Georges Cuisenaire’s numbers in colour. A teaching aid that survived the 1950s

In 1952, a Belgian primary school teacher, Georges Cuisenaire, published Les nombres en couleurs, a booklet in which the author describes his invention and explains a corresponding method for teaching elementary arithmetic. Cuisenaire materialized the numbers from one to ten by means of rods of corresponding lengths and in different colours. The rods provided a concrete tool for exploring and gaining insight in basic concepts and skills, such as the four basic operations, finding divisors and multiples, working with fractions, the decimal system, arithmetic sequences, and area and volume calculation. From the mid-1950s, with the support of Caleb Gattegno, the Cuisenaire rods broke through worldwide. In subsequent years, empirical research into the effectiveness of the Cuisenaire’s method was initiated and in several countries Cuisenaire Associations were founded. In the late 1960s and 1970s, a number of attempts were made to use the material for the teaching of typical modern mathematics contents to (very) young children, but the use of the rods in this context was sometimes far-fetched and did not break through. Keywords: CIEAEM, Cuisenaire rods, numbers in colour, teaching aid, teaching material, Georges Cuisenaire, Caleb Gattegno

Actors in the changes of ICMI: Heinrich Behnke and Hans Freudenthal

The years after WWII up to the late 1960s were crucial in the evolution of ICMI (International Commission on Mathematical Instruction) for both the settlement of some institutional aspects (mainly concerning the relationship with mathematicians) and the establishment of new trends of the activities. By referring to unpublished documents, this paper focuses on the role of two key figures in those years: Heinrich Behnke and Hans Freudenthal. As ICMI Secretary and later as President, Behnke tried to reshape the newborn ICMI after WWII and clarify the relationship with mathematicians. His action was completed by Freudenthal, who, as ICMI President, broke with the past and promoted initiatives that fostered the development of mathematics education as an academic field and the independence of ICMI from the community of mathematicians. Keywords: history, ICMI, mathematicians, mathematics education

From experimental to theoretical geometry in new pedagogical movements at the turn of the 19th and 20th centuries (1872-1906)

There exist many historical works on the new pedagogical movements in the beginning of the 20th century, at the level of one country and at the international level also. Our purpose is to focus on teaching of geometry with comparing situations in four countries: United Kingdom, France, Germany and United States. We show that, behind the agreements, there are deep differences in relation with questions posed by geometrical teaching. We use two kinds of materials, discussions and textbooks, and we specially examine the questions on parallels definitions and their introduction in teaching. Keywords: laboratory method, concrete geometry, experimental geometry, intuitive geometry, practical geometry, rational geometry, Émile Borel, Carlo Bourlet, John Dewey, George Halsted, Julius Henrici, Adelia Hornbrook, Jules Houël, Charles Méray, Eliakim Moore, John Perry, Peter Treutlein.

Scientific novelties implemented into teaching mathematics in secondary schools on the Polish territories in the 19th century. The case of descriptive geometry

This article is dedicated to discussing the implementation of the descriptive geometry, i.e. the scientific novelty from the end of the 18th century, in secondary school education on the Polish territories in the 19th century. At that time, Polish lands were under the occupation of three empires: Prussia, Austria, and Russia. Over the time, the policy of the partition empires toward the Poles was changing in intensity. As a consequence, in the 19th century, there were schools on the Polish territories with Polish, Prussian, Austrian and Russian curricula and relevant lecture languages. The article analyses the implementation of descriptive geometry into teaching mathematics in schools located in all three partitions. Keywords: descriptive geometry, history of mathematics education, history of mathematics

Shaping analytic geometry as a secondary school subject. A comparative study

A comparative study exploring textbooks used in two distinct educational systems, Brazil and Portugal, was performed focusing on the ways in which analytic geometry was developed as a secondary school subject. Our analysis concentrates on textbooks from the late 19th century until the middle of the 20th century known to be used in schools. Keywords: history, analytic geometry, textbooks, mathematics education

Algebra in Swedish mathematics curricula (1930-2000)

This paper can be seen as an overview concerning algebra in the Swedish mathematics curricula in the period of 1930-2000; the prescribed content of the algebra teaching is described and the basic ideas about how to teach algebra. The analysis is, however, focused on how the syllabi and commentary materials gave teachers more or less freedom to plan and carry out their teaching. The analysis is based on Basil Bernstein’s concepts of classification and framing.

Friedrich Fröbel’s conception of crystallography and the mathematical education at the 19th century kindergartens

What is the connection between crystallography and kindergarten activities? The paper aims to show the ways Friedrich Fröbel, the founder of the modern kindergarten, was influenced by crystallography at the beginning of the 19th century. This influence explicitly manifested itself not only in a transfer of concepts from the German crystallographer Weiss, but also in a transfer of images, whose source was the French crystallographer Haüy. Fröbel, having a peculiar conception of mathematics, reshaped concepts and images from crystallography already during the 1810s, and conceptualized his playful objects with them for children. Keywords: Crystallography, Mathematics at the 19th century kindergartens, Friedrich Fröbel, René-Just Haüy, Christian Samuel Weiss.

Back to the future – a journey from current education reforms to reformations in the past

Learning from history does not automatically mean that history prevents us from repeating mistakes. We cannot see what happens in the future, even with the most profound knowledge of the past. Although it is not possible to make such causal connections, the study of structural components, which recur and make up patterns, can certainly contribute to sharpening political judgement. How can the teaching of the history of mathematics education then help to support an understanding of possible courses of individual actions without indoctrination through the political or even ideologically influenced production of time references? The paper presents the concept of a lecture course in mathematics education, held at the University of Mainz. We take as a point of departure the everyday experience of our prospective mathematics teacher with various current education reforms and present seemingly similar processes during former reforms. Here we limit ourselves to reforms during the 19th and 20th century.

Mathematical studies in the 18th century, in the work of François René de Chateaubriand

In the 17th century, the youngest son of a noble family would follow a career as an officer in the army or as a priest. It is not surprising that Viscount François René de Chateaubriand (1768-1848) decided to prepare the exam to join l’École navale (Naval Academy), at that time École des gardes de la Marine. In fact, he did not decide this by himself, but followed the steps of his father, Count René-Auguste de Chateaubriand, ship-owner, navigator, and merchant. In this article, we will explore the book Mémoires d’outre tombe, following the young Chateaubriand in different schools, such as the Collège de Dol and the Collège de Rennes. At that time, the mathematician Étienne Bézout (1730-1783) was the almighty examiner for the entrance to l’École Navale. The memories of Chateaubriand introduce us to the way the scholars of the year 1780 studied their “Bézout” to improve their mathematical level. Keywords: Naval Academy, Bézout

Descriptive geometry in middle school mathematics teaching in Japan (1905-1946)

Teaching of descriptive geometry began in 18th-century France and became widespread in tertiary and secondary education worldwide throughout the 19th century. Until the 20th century, educators often described two aims of descriptive geometry – technical education and mathematics education. In Japan, descriptive geometry was introduced into engineering and artistic higher education after the Meiji Restoration of 1868. Descriptive geometry became part of the general secondary school curriculum in the 1880s, but it had been taught under the auspices of arts and crafts education rather than mathematics. In the early 20th century, Japanese mathematics educators began to focus on descriptive geometry as a way to reform solid geometry. When Japan’s secondary school curriculum was revised in 1942, descriptive geometry was included in solid geometry and mathematics for the first time. Although this curriculum lasted only until 1946, it was the fruit of many educators’ labors and is worthy of examination. This paper examines several books and documents from the early 20th-century Japan and shows that there was a technical, mathematics-oriented debate about the aim of descriptive geometry teaching as seen in Europe. Keywords: descriptive geometry, solid geometry, secondary school, middle school, Nobutaro Nabeshima, Minoru Kuroda