Abstract
Book review of “L’enseignement des mathématiques” edited by Piaget, written by Beth, Choquet, Dieudonné, Lichnerowicz, Gattegno & Piaget, and published in 1955. An assay written during Spring 2009 at Texas A&M University for EDCI 689604, “Knowledge and Research in Curriculum & Instruction.” The translations are mine.
Introduction
This book is a collection of chapters about the teaching of mathematics written by six scholars. Each author wrote their chapter independently of the other authors except for Jean Piaget, the famous Swiss psychologist and constructivist, who wrote the first chapter after having read all the chapters submitted by the other authors (p. 5). The authors were founding members of the “International Commission for the Study and Improvement of Mathematics Education”^{1} (CIEAEM).
The other five authors were (1) Evert Willem Beth, Dutch logician and philosopher of science and professor at the University of Amsterdam^{2}, (2) Jean Dieudonné, mathematician and member of the Bourbaki project^{3} and professor at Northwestern University, (3) Andrè Lichnerowicz, French mathematician and mathematical physicist^{4} who was influential in the new math reform and professor at the Collège de France, (4) Gustave Choquet, French mathematician^{5} and professor at the Université de Paris, and (5) Caleb Gattegno^{6}, scholar of pedagogy, inventor of the Geoboard and professor at the London University. Gattegno founded CIEAEM in 1951. The academic positions of the authors were current at the publication of the book in 1955.
Table 1 shows the chapters authors, their number and their pages and Table 2 gives the chapter titles. I give my English translation of the chapter titles as the section titles of this book report.
The mathematical structures and the operational structures of intelligence
The first chapter of the book written by Jean Piaget is a study of how the mind becomes aware of mathematics. The author starts with a question that is the “fil rouge” for the whole chapter (p. 11)
…si les connexions mathématiques sont engendrées par l’activité de l’intelligence ou si cellci découvre cellesla comme une réalité extérieure et tout faite.^{7}
Piaget tries in this chapter to connect the psychology of learning mathematics with the fundamental structures of mathematics as per Bourbaki. They are the algebraic structures (groups), the order structures (networks), and the topological structures (p. 14). Has he succeeded? He certainly has written a very eloquent article, but it is my impression that he stretched the point to fit his hypothesis.
Reflections on the organization and the method of the teaching of mathematics
The second chapter of the book written by Evert Willem Beth is about the relationship between secondary and postsecondary teaching programs. The author points out some of the differences between these two institutions of learning. One of them is that in secondary education only a minority of the students is adept at learning mathematics, while this problem would not exist in postsecondary education. We can remark the distance of this opinion from contemporary teaching in the USA.
The author also pointed out how mathematics has become more abstract, complicated and rigorous and how this created a gap between secondary and postsecondary study of mathematics. Beth then digresses on the relationship between the fundamentals of mathematics, logic and psychology. Beth refers in addition to Piaget to another Dutch teacher, mathematician and philosopher Gerrit Mannourij ? (?)^{8} and his psycholinguistics (p. 45).
The abstraction in mathematics and the evolution of algebra
The third chapter of the book written by Jean Dieudonné starts pointing out that mathematics shares with metaphysics the interest in abstract entities, detached from the concrete objects of our experience of the senses (p. 47). The author connects that with the natural aversion of the majority of the people to the study of mathematics and then connects this situation to the tendency in schools to mask the abstractness of the subject as long as possible. He considers this to be a “grave erreur.” Dieudonné then asks about the purpose of the teaching of mathematics. He states that obviously it is not to learn a number of theorems on this or that subject, but because it is an excellent way to think logically and rigorously. The author thinks that the power of mathematics is power of abstract thought.
The author then dives into the history of mathematics and how important abstraction is in the development of this discipline
Introducing the spirit of modern algebra in elementary algebra and geometry
The fourth chapter of the book by Andrè Lichnerowicz starts by presenting a dilemma in the teaching of mathematics. How to keep the subject, relevant, thus close to the students and at the same time show the depth of the discipline. Recently mathematics experienced a rapid and radical transformation that created a disconnect between how mathematics is learned as a tool in secondary school and modern mathematics as it is taught in universities.
At this point the author presents an alternative way of understanding algebra to show the reader that mathematics can be approached in completely different ways.
On the teaching of elementary geometry
The fifth chapter of the book written by Gustave Choquet is about the teaching of elementary geometry. The author presents the problem of where to start teaching the subject. Choquet stated that often the geometry are inferior to The Elements with regard to the foundations of geometry.
The author then proceeds to critique some geometry textbooks even though he does not refer to any one by name. In addition Choquet discusses the plurality of axiomatic systems in classical geometry and then as an example builds an axiomatic system devoid of figures.
The pedagogy of mathematics
The sixth chapter of the book written by Caleb Gattegno starts with a positivistic statement. The author claims that
nous traiterons de la pédagogie mathématique comme une science. C’estàdire que nous nous attaquerons aux aspects de le pédagogie qui sont communicables à tous et susceptibles d’être traduits en techniques impersonnelles.^{9}
The author then qualifies this statement by referring to the personal nature of the teaching, but he is still confident that it is possible to arrive at the truth.
Footnotes
^{1}Original name “Commission internationale pour l’étude et amélioration de l’enseignement des mathématiques,” http://www.irdp.ch/cieaem
^{2}http://en.wikipedia.org/wiki/Evert_Willem_Beth
^{3}http://wwwhistory.mcs.standrews.ac.uk/Biographies/Dieudonne.html
^{4}http://en.wikipedia.org/wiki/Andr’e_Lichnerowicz
^{5}http://en.wikipedia.org/wiki/Gustave_Choquet
^{6}http://en.wikipedia.org/wiki/Caleb_Gattegno
^{7}whether the mathematical connections arise because of the activity of the intellect or they are discovered as a preexisting external reality
^{8}Also spelled as Mannoury.
^{9}we will treat mathematic pedagogy as a science. That is, that we will look at aspects of pedagogy which can be communicated to all and can be implemented as impersonal techniques.
Author  Chapter  Pages  




le Bureau  Préface  5 – 9  
J. Piaget  1  11 – 33  
E. W. Beth  2  35 – 46  
G. Choquet  3  47 – 61  
J. Dieudonné  4  63 – 74  
A. Lichnerowicz  5  75 – 129  
C. Gattegno  6  131 – 173  




Table 1  Chapters 
Chapter  Title 


1  Les structures mathématiques et les structures opératoires de l’intelligence 
2  Réflexion sur l’organisation et la méthode de l’enseignement mathématique 
3  L’abstraction en mathématiques et l’évolution de l’algèbre 
4  Introduction de l’esprit de l’algèbre moderne dans l’algèbre 
et la géométrie élémentaires  
5  Sur l’enseignement de la géométrie élémentaire 
6  La pédagogie des mathématiques 


Table 2  Chapter titles 