The concept of Culture in the study of Mathematical Practiceand Mathematics Education

1. The concept of Culture in the study
of Mathematical Practice
and Mathematics Education
Karen François & Brendan Larvor
Third International Meeting of the Association
for the Philosophy of Mathematical Practice
Paris, November 2015

2. A word from our sponsors
Strategic research project on Logic and
Philosophy of Mathematical Practices of the
Centre for Logic and Philosophy of Science.

3. The Promises
We will argue that
• there is no commonly agreed, unproblematic
conception of culture available to researchers and
students of mathematical practices
• Rather, there are many imperfect candidates.
• There is a tension between the material and ideal
aspects of culture that different conceptions
manage in different ways.
• A properly philosophical conception of culture
should include the normative/descriptive and
material/ideal dyads as dialectical moments.

4. The Promises
• We will suggest that the field of study of
mathematical cultures is less well developed than
the number of books and conferences with the
word ‘culture’ in their titles might suggest.
• Secondly, we turn to the research field of
mathematics education to explore the ‘cultural’
turn into this research field and to analyse how
the concept of culture is used in this context.
• We finally will suggest which of the approaches
to the concept of culture is most promising for
philosophy of mathematical practices, and
mathematics education.

5. Culture: a good thing
According to Matthew Arnold, culture is:
…the great help out of our present difficulties;
culture being a pursuit of our total perfection
by means of getting to know, on all the
matters which most concern us, the best
which has been thought and said in the world,
and, through this knowledge, turning a stream
of fresh and free thought upon our stock
notions and habits...
Culture and Anarchy, preface (Cambridge University Press 1978 reprint of the
second (1875) edition, p. 6).

6. Culture: an objective reality
“Culture consists of patterns, explicit and implicit,
of and for behavior acquired and transmitted by
symbols, constituting the distinctive
achievements of human groups, including their
embodiments in artifacts; the essential core of
culture consists of traditional (i.e. historically
derived and selected) ideas and especially their
attached values; culture systems may, on the one
hand, be considered as products of action, and
on the other as conditioning elements of further
action.”
Kroeber & Kluckhohn 1952 p. 181

7. Ideas & Observables
Ideas without
observables are inert;
observables without
ideas are
unintelligible

8. Culture and Practice
“What unites a scientific community need
not be a set of beliefs. Shared beliefs are
much less common than shared practices…
because shared beliefs require shared
practices, but not vice versa.”
(Netz 1999 p. 2)

9. Normative-Descriptive
Normativity is unavoidable, even in those
studies that attempt to use resolutely
descriptive, value-neutral conceptions of
culture. This is because our interest as
researchers into mathematical practices is in
the study of successful mathematical practices
(or, in the case of mathematical education,
practices that ought to be successful).

10. Conferences
• 27-29 May 2010. Mathematics as Culture and Practice, Bielefeld,
Germany.
• 2-3 December 2011. Mathematics as Culture and Practice II,
Greifswald, Germany.
• 2012-2014. Mathematical Cultures: workshop series funded by the
UK Arts and Humanities Research Council and the London
Mathematical Society:
– Mathematical Cultures 1: Contemporary mathematical cultures,
London, England, 10-12 September 2012.
– Mathematical Cultures 2: Values in mathematics, London, England,
17-19 September 2013.
– Mathematical Cultures 3: Mathematics in public culture, London,
England, 10-12 April 2014.
• 9-12 November 2012. Cultures of Mathematics and Logic,
Guangzhou, China (the origin of this book).
• 22-25 March 2015 Cultures of Mathematics IV New Delhi, India.
? 2017 Cultures of Mathematics V Buenos Aires

11. Mathematics Education
Two distinct strands:
• Critical Mathematics Education (northern
Europe)
• Ethnomathematics (Brazil)

12. Mathematics Education
Difference with mathematical practices
–Producing mathematics
–Reproducing mathematics
–(although there is an overlap: some
reproducing math practices and some
producing activity in the learning process)

13. Explicit normative dimension
• Math Education is concerned with best
practices (as studied in the different learning
theories)
• Normativity is also reflected in the ‘World
Declarations’ by UN/UNESCO/OECD

14. Explicit normative dimension
UN Universal Declaration of Human Rights
(1948 )
• “Everyone has the right to education” and
• “higher education shall be equally accessible
to all on the basis of merit”
Article 26

15. Explicit normative dimension
UN Universal Declaration of Human Rights
(1948 )
The notions of rights and equality are at the
core of the declaration. Moreover the
program on education has not only to do with
the individual right, it is at the same time
directed to a societal achievement of the
maintenance of peace.

16. Explicit normative dimension
UNESCO statement on education in general
(1990)
…education is key to social and economic
development. We work for a sustainable world
with just societies that value knowledge,
promote a culture of peace, celebrate diversity
and defend human rights, achieved by
providing education for all.

17. Explicit normative dimension
OECD statement on mathematical literacy
(1999)
Mathematical literacy is an individual’s
capacity to identify and understand the role
that mathematics plays in the world, to make
well-founded judgments and to use and
engage with mathematics in ways that meet
the needs of that individual’s life as a
constructive, concerned and reflective citizen.

18. Explicit normative dimension
• What is the most successful educational
practice?
• To achieve the UNESCO goal “education for
all” / increase mathematical literacy
• We can observe a shift in the attention of
education theorists from the individual to the
social

19. Origins of Math Ed as a field
• Educational Studies in Mathematics (1968)
(WoS journal)
• The first International Congress on
Mathematical Education (ICME) conference
(1969) Lyon France (the 13th ICME Hamburg
2016)

20. Origins of Math Ed as a field
• Focus on the psychology / the individual
• Initially psychology became one of the most
important perspectives from which
mathematics education was analyzed and
investigated.
Cognitive learning theory (Piaget)
• International Group for the Psychology of
Mathematics Education (PME) (1976)

21. Origins of Math Ed as a field
The social turn
• For two decades there has been a growing
interest in the socio-cultural aspects of math
education.
• The cultural turn became sedimented by the
foundation of the Mathematics Education and
Society (MES) (1998)
• Focuses on the social, political, cultural and
ethical dimensions of mathematics education

22. Origins of Math Ed as a field
The social turn
François, K.; Coessens, K. & Van Bendegem,
J.P. (2012). The Interplay of Psychology and
Mathematics Education: From the Attraction
of Psychology to the Discovery of the Social.
Journal of Philosophy of Education, 46(3), 370-
385.

23. Origins of Math Ed as a field
The social turn
• 1969: International Congress on Mathematical
Education (ICME)
• 1976: Psychology of Mathematics Education
(PME)
• 1998: Mathematics Education and Society
(MES)

24. The programme of CME
• CME is concerned with the social and political
aspects of the learning of mathematics.
• It is concerned with providing access to
mathematical ideas for everybody
independent of colour of skin, gender and
class.
• It is concerned with the use and function of
mathematics in practice, being an advanced
technological application or an everyday use.

25. The programme of CME
• It is concerned with the life in the classroom,
which should represent a democratic forum,
where ideas are presented and negotiated.
• It is concerned with the development of
critical citizenship (Skovsmose & Borba, 2004,
p. 207).
• No explicit use of the concept ‘culture’

26. The programme of CME
Looking at the level of the classroom:
• Background – Foreground distinction
(Skovsmose)
• Situated Learning concept (Lave & Wenger,
1991) introducing the notion of culture

27. The programme of CME
Background
• “that socially constructed network of relationships
and meanings which are the result of the learner’s
lived past history” (Skovsmose, 1994).
• These backgrounds cannot be conceptualized as
fixed categories into which any student
or group of students may be fitted. They cannot be
thought of as being uniform.
• Students’ background knowledge is characterized by
diversity the teacher should address.

28. The programme of CME
Foreground
• the concept foreground as ‘the set of opportunities
that the learner’s social context makes accessible to
the learner to perceive as his or her possibilities for
the future’. Skovsmose (1994)
• Foreground has to do with the student’s possibilities
in future life, not the objective possibilities as
formulated by an external institution but the
possibilities as the student perceives them (Vithal &
Skovsmose, 1997, Skovsmose, 2005).

29. The programme of CME
Situated Learning concept (Lave) introducing
the notion of culture
• recognizes that learning styles and learning
processes can differ over cultures, since
learning is not only in the head, but happens
in and through the interaction between an
individual and his or her social, historical and
cultural environment.
(Lave & Wenger, 1991)

30. …Transition…
“…recognizes that learning styles and learning
processes can differ over cultures…”
• This is where ethnomathematics comes in!

31. Ethnomathematics
• Has its roots in Brazil, São Paulo
• D’ Ambrosio is named the intellectual father
• Geopolitical background: Post-colonial
countries
• Anthropology of mathematical practices
• Criticizes the import of western academic
mathematics, the uniform curriculum, the
alien curriculum (as if the learning is only in
the head)

32. Ethnomathematics
• Shifted meaning from ‘exotic’ to generic term
(François, Van Kerkhove 2010, PMP publication)
• Study of ‘mathematical’ practices of non-
literate people (Ascher & Ascher, 1986, 1988)
• To give value to those practices (as complex
‘mathematical’ practices)

33. Ethnomathematics
…and education
• Criticizing the import of alien curricula
• Wondering that the curriculum is not working
as expected!
• Became a research field concerned with social
justice and equality, taking into account the
cultural dimension

34. Ethnomathematics
The generic term
I call mathema the actions of explaining and understanding in
order to survive. Throughout all our own life histories and
throughout the history of mankind, technés (of tics) of
mathema have been developed in very different and
diversified cultural environments, i.e. in the divers ethnos. So,
in order to satisfy the drives towards survival and
transcendence, human beings have developed and continue
to develop, in every new experience and in diverse cultural
environments, their ethno-mathema-tics
(D’Ambrosio, 1990 , p. 369)

35. The concept of culture in the Math
Education research field
CME: no need to focus on ‘culture’
• The CME program originates from Western
geopolitical background
• Not confronted with alien curricula
• Is focusing on social inequalities (class, gender,
…)

36. The concept of culture in the Math
Education research field
EM: uses the concept of culture
• Origin in post-colonial geopolitical background
• Confronted with alien curricula, oppression
and hegemony
• Freire, P. (1968). Pedagogia do Oprimido
(Pedagogy of the Oppressed)
• Makes clear that western culture is indeed
also a ‘particular’ culture (even when it
becomes a universal culture)

37. How to use the concept of culture
• Bishop (1985) distinguishes five *interwoven*
significant levels at the research on the social
dimension of mathematics education going
from a macro perspective (culture) to a micro
perspective (the individual).

38. How to use the concept of culture
Bishop’s Levels
• The cultural level
• The societal level
• The institutional
• The pedagogical level
• The individual level

39. How to use the concept of culture:
philosophy of maths practice
• According to Ken Manders, `at its most basic,
a mathematical practice is a structure for
cooperative effort in control of self and life';
[Mancosu, 2008, 82, emphasis in original].
• Control to what end? With what authority?
Cooperation with whom? By what means?
…These questions demand cultural answers

40. How to use the concept of culture:
maths education
• If we discuss math practices (from one culture e.g.
the western one) then culture seems to be an
irrelevant issue
• If we discuss math practices with Ethnographers, or
with people from Vanuatu, the cultural notion (and
the hegemony) comes in
• Can we map this back to our own societies?
• Some mathematics education concerns e.g. gender
seem to demand cultural analysis

41. Who was that?
Karen François Karen.Francois@vub.ac.be
Brendan Larvor b.p.larvor@herts.ac.uk