Steve Willoughby is that rare mathematics educator who has always been respected by research mathematicians. Although he would not describe himself as anything except a mathematics educator, his three professorial appointments (Wisconsin 1960--65, NYU 1965--97, and Arizona 1997--2002) have all been in the department of mathematics and, at Wisconsin and NYU, also in the school of education. In addition, he was President of the National Council of Teachers of Mathematics in 1982--84, and he was Chairman of the Council of Scientific Society Presidents, whose members are the presidents of just about all the main scientific societies in the USA. Besides being the author of many mathematics education papers, he is also the lead author of the Real Math textbook series, which was, he claims, "arguably the most innovative and influential elementary school mathematics textbook series of the last half of the twentieth century" (p.332).

I recount all this in order to stress that Steve Willoughby has enough stature to give all mathematicians reason to heed what he says. This memoir, after introductory chapters on his family and background, covers his student years at Harvard and Columbia, his faculty years in Madison, New York and Tucson and his experiences in professional societies and publishing. It is readable and opinionated. Overall the reader will get a good sense of the issues in school mathematics education, particularly elementary school mathematics education in the USA over the past half century, and the disputes that have culminated in the current "math wars".

In this review I will concentrate on what Willoughby says about the three crucial T's of math education: teachers, testing and textbooks. Although knowledge about the three T's may not quite encompass all you need to know about American precollege education, an understanding of the issues involved in these three areas makes it rather easy to understand other areas such as politics.

Although strictly curricular matters (should calculators be allowed in elementary school? should long division of polynomials be taught in secondary school?) dominate the math wars, the most serious problem in American school mathematics education, now and for the foreseeable future, is the lack of mathematics expertise in too many American elementary school teachers and secondary school mathematics teachers. There are various reasons for this (poor salaries, poor working conditions in schools, many other opportunities for mathematically-trained people, etc.) but Willoughby highlights one that is seldom discussed, namely, the tendency of college and university mathematics faculty to discourage any student with a real talent for mathematics from becoming a teacher. In his last year as an undergraduate at Harvard, Willoughby mentioned to Garrett Birkhoff that he intended to become a high school teacher. Birkhoff responded that he "needn't do that; the mathematics department would give [him] a full fellowship to work towards a Ph.D." (p.57). Willoughby comments:

I wonder how many readers of this review recognize themselves or their colleagues in this quote. Two other relevant quotes in this context are:

Willoughby is a strong supporter of specialist mathematics teachers in the upper elementary grades (4--6). This reviewer thinks that specialist teachers in K--3 would also be a good idea. However, using specialist teachers of mathematics in American elementary schools is a reform that has little support at the present time.

Testing is at the center of the American educational debate today. The No Child Left Behind Act mandates a testing regime in elementary and middle schools far more intense than in any other country of the world. Already mathematicians who should know better trumpet rising test scores in some states as evidence that curriculum changes they have promoted are bearing fruit. Willoughby uses his "fifty-five years of involvement … with standardized tests" (p.112) to make some insightful remarks about the testing culture in the U.S.A.

One of the most amusing passages in the book recounts Willoughby's experience at the Wisconsin High School, the laboratory school of the University of Wisconsin. In addition to his appointments in the Department of Mathematics and the School of Education in Madison, he was also head of the mathematics department at Wisconsin High School. After his first year there, a faculty member of the university department informed all his colleagues that the scores in a "state-wide high school mathematics test had deteriorated seriously from the previous year" (p.161) and wondered why this had happened. Willoughby, who believed that the test was "hogwash", said that he would arrange "that our scores will be the highest in the state this year" (p.162). Although perhaps at some cost to the mathematical education of students at Wisconsin High School, Willoughby made sure that the students had enough instruction on the contents of the test and on test-taking skills that the scores that year were not just better than when his predecessor was there but actually first in the state.

It is, I suppose, unsurprising that politicians and school administrators look at rising test scores to validate changes that have been made:

Still, I continue to be surprised that professional mathematicians are also prone to the idiocy of a belief in standardized testing. As Willoughby notes elsewhere:

Many readers of this journal have been involved with writing college textbooks or monographs at some time. They are used to having complete control over the content of their books, perhaps deferring to publishers on such matters as design and title. One who has had no experience with precollege textbook publishing might think that things must be similar there. Wrong! Editors at publishers exert much more control over content than college book authors would ever allow even when these editors are mathematically incompetent. (Willoughby mentions an editor who thought that "real number" had been used too much in a manuscript and changed about half the usages to "actual number".)

Willoughby's first experience with precollege textbooks occurred when he was just finishing his Ph.D. and was asked to be coauthor on a first-year algebra book. He was one of six authors listed in the book although only three (the junior ones) had actually done any writing. When the book was published, Steve discovered that some of the material he had written had been "substantially changed" and that errors had crept in as a result. Who had done this? One of his coauthors whom he had, in fact, never met. Later, when the same publisher asked him to become lead author on an elementary mathematics series they were planning, he agreed "on the condition that the contract would say that I had final control of content and pedagogy" (p.151). So another author was chosen who, Willoughby notes, arrived at his office for a meeting some years later driving a Rolls-Royce.

By 1972 Willoughby had had discussions with seven publishers
about an elementary mathematics textbook series. All but one
"promised that I would become very rich, very quickly if I would
agree to allow my name to appear on the books and do a very small
amount of work (the less the better, it seemed)" (p.302).
Finally Willoughby signed a contract with Open Court for the
series that would become Real Math. The complete Real Math K--6
series was published in 1981 and the K--8 series in 1985. In 1986
the commission in California charged with making recommendations
on textbooks "voted 5 to 1 to make Real Math the

The lesson from this is that textbook adoptions in large states are so lucrative that big publishers will pull out all the stops to get on adoption lists. And although most would like "to produce high quality materials and improve education" (p.333), the reality is that their "principal goal is to make money" (p.333). Willoughby believes---correctly but, no doubt, naively---that "eliminating all state adoptions would be an effective way to improve education" (p.334).

[**1**] A. L. Amrein and D. C. Berliner, High
Stakes Testing, Uncertainty and Student Learning, *Education
Policy Analysis Archives* **10**(18) (2002) 1-71;
`http://epaa.asu.edu/epaa/v10n18`

[**2**] A. Kohn, *The Case Against Standardized
Testing*, Heinemann, Portsmouth, NH, 2000.

[**3**] E. Fernandez, Being a mathematics
educator in a department of mathematics, *Focus*
**23**(1) (2003) 18.