Alf Coles's Reading List

"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. 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."

"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. 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… 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 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."

"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."

"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. 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."

"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. 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. 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."