Imagine for a moment a world without numbers. Many things would not be possible. The Olympic games might still exist, but there would be no world records to be beaten as there would be no way of comparing today’s races with those of the past. Buses might still run, but catching one would be pot luck as there’d be no timetables to say when the next was due. And there’d be no money, so we would have to barter goods and services rather than pay for them.
Something as simple as ordering enough pies for school dinners would be rigmarole. You’d have to tally up how many people there were, present the tally to the baker who would then have to match pies one-toone with each tally. Far easier to ring up and order 25.
Beyond this, mathematics shapes how we think about the world in a myriad of ways that cannot be counted. It is not just a tool for solving problems.
I’m using the term ‘tool’ to describe something that extends what I can do naturally.
Physical tools are obvious and abundant. I could not, for example, frame and hang a photograph on the wall without scissors, a glass cutter and a hammer – not to mention a host of other tools.
But it was the Russian psychologist Vygotsky who introduced the importance of psychological tools – tools of the mind that enable us to do things we otherwise could not. And mathematics is a powerful psychological tool.
Take the simple example of having, say, six eggs in a box, putting another five eggs into the same box and wondering how many eggs there are in total. The answer can be found physically by opening up the box and counting the eggs. But mathematics allows us to find the answer purely mentally (psychologically), and in many different ways. I can mentally try and picture the six eggs and imagine five more being added one at a time, counting up to 11. Or I can imagine the six eggs as five plus one, so five combines with five to make 10, and one more is 11. Or I may just know that six plus six is 12 and so six plus five is going to be one less.
The power of mathematics as a psychological tool becomes clearer as we think about larger numbers. Suppose there are 199 eggs in the box and another 201 are added. Now counting 201 on from 199 is not only impractical (and highly prone to error), but most seven-year-olds can reason that taking one off the 201 and adding it to the 199 quickly leads to realising the calculation is equivalent to 200 + 200.
A paper and pencil provide a powerful extension of the mind; it involves using a tool that is neither totally physical nor totally psychological. Using paper and pencil to calculate 37 x 28 involves making marks on the paper – a physical act – and doing some mental calculation (eight times seven and so forth) – a psychological act.
Sometimes, when children have done a lot of work on ‘standard’ written methods, they will try and ‘do’ a mental calculation as if they had paper and pencil in front of them, even when these tools are not provided.
Visiting a secondary school recently, I asked some 14-yearolds if they could mentally figure out what 40 x 0.8 would be? One girl, and a mathematically able one, was tracing in the air with her finger. When I asked her what she was doing, she explained that she was trying to picture the calculation as if it were set out on paper: 40 at the top with 0.8 underneath, then doing eight times zero is zero, eight times four and so forth.
I suggest that when pupils, like this girl, try to picture in their minds what they would do on paper, then mathematics hasn’t really become a tool with which they can think. Too much emphasis, too soon, on setting out calculations in a formal way can hamper this development of mathematical thinking. But there is another way that paper and pencil can support the development of psychological tools.
The Dutch Realist Mathematics Education at the Freudenthal Institute has done much ground breaking work in this area. Number lines for addition and subtraction and arrays for multiplication and division are now often used in classrooms. But I’ve seen them used as physical tools – ways of doing paper and pencil calculations that are nontraditional – rather than stepping stones to becoming mentally proficient, as originally intended.
The Dutch researchers distinguish between:
It’s easiest to understand these distinctions by looking at some examples.
Suppose children were asked to calculate 35 + 29. Ways they might do this would include:
As children explain such methods, the teacher can represent these on an empty number line.
In the language of the Dutch research, the teacher is creating a ‘model of’ what the children describe, translating their words into a visual image. When the teacher does this, she does not expect children to be working with an image of a number line in their heads. Over time, if the children are given plenty of experience in explaining their solutions to calculation and the teacher consistently creates ‘models of’ these on the number line, then they will start to use the number line themselves. They will start to draw number lines to help them figure out answers. There is a shift from the teacher providing a ‘model of’ the solution to learners taking this on as a ‘model for’ them to use themselves.
The Dutch research shows that, eventually, if the model is a good one (as the empty number line is proven to be) then learners come to be able to work with the model, but without needing to make actual marks on paper: the model is imagined and has become a ‘tool for’ thinking with.
The power of arrays as tools for thinking
A similar movement from ‘model of’ to ‘model for’ can be developed in the use of arrays to support understanding of multiplication. For example, given a calculation like 4 x 39, children might explain methods such as:
A teacher’s model of each of these explanations would look something like this. And again, over time, children will take up using the model for themselves and, with sufficient experience, can imagine it and use it as a tool for thinking.
Another way the array can support thinking about multiplication would be working on pairs of calculations like 6 x 8 and 12 x 4. The answer to both of these calculations is 48. Some children will think this is just a coincidence, others may have an idea about doubling one number and halving the other. Using the array to model what is going on here can help children begin to understand the underlying mathematical structure that allows the ‘double one number, halve the other’ rule to work.
As they work on similar examples and move to creating arrays as models for showing equivalence, children are beginning to understand mathematical structure, and developing the thinking tools for solving other problems.
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