Informally taught ideas on sets and equivalence relations underlie almost all school mathematics. Much of this work can be done on a visually intuitive basis using Venn, Carroll or tree diagrams, and it proves to be an efficient means for the refinement of children’s language and logical thinking. Moreover, not only are such ideas vital for the formation of number concept and counting, but they assist greatly in processes of geometric classification (congruence, similarity etc) and the specification of sample spaces in elementary work on probability.

As for number concept and counting, young children undertake many pre-numerical activities on classifying and sorting objects into sets, and this is very much in harmony with the widely accepted foundational work on number, done by Cantor and Russell in the early 20^{th} century. A natural number, then, is an equivalence class of sets, that are in one-to-one correspondence. For example: 3= [{cat, dog, pig}], and the *Collins Dictionary of Mathematics* illustrates this with slightly different notation:

3 = |{knife, fork, spoon}| = |{cat, dog, pig}|

In practice, a particular natural number is specified either by the appropriate word or symbol or by referring to a specific class representative, so that, when a teacher says ‘Show me three’, a child will respond by pointing to some group of three objects (usually three fingers!), and ‘threeness’ is a property of any such representative of this equivalence class. A natural number, of course, is not the same thing as the symbol that represents it, because ‘four’, for instance, can be represented by various symbols, such as 4, IV or 100 in binary form; but this is not to deny the existence of the concept of ‘four’.

Teachers will also provide other graphically stimulating visual aids (and practical activities) seeking to establish the link between ordinal and cardinal number, usually by means of sequences of sets, such as:

* * {cat} →{cat, dog}→{cat, dog, pig}→{cat, dog, pig, goat}→ etc.

1 2 3 4 …

However, not only is this a correct pedagogical progression, but it is also compatible with formal definitions of ordinal number, as in the Collins Dictionary , which says that ordinal number is a measure of a set that takes account of the order as well as the number of elements.

As for the book under review, it seems to be partly addressed to educational psychologists and to those concerned with the development of the primary mathematics curriculum. Unfortunately, I have spent some fruitless hours trying to relate its contents of to the tried and tested viewpoints expressed above. For example, the author presents us with the over-arching concept of numerosity, a notion that I am still struggling to comprehend. Here are a few of the author’s statements on this subject:

- “The numerical properties of collections are termed
*numerosities* and distinguishing that term from number in its symbolic sense will help to keep the relation between numbers and quantities clear.” (p. 4)
- “Numbers exist only as symbols, whereas quantities are properties of things.”
- “Sets are not merely physical collections of things, just as piece of string is not in itself a length. A collection of objects becomes a set, just as a piece of string becomes a length, only when it is considered as a quantity to be evaluated in relation to other quantities…” (p. 7).

And then, having searched the book’s index to ascertain the author’s explanation of ordinal number, I found the following comment on the top of page 23:

An intriguing possibility is that the transition from distinguishing only between same-numerosity and different numerosity pairs of arrays to also distinguishing between pairs in which the second array is less than versus greater than the first corresponds to the emergence of genuinely numerical representations. After all, *ordinality* is what distinguishes quantitative inequality from non-quantitative difference.

Having also looked at the chapter 6 (Understanding Fractions), I came to the conclusion that, in relation to primary mathematics, this book clarifies very little and obfuscates many aspects of elementary mathematics that are amenable to lucid exposition. It therefore comes highly un-recommended.

Peter Ruane (ruane.pandh@blueyonder.co.uk) was Senior Lecturer in Mathematics Education at Anglia Polytechnic University, England. His research interests lie within the field of mathematics education and the history of geometry.