The declared intention of this book is to offer insights to those who wish to learn mathematics, or to teach it. It is based upon the premise that there exists something referred to as a 'glass wall', which is said to be an ongoing impediment to mathematical understanding.

On reading the book, one soon realises that by the word *mathematics* the author really means basic arithmetic, although there is some mention of geometrical ideas. This is compatible with his statement in the introduction that the text is intended to include teachers involved with primary education. However, primary mathematics, in many countries, includes many more themes than arithmetic (algebra, probability, shape and space, problem-solving skills etc). Nonetheless, Frank Smith uses his comments about children's difficulties with basic number work to form various generalisations about the learning of mathematics as a whole.

My chief reservations about this book fall into three categories, as follows:

- I still don't know what is meant by a 'glass wall', since the concept is never properly defined. Why not use alternative metaphors such as 'brick wall' or 'dense fog'? Glass walls are transparent and one can gain much understanding of the outside world by gazing through the windows of a train or those of one's own home. To my mind, it isn't clear whether the author believes that this entity arises from inappropriate teaching or whether it is due to the inherent nature of mathematics itself. To what degree is it due to both of these factors?
- In the teaching of mathematics, it is soon realised that some concepts and skills are very difficult to teach from the point of view of children's understanding. For example, how can one establish pupils' understanding of the traditional algorithm for doing long division? What physical models can possibly show that 1/3 = 0.3333...? Can the multiplication of negative numbers have a place in a child's mathematical conceptual framework? Can it ever made comprehensible?
In other words, mathematics can sometimes seem difficult because much of it *is* difficult, but this, for many learners, is part of its appeal, which is something not fully explored in the book.

- Much progress, in the world of mathematical education, has been made by basing the teaching of it upon practical work with appropriately chosen apparatus. For example, Dienes base 10 material is an excellent medium for developing of understanding of place value. Geostrips are a wonderful teaching aid for the exploration of 2d polygonal figures etc, etc. However, Frank Smith makes many comments of the following sort:
The world of mathematics doesn't arise from the physical world... except to the extent that it has its roots in the human brain, and it can't be made part of the physical world'

If, by this, he means that a concept is not the same thing as an object representing it, I can see some truth in it. Otherwise, it is not the sort of precept upon which to form an approach to the learning of primary mathematics. But then there is the following contradictory statement of page 13:

The structures of mathematics do not need a human brain or a physical world to support them.

Overall, I find that many of the issues about the learning of mathematics are discussed in a way that tends to obfuscation rather than clarity. I offer two more examples by way of illustration.

In the formation of children's concept of cardinal number, there is a range of practical activities that are based upon the Russell definition, which says that 'a natural number is an equivalence class of finite sets under the relation "is in 1-1 correspondence with".' Therefore, to establish the concept of three, teachers will direct children's attention to a wide variety of class representatives, such as a set of three bananas, a set of three blind mice, the three bears, the three wise men etc. Yet what are readers to make of statements like that on page 35, where it is said that:

Numbers don't derive their meaning from anything in the physical world, but from something in our mind...

And, on page 4:

But in fact, holding up three objects to illustrate the meaning of the word *three* explains nothing at all...

Finally, I refer to the discussion of the topic of fractions, discussed in chapter 10, called 'Numbers between Numbers'. What on earth is one to make of the following three statements, all made within a few pages of one another?

... fractions are numbers and can be treated in exactly the same way as whole numbers. (p. 93)
Ratios aren't numbers- they are relationships between two numbers. (p. 96)

To sum up, a fraction is a ratio or proportion of the numerical distance between one number and the next. (p. 96)

Perhaps this is an example of what the author meant when he included, in the title, the clause 'Why mathematics can sometimes seem difficult'!

P. N. Ruane ([email protected]) is Senior Lecturer in Mathematical Education at the Anglia Polytechnic University, Essex, England. His research interests lie within the field of mathematics education and the history of geometry.