Yes, but why? (book review)

Ed Southall of the Unversity of Huddersfield ( @solvemymaths on Twitter) has written a book called "Yes, but why? Teaching for understanding in mathematics."

I have to start off by saying that the publication of this book makes me very angry.

I admit that there is some risk of that giving the wrong impression, so I should explain why.

The book itself is not the problem: it has received many enthusiastic endorsements, and this review is in fact one of them. What makes me angry is that there is a need (and there really is a need) for anybody to point out that mathematics teaching should involve the students being led to an understanding of the material. Unfortunately, far too much school mathematics is taught as a bag of tricks which have to be memorized and applied in well-drilled situations. I also admit that I don't teach in school, so I don't have direct evidence of this: however, I do teach at the undergraduate level, and the indirect evidence is overwhelming. Of course, it isn't all students, but it is far too many.

So I repeat: I'm angry that the book serves a purpose. Ideally, my review would have consisted of

This is an entirely superfluous work. How could anybody even think of teaching mathematics in any way other than for understanding?

Alas that this is not the case, and the book is addressing an extremely serious problem. As a consequence of the 'bag of well-drilled tricks' approach to teaching, far too many students come to the conclusion that mathematics is a collection of more-or-less arbitrary rules and methods which they have to memorize, but which have no meaning or value other than to enable them to get at least a grade C (or its equivalent in the new version) in GCSE and preserve the school's precious place in the league tables. It is onerous, boring and pointless. Except for a relatively fortunate minority, for whom the quality of teaching is probably largely superfluous, it does nothing to inspire a wish to take the subject further.

So, what should be done about it?

Ed Southall has provided a sourcebook for teachers and trainee teachers with a wealth of useful material, all aimed at helping the teacher to help the student to understand what is going on.

It would have been all to easy to produce a book of careful proofs of all the results used in GCSE mathematics and the instruction to teach these to the students. If I'd been tasked with trying to solve the problem, I might well fallen into that trap. Fortunately, the book was written by somebody with a much more sensible plan.

The book begins with basic arithmetic and works through the algebra, geometry and statistics required for GCSE. But rather than stating and proving theorems, the results are explained by means of examples accompanied whenever possible by explanatory diagrams. By means of these illustrations, the student can come to an understanding of the mathematics, which can subsequently be developed into formal proof as and when that is appropriate.

The great advantages of this approach are: when something is actually understood, it is much easier to remember (in particular it doesn't have to be memorized); and it is then easier to see what piece(s) of mathematics can be applied to which problem.

Of course, good intentions are one thing, implementing them is another. Fortunately, the implementation matches the intention. The examples are well-chosen, the illustrations clear and relevant, and the combination should furnish any trainee teacher (and many experienced ones!) with a wide variety of material to help them teach the material so that the student understands it. The text is enriched by inclusion of 'Teacher tip' boxes providing explicit advice, 'break out sections' with additional material for investigation, and explanations of the historical development of some of the mathematics.

But what if your teaching is at a higher level than GCSE, say A level, or undergraduate? I'd still recommend the book, for two reasons.

1. It is often the case that a student who appears to be struggling with more advanced maths is actually being held back by problems with GCSE level material. This book will be valuable for those of us unaccustomed to teaching the material, as it provides strategies for explaining concepts we tend to take for granted.
2. The examples remind us that students at whatever level we teach are liable to learn better from the approach in which well-chosen examples are used to provide a framework for the understanding to develop on. (This is a salutory reminder for those of us who perhaps coped well with the definition-theorem-proof exposition rather than just surviving it, and who may be inclined to assume that their students will find it equally congenial.)

So I agree with all the previous responses I've seen to this new book: it's a valuable addition to the reading list of anybody who teaches mathematics from early secondary school onwards.

Of course, I have a few (very few) quibbles. I list these mostly in the hope that they might make a second edition of the book even better than the current one.

• p39 It might be helpful if the discussion of \(20-3-4-5-6\) were to introduce the convention of left-associativity explicitly rather than just using it without comment.
• p53 onwards, mathematical arguments are sometimes presented as sequences of equations without any explicit statement that each equation is in fact equivalent to the next. I think it would greatly help the students to realize that they really have to present an explanation of what is happening, and this includes the logical connection between each line and the subsequent one. Being loose about this at the early stages can lead to considerable problems later in the student's mathematical development, where (for example) solutions may be lost or gained in the attempt to solve equations if the logical connections are not dealt with carefully.
• p95 It is common to state that the angle at the circumference is half the angle at the centre: this might be a good place to consider what happens if the point on the circumference is not on the same side of he chord as the centre, as the theorem statement doesn't usually specify.
• p98 Two triangles are argued to be similar on the basis have a common hypotenuse and he same length of base: but this only gives a matching two sides and the non-included angle, which does not imply (in general) that the triangles are similar.
• pp149f It might be helpful to explain that the top right hand box in the diagram at the top of p150 contains a 75 because 25% off gives 75% of the original price.
• p237 Anscombe's quartet is much stronger than just having the same mean; the means and variances of the \(x\) values are all the same, as are those of the \(y\) values, and all four data sets have the same line of best fit. It's an important example of how totally different types of data may be summarized by the same descriptive statistics, and deserves at least a little more than it gets here.
• p306 In the 'Teacher tip' box, the degrees symbol is missing from the angle.