Philosophical inquiry in maths 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. Do we say the "things to say" that are suggested? 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.