Ivor collects pencils. Ivor’s mum gave him 7 pencils. His dad gave him 4 more. Ivor gave 6 of his pencils to his sister. Did Ivor end up with more or fewer pencils than he started with? How many?

Both of these ‘problems’ (yes, problems is in scare quotes because they are the type you only meet in school maths, but bear with me, they have a purpose) can be answered by calculating 7 + 4 − 6 (and yes, there are other ways).

Yet children find the second problem much harder than the first. In fact, since we are not told how many stickers Ivor started off with, some children will say that the second problem cannot be answered. Why do they find it more difficult?

The difficulty is in the nature of the answers. In the first problem, five represents a quantity - the number of stickers Ivor ends up with. But the answer five to the second problem represents a relationship - how many more stickers Ivor had than when he started.

We often switch between using numbers as quantities and numbers as relations without paying attention to the distinction. But awareness of the distinction is important in supporting children’s developing number sense.

Take a simple calculation like:

7 − 3 = 4

If we read the ‘−’ as ‘take away’ then the answer four is a quantity - it is the quantity remaining when three things (stickers, marbles, zingbats) are removed from seven. When we ‘take away’ we can hold up the particular quantity that is left behind.

Reading ‘−’ as ‘difference between’, the four then represents a relationship - how many more there are in a collection of seven objects then there are in a collection of three. Unlike ‘taking away’, when comparing two collections there is no particular quantity from the larger collection that specifically makes up the difference. Four expresses an abstract relationship between seven and three.

I often invite teachers to show what picture they might draw to help a child understand 7 − 3. Almost invariably people draw seven objects and strike out, or somehow bracket off, three of them.

It is very rare for someone to draw a set of seven and another set of three and point up the difference. Even adults seem to prefer the concrete, the quantity, over the abstract, the relationship.

To fully develop children’s number sense we need to make sure that they get a rich diet of operating with numbers as relationships as well as quantities. I’ll be exploring this in future blogs.

Meanwhile, if you would like to read more about the research behind this, there is a great, free, set of reports - *Key Understandings in Mathematics Learning* - that you can download from the Nuffield Foundation website.

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Visiting a class once, my attention was caught by a boy, call him Tony, who at every opportunity for ‘hands up’ was as energetic in his ‘pick me, pick me’ actions as donkey in Shrek.

Chatting after the lesson, the teacher was bemused over Tony’s willingness, as he wasn’t that mathematically confident.

Having watched the lesson unfold it was clear that the teacher did what many teachers do: wait for several children to put their hands up before selecting someone to answer. I think Tony had figured out that the easiest way NOT to get picked was to be the first person to volunteer the answer.

The routine of ‘hands up’ is so ubiquitous in maths lessons that it can seem as natural as the sky looking blue, water feeling wet and Cliff Richard staying young.

But just because ‘it’s always been that way’ isn’t a good reason to carry on with some classroom practices. And there is good evidence that ‘hands up’ may not be the best practice.

How do you know if you are good at maths?

Dropping into a maths class I will often ask the children this question. There is a remarkable degree of agreement in the answers, which are usually along the lines of:

The practice of ‘hands-up’ for answers perpetuates the belief of speed as being the essence of being good at maths. Children who need a bit longer to figure things out simply don’t have the time they need or are discouraged when others around them get a hand up first.

The psychologist Carole Dweck has looked at the downside of such an attitude to learning and one of her perhaps surprising findings is that such beliefs hamper successful students as much as the less successful.

If you’ve been successful at school maths because you can do what is set without much effort, then what happens when the time comes (and it will) that you meet some maths that isn’t easily done and requires more effort? Dweck’s research shows that many learners then think along the lines of ‘perhaps I was wrong, I ‘m not that good at maths after all’ and conclude that maybe maths isn’t for them.

Behind this, Dweck argues, is whether learners view mathematical ability as a gift—something that you either have or not - or whether they view it as something that can be developed through practice and effort - a growth mindset.

The message is clear over-emphasis on quick answers perpetuates the ‘gift’ view of mathematical ability. ‘ Hands up’ leads to everyone losing out mathematically: the lower attainers in the short term, the higher attainers long-term.

A quiet ’thumbs up’ (held still against the chest) can still signal to the teacher when learners are ready, provide a calmer space within which to think, and dispel the myth that mathematical ability is all about speed.

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ODD ONE OUT

23 20 15 25

Which number is the odd one out?

Most people’s immediate response to ‘Odd one out’ is that 20 is ‘odd’ because it is ‘even’ (pun intended!). A moment’s reflection can open up a seemingly closed problem like this to some creative thinking.

Can we find a reason for each of these numbers being singled out as the odd one?

And how many different reasons for each number?

Here are some starters for 25. It is the only number in the set that:

is a perfect square number

is a factor of 125

has an odd number of factors

has digits that add to seven

….

Thinking creatively can engage learners in mathematical reasoning with the different reasons given acting as springboards for further inquiry:

Can you find some other numbers with exactly three factors?

How many numbers under 100 have digits that add to 7?

Creativity as a key to deeper learning?

One popular argument for encouraging creativity is that we live in times of rapid change and being creative helps learners cope with such changes. Over and above this the work of the psychologist Ellen Langer suggests that creativity can have other payoffs.

Langer has extensively researched what she calls the conditionality of knowledge: that what we hold to be ‘true’ relies on certain conditions (in this case 20 is the odd one out under the conditions of odd and even numbers).

Langer’s research on Mindful Learning shows that treating knowledge as ‘conditional’ rather than ‘absolute’ not only leads to creativity but also to deeper learning of the knowledge being explored. In other words, justifying a range of answers may help kids become mathematically smarter.

I discuss these ideas more fully in my new book ‘Transforming Primary Mathematics’

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I was lucky this week to hear the wondrous Ellen Langer give a talk. If you haven’t come across it, then I highly recommend her book ‘The Power of Mindful learning’. She spoke for over two hours and it was pure enchantment.

One of the key findings from Prof Langer’s research of over 30 years is that learning is more powerful if we are mindful of the conditionality of most ‘facts’. For example, in a lesson on fractions, a diagram like the one above is usually introduced as the pale blue shows ½. In contrast, Langer argues that presenting the diagram as it ‘could’ represent ½ makes the learning mindful and deeper.

Part of the argument for this is that the ‘fractions’ are not in the diagram itself but they are the result of someone somewhere at sometime deciding how the diagram should be read. In other words, there was a choice to be made over what mathematics the diagram could represent - not what it does represent.

By re-introducing this choice and conditionality into the teaching of it, the learner is encouraged not simply to take the mathematics for granted, nor to just commit something to memory. In contrast they engage in mathematical thinking.

So what else could the pale blue represent?

Well, it could be different if we are mindful of not taking for granted the condition that the ‘unit’ has to be four small squares. Under the condition of the whole picture as the unit, then the pale blue could be ¼.

Or we could set up the condition of comparing the pale blue to the dark blue and so the pale blue could be ⅓. The possibilities do not end there.

Of course this does not mean telling children that ‘anything goes’ and they are free to choose any reading they like. But by implicitly acknowledging in our teaching the human agency in mathematics we could help more learners come to see mathematics as a human endeavour worth engaging with, rather than something to keep at arm’s length.

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*At the age of eleven, I began Euclid, with my brother as my tutor. This was one of the great events of my life, as dazzling as first love. I had not imagined there was anything so delicious in the world. *

Bertrand Russell

Running a teachers’ professional development session recently, I was sharing some children’s mathematics. The children had investigated constructing arrays that allowed them to calculate products in a variety of ways. For example, one child explored how nine times twenty-two could be modelled from an array for ten times twenty-two by slicing off a ‘strip’ to reduce one side to nine.

The array presents a powerful model of why what might become a rule of thumb (to multiply by nine, multiply by ten and subtract the original number) is grounded in mathematical structure.

I got the teachers exploring 75 x 89 in similar ways, for example starting with an array for 100 x 89 and then working with the array and using halving to get to 75 x 89.

Towards the end of the session I got the ‘what’s the point?’ question.

This question often hangs in the air but there is not always someone brave enough to voice it. I asked for elaboration and the teacher declared that such activity was meaningless and had nothing to do with ‘real-life’. Taking a risk I asked the group (of around 120) whether anyone had actually found the activity meaningful. Thankfully around 2/3 of the room put their hands up. The questioner persisted that the activity was meaningless, despite my pointing out that since that clearly was not the case for the majority of people in the room, the activity itself could not be regarded as meaningless, only some people’s response to it.

We talk about mathematical tasks as having ‘meaning’ or being ‘rich’ but course tasks do not have ‘meaning’ or ‘richness’; these are qualities that people engaged in the tasks bring to them.

The discussion with the teachers moved in the direction that schools are advised to always relate mathematics to ‘real-life’, coming back to the utilitarian argument for mathematics. I’m not convinced that ‘utility’ actually does succeed in making mathematics more meaningful for more children. For some, yes, but for many children playing around with ideas can be ‘meaningful’ in much the same way that people find meaning in playing the piano, painting an abstract, or reading poetry; activities each of which could equally be argued as having little immediate impact on people’s lives.

The idea that someone might engage with playing with mathematical ideas simply for the intrinsic pleasure of the play itself is alien to many. But that’s not the fault of with mathematics itself, and all to do with the way we continue to teach it.

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The Age

The novel and movie, Never Let Me Go, paints a chilling picture of children whose destinies are determined from birth. Without spoiling the plot for anyone not familiar with it, the story revolves around a group of children who are educated to accept their roles in a society that treats (some) humans as commodities, bred only for the service of others.

Never Let Me Go is set in a fictional parallel world, but the rhetoric from England, far from being science fiction, suggests schooling there is heading towards the grooming of children for particular destinations.Education Secretary Michael Gove recently announced a new education league table for England, showing which state schools are preparing children to be "university-ready" — numbers of children going on to graduate with an honours degree. (Presumably everyone from a private school graduates with honours.)

Why not go further and shrink-wrap kids, slap on an "oven-ready" — oops, "university-ready" — label and ship them off? Any rejected as "non-university-ready" could be ground down and turned into "worker nuggets".

In the 20-odd years since the introduction of England's national curriculum (not the United Kingdom — no grudges against the Scots, Welsh or Northern Irish, please) the pressures on schools, parents and, particularly, children, have been worse than anyone predicted. As Australia gets set to have its own national curriculum, what might be learnt from England's follies?

A national curriculum on its own may be no bad thing. The rot sets in with the paraphernalia for measuring results and "standards". National testing is not evil but when the results were used to put England's schools into league tables, the mischief started. At middle-class dinner tables from London to Manchester, parents anxiously discuss whether they sent their children to the "right" school, or plot the "best" catchment area to move to. Parents without the savvy, mobility, or SUV to get the kids to school, are left with little choice and often with schools that go into a tailspin as their results drop.

Will data from My School become more public and a similar tiering of the education system develop in Australia?

Now, yet another review of England's national curriculum (new government, new curriculum) came with Mr Gove announcing that England had "sunk in international league tables and the national curriculum is substandard". One review panel expert declared the national curriculum "on the front line of the battle for the UK's future competitiveness". Ah, the Cold War may be a distant memory but England still has a battle to fight.

The saddest aspect in all the talk of "university-ready" children and global competition is the scant sense that education might provide a good experience for children and adolescents in the here and now. Everything is about getting them ready for the next stage — what the universities need, what the global economies want. Just like the youngsters in Never Let Me Go. What happened to thinking about what a six-year-old needs to be a six-year-old?

Of course Australia is more moderate, although the opening of the Melbourne Declaration on Educational Goals for Young Australians does state: "In the 21st century Australia's capacity to provide a high quality of life for all will depend on the ability to compete in the global economy on knowledge and innovation."

There is another way. As British academic Robin Alexander puts it, global perspectives are often "contra-national" rather than inter-national. Continuing to see the world in terms of trade and a need for local, national, economic supremacy may no longer be viable or sustainable. Rather than educating children to be global competitors, we would be better off helping them see themselves as global citizens, in the broadest sense of the moral obligation that carries.

Everyone can gain from a global education based on co-operation and collectivity rather than competition and supremacy. An education based on acknowledging that globally we are more interdependent than independent. An education working to help the world be united, not divided. That would be an education system to be proud of globally.

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Then I’d be able to enjoy them again for the first time.

I sympathised with this girl’s comment, overheard on the train home from work. We could all list a multitude of experiences that thrilled and engaged the first time round but lost frisson with repetition. Of course we can’t go back (and if we could return to the first time then we wouldn’t be getting the excitement twice over anyway), but one of the joys of teaching is to witness children’s ‘Aah’ or ‘Aha’ moments - a vicarious re-living of experiences.

Her comment got me mulling over what mathematics I could remember learning that, first time round, had caught my interest. I couldn’t recall much of my primary maths lessons other than doing pages of ‘sums’ and I enjoyed those more for the precision of neatly penning one-digit-to-one-square than for the pleasure of the calculations themselves. Nice graphics, dull mathematics.

Then I recalled a student teacher introducing us to curve stitching. No longer fashionable, so for anyone not familiar with the idea, it involves constructing two perpendicular axes with scales from, say 1 to 9, and then joining up the pairs of points on the axes that total to ten. I remembered forcing a bodkin threaded with thick yellow wool through fibrous unbleached board. As more pairs got joined a little bit of mathematical magic happened: the straight lines combined to reveal a curve.

Curve stitching probably isn’t taught much now as it lacks immediate application and doesn’t appear on international tests. A shame since I think it pinned down some important impressions for me. First, it introduced me to the world of mathematics beyond arithmetic and showed me the beauty in that world. Second, when, years later, I met calculus and differentiation, the traces in my body from, literally, stitching a curve out of straight lines, gave me the physical grounding of the meaning of tangents to curves.

I can’t remember the name of the student teacher and I doubt she chose to teach curve stitching as a foundation for calculus. But we should not underestimate how what we teach today may echo down the years. Another joy of teaching.

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When a bowl or cup is freed from the human hand as a cupped hand of clay, we can take in hand or pass to another hand what the hand has freed from the hand.

Kenneth R Beittel, Zen and the Art of Pottery

Over the holidays I took a pottery course. Our teacher, Jane Sawyer, trained in Japan and introduced us to some classical techniques, beginning with 'wedging' the clay: a strenuous kneading that in her hands produced a beautiful nautilus spiral, but in my hands turned to, well, a knarled lump of clay. But I got a feel for the clay as I rolled and squished and stretched it.

At the potter's wheel, trying to 'centre' my clay I was struck by the centuries of tradition behind replacing the hand with the bowl and got to thinking about tradition in teaching mathematics. Perhaps the 'traditional' algorithms for addition, subtraction, long multiplication and long division are the mathematical equivalent of throwing a pot and teaching these means celebrating our mathematical heritage, joining the long line of those before us who had gone through similar processes.

After lunch, two things made me rethink this reverie. First was when Jane showed slides of Japanese pots, including one of an exquisitely elaborate pot more than 4000 years old. Mathematical algorithms, in that time scale, seem rather less 'traditional' having only been around a few hundred years.

Second Jane mentioned that the hard labour of wedging clay isn't so important now as many potters happily use a machine to do this (which has also opened it up to more women potters). I decided that algorithms are more like wedging than throwing. We have machines that can wedge our calculations, machines that can free us up for the creative part of mathematics: solving problems. We might want children to experience algorithms, to know of their place in the mathematics tradition, but they are not the core of the discipline. And children certainly do not need to spend hours being drilled in them.

As my first wobbly but complete bowl came off the wheel, I felt the same satisfaction that I get from resolving a mathematical problem after several false starts. That's the tradition of mathematics we must pass on.