In *Motivating Mathematics: Engaging Teachers and Engaged Students *David Wells argues that students can be motivated in mathematics, which is historically thought to be a non-motivating and dry subject, by being presented with the “Big Picture.” This is achieved by connecting mathematics to the sciences, the art of problem solving, and having the students create their own proofs. Wells main idea of the book is outlined on page 6, where he states:

One of the arguments of this book, implicit more than explicit, will be that all children can and should be motivated to succeed at mathematics, within their capacities, which I would confidently assert are much higher than usually admitted or recognized.

Throughout the book, Wells draws on his personal experiences. He understands how students think and, more importantly, how they perceive mathematics.

Wells believes strongly that students are typically exposed to mathematics as a collection of rules, definitions, theorems, and calculations, which are learned in a step-by-step, pacing chart, check-off sheet type of manner. There is little to no room for students to see where the mathematics they are learning can be applied or used outside the classroom. In addition, Wells points out that students have anxiety about mathematics. While there are many causes of this anxiety, one of the prominent causes is the overload of the mathematics curriculums. This begins with the crammed syllabi to benefit the stronger students. As Wells points out:

No doubt the textbook authors are not entirely to blame. They are providing what the publishers demand, the publishers publish what teachers demand in order to get pupils through public examinations, which are in turn hamstrung by syllabuses and a curriculum decided, in the final analysis, by the government.

With this push to cram in as many concepts as possible to lesson plans, curriculums, and pacing charts, students are put under more pressure, which generates more anxiety, and are less inclined to be motivated in mathematics.

Wells goes on to talk about the lack of time for students to explore mathematics and about teachers not having enough time for students to have self-discovery activities. Wells presents several examples of how teachers can create lessons and activities to motivate the learning of mathematics. For example, on pages 103–105, Wells talks about patterns in algebra. Patterns in algebra can be difficult to recognize but a large part of mathematics *is* pattern recognition. For example, Wells talks about the cubic and relates it to the quadratic equation \(x^2-6x+7=(x+1)(x-7)\), where it is not evident that the graph is symmetric about \(x = 3\). The cubic equation \(y = x^3 – 6x^2 - x + 30 = (x + 2)(x - 3)(x - 5)\) doesn’t help the student to recognize the symmetry. On the other hand, if we move the origin to the point of inflection \((2, 12)\), the equation will become \(y + 12 = (x + 4)(x - 1)(x - 3)\) or \(y=x^3-13x=x(x-\sqrt{13})(x+\sqrt{13})\). As Wells points out;

This translation of the origin also illustrates invariance and unity-in-variety: the shape of the cubic has not changed and so in a sense all cubics of this shape, despite their different equations, are ‘the same.’

The overall goal of mathematics is proving theorems, so Chapter 6 is on *Proofs in School Classrooms*. Wells starts the chapter by pointing out that with Euclidean geometry, proofs in mathematics have been seen to by students and textbook authors to be hard and “to be approached with care and even trepidation by both pupils and teachers.” Wells strongly believes proofs in mathematics are appropriate at every level of schooling and should be a familiar idea to any age of student. Personally, I totally agree with this viewpoint, as calculations use logic, but are not the sole purpose of mathematics.

Wells talks about *false simplicity*, which is the omission of proofs and other harder problems in mathematics. However, we do no service to our students as it deprives our students of, as Wells points out, the *rich complexity* of mathematics. Wells points out that more and more textbooks are omitting some proofs, which is detrimental to the learning of mathematics. Wells points out;

Textbooks today demotivate because they omit proof as it is ‘too difficult’ but then find that they have little that is challenging or attractive to present to pupils except some very basic ideas and facts with which the pupil can ‘do’ nothing because he or she lacks intuition, lacks familiarity, lacks background and lack appreciation.

As we are in the 21st Century, one would assume textbooks would be more involved with proofs, more complex problems, and more discovery-style problems to get students ready for global competitiveness. In mathematics, however, the opposite is happening. Wells goes on to provide some examples of how to challenge students with proofs through sections on quadratic equations and proof, polyomino puzzles, and magic squares.

Towards the end of Chapter 6, Wells presents Jennifer Kano, a Cape Cod Community College student and her solution to the overlapping maps problem presented to another student, Gary, on page 143. Kano was not satisfied with the solutions to the problem and when she consulted with her professor, David MacAdam, he said it was possible she had a new way of proving the problem. Kano’s proof starts out in the same manner as Coxeter’s proof in *Introduction to Geometry*, but Kano’s difference is the qualification, “as long as the areas outside the rectangles are considered to be part of the map.” A picture of Kano’s idea is on page 148 of the book. This shows that mathematics makes students think about logic and exercises the brain to be an effective thinking machine. Indeed, the mind *is* the greatest weapon!

The creation of more and more new techniques of teaching mathematics can sometimes make for ambiguity and confusion in students’ learning. For example, Wells points out the ambiguity of fractions in textbooks on page 227. When making comparisons between various countries on how students learn mathematics, Wells points out that in the United States progressive math educators promote problem solving, but the success has been limited for many years. Wells points out;

Currently, U.S. math teachers are embroiled in the so-called ‘Math Wars’, in which the NCTM are the voices of reason, struggling against the back-to-basics brigades who accuse them of promoting Fuzzy Math, and even of being in the New New Math, which is ironic because there is no connection at all between the two goals, though many professional mathematicians are now on the side of the back-to-basics just as they promoted the New Math in the 1960s.

It is clear that the standards of yesterday are not the same as today.

In reading this book, I was surprised by some of the findings but I also expected them. By having fewer proofs in textbooks and promoting the changes in the simpler curriculums, we are doing a disservice to our students. We as teachers must be on guard against any misleading calculations and wrong thinking. Mathematics builds upon itself. Each concept must be taught correctly so that the next concepts make sense. We owe it to our students to provide them with the necessary tools not only to solve problems, but also to be better thinkers. To allow self-discovery and struggle when first approached with new concepts, and thereby to create a fun and motivating mathematical environment, is vital to a student’s success.

Peter Olszewski is a Mathematics Lecturer at Penn State Behrend, an editor for Larson Texts, Inc. in Erie, PA, and is the 362 Pennsylvania Alpha Beta Chapter Advisor of Pi Mu Epsilon. He has written several book reviews for the MAA and his research fields are mathematics education, Cayley Color Graphs, Markov Chains, and mathematical textbooks along with online homework softwares. He can be reached at pto2@psu.edu. Webpage: www.personal.psu.edu/pto2. Outside of teaching and textbook editing, he enjoys playing golf, playing guitar, reading, gardening, traveling, and painting landscapes.