The question “What is five?” is a tricky philosophical conundrum, but one which helps set the scene for a review of this book.

It is easier to explain what two sets have in common when they both contain five things - so that you can pair up the elements of the two sets, with none left unmatched. For such a reason, Piaget suggested that understanding of matching (one-to-one correspondence) precedes proper understanding of number, and most primary maths schemes published over the past 25 years include pre-number activities on things like matching. The same theory suggests that counting sets is fraught with difficulty, since this entails matching objects with abstract number-names. If that’s the theory, what happens in practice?

About 10 years ago I was asked to do some maths with groups of children in a special school. The pupils had various learning and behavioural needs. In one activity, I showed them two similar looking piles of cubes. There were 15 cubes in one pile and 14 in the other. I gave one of the piles to John (say) and the other to Jane. Then I asked the children whether John had more cubes than Jane, or vice-versa, or did they have the same number. In every case, the children solved the problem by counting John’s cubes and then Jane’s, remarking that 15 was more than 14, so John had more than Jane. I asked whether they could solve a similar problem (different numbers of cubes) but without counting. After a while, some children suggested joining the cubes to make two “sticks”. Nobody suggested anything like matching John’s cubes with Jane’s, although that is theoretically a less sophisticated strategy than counting.

Later, I tried the same activity with some undergraduate students. The results were more or less identical. Thus, it would seem that an operational approach to number based on counting is significantly preferred to one based on matching, and that children can do an awful lot with numbers without “fully” understanding them in Piaget’s sense.

In this book, Ian Thompson sets out to debunk the whole idea of pre-number activities with sets, in order to propose a different kind of early years number curriculum which builds on counting. There is a grim irony in this, since for years teachers have tended to dismiss young children’s ability to recite the first number names in sequence as rote sing-song, meaning-less and devoid of understanding.

There are many splendid chapters in this collection. For me, the writing of Ian Thompson himself is particularly lucid and thought-provoking. His five chapters range over the status of pre-number activities; the role of counting in deriving new information about numbers from known number facts; bridging the gap between mental and written methods; and practical strategies for developing young children’s counting strategies.

I enjoyed Carol Aubrey’s well-documented account of the discontinuity between home-maths based on counting and school-maths based on sets, and Penny Munn’s thoughtful critique of Martin Hughes’s well-known findings on the gap between children’s ability to visualise and manipulate quantities, and their incomprehension of unconventional sign systems for number.

Thompson concludes with clear proposals for an alternative approach to teaching early number, based on counting, mental calculation and teaching styles that exploit and develop children’s interest and creativity. Along with some relevant and readable research reports, the book includes plenty of clear suggestions for practical action in the classroom. The style of writing throughout is clear and accessible. This is the kind of book that gives educational research a good name.

Don’t be fooled by the comfortable familiarity of the title. This is a radical and influential book, that ought to be read and acted upon, and that will continue to be quoted and discussed for some time to come.

Tim Rowland lectures in primary mathematics education at the Institute of Education, University of London.