ABRAHAM ARCAVI, Weizmann Institute of Science, Israel

“Roles of the history of mathematics in the mathematical knowledge for teaching”

The construct “Mathematical Knowledge for Teaching” (MKT) is receiving much attention in the last two decades, both in research and in the practice of mathematics education. MKT is the mathematical knowledge required to practice and accomplish the work of teaching mathematics. Among others components, it includes: (a) competence with the contents and their underlying ideas (including subtleties or extensions which may go beyond the topic that is required to teach); (b) acquaintance with pedagogical means (material resources, representations, explanations, enlightening examples of various kinds); (c) sensitivity towards and resourcefulness to deal with students idiosyncratic ways of knowing (including their errors, doubts and misunderstandings); (d) familiarity with diverse curricular approaches and alternative ways to introduce a concept or a procedure and versatility in applying them, and (e) having an eye towards knowledge in the “horizon” (i.e. the mathematics that students will need in their future studies).
In this presentation, I propose that history of mathematics can be a bountiful source to address the enhancement of teachers’ MKT and I will try to substantiate this claim through illustrative examples and the morals which can be derived from them. I hope that this will be a modest contribution to the role of MKT in research, in teacher pre-service education and during their lifelong teacher professional development. In addition, the ideas may possibly warrant consideration by historians of mathematics, and thus widen the already existing dialogue between the two communities.



DOMINIQUE TOURNÈS, Université de la Réunion, France

“What history training for future mathematics teachers? Personal experiences and reflections”

In this lecture, I will present my way of training future teachers in the use of history to improve students’ mathematical
learning. The strategy that I recommend, based on a long experience of training in Reunion and Mayotte (two French
islands in the Indian Ocean), includes three components in constant interaction. First of all, there is the manipulation of
artefacts (ruler and compass, puzzles, token abacuses, linkages, nomograms, planimeters…) that the teacher can use in a
process of semiotic mediation to favour the appropriation of the mathematical knowledge embedded in these objects.
This importance given to gestures, procedures and instruments is reinforced by an opening towards ethnomathematics.
Secondly, there is the study of short original texts taken mainly from six major works covering almost completely the
contents of secondary education (Euclid, al-Khwārizmī, Descartes, Newton, Cauchy, Jakob Bernoulli). This work is
done in close connection with specific curriculum items and the design of scenarios for the classroom. Finally, the
future teachers are responsible for designing pedagogical sequences inspired by history and experimenting with them in
their classes during the internships. This devolution phase, which is the subject of a priori and a posteriori didactic
analyses, seems to me essential for a sustainable integration of the training’s achievements. The main objective of this
triptych of activities is not to train future teachers in history, but simply to train them in the teaching of mathematics by
deepening their knowledge of the discipline on the cultural, epistemological and didactic levels.


MARIA ROSA MASSA-ESTEVE, Universitat Politècnica de Catalunya – Barcelona Tech (Spain)

“The Use of Original Sources in the Classroom for Learning Mathematics”


The teaching and learning of the history of mathematics contributes to the comprehensive education of students, whether they are future mathematicians or engineers or future teachers. The use of the history of mathematics, as an implicit and explicit resource, makes it possible to improve the teaching of mathematics and the comprehensive training of students. History can be used as an explicit resource to introduce or to better understand certain mathematical concepts and methods through the analysis of selected historical sources in the classroom.
In addition, the history of science, and specifically the history of mathematics, is a very fruitful tool to convey to students one perception of Mathematics as a useful, dynamic, humane, interdisciplinary and heuristic science, while complementing the thematic study of the parts of mathematics.
History shows that societies develop as a result of the scientific activity undertaken by successive generations, and that mathematics is a fundamental part of this process. Thus, students with a knowledge of the history of mathematics are able to understand mathematics as an intellectual activity that can be useful for solving the problems of society in each time and place.
Nevertheless, my talk will be focused on other uses in the classroom; that is, on activities based on the history of mathematics for learning mathematics. In the practical activities, when faced with a mathematical text, students are motivated to make it their own and create their own knowledge, which is the best way to learn mathematics. At the same time, they can recall or review theorems, formulas or mathematical rules from another perspective. The practical activities using original sources from the history of mathematics can provide students with a greater comprehension of the foundations and nature of this discipline, as well as a deeper approach to the understanding of the mathematical techniques and concepts used every day in the classroom.
Therefore, the aim of my lecture is to reflect on the use of original historical sources for learning mathematics through practical mathematical activities. In the first half I would like to address this way of introducing the history of mathematics by providing the analysis of some new resources and ideas, and in the second part I will analyse some practical activities drawn from the transformation of mathematics in the 17th century. The crucial aspects of this period will enable improvements in the mathematical education of students by learning new mathematical ideas, procedures and proofs.


MICHEL ROELENS,UCLL Hogeschool, Campus Diepenbeek, Belgium

“Algorithms before computers”

In the new curricula that are now gradually being implemented in secondary schools in Flanders, the Dutch-speaking part of Belgium, there is a shift towards mathematics as a part of STEM (Science, Technology, Engineering, Mathematics) and towards computational thinking in mathematics: programming, logic and electronic gates, graph theory, and for some pupils linear programming. Meanwhile, the new curricula contain less geometry and fewer references to history and art. With the words of Man Kung Siu (2002) in mind, we could call it a shift towards more algorithmic mathematics and less dialectic mathematics.
As a first reaction, I welcome the introduction of graph theory, but I regret the loss of some beautiful parts of geometry. Instead of complaining, I try to relate the new subjects with geometry and the history of mathematics. Some young students might think that algorithms are just a way to prepare computer programs. But, as Jean-Luc Chabert et al. have extensively documented , there have always been algorithms in mathematics, long before there were electronic computers that could be programmed. Moreover, geometrical and visual thinking can be used to discover many algorithms.
In my talk, I would like to illustrate historical and geometrical aspects of algorithms with some examples.
The calculation by hand of the digits of a square root, starting from the geometrical problem “given the area of a square, find, step by step, the length of the side”. This algorithm has roots in Babylonian, Greek and Indian mathematics. Even if the practical use of this algorithm has been made superfluous by computers and calculators, discovering the algorithm is still an exciting activity.
The proof that a connected multigraph in which the degree of each vertex is even has an Eulerian tour (Hierholzer, 1873) . This proof consists of an algorithm for constructing such an Eulerian tour.
In addition to this talk, I also propose a workshop “Graph theory algorithms before computers”, in which participants can discover some more historical algorithms in graph theory.
Ana Millán Gasca | Roma Tre University, Rome, Italy - Academia.eduTHEME 5

ANA MILLÁN GASCA, Università di Roma 3, Italy

“A hidden thread: ideas and proposals on children’s mathematics education in history”

A lively world of ideas and proposals regarding the initiation of children to mathematics can be followed in Europe and the countries of European educational tradition since the 18th century. This cultural thread has involved mathematicians and educators, women and men; it’s addressed to boys and girls, to the high classes and to the working classes; it’s linked to the evolution of views and
social practices regarding childhood as well as the self education of adults; entrepreneurial initiatives were prompted by these ideas and proposals (books for teachers and parents, books, picture books and boxes/rational toys), and at the same time their supporters were also encouraged by political visions of a modern, equal society. And yet it’s a hidden thread, because historians of education often overlook mathematics in sources (from the iconic Johann Pestalozzi to the founder of experimental pedagogy Wilhelm Lay) and historians of mathematics and of mathematics education have disregarded this basic level as little relevant for the reconstruction of the mathematical universe throughout the ages. Moreover, in this sphere, the national and linguistic specificities require a recognition of the evolution in countries or cultural areas as a basis for comparative analysis. The situation is changing in recent times, also because of the challenges of present mathematics education in childhood for an inclusive, equal society as well as for our digital future. I present a tentative timeline – starting from the origins of the European conception of numeracy as an educational challenge linked to the diffusion of Indian figures in the Italian Medieval scuole d’abaco –; some historiographical aspects; some recent contributions, ideas and proposals that appear as prominent, original or influential; and the open questions in front of us.

MARIA TERESA BORGATO, Università degli Studi di Ferrara, Italy

“The History of Mathematics in Italy through the ages: sources, correspondences, and editions”.
The study of primary sources is the starting point for the research in the history of mathematics, and increasing those resources is one of the most significant and enduring contributions that scholars can make to the discipline. In past decades, Italian historians of mathematics have contributed to it with critical editions of works, and the publication of numerous inedited documents, manuscripts and correspondences. Correspondences, in particular, allow us to investigate the circulation of scientific thought and the origin of mathematical ideas. In my talk, I will present some of these contributions, which go from the late Middle Ages and the Renaissance, to the first half of the twentieth century. In the teaching of mathematics, quoting meaningful passages from these sources can be a useful strategy to introduce arguments and concepts, or to raise awareness of historical development. In the related workshop, a certain number of examples and suggestions, suited to this purpose, will be illustrated in more detail.
The plenary will be extended by two workshops about Abacus mathematics and Archimedean tradition
The resolution of algebraic equations of third and fourth degrees
The pre-Newtonian systems of the world
The foundation of infinitesimal calculus
The science of waters as the main field of applied mathematics
Re-launching Italian education and research after political unification
Special lecture
University of Minho, Portugal
Ethno + mathema + tics: The legacy of Ubiratan D’Ambrosio

Ubiratan D’Ambrosio has been considered as the father of ethnomathematics. Above all, in his conception of the term Ethnomathematics. In his own words: ‘the adventure of the human species is identified with the acquisition of modes, styles, arts and techniques (tics) of explaining, learning, knowing and coping with (mathema) the natural, social, cultural and imaginary environment (Ethno)’. Having worked in the fields of Mathematics Education (for which received the Felix Klein medal from ICMI) and the History of Mathematics (he also received the Kenneth O. May medal from ICHM), he considered ethnomathematics as a subfield of both the History of Mathematics and Mathematics Education, enriched by its connections with cultural studies and political domains. In this lecture, I will try to highlight Ubiratan’s most important ideas regarding ethnomathematics and discuss some of their implications for both Mathematics Education and the field of History of Mathematics.