Philosophical inquiry in mathematics classrooms

Andrew Day (2014). The Numberverse: How numbers are bursting out of everything and just want to have fun. Carmarthen: Crown House Publishing.

Your answer to "What is mathematics?" has a huge bearing on how you teach mathematics. Numberverse answers the question through the subject's links with philosophy and, in the first few chapters at least, presents classroom maths as a philosophical inquiry of meaning. On that basis alone, the book is to be welcomed as providing a fresh perspective on how concepts can be introduced to primary pupils.

Numberverse has three intertwined parts. Firstly, at the start of each section, the reader is given a short introduction to the topic. This might cover tales of its historical development with which the teacher can enrich lessons; for example, we read about progressively more accurate estimates for π and the maths behind the design of arches. Alternatively, the introduction might discuss ways to teach topics in maths. Some of these are more successful than others. The suggestion for introducing fractions would overcome misconceptions I see survive into secondary classrooms; but countenancing the "adding zeros" trick for multiplying by powers of ten does not help pupils develop a conceptual understanding of place value.

The second part of each section - "things to do" - gives a precise classroom-tested script for teachers to initiate inquiry and an activity to follow. There is a diverse range of stimuli and activities: deep philosophical questions about numbers, prose and poems, standard investigations and problems to solve. My favourite is: How many squares can you form with four strips of paper and two half strips? Of course, stated in this way the problem might not provoke much curiosity in primary pupils. And that, for me, is the key message of Numberverse: "the genius in teaching is making people ready to be told" (p. 184). Draw the pupils in, arouse their curiosity through discussion, and, when they perceive a need for new knowledge, tell them.

The third part of the book will be of great interest to all inquiry teachers. In our classrooms, do we say the recommended "things to say"? Do we agree with the key words that form Numberverse's lexicon of inquiry teaching? Last year, I concentrated on holding the big picture in focus so students could link their exploration to the purpose of the inquiry. Is that the same as the key word "anchoring"? Perhaps not, but Numberverse challenged me to consider why not.

I also found myself considering the extent to which maths can be learnt through philosophical inquiry (as opposed to mathematical inquiry). At the start of the book, Andrew Day writes that the teacher is "controlling the process completely ... but not controlling the content" (p. 18). While I would expect pupils – certainly those in my secondary classes – to be involved in directing the process, I also think that Numberverse, on my reading, does not hold throughout to the second part of the statement. The axiomatic nature of maths does require the teacher to control the content to an extent.  

Within that "to an extent" resides the crux of classroom inquiry. The skill of the inquiry teacher lies precisely in finding the balance between eliciting students' existing knowledge and encouraging them to engage with new knowledge. It is the spirit of continuous engagement with pupils' understanding that shines through Numberverse. For that reason, the book is recommended reading for all teachers of mathematics.

   Andrew Blair, August 2014

Andrew Day comments on the review:

Thanks for the review. Yes, I do like it, and am very interested by your comments. I think you have understood the main ideas of the book really accurately, and your reservations about some of it are well-founded. My comments in response to some of your points are:  

In the chapter on place value, I meant to express as much doubt as certainty about how to teach it. I agree that teaching children to 'add zeroes' when multiplying by factors of 10 wouldn't help them to understand place value, and could even have a negative effect if not balanced by a genuine concept of place value. The point I was trying to make there was that 'banning' children from using certain tricks or rules of thumb can backfire - better to allow them at some stage but explain to the class that they are just tricks? Also, I was taught that 'the digits move to the left' when you multiply by 10 and the decimal point 'never moved'. I still can't see why saying the digits move is truer than saying that the decimal point moves? Happy to be enlightened on that one, though.

One thing I now wish I'd put in is that children are often introduced to the concept of place value when learning subtraction (when the subtracted digit in the units column is higher than the digit above it). It would help them a lot to explore place value a bit BEFORE then, so that they already know that a number in the tens column can be 'cashed out' into units. That is why I put forward the idea that the teaching of bases - at least introducing the idea that the way we represent numbers is one possibility among others - might... MIGHT be appropriate at a younger age. I just don't know.

I'm glad you think that the Things To Say and Key Words are up for discussion. I hesitated before adopting that approach as I don't like to be prescriptive. But in INSETS and CPD generally I find a lot of teachers - especially at primary level, where I usually work - get limited benefit from abstract explanations, even when they understand them fully. In order to get the teachers to put the ideas into practice I have to say 'do this' more than I would like to. Having said that, I have also found that certain instructions and questions, such as 'Try again' 'Why is that?' 'Show us' and comments such as 'Isn't that weird?' and 'Good thinking' are very effective ways of steering the class and can be used a lot. The idea behind the Key Words is that many teachers already know them, or will encounter them soon enough, and it might help to say how my ideas relate to them. There are a number of over-simplifications in the book (partly deliberate, for ease of digestion, partly the result of the limits in my own mathematical knowledge).

Anyway, I'm very grateful for your support and interested in the work you do, Andrew. So thanks again.