Within these books there is a sense of mathematics as a subject over which we can actually have some control, as something that is much closer to the humanities and aesthetic subjects where the role of the community plays a part. Following that through into the classroom, in some of these books there is a portrayal of classrooms or classroom cultures where children can be operating as mathematicians. When someone has just made a slip I correct it and move on. Exploring what might have happened for 28 times 28 to have been the answer is actually a useful thing to do.

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Obviously there are mistakes. But at a more conceptual level, if there are mistakes that might be considered errors of understanding, rather than seeing those as deficits, it seems to me much more productive and actually a much truer way of viewing the situation to just say that these people are operating under different rules and we can then explore that difference. Is it badly taught or are some of us just not good with numbers?

I would distinguish being good with numbers and being good at maths. It does seem to be the case that some people feel much more comfortable working with numbers than others. At higher levels, being able to think through complex problems in a systematic way and not jumping to quick conclusions are much more important skills.

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For some people, it can be an escape, a safe place. What do you make of the concern that the West or the US in particular — is falling behind in maths, compared to Asia? What little I do know suggests that the US takes quite a rote learning approach to teaching mathematics, with a focus on memorization. I have been to Japan to see some of the maths teaching there, and there was a much more conceptual focus compared to this image of teaching via memorization. In other words, the lessons I saw in Japan had a strong emphasis on children problem-solving, and on sharing approaches to problems.

There is always an interesting balance between developing a conceptual understanding and developing a fluency with procedures, and the Japanese children were developing a high level of skill, but in the context of solving problems that made sense to them. That would be one story for me of why there are some very good mathematicians coming out of Japan.

Then of course there are huge cultural differences in terms of what families value and the general approach to education. When I work with primary school teachers in the UK that seems to be a very common view of the subject. Those who are not are given special support. But there is evidence that if you are put into one of these bottom groups that actually you never get out of it. You will never then get contact with people who are getting excited about mathematics, and step 19 becomes a bit pointless and meaningless to you.

You just flop around there and make slower and slower progress in relation to your peers and that carries on up to age Then, in the UK at least, your life chances are very highly correlated to your attainment at age It seems to me that an alternative approach is possible. Certainly my experience confirms that. You can actually jump straight from step 19 to step 26 and pretty much fill in all the other steps along the way.

What do you mean by offering images of what you could do with step 19?

Can you give an example? Actually a perfect example of this came up the other day. The curriculum in the UK focuses first on the numbers one to ten and then 11 to I first came across this book at university in a course on the philosophy of mathematics. Looking back, it was one of my first experiences of how maths could be different to how I was taught it. In the book, Lakatos takes a particular area of mathematics to do with shape and recreates an imaginary dialogue where he and the characters in the book go through this extraordinary process of developing what mathematicians would call conjectures.

So they are developing predictions about how a particular property of a shape might operate, or which types of shape it might be true for. It was very important in the development of my own thinking, just about how mathematics can then become alive. On Amazon. Again, it puts across this more human side of the subject, where this is about mathematicians, as a community, deciding what the standards are for rigour and proof.

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One of the misleading things about the subject is that because a lot of this work happened a long time ago — and those standards have been agreed on for hundreds of years — it can appear there was never any choice about them. Actually, there was a lot of choice and a lot of debate about how things should be defined. The connection with how misleading the presentation in textbooks is, is important as well. Retrieved 25 October American Mathematical Monthly. Melody Maker. Archived from the original on 7 February Retrieved 20 October Anderson, John R.

Anders; Glaser, Robert Brookings Papers on Education Policy 1 : — Auslander, Maurice; et al. Sriraman, Bharath ; English, Lyn Theories of Mathematics Education. Strogatz, Steven Henry ; Joffray, Don Princeton University Press. Areas of mathematics. Category theory Information theory Mathematical logic Philosophy of mathematics Set theory.

Abstract Elementary Linear Multilinear. Calculus Real analysis Complex analysis Differential equations Functional analysis. Combinatorics Graph theory Order theory Game theory. Arithmetic Algebraic number theory Analytic number theory Diophantine geometry. Algebraic Differential Geometric. Control theory Mathematical biology Mathematical chemistry Mathematical economics Mathematical finance Mathematical physics Mathematical psychology Mathematical statistics Probability Statistics. Computer science Theory of computation Numerical analysis Optimization Computer algebra.

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