This book, although indicated to be about improving mathematics and science education, appears in the Springer Series of *Studies in Computational Intelligence* rather than in one of the Springer series on mathematics education. It is unusual in that it is not an edited “chapter book” with many different authors who specialize in mathematics education research. Indeed, the more I perused the book, the more I came to the conclusion that it is really about the possibilities of applying interval and fuzzy techniques to education-related problems, some more closely connected to teaching than others, than it is about traditional mathematics education research.

In the interest of full disclosure, both authors are colleagues, the first now in the Department of Teacher Education, University of Texas at El Paso (UTEP), and the other now in the Mathematics Department at New Mexico State University; however, I have never published with either of them, nor had lengthy discussions with them about mathematics education research. Indeed, I was unaware of the book’s existence until the second author recently offered me a copy of the book from her complimentary boxful. She mentioned that although she was in on the early stages of the planning and writing, the book is largely the work of the first author, Olga Kosheleva, who has an M.S. in Mathematics and Applied Mathematics from Novosibirsk University and an M.S. and PhD in Computer Science from UTEP. She has also taught in the UTEP Department of Electrical and Computer Engineering. Perhaps as a result, this book has a distinctive theoretical and computational flavor that begins with an introduction on the need for interval and fuzzy techniques in mathematics and science education.

In this regard, it is noted that experts on teaching rarely make precise statements or recommendations such as “Use 13 min. of your class explaining new material, then call 4 students to the board and let them solve each problem for 17 more minutes…” Rather, they say “First, use about a third of your class explaining the new material, then call on a few students to solve the problems on the board…” (p. 3). Thus, the need for a precise reformulation of such expert knowledge, and hence, the need for fuzzy techniques, which were invented in the 1960s by Lotfi A. Zadieh, to whose memory this book is dedicated.

The existence of numerous papers using fuzzy techniques in education is noted; however, most of the papers referenced in this book first appeared in proceedings of conferences on computational intelligence, fuzzy systems, or informatics or in publications like the *IEEE Transactions on Education*, rather than the traditional mathematics education research literature. The book is an attempt to present a unifying view of the papers on fuzzy techniques in different aspects of education, with quite a number of the referenced papers being published initially by the first author, and her husband, Vladik Kreinovich, a Professor of Computer Science at UTEP.

The book has four parts, on: 1) motivating students; 2) ordering the material for teaching; 3) selecting an appropriate way to teach each topic, and 4) assessing students, teachers, and teaching techniques. Altogether there are 38 chapters, with 24 of them with titles beginning, “How to …”. For example, Chapter 24 in the third part is titled, “How to Divide Students into Groups so as to Optimize Learning.” This chapter, like many of the others has a distinct mathematical flavor, containing 15 propositions, with proofs of 7 indicated at the end of the chapter.

For example, after defining \(d_i\) to be the degree of knowledge of the \(i\)-th student in a class and the average grade as \(a\stackrel{\text{def}}{=} \frac{1}{n}\left(\sum d_i\right)\), the first proposition is stated.

**Proposition 24.1. ***To maximize the average grade a *[*of a class*], *we divide students into pairs as follows:*

*we sort students by their knowledge*, *so that *\[d_1\leq d_2\leq\dots\leq d_n,\]
*in each pair, we match one student from the lower half *\(L_0\stackrel{\text{def}}{=}\{d_1,d_2,\dots, d_{n/2}\}\) *with one* *student from the lower half* \(L_1\stackrel{\text{def}}{=}\{d_{(n/2)+1}, d_{(n/2)+2}, \dots, d_{n}\}\).

Nothing is said in this chapter about how one gets information on the degree/state of knowledge of each student. Given the definition of \(a\) as the average grade, the \(d_i\) are probably the current grades of each student in the class. Other optimization criteria, rather than average grade, are considered: maximizing retention (i.e., minimizing the number of students who fail the course) and maximizing the best grades so that one can maximize the number of students who can get into “Ivy League colleges” (pp. 210–211).

The above ideas on group work contrast greatly with some of the more traditional mathematics education research literature. For example, one of the results I remember on group work in mathematics education is that for effective learning one should *not* put high, medium, and low ability students together in a group, but rather put high and medium ability students together or medium and low ability students together so they can communicate. Another result I remember is that the individual in a group who learns the most is the one that does the explaining; the one who learns the next most is the one who receives the explanation; and that just telling an individual that an answer is correct or not will not be very helpful in promoting learning or understanding. [For more information, see Elizabeth Cohen, “Restructuring the Classroom: Conditions for Productive Small Groups, *Review of Educational Research*, Vol. 64 (1994), pp. 1-35.] Furthermore, the above optimization approach takes no account of the prevalent view in current mathematics education research that students are not corn plants so the “agricultural model” of research does not work well. Students come with histories, whereas corn plants do not.

Not all chapters are strictly about teaching. For example, Chapter 16, titled, “How AI-Type Uncertainty Ideas Can Improve Inter-Disciplinary Education and Collaboration: Lessons from a Case Study,” is about how to make interdisciplinary research collaborations more productive. The particular case study concerns research communication and collaboration between a geoscientist and a computer scientist solving a problem while accessing data from a variety of databases with differently formatted information. In effect, each expert spoke his own technical language and considering a watered-down “toy problem” did not help. As it happened, the difficulties were solved by communicating in an intermediate domain, in this case, solar astronomy, with which both researchers were familiar, but neither was an expert. It is pointed out that this particular case study is not unique, and the authors provide an explanation, via a formula, for why it works, in terms of degrees of understanding, \(d_1\) and \(d_2\), of the third domain of communication of the two researchers, namely, \(d=[\min(d_1,d_2)]/[\max(d_1,d_2)]\). When one communicator formulates a message in his/her own domain terms, \(\min(d_1,d_2)\) is close to \(0\) and \(\max(d_1,d_2)\) is close to \(1\), so \(d\) is close to 0. However, when a communicator describes his/her message in the language of a third common domain in which \(d_1\) and \(d_2\) are close, \(\min(d_1,d_2)\) is approximately equal to \(\max(d_1,d_2)\), and so \(d\) is approximately \(1\).

I can see why this book was published in the *Studies in Computational Intelligence* series, as it takes a very quantitative, if fuzzy, view of many mathematics teaching problems. The organization of the book is clear — one can easily tell from the title what each chapter is about. The references are listed numerically at the end of the chapter, and in this sense, each chapter seems to be a “standalone” discussion of a particular topic. There is plenty of mathematics throughout the book, but there seems to be little data to show that the suggested examples of motivating students actually *do* motivate any particular group of students, or that the suggested ways of dividing students into groups actually improves learning. It is definitely not a practical “how to” book, nor does it claim to be. Still, as an intellectual exercise in how one might apply fuzzy techniques to teaching problems, it is interesting.

So, who might be interested in this book? Not teachers of mathematics, at any level, seeking practical advice. Given that the references come from conference proceedings on computational intelligence, fuzzy systems, or informatics and publications like *IEEE* *Transactions on Education*, academics who attend such conferences seem to be the most likely readers, as well as any mathematicians interested in potential, if somewhat theoretical, applications of fuzzy sets and interval techniques.

Annie Selden is Adjunct Professor of Mathematics at New Mexico State University and Professor Emerita of Mathematics from Tennessee Technological University. She regularly teaches graduate courses in mathematics and mathematics education. In 2002, she was recipient of the Association for Women in Mathematics 12th Annual Louise Hay Award for Contributions to Mathematics Education. In 2003, she was elected a Fellow of the American Association for the Advancement of Science. She remains active in mathematics education research and curriculum development.