*Question (central to Mathematics, education, and other endangered species)*: What is mathematics education for?

*Answer 1* (attributed to Underwood Dudley): to train the mind.

*Answer 2* (attributed to Jere Confrey): to be a “draconian” filter for further studies.

*Possible answer the author squarely rejects*: for students’ use in everyday life.

I estimate that these two affirmative answers and the rejection of the third are sufficient to convey the controversial nature of many of Dr. Shlomo Vinner’s reflections on his 40 years as a mathematics educator. While the tone of this book is not strident, Dr. Vinner takes this opportunity to lay out a number of claims that challenge common viewpoints among the intended audience of mathematics education researchers and mathematics teachers.

The following examples further illustrate the wide range of topics he touches upon with a similar inclination toward possible controversy:

- Not all students should be required to take mathematics beyond the bare minimum if their talents and interests lie elsewhere.
- It is more important to compel your students not to smoke than to teach them canonical mathematical facts.
- “Secular thinking is much simpler than the [religious] believers’ thinking since it does not have to cope with [previously-described] problems” (p. 16). “The real question is not whether God exists. The real question is whether God is needed. And the answer to this question is Yes (with Capital Y)” (p. 17).
- Not all student responses to mathematical questions should be viewed as “conceptual” (i.e. sense-making or rational) since many are produced spontaneously through loose associations and were not measured through reflection (the book’s subtitle is a reference to this issue). The author calls the reasoning behind such responses pseudo-conceptual.
- Pseudo-conceptual reasoning is inconsistent with the intent of instruction but is nevertheless frequently promoted by mathematics instruction itself.
- “Teaching is tough when you have students who are not interested in meaningful learning but only in passing exams” (p. 62).
- “Many people do not have the ability, as well as the willingness, to examine their generalizations and to change their minds about them” (p. 80). “As to us, mathematics educators, we stick to our belief that it is possible to teach mathematics in a meaningful way. This is in spite of the fact that there are infinitely many counterexamples” (p. 82).
- “The most beautiful rhetoric for teaching mathematics which I know is the NCTM (1989) rhetoric… This is not the place to elaborate at length about how misleading these claims are” (p. 111).
- “Observing [elementary teachers] in their classes indicates that in most cases they are dedicated people. They do their best to teach mathematics. Sometimes their best is not good enough mathematically. However, it is useless and pointless to request more than their best” (p. 117).

This string of bold and candid ruminations represents one of the more unique features of the book. Many of the justifications are highly personal. As one might expect when this much territory is covered in a mere 130 pages of text, these topics are not all explored in great detail.

One of the more compelling ideas proffered in the book is that, since the pseudo-conceptual kinds of connections are unavoidable in human reasoning, the main solution to the issue, and thus the goal of mathematics instruction is to foster metacognitive abilities to inhibit these weak connections and move toward more careful and reflective reasoning. Dr. Vinner similarly valued this point since he made it the subtitle of the book, and I applaud that choice.

On many other accounts, the book failed to resolve some of the internal tensions between its various claims. The narrative often alternates between a critique of the status quo and a reasoned resignation to the current state of affairs. For instance, Dr. Vinner decries using mathematics in standardized testing as a filter, but acknowledge this is unlikely to change. He thus recommends that mathematics instruction do the bare minimum necessary to help students get through this ordeal. The book classifies mathematics as part of scientific and rational thinking (as opposed to pseudoscientific and religious thinking) that is useful to help become an educated person, but he questions at various points whether all people are truly capable of mathematical thinking. Dr. Vinner thinks mathematics instruction is the best way we have to train students in rationality, and yet downplays the efficacy of current instruction to accomplish this goal. The book critiques the mathematical knowledge of teachers, suggesting that many are unready or unwilling to guide students toward conceptual mathematical reasoning rather than pseudo-conceptual reasoning. However, he does not fault teachers for this and suggests that their other contributions to students’ development as functional human beings make up for the mathematics-related difficulties. I appreciate the way Dr. Vinner balances his dissatisfaction with the current state of mathematics education with accounts of the structural factors keeping it that way. It unfortunately leaves the reader with many conflicting impressions of what can/should be done.

Another tension that many readers will recognize is that the book attempts to maintain a sympathetic tone toward students and teachers and yet consistently speaks with an evaluative tone that questions people’s fundamental abilities as learners. Dr. Vinner does not address the equity-related concerns that the mathematics education literature raises about claims that students are incapable of learning mathematics and should be discouraged from such study. I find this fact particularly curious given that he emphasizes the moral aspects of education as more important than the esoteric mathematical aspects.

In summary, this book provides the reader with a range of bold claims to consider and tensions to work through. Consistent with its reflective and personal nature, it does not provide a thorough treatment of the varying viewpoints on the central topics. While Dr. Vinner is certainly courageous in sharing his viewpoints that challenge common assumptions, he is reasoned enough not to claim he has a simple solution to the challenging tensions that he discusses. Maybe this is a demonstration of his central point that we should move from our initial intuitive associations to reflective inhibitions to help us become more rational and even more moral human beings.

Dr. Paul Christian Dawkins is a researcher in undergraduate mathematics education who focuses on the teaching and learning of proof-oriented mathematics, especially Transition to Proof, Real Analysis, and Geometry.