Toward a Cognitive Historiography of Mathematics Education Iason Kastanis a & Nikos Kastanis b b) Aristotle University of Thessaloniki, Greece a) Universidad de Barcelona, Spain

Overview Introduction Cognitive consideration in contemporary mathematics contexts: In mathematics education In history of mathematics A look into the methodology of the cognitive history of mathematics The practice in the historiography of science An example in history of mathematics: the abacus cultural practice The practice as means in the historiography of mathematics education Conclusion

Motivation In 1999 published the book Cognitive history of mathematics, it is something that does exist it is something that does exist. So, there should also be a cognitive history of mathematics education But, what is a cognitive history of mathematics? And what about a cognitive history of mathematics education cognitive history of mathematics education?

The cognitive consideration is favored today in mathematics contexts In Mathematics Education

discourses of practice James Greeno states that the cognitive approach emphasizes: conceptual structures, representations of information, conceptual changes, and

In History of Mathematics page 7

cognitive It is obvious that there are cognitive concerns Mathematics Education and inquiries in the Mathematics Education and History of Mathematics the History of Mathematics. methodology But, what is known about the methodology cognitive of the cognitive history of mathematics?

Unlocking the historiographical door of mathematics “The historians need to analyze scientific action the concrete scientific action rather than an abstract life of mathematical ideas.” Page 13 This is a position of the German historian, Moritz Epple.

practice of mathematics “The practice of mathematics […] is a complex of actions, such as defining, conjecturing, proving, etc. These mathematical actions are immersed in communicative and social actions like publishing, giving talks, applying for positions, organizing meetings, and the like.” As Epple further clarifies: This is a new approach to the history of mathematics, and it can be applied to various case studies, for example:

In their history-oriented studies, Epple and Kjeldsen research practices analyzed the distinct research practices that led to new theories of modern mathematics. They noticed the local character of the respective research activities local traditions that expresses the local traditions mathematical practices of mathematical practices, distinct mathematical that is the distinct mathematical cultures cultures. On the other hand, Matthew Jones discursive practices focused on the discursive practices that were developed within the broader cultural and cognitive context of 17 th century mathematics.

practice The practice in the historiography of science Since the 1960’s, the ideas of Thomas Kuhn dominate the historiography and epistemology of science. Kuhn and his followers “reorient the philosophy of science scientific practices toward an account of scientific practices rather than scientific knowledge”.

Around 1980, two movements emerged in the epistemology of science: Sociology of Scientific Knowledge the Sociology of Scientific Knowledge, Cognitive Science and the Cognitive Science Kuhnian epistemology Sociology of Scientific Knowledge Cognitive Science Steven Shapin Nancy Nersessian

The position of Sociology of Scientific Knowledge: -science is a product of the historical interactions of intellectual groups, rather than [rather than a rational inspiration of individual human mind], -there is a shift from conceiving of science as knowledge practice to conceiving of science as practice.

History of mathematics from the point of view of Sociology of Scientific Knowledge

The position of Cognitive Science: -there is a interplay between the case studies historical scientific practices of historical scientific practices and the corresponding problem-solving ways of thinking, human reasoning representations human reasoning and representations, scientific activities emerge cultural and -the scientific activities emerge within the cultural and social environment social environment of a specific historical period and region.

History of mathematics from the point of view of Cognitive Science

Historical approach to mathematics with elements from both Cognitive Science and Sociology of Scientific Knowledge

practice A look at practice from the historiographical point of view cognitive practice historico-psychological viewpoint Practice: Motivations/goals Toolsmeans — Tools/means — Products/consequences Sociocultural viewpoint Social conditions, Cultural values, Legitimacy of means of practices in Historical and Local Contexts Interests, Choices, Representations, Ways of reasoning practices in a Historical Case

Components of cognitive practice cognitive practice Social and Cultural Trends for Scientific Changes in a Historical Period Institutional, Professional, Normalizational Possibilities/Limits in a Cultural Context Epistemological, Methodological, Discursive Resources in a Historical Context

An example from the history mathematics: the abacus cultural practice norm system practice “The norm system which governed the practice of abacus mathematics not identical of abacus mathematics was not identical with that of Greek-inspired Humanist and university mathematics, and could not be already because the practices they governed were different in spite of similarities.” Jens Høyrup points out that “As early as 1900, it is true, Moritz Cantor had spoken of the existence throughout the two coexisting “schools” 15 th century of two coexisting “schools” of “clerical” mathematics, one “geistlich” (“clerical”, that is, universitarian universitarian), the other “weltlich oder “secular or commercial” kaufmännisch” (“secular or commercial”, supposedly derived from Leonardo Fibonacci’s work).”

The distinctive professional normalization formed different practitioners scientific identities: that of mathematics practitioners and scholars that of mathematics scholars. But, this diverse identities reflected, also, different discursive practices: folk a folk mathematical discourse, from one side, learned and a learned mathematical discourse, from the other. different This historical case shows the existence of two different cultural practices cultural practices in late Middle Ages and early different Renaissance, which corresponded, also, to two different mathematical educations mathematical educations.

The practice as means in the historiography of mathematics education Already, such historical approaches have appeared in the history of science education, e.g. the works of Kathryn Olesko and David Kaiser.

In history of mathematics education, very close to this historiographical kind are the publications of Gert Schubring and Lewis Pyenson.

Schubring and Pyenson developed, systematically, the epistemological, cultural, and institutional approaches to the historical cases of mathematics education. These are shared with the contemporary cognitive historiography of mathematics. Warwick’s book Masters of Theory, Cambridge and the Rise of Mathematical Physics, makes a turn from the standard perception of “focusing mainly on the history mathematical of mathematical innovation[s]” toward mathematical practiceslocal pedagogical practices and their dependence upon local pedagogical contexts contexts.

Warwick is inspired by the contemporary tendency of the historiography of science and science education, which he applied in his analysis of the emergence of mathematical physics in the context of Cambridge University in the 19 th century. values institutional conditions pedagogical practices He shows “how a system of values that holds in the mathematics and science education, the institutional conditions of pedagogical practices, the coordination of secondary and higher education, and the assignation of social prestige new scientific careers local scientific culture social prestige to the new scientific careers and to the science teaching led to transform the state of a local scientific culture”.

Two interesting remarks in Warwick’s analysis: “pedagogical revolution” The first one refers to the “pedagogical revolution” by the development and impact of such pedagogical devices as problem solving on paperwritten examinationseducationally orientated treatisesend- of-chapter exercises face-to-face training in problem solving on paper, written examinations, educationally orientated treatises, and end- of-chapter exercises. appearance of blackboard in teaching The second remark concerns the appearance of blackboard in teaching. He presented a related 1850 illustration:

questions These remarks, naturally, generate related questions central to the cognitive historiography of mathematics education: When did the use of exercises in the mathematical textbooks of various countries or cultural communities begin? How and why did it spread? Which pedagogical necessities or pedagogical theories motivated the practice of blackboard? How did this pedagogical tool spread to various countries? These questions concern the history of mathematics education. And they are related to pedagogical practices.

This new perspective on history of mathematics and mathematics education is fully compatible with social constructivism social constructivism, nowadays dominant in the epistemology of mathematics and mathematics education. And its momentum in contemporary historiography of science and mathematics is very strong.

Thank you for your attention