Then I ask, “And what is division?” She immediately traces a horizontal line in the air and pokes out one imaginary dot above and one below. “It’s this” she says, in the slightly questioning tone of someone who thinks this is obvious but now they’ve come to say it out loud isn’t sure it is the response I am looking for.

We had no time for more, but as we hustled away from the table to get dressed and ready for the day, I thought about this interaction. It stuck with me. That response had triggered a really strong emotional response in me that groaned “Oh no” and made my heart sink. But why? Is that answer problematic? What could she have said that would have triggered a different response in me – a more positive one?

So, I thought about it. I let it sit in the back of my mind for the next few weeks as my daughter came home telling me of methods she had learnt, pictures she had been taught to draw (of “groups” and “arrays”), and these other things called “times tables.” I even allowed this quiet obsession to take the lead in a workshop session in a recent conference, where I asked the participating teachers, researchers and designers that same question: “What is multiplication?” I invite you now to pause and consider how you might respond (maybe the following three prompts are helpful here):

The first thing that I think is…

An image that comes to mind for me is…

Words I associate with multiplication are…

What I notice when I consider my own, and others’ responses are, firstly, that this is not an easy nor obvious question to answer. As is often the case, we tend to rely on analogy: multiplication is like this. To this end, examples tend to form the basis of what we say: examples of multiplication facts, statements, or number sentences; examples of particular strategies or methods for calculating; and perhaps, though less commonly in my experience, examples of images, pictures and representations.

This led me to postulate that my original question was not a good one, or indeed the “right” one. Perhaps it would be better to acknowledge different perspectives on multiplication (and division) and think about specific questions that I might hear (and have to respond to) that “belong” in some way to those perspectives. I choose here to frame these by posing two big questions of my own thinking:

**A) Am I thinking about multiplication and division as procedures or operations?**

If so, maybe I might hear/ask:

- How do I multiply/divide these numbers?
- How do I write down a multiplication or division calculation?
- What do I get if I multiply/divide these numbers?

**B) Am I thinking about multiplication and division as processes, concepts and/or ideas?**

If so, maybe I might hear/ask:

- What’s multiplication (or division) about?
- What happens when we multiply/divide?
- How are multiplication/division different from addition/subtraction?

The second of these perspectives (B) interests me the most, and there perhaps is the root of my unease with my daughter’s original response, which seems to sit squarely in (A). I find this a fascinating, challenging, and surprising set of questions, and I continue to think about them a lot.

This morning (many weeks after the initial interaction), my daughter is at home and I ask her if she would like to draw, write or make something that represents multiplication. She says yes, but that she doesn’t really understand what I want her to do. I ask her to tell me something she remembers about multiplication; she says, “Ten times ten is a hundred,” and writes down 10 x 10 = 100 on a piece of paper. We are in the kitchen and have a number of things available on the table at this point (largely because I have failed to tidy up): pens, paints, paper, various objects, Duplo, and a large tub of coloured centimetre cubes. I point to the written multiplication fact and say, “Could you use something to represent that for me? Something I could take a photo of?” and I leave her to it.

A little while later, she comes to find me. She is visibly excited to reveal her handiwork: take a look below and maybe take note of your initial reactions!

She seemed to enjoy this, so I ask a follow-up question: “Could you find another way, a different way to represent it for me?”. After a little more time on her own, she finds me and shows me this:

And finally, I ask, “Could you use something else, like “groups” to represent it for me in another way?” This takes a little longer. I hear an exasperated “Oh, I need a lot of cubes!” followed by fistfuls of cubes being scooped out onto the table. Eventually, I get the call: “I’ve done it!”

And this is what my wonderful child has created:

I encourage you, as I did, to take a good long look at each of these. Perhaps ask yourself these questions:

- What decisions has she made in her attempt to complete the (somewhat ambiguous) task I posed?
- What has she given attention to in each creation?
- What do you think each image suggests about the ways in which this child sees multiplication?

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