This book has been written with the parent of school-aged children in mind. It might also be read by educators and college students looking back on their K--12 math education.
The book is approximately 250 pages, complete with many references to other books and websites, appendices containing solutions to puzzles in the text and various mathematics curricula, endnotes, and an index.
The book begins by describing how poorly students do in the United States in international mathematics assessments. Some history and trends are given as well. Also, general attitudes toward math are outlined. The book then uses several chapters to describe what is currently going on in mathematics classrooms across America and what should be going on. This is done largely in the context of the "Math Wars" where the author pits the "Traditionalists" against the "Reformers." The book ends with advice as to what parents can do to change the situation and involve their children in mathematics.
In addition to the general outline above, there is material on "tracking" students, which the author believes is dangerous, and there is material on the differences between boys and girls with respect to mathematical thinking and how they are treated differently. Throughout the book, several mathematical problems and puzzles are given as examples of the types of problems students should be working on. Solutions are given in the back of the book.
Much of what the author writes concerns her own personal research (as a Professor of Mathematics Education). Many case studies are examined, and classroom scenarios desribed.
The author does seem passionate about her case, and much research is cited to help substantiate her claims. However, despite the good intentions of the author, I believe the book fails to accomplish anything. The major reason for this is that the author never really describes or delves into mathematical content. Just what is it that children in K--12 supposed to be learning in the math classroom? What topics, problems, etc. should be learned? What should students be preparing for in college? I'm talking about specifics like factoring, solving systems of linear equations, using similar triangles to solve a problem, etc. She does emphasize number sense (in a limited way), as well as patterns, but that's about it. Without knowing exactly what it is students should be learning, no one would be able to judge whether they believe her approach is superior.
Another reason this book fails, is that it puts forth a false dichotomy between the traditional math teacher and the reform math teacher. It's as if there is no spectrum or different ways of teaching. According to the author, either you are a dangerous traditionalist at the root of all math problems in this country, or you are a mathematical reformer. In her narrow outlook, all traditional math teachers do is tell students to memorize algorithms, make them rehearse in solitude through multitudes of repetitive problems. They never prove or justify or explain anything, and they discourage all questions from students on these issues. And according to her book, if as a teacher, you spend much of your time presenting a lecture at the chalk board, then you are a traditionalist. If this is the case, then no traditional math teacher in this country is actually a mathematician, good at math, and interested in math. This is because all mathematicians know that at the heart of mathematics lies proof, justification, understanding, linked concepts, logical reasoning, and problem solving.
Actually, there probably is quite a large percentage of K-12 math teachers who are not mathematicians at heart. In my limited experience of teaching both elementary education and high school education math courses to future teachers, the vast majority of my students have been terrible at math, but worse than that, they had no interest in math at all. I have heard similar information from other people. I also remember finding out that the elementary education students at the university where I was a grad student and teaching assistant had the absolute lowest standardized test scores (ACT, SAT, and GRE). So, I would not be surprised to find out that there are many teachers out their that fit the author's description of a traditionalist. But this does not mean that there isn't an entire spectrum of possible ways to teach.
In any case, I don't really trust the author to know what mathematics is really all about and to know how it should be taught. There are a few reasons for my mistrust. First, it is clear that the author isn't very knowledgeable about mathematics. There are a few times in the text when she says something that lets a little bit of mathematical ignorance peek through. For example, when talking about multiplying binomials, such as finding the product (x + y)(a + b), she (more than once) referred to a binomial (a two-termed polynomial) as a "binomial distribution" (which is quite another thing). She also talked about going back to school (recently, as a professor) to take some "advanced math" again. Her idea of advanced math appropriate for the Marie Curie Professor of Mathematics Education at the University of Sussex seemed to be an introductory course in statistics. Another thing that struck me as odd was that she used the word "pneumonic" (which means pertaining to the lungs or pneumonia) when she meant "mnemonic" (a memory aid). I would have thought a professor of education would know the difference.
In any case, the author's view of what math students should be doing with their time seems a little bit limited. She is very fond of patterns. Now she does say that she uses the word "pattern" in the broadest possible sense when she claims that math is all about patterns, but then she seems quite fixated on math problems such as taking a sequence of geometric objects built from squares or cubes or something and having students count and express how many squares or cubes make up the object at each stage. Such number patterns really aren't what much of mathematics is all about. She describes a summer course she taught in which high school students seem to have spent most of their time exploring such patterns, communicating their ideas, and having a good time. She gave students A's and then laments how most of them went back to their "traditional" classes to receive F's again. I'm not sure, but maybe that was because playing with blocks didn't help them learn how to simplify algebraic expressions or solve quadratic equations. This comes back to the idea of what students should be learning. If simplifying a difference quotient isn't important than say so. But please explain.
The author should be commended for stressing how important it is for students to think and not simply memorize or mimic in a math class. I myself find it almost incomprehensible that there are people out there who don't agree. She is in favor of teaching number sense to young children and she is in favor of giving more complex problems to older children. I agree, but I'm not so sure that her methods of teaching will produce the results I would like to see. I teach at a two-year college, everything from pre-algebra to differential equations and linear algebra. I admit I'm not a K--12 teacher, but many of the classes I teach are at the level of 6th through 12th grade. I must say that the more I ask my students to think, the more they resist and the more they hate math. (Actually, I constantly expect them to think.) The harder the problem I ask, the more they resist and the more they hate math. I get nothing but complaints when I expect my students mathematical knowledge and skills to be flexible. My students' grades drop, I get frustrated, and I think about leaving the profession. This doesn't mean I turn to asking my students to memorize and learn things by rote. I won't compromise (well except a little bit every once in a while during my office hours when I just have to go home and a student just doesn't want to learn and I just give them the answer). My experiences, however, may be showing nothing except that my students have been given nothing but the rote method for 12 years before coming to me and by that time, they are incapable of changing. This would seem to vindicate the author.