# Mathematics Rebooted

Lara Alcock has a degree in mathematics, but her research is about mathematical education and mathematical thinking. She has written two books on these subjects already and won several awards for teaching mathematics. The present book is a bit difficult to classify. It is not a book about her research, neither is it a textbook in mathematics. It is also not a popular math book, or is it? That depends on how you define a popular book about mathematics. It is not a collection of puns, puzzles, paradoxes, and all these other topics that are usually found in such books. On the other hand, you could call it a popularizing book about mathematics because it does not require much prior mathematical education to read. In her introduction she describes the readership as those people who have some affection for mathematics, but who lost track at some point in the past. It could for example be a help for teachers or parents who have to help or teach children and who themselves are missing some mathematical way of thinking. Some people, usually at a later stage of life, start to learn a foreign language, or read history books, or biographies, etc. Why should they not learn some mathematics? If they want to pump up their literacy, they could as well improve on numeracy or polish their knowledge of mathematics.

So what does the book contain? It is some kind of a textbook, but it is a freewheeling kind, not restricted by any kind of prescribed rules of what should or should not be included. There are no delimiting containers of algebra, calculus, or geometry. A simple idea brings along another idea, which leads to a different topic etc. Neither is it just entertainment. There are claims, even some sporadic theorems, and there are proofs (sometimes graphical or geometrical to make them more intuitive).

The book has five chapters, which are five threads of ideas. The first chapter is called Multiplying. It starts with the simple idea that by visualizing n rows with m elements makes multiplication easy and immediately proves the commutativity and distributivity. Also identities like $(a+b)^2=a^2+2ab+b^2$ are easily verified geometrically. The area of a triangle and of course also the Pythagoras theorem have geometric proofs as well. But then there is a trail from the Pythagoras theorem to Pythagorean triples and this in turn leads to the last theorem of Fermat. Along the way, suggestions are made for exercises to be elaborated further (Alcock refrains from formulating them as formal exercises, she just suggests to think a bit longer about some problem).

The other chapters are somewhat similar in nature. In the second chapter, entitled Shapes, the starting point are tessellations of the plane. Again by a geometric proof, it follows that the sum of the angles in a triangle is 180°, and by subdividing a polygon in triangles, the formula can be generalized to polygons. But considerations of regular and semi-regular tessellations also lead to symmetries and Penrose tiling. An interesting remark here is that Alcock also stresses the fact that, even with all the formulas included, the mathematics read as sentences. Thus that one is reading mathematics, just like one would be reading some other formula-free text. Another lesson learned is that mathematicians are often interested in generalizing some result, more than just applying it.

Chapter three is called Adding up. The goal is to arrive at infinite sums, but the starting point is adding fractions. But once more there are easy visual ways to show how to add a finite number of integers. This gives classical formulas for the sum of the first n integers, or the odd or the even ones. This is also the place to introduce proofs by induction. Furthermore she tackles convergence and divergence of series, with the geometric and the harmonic series as a particularly interesting cases.

The chapter on Graphs is about plotting functions in a coordinate system. The motivation here is a word formulation of an optimization problem with several (linear) inequality constraints. Plotting the constraints shows easily where the target function will be optimal. Further explorations lead to circles and polar coordinates, but curiously enough the sine and cosine do not appear because only Cartesian equations are used. A glimpse is given at these topics in three dimensions.

The title of the last chapter is Dividing. It starts with explaining our positional number system. This can lead to rules about divisibility like a number is divisible by 3 if and only if the sum of its digits is divisible by 3. When rational numbers are represented as ratios of integers, the prime factorization is needed to simplify them. A bit more advanced is a discussion of rational and irrational numbers and issues of countability.

In some concluding remarks Alcock reflects on what has been learned by reading this book (she has already summarized the main points after each chapter). She also gives suggestions for further reading, many of them are the more mathematical popularizing books, but also some books inspired by research on and experience in mathematical education.

Anybody interested in mathematics (those who want to learn and those who want to teach) will benefit from reading this book, but in my opinion it is particularly of interest for beginning teachers in secondary schools, especially when they were not explicitly trained in teaching mathematics. Parents too who want to follow up their children can be classified in this category. Furthermore there are those who are sincerely interested in jacking up their long forgotten mathematical knowledge.

**Submitted by Adhemar Bultheel |

**20 / Feb / 2018