Proofs and Refutations: The Logic of Mathematical Discovery
by Imre Lakatos

2

The Common Sense of Teaching Mathematics
by Caleb Gattegno

3

Calculus by and for Young People
by Donald Cohen

4

Teaching Mathematics: Towards a Sound Alternative
by Brent Davis

5

Starting Points for Teaching Mathematics in Middle and Secondary Schools
by Banwell & Saunders and Tahta

What is the best way to teach maths? Maths teacher and author Alf Coles recommends books that offer some clues.

Alf Coles

Alf Coles studied maths and philosophy at Oxford University. He went on to spend 15 years teaching maths in secondary school, including 7 years as head of a maths department. He is now a senior lecturer in education at Bristol University. He is currently working on a project focused on tackling underachievement in primary school mathematics in collaboration with the charity "5x5x5=creativity" and funded by the Rayne Foundation. His book, Being Alongside: For the Teaching and Learning of Mathematics, analyses the teaching and learning of students and staff at the school where he worked.

These books chart my own thinking about maths teaching and also the subject mathematics. They encapsulate the journey I’ve gone on, from seeing maths as this very right/wrong subject where the only way to approach teaching it is to tell people how to perform some sort of procedural algorithm, and then get them to practise it. That image of teaching is probably many people’s experience across the world of learning mathematics, but through these five books I have come to a very different view both of the subject of mathematics and the potential for how it can be taught.

Tell me more.

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. There isn’t some absolute truth of the matter, because actually mathematicians argue about definitions and about proofs and there’s a human dimension to it.

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. Children can be creative and exploratory in their learning of mathematics, and that doesn’t have to come at the expense of their attainment on standard tests or their learning of those algorithms and procedures that are commonly the focus of maths teaching.

So if you ask “What’s 2+2?” and a child answers “3” is that allowed?

When someone has just made a slip I correct it and move on. But often when people make what might be classified as an error by the wider community, they’re just operating under a different set of assumptions. So rather than just saying “No, that’s wrong,” it’s often quite interesting to explore where their answer has come from. To give you an example from my own teaching, an activity I often organize at the beginning of a class — as a mechanism for everyone to learn each other’s names — is that a first person stands up and says “My name is John.” A second person then has to stand up and say, “You’re John and my name is Fred.” The third person has to stand up and say, “Your name is John, your name is Fred, and I’m Alf.”— and it goes on like that. Maybe there are 28 people in the class and at the end I say, “So, if we consider what’s just happened – how many names have been said altogether?” Clearly, the first person said one name, the second person said two names and the third person said three names and so on up to 28. But doing this activity with 11-year olds, someone will almost always say “The total number is 28 times 28.” That’s clearly wrong, but, in a sense, it’s an answer to a different thing that might have happened. Exploring what might have happened for 28 times 28 to have been the answer is actually a useful thing to do.

Why do they think it should be 28 times 28?

That would be the answer if everybody had said everybody’s name. That distinction is then quite useful and generally helps the class realize what they’ve got to do, which is to add up one plus two plus three all the way up to 28. So the perspective is not so much that there aren’t right and wrong answers, but that, in a sense, the answers are dependent on the set of assumptions you are working under. Obviously there are mistakes. We can just make mistakes in carrying out a process, and I’m not shying away from that. 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.

Why do some of us, like me, do fine with most of our subjects at school, but come out feeling we’re really bad at maths? 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. But that doesn’t seem, to me, at the essence of what it is to be a mathematician or to be successful mathematically. What is valued and prioritized, particularly in the early years of school, is basically arithmetic and it doesn’t bear a strong relationship to the kind of skills you need to be successful at maths at ‘A’ level and beyond. I’m not suggesting they’ve got things wrong at primary school, you have to develop your sense of number and so on, but I think an emphasis on the quick recall of facts — which is often a characteristic that is used to distinguish between people who are good at maths and people who are not – isn’t a very important skill for higher levels of mathematics. 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.

In answer to your question about why so many people feel differently about maths than other subjects, I do believe that when maths is presented in a very right/wrong manner, then, if you’re an adolescent and the rest of your experience is very fuzzy, it’s hard to spend time in this world where everything is black-and-white. When you’re interested in negotiating boundaries and navigating uncertainty and the relationships within that, why would you want to bury yourself away in a black-and-white world which clearly has no resemblance to your lived experience? It’s not my idea, but some people have suggested that as an adolescent, to really get into maths you’ve got to have had some kind of psychological damage. For some people, it can be an escape, a safe place. If you don’t need that safe place, why would you be interested in maths?

What do you make of the concern that the West or the US in particular – is falling behind in maths, compared to Asia?

I don’t have a lot of experience of education in the US. What little I do know suggests that the US takes quite a rote learning approach to teaching mathematics, with a focus on memorization. There have been big moves to try and shift that, but I understand that’s been one of the problems in the past. 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. But I’m not in any sense an international expert, so I’m very hesitant to make these kinds of generalizations…

What about your own pupils – you’ve taught secondary and been involved in primary school teaching as well. What’s your general feeling about why children fall behind at maths?

It’s very easy to conceive of mathematics as this building block subject where if you want to learn something at, say, step 20, you have to have steps 1-19 securely in place. When I work with primary school teachers in the UK that seems to be a very common view of the subject. One of the implications is that if a pupil starts falling behind — you start working on step 23 and they haven’t got to step 20 — then they need to be taken out of that class and supported to try and get to step 20. One of the effects of that in the UK is that it’s not uncommon to have a class of children aged 4-5 already split up into people who are high attaining at maths and those who are not. 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. While your peers are then going on to steps 21, 22, 23, 24, 25, 26, you’re still struggling around at step 19. 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 16. Then, in the UK at least, your life chances are very highly correlated to your attainment at age 16.

It seems to me that an alternative approach is possible. If you think about your learning in almost any other sphere, it doesn’t seem right that it should be this very linear, step-by-step process. Perhaps what’s actually more important than whether you understand step 19 is how you’re feeling about the subject. If you’re actually feeling positive and you’re in an environment where some people are excited about the subject — and you’re offered images of what you’d be able to do if you did master step 19 — then maybe that’s more important than trying to fill in every single block. Certainly my experience confirms that. When I’ve worked at secondary school with people who have come to the school underachieving compared to the norm, what will get them through is if they actually change their emotional relationship with the subject. If they begin to start seeing themselves as mathematicians, and start feeling they have some control over the subject, that it’s not just this thing that’s outside them where somebody else tells them whether it’s right or wrong, but that they’ve got some agency, then that can completely switch around, in a very rapid period of time, what they’re able to attain. 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. It’s at the primary level, so students aged 6, 7, 8. The curriculum in the UK focuses first on the numbers one to ten and then 11 to 20. And a teacher was talking about some of her students getting “stuck in the teens” — as though their understanding of mathematics had somehow got stuck after the number 10. Well, the numbers 11 to 20 are the only irregular part of the whole system and, actually, if you’ve understood the numbers 1-10, you can start immediately working with 100s, just by adding that one single word. If you know one and one is two, it’s actually not hard to figure out that 100 plus 100 is 200. That would be an example where if someone is stuck in their understanding of the teen numbers, I don’t think it’s particularly useful to then really focus on that. It’s much better, in my view, to try and work with them on the larger numbers and to see the patterns that are there in the rest of the system. That will help them sort out what’s happening in the teens, without even consciously focusing on it.

OK let’s find out a bit more about all these ideas by going through your book choices. Your first one is by Imre Lakatos and it’s called Proofs and Refutations.

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. There’s this amazing description of a process which, for me, is at the heart of doing mathematics, which is coming up with conjectures, coming up with predictions and then the search for counter-examples that don’t fit that conjecture, and, as a result of that, the refinement of the conjecture itself. It’s quite a scientific process: you come up with an idea and you try and refute it.

There’s a phrase, “mathematizing”, which is about trying to capture the process of doing mathematics as a mathematician and this book is just a beautiful description of that. It was very important in the development of my own thinking, just about how mathematics can then become alive. OK, other people may have gone down these steps of logic before, but that doesn’t make the process any less valid. Those ended up being guiding ideas for me as a teacher in my classroom, trying to set up situations where the children can come up with conjectures, can come up with predictions and they can work on testing those ideas, and trying to find examples that don’t work and so on.

On Amazon.com one of the reviewers of this book said it showed how misleading the textbook presentation of maths really is. Another reviewer pointed out that the book shows that even famous mathematicians got some of these things wrong, and that things that seem obvious now, weren’t obvious at all at the time…

That’s absolutely right. 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. 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. Generally, in a textbook, a mathematical theorem will be presented at the beginning of a chapter and then there will be some very neat proof offered next. You get no sense at all, really, of the toing and froing that took place to arrive at this, or how this theorem has come about, or even, in some cases, what problem it was there to solve. It’s as if, in the textbook, you get this very, very condensed nugget of a theorem and its proof and the whole messy genesis of it is somehow airbrushed out.

Let’s go on to your second book, The Common Sense of Teaching Mathematics.

I could have chosen almost any book by Caleb Gattegno and in fact one of the exciting things is that most of his books are now available for free on a website, www.calebgattegno.org. Gattegno was a maths educator who talked about, in his own practice, teaching the entire five-year secondary maths curriculum in 18 months — and teaching it to mastery. What he meant by mastery was that students were as good at performing these things as he was. Gattegno did a lot of work in teaching languages, and he basically felt that learning mathematics was like learning a language. He looked at how young children learn languages, and he developed a curriculum which, certainly in his own hands, demonstrated it was possible to access all those powers we have as learners, for learning our mother tongue, in the learning of mathematics. Hence the speed and skill which he was able to impart to his students.

He advocated a route through this whole debate about “should I teach for understanding or should I teach rote learning?” What he did was devise activities where children could very quickly gain a symbolic mastery and then through a more creative exploration within those symbols, would develop their own understanding. In a sense, to quote a man called Dick Tahta who is actually the author of one of my other books, the teacher takes care of the symbols, the teacher is there to design activities which will mean the children get a sense of how these symbols work, and the sense takes care of itself. We leave it up to the children to actually make their own connections about how these things work. So it’s not denying the importance of understanding, but it’s suggesting that if that’s what you focus on right from the beginning, that’s quite an inefficient way to learn mathematics. By devising game-like activities which involve these mathematical symbols, you can get children into using symbols in a very sophisticated way, and through that use, develop their understanding.

So to try and ground that suggestion, again going back to working with primary children, Gattegno from a very early age would get children working with very large numbers. He had a chart where it would be possible to get children working with hundreds and thousands and tens of thousands and writing those symbols accurately from a very early age. I don’t think it matters what the children understand by being able to say, age 4, 10,000 plus 10,000 is 20,000. It doesn’t really matter what that means – they can get excited by using these symbols and names. I’ve certainly seen that. They do get very excited. In a sense the understanding or the meaning of those, and the relationships between those words develops over time. It’s not like I need to understand what 10,000 means before I can start using it and playing with it.

This book was written in the early 1970s – has his teaching had a big impact?

At that time he was a well-known figure, he was written about in Time magazine and I don’t know if you know the Cuisenaire rods? Cuisenaire was a Belgian teacher. They’re these different coloured rods starting with a white cube, going up to an orange one that is length ten. Gattegno was involved in the promotion of those materials, and I think at one stage every primary school in Canada had them and they were very well used in the UK. He certainly has had influence, but I don’t think many people today would recognize his name. It’s not clear to me why that has happened. Partly, it’s his writing. The book I’ve chosen here is one of his more accessible ones, it’s quite curricular focused. When he writes in a more general way about mathematics, it’s almost impenetrable. He has a whole system of thought about what it is to be human and learning, and really I’ve only been able to access his work through working with it in a group, with groups of people who are interested in the same thing. It’s not an easy approach. What he’s advocating depends a lot on the awareness of the teacher. He developed these materials – like the rods – and I think they can be spectacularly successful, but they’re not easy to use. They’re not the kind of thing that you can just pick up and go into a classroom and from day one expect to be successful…

Isn’t that one of the problems, that for someone to be a really good teacher they have to be rather brilliant themselves? They have to be completely on top of the subject matter, in order to be able to explain it simply to their pupils?

One answer I have to that links with a research project I’ve been involved with at a primary level, where teachers have been getting into this idea of positioning the students as “becoming mathematicians”. This has allowed some of these teachers to let go of their worries about not necessarily knowing what mathematics will come out of a particular activity. The teachers can place themselves alongside the students and say, “Well, actually, as a mathematician I’m not sure what the outcome will be, but maybe as a mathematician this is the question I’d ask, or I’d try to pursue that pattern.” On one level I think you’re right, but there are alternatives. The learning of mathematics can turn into something where the teacher is not needing to offer a very clear explanation, but a process of exploring and discovering things together with the children.

I like this idea of children defining themselves as mathematicians, because I do think a lot of the problems with maths come from people writing themselves off and saying “oh I’m no good at maths” and giving up. Whereas if you say “Right! I’m a mathematician!” you feel good about yourself. “Maybe I could be the next Fermat…”

I think you’re right. In this primary project, the kids have been loving even just trying to say the word mathematician. It’s powerful, I think.

OK so tell me about Calculus by and for Young People. I love the subtitle: “Ages 7, Yes 7 and Up”

This is a wonderful, very short book by a man called Don Cohen. It’s just an image, for me, of what’s possible. If we do turn the situation around and get children interested in exploring ideas and feeling confident enough to follow through on those ideas, then, actually, things that are said to be incredibly complex and you can’t do them till you’re 16, Don Cohen — as Gattegno did in his own work — has shown that to be nonsense.

So Don Cohen gets children excited about ideas of infinity. One of the activities he does with very young children is getting them to consider what happens if you add 1/2 to a 1/4 and then add on an 1/8 and 1/16 and a 1/32 and so on to infinity. He does this by getting people to imagine a square. Picture a square in your mind and imagine shading half of it. Then shade a quarter — a half of the bit that’s unshaded. Then you shade an eighth – again, a half of what’s unshaded. And then you shade a sixteenth, again you’re shaving a half of what’s unshaded. Quite quickly, even young children can get a sense that they can carry on shading half of what they’re left with forever. They’re filling up more and more of this square, but there’s always going to be a tiny bit left over. They get a sense that there might be this infinite process.

Those are extraordinary ideas, I think. On one level quite counter-intuitive, but incredibly exciting and certainly I’ve worked with children of 9,10,11 on them, and there’s no way they can’t get their head around them. In this little book that’s developed and he does go into calculus, he gets children finding areas under straight lines and generalizing. It’s an exciting book, and for me just an extraordinary image of what might be possible in the classroom.

On to your next book, Teaching Mathematics: Towards a Sound Alternative. I have to confess I tried reading it but I still have absolutely no idea what it’s about — please tell me.

This book was an important part of my own research journey. It’s by a man named Brent Davis and it’s a write up of his PhD thesis. What he got interested in was trying to say something about the listening of the teacher. I just found that such an appealing idea. From when I first read it in 1999, the idea of the importance of listening has been a recurrent theme of interest for me, both in my teaching and in my research. In the book, Davis distinguishes an evaluative form of listening — where as a teacher what I’m basically doing is saying this response is right or wrong – and contrasts that to a non-evaluative form of listening. He then further differentiates that, but I don’t think that’s so significant.

What seems to be powerful — from when I started looking into this idea and doing video recordings in classrooms trying to analyse the listening that was taking place — is that it really did seem possible to distinguish between what happened when a teacher evaluated a student comment (basically saying “Yes I agree” or “No I don’t“ or some variation of that) and the dialogue that ensued, compared to when a teacher made a more non-evaluative response, for example, inviting someone else to comment, or writing the answer on the board and asking if anyone had a different answer. In the video recordings I took, the classroom somehow became alive as soon as the teacher stopped evaluating. A space was created for other voices to be heard. If you imagine a whole class discussion going on, very often the pattern of dialogue will go “Teacher-student-teacher-student.” What I got interested in was when it might go “Teacher-student-student” or “Teacher-student-student-student-teacher” and so on. In the small sample of classroom data I took, the only times the ping-pong pattern was disrupted was when the teacher was responding and listening in a non-evaluative way.

Which ties in with your comment at the beginning about making maths more like the humanities, where something is put out there by the teacher but then everybody discusses it. Can you give an example or anecdote?

In my Master’s dissertation I looked at one particular teacher and tried to analyze the listening in her classroom. Over the year in which I was taking videos, there was a real shift. The result was that the role of the teacher became much less important. There was this amazing lesson that I saw, where the teacher just wrote 1/2 plus 1/3 up on the board. In a sense that’s a closed activity, in most traditional classrooms it would just be learned by rote. But because this teacher had established a classroom environment where the children were expecting to have their say and question what each other thought, this extraordinary discussion ensued about how you might figure out the answer. It led to some really deep mathematics being discussed about fractions, and the children doing some really interesting work on different sums that might come to the same total. It feels to me like the difference here is in the classroom environment that is created as a result of the teacher taking different approaches to listening.

We’d better get to your last book, Starting Points: For Teaching Mathematics in Middle and Secondary Schools.

This, for many years, was my Bible of teaching. I got to know about it through Laurinda Brown, with whom I have worked and co-researched for many years now. The third author, Dick Tahta, who is dead now, was a close friend of Laurinda’s. I knew him in the last ten years or so of his life and he influenced me hugely. It’s a very simple book on one level. It begins by talking about ways of working in the classroom and there’s a section on the role of the teacher. He asks, “What is the role of the teacher? How can he or she create the conditions in which creative and independent work can take place?” That, in a nutshell, is what I’m interested in in my work in mathematics.

Most of the book is just page after page of starting points for mathematical activity, which generally have something visible or tangible or something the children can just do straight away, but from which they are able to ask questions, spot patterns, make conjectures. Dick Tahta was certainly very influenced by Gattegno in this. They are just very rich starting points for children to get into the process of thinking mathematically. As a teacher wanting to develop a more creative approach to my own teaching, it was just an invaluable resource. Time and again, these activities would end up getting kids energized and excited and engaged and asking questions.

Can you give a specific example?

One is a game called the function game, where he imagines the teacher having some rule in mind. So with quite a young group, the rule I might have in mind is “add one.” So I might write the number three on the board and a little arrow (this is all ideally done in silence) with a four next to it. Underneath, I might write seven with an arrow and then eight. Then maybe I’d write six underneath, with an arrow, and hand the pen over to the children – or motion with my pen towards the class and try and get someone to come up to the board and fill in what they think the answer it. With older children the rule might be square the number – you can make it as complex as you like. I then give them feedback – so if a child comes up and doesn’t write seven, I draw an unhappy face. It’s the kind of activity which children can get very engaged in and can just lead in so many different directions.

Do you feel you’ve had results with your teaching methods?

At the school where I used to teach our results were pretty good age 16. We had a lot of people coming through to do ‘A’ level and it was quite often the most popular ‘A’ level choice [Editor’s note: in the UK, children after age 16 choose only three subjects for their last two years of school, at the end of which they sit ‘A’ level exams in those subjects]. We were the only school in the local area to teach Further Maths as an ‘A’ level, which at that time was a subject in major decline outside public [private] schools. Those things for me were indicators that we were successful in developing in quite a few children a sense that mathematics was something they could be engaged in and excited about. I was teaching in a school with quite a disadvantaged catchment area, and we had a couple of people getting into Oxford to read maths and quite a lot of people doing maths-related university degrees. So yes, there were some good responses from the children.

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Alf Coles

Alf Coles studied maths and philosophy at Oxford University. He went on to spend 15 years teaching maths in secondary school, including 7 years as head of a maths department. He is now a senior lecturer in education at Bristol University. He is currently working on a project focused on tackling underachievement in primary school mathematics in collaboration with the charity "5x5x5=creativity" and funded by the Rayne Foundation. His book, Being Alongside: For the Teaching and Learning of Mathematics, analyses the teaching and learning of students and staff at the school where he worked.