This book, published in 1973, is a classic in the mathematics education literature. Many of Freudenthal’s ideas are current today, especially those of Dutch Realistic Mathematics Education (RME). (For example, Sean Larson and colleagues have a method of teaching abstract algebra that is based on the idea of guided reinvention. See “The Teaching Abstract Algebra for Understanding Project: Designing and Scaling up a Curriculum Innovation”, *Journal of Mathematical Behavior*, **32**(4), 691–790, December 2013.) The crucial ideas of *reinvention *(now often referred to as *guided reinvention*) and *mathematization *are discussed in detail in Chapters 6 and 7. For guided reinvention to work, the teacher, or the curriculum developer, should ensure that the learner will regard the knowledge as his/her own. Freudenthal saw guided reinvention as an elaboration of the Socratic Method, but one in which students are much more active.

That said, how can one adequately summarize, or even indicate, the contents of this huge volume? There are 19 chapters plus two appendices. They range over the aim of mathematics instruction, the Socratic Method, mathematical rigor, logic, number, sets and functions, geometry, analysis, probability and statistics, and more, with the first of the two appendices devoted to a critique of Piaget’s views on the development of mathematical notions in children.

The introduction itself is a gem, written in typical forthright but opinionated Freudenthal style. Of the psychological literature and its uses in education, he writes,

To be honest I should say that I feel there is no need to embellish low-key education using high-brow psychology, in particular if the cited literature is far removed from educational preoccupations. If others prefer this procedure, then indeed, I feel the need to oppose it. … Except for some general ideas I did not take my empirical material from psychology. My most direct sources are textbooks, didactical designs, actual lessons, as well as observations with individual children; indirectly my main sources were talks and discussions with teachers. (pp. v-vi).

Freudenthal continues his own assessment of this book as follows,

In spite of all the detailed investigations in this work this book is above all a philosophy of mathematical education. I am not the first to have written such a book. The least one should have learned when studying his predecessors is that one has come to terms with their ideas. The scientific character of a book like the present one is not measured by the number of footnotes but by the thoroughness of this preliminary discussion. … This book does not contain any essentially new material compared with my earlier papers; … Here I have taken the opportunity of rearranging my old ideas. (p. viii)

However, if one has not read any of Freudenthal’s earlier papers, this book is a good place to encounter his educational ideas for the first time.

Though much of this was written during the “New Math” Era of the 1960s, Freudenthal, while not in favor of traditional mathematics instruction, was also definitely *not* a proponent of the “New Math”. As he wrote in Chapter 11 on the number concept, “The reader can testify that I am not a faithful believer in tradition. But I deeply regret serious innovators behaving as if there had not been any didactics of arithmetic before the advent of the New Mathematics. This wild tribe of charlatans take it as their licence [sic] to indulge in all manifestations of insanity.” (p. 195). From this one short critical excerpt, one can see that Freudenthal was not shy of expressing his opinions, often in rather pointed ways.

Freudenthal was a prolific yet reflective author, who published over 380 titles in a variety of languages, many listed in the second appendix of this book. However, he did not “grind out” his many publications with abandon. As he explained in his 1987 life sketch, *Schrijf dat op, Hans* or *Make a Note of That, Hans*, “Every well-turned phrase I write is considered, weighed and turned around until I have achieved the ultimate level of spontaneity.” Productive to the end, only a week before his death in 1990, he was putting the finishing touches on his magnum opus,* Revisiting Mathematics Education. China Lectures, *which summarizes his life’s work.

Freudenthal had a strong belief that mathematics is a human activity, that is, that mathematics is a process. He disliked what he referred to as the “anti-didactical inversion,” that in traditional mathematics instruction, the results of the mathematical activity of others is taken as a starting point, rather than teaching the activity itself. Freudenthal also attacked Chevallard’s idea of *transposition didactique*, in which the expert knowledge of mathematicians is taken as a point of departure.

I don’t think this book needs to be read from beginning to end. One can just pick up a chapter anywhere and get something out of it. For example, in Chapter 7 on mathematizing, one finds Freudenthal’s objections to mathematical problems as normally posed. He states that a [textbook] mathematical problem usually describes “the nucleus of a situation in too abstract a manner. The problem should grow out of the situation, and the child should learn to recognize the problem in the situation. Raising a problem is mathematics, too.” (pp. 134–5). In his rather long Chapter 19 on logic, one finds, “The mathematics student at university should in any case be accustomed to quantifiers as early as possible. Nothing is to be gained by delaying it. The earlier the student is vaccinated with quantifiers the better will he stand the quantifier fever.” (p. 656).

This is not a textbook, except possibly for some mathematics education graduate students who would benefit from knowing where some present day ideas come from. Below I provide some background on how Freudenthal transitioned from research in mathematics to an all-consuming interest in mathematics education.

**Background Information**

Freudenthal is perhaps best known to the mathematics community as a topologist. Born in Germany in 1905, Freudenthal entered nearby University of Berlin in 1923 and did his Ph.D. in topology because it was then a “hot topic.” In 1930, because of his interest in intuitionism, Freudenthal was recruited to Amsterdam as Brouwer’s assistant. He gave courses on algebraic fields, group theory, measure theory, complex analysis, linear operators, topology, and elementary mathematics from an advanced standpoint — topics that were not then part of the standard curriculum. He published papers on topological groups, Lie groups, almost periodic functions, algebraic topology, and functional analysis, as well as intuitionism. These gained him an international reputation.

In 1937, with the retirement of several professors, Freudenthal was given regular teaching duties. He drastically changed a two-year analysis course, choosing to use the abstract approach of Bourbaki, introducing the language of sets from the outset and seeing metric spaces as the natural setting for some basic theorems. During World War II, he was suspended from his duties, leaving him time to pursue his own interests, punctuated by several rather unnerving experiences — a few arrests of short duration, a period in a work camp, and a period of hiding. When the war was over, Freudenthal was offered the Chair of Geometry at Utrecht, which gave him the chance to modernize their mathematics curriculum.

Unlike a number of other mathematicians, Freudenthal did not turn to education only as he got older. His interest was sparked early in the 1940s by lessons given his two young sons — these caused him to reflect on arithmetic. He eventually read everything on the subject from schoolbooks to didactic theories. In 1942, from observations of his children’s learning and his knowledge of mathematics and the history of mathematics, Freudenthal wrote the first chapter of a work to be titled, *Didactics of Arithmetic*. In it, one can see the origins of many ideas that were to occupy him for the next half century.

From 1945 on, Freudenthal participated in secondary education meetings of the Dutch Mathematics Study Group of the Werkgemeenschap voor Vernieuwing van Opvoeding en Onderwijs (WVO). These became his “college of education.” During the 1950s, he directed the Ph.D. dissertations of Dieke van Hiele-Geldolf and Pierre van Hiele on geometry. Pierre was inspired to create his level theory by Dieke’s observations of her lower secondary pupils’ learning of geometry — her pupils’ learning appeared to be discontinuous, with insight coming in jumps. At the lowest level, they proceeded intuitively and informally, acting on objects. These actions themselves become the objects of subsequent mathematical considerations. Freudenthal, while noting that the van Hieles deserved all the credit for their discovery, saw levels as relative rather than absolute, attributed a rise in levels to reflection, and applied the theory to other areas of mathematics learning.

A Dutch group called Wiskobas (or mathematics in primary school) was formed in 1968. The Institute for the Development of Mathematics Instruction (IOWO) — now called the Freudenthal Institute — was established in 1971 to modernize the Dutch school curriculum. He served as Director until 1976. Wiskobas promptly became a department of IWOW with Freudenthal the group’s most prominent member. It was he who put Wiskobas on the track of realistic mathematics, steering the group away from both traditional arithmetic and the New Math. He urged the incorporation of everyday reality into mathematics education as a source of learning, not just for applications. Freudenthal and Wiskobas emphasized the importance of rich thematic contexts and long-term learning strands such as basic skills and column arithmetic, ratios and fractions, and measurement and geometry.

Freudenthal criticized the New Math because it employed an anti-didactical inversion, meaning that the “smoothed over” end products of the historical learning process, the definitions and axioms, were taken as the point of departure. Realistic mathematics education reverses this idea. It begins with rich, context-laden problems upon which students reflect, gradually progressing from concrete to operational to abstract.

From 1967 to 1970, Freudenthal was President of the International Commission on Mathematics Instruction (ICMI). At the first International Congress on Mathematical Education (ICME) in Lyons in 1969, Freudenthal asked E. Fischbein to organize a round table devoted to psychological problems of mathematics education. This led, in 1976, to the creation of the International Group for the Psychology of Mathematics Education (PME), which meets every year in various parts of the world — this summer in Vancouver in conjunction with PME-NA, its North American Chapter. In 1968 Freudenthal founded what has become the most prestigious European journal for mathematics education research, *Educational Studies in Mathematics*. His contributions to mathematics education research were so great that the ICMI now presents a medal named in his honor — the *Hans Freudenthal Award*, presented biennially, which recognizes a major cumulative research program in mathematics education.

Freudenthal was critical of fads in education. He criticized mastery learning, which was in vogue in the 1970s, and took Bloom’s taxonomy and learning objectives to task, remarking that these could not possibly describe everything worth teaching and learning. During the New Math era, he was concerned that counting was out and equipotent sets were in at nursery school. He also disagreed with Piaget’s idea that cognitive development in children took place from poor (abstract) to rich (concrete) structures, arguing that it was the reverse. He criticized Piaget’s experiments for the artificial character of their questions and for his pre-determined interpretations of students’ reactions.

Earlier than most, Freudenthal emphasized the importance of having students engage in group work and reflection to reach new levels of insight, of having students re-invent mathematics under guidance, and of using research, particularly observations of individual students’ thinking processes, to inform curriculum development. He once said, “Yes, I was a drummer, but to deaf ears… But don’t think I have regrets. An enfant terrible should be proud of his vices.” Fortunately, not all ears were deaf and the drummer’s legacy lives on.

Much of the information on Freudenthal’s life and work was adapted from the *UME Trends* article, “The Legacy of Hans Freudenthal”, May 1994, by Annie and John Selden.

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.