Toward a Cognitive Toward a Cognitive Historiography of Historiography of

Mathematics EducationMathematics Education

Iason Kastanisa & Nikos Kastanisb

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

MotivationIn 1999 published the book

Cognitive history of mathematics,it is something that does existit is something that does exist.

So, there should also be a cognitive history of mathematics educationcognitive history of mathematics education

But, what is a But, what is a cognitive history of mathematics?cognitive history of mathematics?

And what about acognitive history of mathematics educationcognitive history of mathematics education?

The cognitive consideration is favored today in mathematics

contextsIn Mathematics Education

discourses of practicediscourses of practice

James Greeno states thatthe cognitive approach emphasizes:

conceptual structures, representations of information,

conceptual changes, and

In History of Mathematics

page 7

It is obvious that there are cognitivecognitive concernsand inquiries in the Mathematics Education Mathematics Education and the History of MathematicsHistory of Mathematics.

But, what is known about the methodologymethodology of the cognitivecognitive history of mathematics?

Unlocking the historiographical door of mathematics

“The historians need to analyze the concrete scientific action scientific action rather than an abstract life of mathematical ideas.”

Page 13

This is a position of the German historian, Moritz Epple.

“The practice of mathematics 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 analyzed the distinct research practices research practices that led to new theories of modern mathematics. They noticed the local character of the respective research activities that expresses the local traditions local traditions of mathematical practicesmathematical practices, that is the distinct mathematical distinct mathematical culturescultures.

On the other hand, Matthew Jones focused on the discursive practices discursive practices that were developed within the broader cultural and cognitive context of 17th century mathematics.

The practicepractice in the historiography of science

Since the 1960’s, the ideas of Thomas Kuhndominate the historiography and epistemology of science.

Kuhn and his followers “reorient the philosophy of science toward an account of scientific practices scientific practices rather

than scientific knowledge”.

Around 1980, two movements emerged in the epistemology of science: the Sociology of Scientific KnowledgeSociology of Scientific Knowledge,

and the Cognitive Science Cognitive Science

Kuhnian epistemology

Sociology ofScientific

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 thanrather than a rational inspiration of individual human mind],

-there is a shift from conceiving of science as knowledge to conceiving of science as practicepractice.

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 of historical scientific practices historical scientific practices and the corresponding problem-solving ways of thinking, human reasoning human reasoning and representationsrepresentations,

-the scientific activities emerge scientific activities emerge within the cultural and 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

A look at practicepractice from the historiographical point of view

cognitive practice

historico-psychological viewpoint

Practice:Motivations/goals— ToolsTools/meansmeans— Products/consequences

Socioculturalviewpoint

Social conditions,Cultural values,

Legitimacy of means of practicesin 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,

NormalizationalPossibilities/Limits

in a Cultural Context

Epistemological,Methodological,

Discursive Resourcesin a Historical Context

An example from the history mathematics: the abacus cultural practice

“The norm system norm system which governed the practicepractice of abacus mathematics of abacus mathematics was not identicalnot identicalwith 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 15th century of two coexisting “schools” two coexisting “schools” of mathematics, one “geistlich” (“clerical”“clerical”, that is,universitarianuniversitarian), the other “weltlich oder kaufmännisch” (“secular or commercial”“secular or commercial”, supposedly derived from Leonardo Fibonacci’s work).”

The distinctive professional normalization formed different scientific identities: that of mathematics practitionerspractitioners and that of mathematics scholarsscholars.

But, this diverse identities reflected, also, different discursive practices: a folkfolk mathematical discourse, from one side, and a learnedlearned mathematical discourse, from the other.

This historical case shows the existence of two different different cultural practices cultural practices in late Middle Ages and early Renaissance, which corresponded, also, to two different different mathematical educationsmathematical educations.

The practice as means in the historiography of mathematics

educationAlready, 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 of mathematical innovation[s]” toward mathematical mathematical practicespractices and their dependence upon local pedagogical local pedagogical contextscontexts.

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 19th century.

He shows “how a system of valuesvalues that holds in the mathematics and science education, the institutional conditions institutional conditions of pedagogical practicespedagogical practices, the coordination of secondary and higher education, and the assignation of social prestige social prestige to the new scientific careers new scientific careers and to the science teaching led to transform the state of a local scientific culturelocal scientific culture”.

Two interesting remarks in Warwick’s analysis:

The first one refers to the “pedagogical revolution” “pedagogical revolution” by the development and impact of such pedagogical devices as face-to-face training in problem solving on paperproblem solving on paper, written examinationswritten examinations, educationally orientated educationally orientated treatisestreatises, and end-of-chapter exercisesend-of-chapter exercises.

The second remark concerns the appearance of appearance of blackboard in teachingblackboard in teaching. He presented a related 1850 illustration:

These remarks, naturally, generate related questionsquestions 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 constructivismsocial 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