Michael N. Fried

Can Mathematics Education and History of Mathematics Coexist

Despite the wide interest in combining mathematics education and the history of mathematics, there are grave and fundamental problems in this effort. The main difficulty is that while one wants to see historical topics in the classroom or an historical approach in teaching, the commitment to teach the modern mathematics and modern mathematical techniques necessary in thepure and applied sciences forces one either to trivialize history or to distortit. In particular, this commitment forces one to adopt a “Whiggish” approach to the history of mathematics. Two possible resolutions of the difficulty are (1) “radical separation” – putting the history of mathematics on a separate track from the ordinary course of instruction, and (2) “radical accommodation” – turning the study of mathematics into the study of mathematical texts.

Special Issue on History and Philosophy of Mathematics in Mathematics Education

Although Science & Education has published papers specifically concerning mathematics over the years—some quite significant for education—mathematics has never received the spotlight. Indeed, when the journal was established it bore the subtitle, Contributions from History, Philosophy and Sociology of Science and Mathematics, but so few papers on mathematics were received in those years that the word ‘‘mathematics’’ was duly dropped. Yet within the community of science and mathematics educators, there is no lack of interest at least in the contributions of history and philosophy of mathematics to mathematics education, particularly the history of mathematics. Besides publications such as those we have listed at the end of this introduction, one can point to the activities of the International Study Group on the Relations between the History and Pedagogy of Mathematics (HPM) which dates back to 1972, and which almost from the start became an affiliated study group of the International Commission on Mathematical Instruction (ICMI), the largest and oldest international organization for mathematics education. International conferences of HPM have been held in Europe, Canada, Australia, Taiwan, Mexico, and Korea in association with the quadrennial International Congress on Mathematical Education (ICME). The European Summer University on History and Epistemology in Mathematics Education (ESU), now held every 4 years as well, is another regular venue for meetings dealing with history in mathematics education, as is the working group on History in Mathematics Education at the biennial Congress of European

High-stakes assessment as a tool for promoting mathematical literacy and the democratization of mathematics education

Abstract The extensive use of National Completion Examinations in Mathematics (NCEM) as a critical filter for educational and social advancement highlights the tension between society’s ambition for high achievement and its ambition for general mathematical literacy. This paper presents an innovative model for a NCEM that demonstrates how such examinations can serve to promote mathematical literacy for all, without losing its ability to promote high achievement. In the course of the paper, the meaning of mathematical literacy for all is also discussed; it is stressed that at the heart of this notion lies students’ openness to mathematics, so that the success of a program directed towards mathematical literacy for all is reasonably measured by the number of students who persist in their study of mathematics and who develop a confident attitude towards it. Data were given suggesting that the new model for the NCEM is successful in this respect.

History of Mathematics in Mathematics Education

This paper surveys central justifications and approaches adopted by educators interested in incorporating history of mathematics into mathematics teaching and learning. This interest itself has historical roots and different historical manifestations; these roots are examined as well in the paper. The paper also asks what it means for history of mathematics to be treated as genuine historical knowledge rather than a tool for teaching other kinds of mathematical knowledge. If, however, history of mathematics is not subordinated to the ideas and methods at the heart of the usual mathematical curriculum – algebraic equations, functions, derivatives, analytic geometry – if it becomes an essential subject for mathematics, then the attempt to find a place for history of mathematics requires refining our understanding of the nature of mathematics education itself. Thus, the paper asks not only how can history of mathematics be incorporated into mathematics education but also how the idea of mathematics education may need to be adjusted to accommodate history.

The co-development and interrelation of proof and authority: The case of Yana and Ronit

Students’ mathematical lives are characterized not only by a set of mathematical ideas and the engagement in mathematical thinking, but also by social relations, specifically, relations of authority. Watching student actions and speaking to students, one becomes cognizant of a ‘web of authority’ ever present in mathematics classrooms. In past work, it has been shown how those relations of authority may sometimes interfere with students’ reflecting on mathematical ideas. However, “…by shifting the emphasis from domination and obedience to negotiation and consent…” (Amit & Fried, 2005, p.164) it has also been stressed that these relations are fluid and are, in fact, asine qua non in the process of students’ defining their place in a mathematical community. But can these fluid relations be operative also in the formation of specific mathematical ideas? It is my contention that they may at least coincide with students’ thinking about one significant mathematical idea, namely, the idea ofproof. In this talk, I shall discuss both the general question of authority in the mathematics classroom and its specific connection with students’ thinking about proof in the context of work done in two 8th grade classrooms.

Otto Toeplitz's 1927 Paper on the Genetic Method in the Teaching of Mathematics

“The problem of university courses on infinitesimal calculus and their demarcation from infinitesimal calculus in high schools” (1927) is the published version of an address Otto Toeplitz delivered at a meeting of the German Mathematical Society held in Dusseldorf in 1926. It contains the most detailed exposition of Toeplitzs ideas about mathematics education, particularly his thinking about the role of the history of mathematics in mathematics education, which he called the “genetic method” to teaching mathematics. The tensions and assumptions about mathematics, history of mathematics, and historiography revealed in this piece dedicated to educational ideas are what make Toeplitzs text interesting in the study of historiography of mathematics. In general, the ways historiography of mathematics and teaching of mathematics, even without an immediate concern for history, are deeply entangled and, in our view, worth attention both in historical and educational research.

The Use of Analogy in Book VII of Apollonius’ Conica

Argument Apollonius of Perga’s Conica , like almost all Greek mathematical works, relies heavily on the use of proportion, of analogia . Analogy as the assertion of a resemblance, however, also plays a role in the Conica . The homologue , which Apollonius introduces in Book VII, is a striking example of analogia in both senses. On the one hand, the homologue is defined by means of a precise proportion relating the diameter and latus rectum of a conic section to fixed segments along the diameter. On the other hand, Apollonius’ use of the homologue in Book VII makes it clear that he meant it truly to evoke the latus rectum in the reader’s mind; in this way, Apollonius treats the homologue as an image of the latus rectum . The example of the homologue thus suggests the possibility that, in Greek mathematics, proportion was not only a vital manipulatory tool but also a means of making images.

Mathematics and Mathematics Education: Beginning a Dialogue in an Atmosphere of Increasing Estrangement

In 2009, Norma Presmeg wrote a piece for a special issue of the ZDM on interdisciplinarity. Presmeg’s paper presented her view of the general spirit of and possibilities for mathematics education research. This prompted a dialogue on the state of mathematics education by Ted Eisenberg and Michael Fried, published in the same issue of ZDM. This paper gives an account of that dialogue and the symposium in honor of Ted that arose out of it; in doing so, it also further elaborates on the themes that motivated this book.

History of Mathematics, Mathematics Education, and the Liberal Arts

This paper considers how the history of mathematics, if it is taken seriously, can become a mode of thinking about mathematics and about one’s own humanness. What I mean by the latter is that by studying the history of mathematics rather than simply using it as a tool—and that means attempting to understand it as an historian does—one becomes aware of how mathematics is something human beings do that therefore informs our human identity. In this way, the history of mathematics in mathematics education has the potential to make us fuller human beings, which is at the heart of the educational tradition known as the “liberal arts.” By considering the nature of the liberal arts, we may understand better the meaning of the history of mathematics in mathematics education and, indeed, the meaning of mathematics education tout court.

Handbook on the history of mathematics education

It is quite true what philosophy says; that life must be understood backwards … [but] must be lived forwards. (S. Kierkegaard, Journals IVA 164, 1843)Roughly speaking, mathematics education researc...