Professor Boaler has become a standard bearer for progressive education in mathematics. She believes in mixed ability teaching and is opposed to textbooks, “procedures” (standard methods of calculation) and homework, which she considers inequitable as some children don’t have anywhere quiet to do it. She has been quoted as saying that multiplication tables cause “crippling anxiety”, and is opposed to testing them, though she does not say that they are “not worth knowing”. This is an important point, to which I'll return. If they are worth knowing, how should we teach them?

Many of the teaching tasks in this book are quite brilliant, and I have started to use them. They are interesting and challenging, promoting mathematical reasoning and the flexible application of knowledge. Her criticism that textbooks interfere with this by presenting only the most simple examples – eg only hexagons that are regularly shaped – is often true. She is also correct in saying that school systems too often put sheer speed before understanding, and that this penalises some gifted mathematicians, including the Fields Medal winner Laurent Schwartz. It also penalises slower-learning children in mixed ability classes, as they do not receive the additional guidance and practice they need in order to learn effectively. Professor Boaler tries to combat this by using tasks with a low starting point and a great deal of extension - "low floor, high ceiling". This is a good idea in itself, and worth developing, whether the groups taught are of mixed ability or not.

Professor Boaler's politics are expressed in her reply to emails, Viva la revolution, and her methods are based on the Hollywood principle, "first simplify, then exaggerate". She starts in the introduction, where she describes the OBE as the “greatest honor bestowed in England, given by the queen”. It isn't, and Professor Boaler must know this. The OBE is third in rank out of four in the Order of the British Empire, and at least two other civil honours - Companion of Honour and Order of Merit - rank higher than any of these.

The same fast and loose attitude applies to the presentation of studies and data. A UK school dubbed Phoenix Park is compared with another, Amber Hill (amber – slow down and stop) on the basis of work that took place in the 1990s, though the dates don’t appear in the text. An interesting critique of this study is here. Professor Boaler’s flagship “Railside School” in California, is equally controversial. Her account of the work has been attacked for selective use of data by a group of California academics, whom Professor Boaler has in turn castigated on her university website for alleged bullying and intimidation. I hate bullying, but in a 2008 paper cited in a US Supreme Court case, she concedes her critics’ main point – Railside pupils did very well in algebra, but not so well in other aspects of maths. Professor Boaler says that this may be because of the pupils’ lower attainment in “language arts” – English, to us – but the available evidence suggests that it might equally well be the result of paying more attention to algebra than to other aspects of maths, particularly calculation.

On reflection, this is the basis of Professor Boaler’s “revolution”. She wants to replace arithmetic with algebra, and a little geometry, as the foundation of school mathematics. The trouble with this is that the mathematics of civil society. politics, law and everyday life, from the annual budget to a gym subscription to a supermarket offer, are based on arithmetic, and every citizen needs to know how to use it. So why has she not devoted her considerable ability as a teacher to ensuring that everyone has these skills?

A final point on brain research. Professor Boaler cites the well-known study of London taxi drivers, showing enlarged hippocampus in London Black Cab drivers following the knowledge of London test, a finding that was not repeated in bus drivers who had only learned simple bus routes. She also points out that neurons fire when children make mistakes, and concludes that making mistakes is a positive thing. This is misleading. Kandel's Nobel Prize winning research shows that repetition results in the growth of new connections in cells, while it is established that repeated activity, such as learning music, results in layers of myolin being deposited or formed along connections that speed things up. The fact that neurons are firing indicates no more than that the person is alive and that the brain is active. Some mistakes are, no doubt, creative. Mistakes on simple number facts, such as the 2x table, are anything but.

Multiplication tables are a good example. If they are worth knowing, they are worth teaching, and most schools do not do this well. Some schools teach children to count in multiples instead of teaching the table, and some as a preparation for it. In either case, I have always found that this makes learning the table more difficult, and see it as an problem of co-ordination rather than maths, as nearly everyone can add two to a given number. What they can't do without practice is get used to one column advancing one step at a time, while another advances in multiples. My approach is to build the necessary co-ordination by teaching 2x very carefully, with plenty of extra practice where needed to make it automatic. I don't move to 3x until 2x is very clearly in place, as this provides a template for the others. I then find I can teach the remainder in three to six months, and that they help pupils enjoy maths.