Learning from others

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The cultural dimension in (research into)
mathematics teaching and learning
Mathematics is a cultural construct
Paul Andrews, University of Cambridge Faculty of
Education
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Apologies
A comment about my slides and my use of English.
I am confident I can solve the problems we are
given in mathematics. I am sure I can solve the
problems? I am certain can solve the problems? I
am convinced I can solve the problems My talk is
something of a fraud in that it is about school
mathematics, not academic mathematics and
certainly not technology. However, I hope to show
how cultures, however they are defined, influence
greatly both what is taught and how it is taught
in schools. However, technology is not immune to
the influence of culture, as Silva Kmetic's paper
yesterday afternoon highlighted well.
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Some history
Since the late middle ages the ways of life of
all European countries have been influenced by
great intellectual, economic and social change.
The Renaissance and the freedom to think beyond,
or even challenge, the church's
teaching Reformation and the impact, in
particular, of protestantism on northern
Europe. Enlightenment, Thomas Paine, and the
rights of man. Industrial revolution, increasing
mechanisation and the desertification of the
countryside.
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This talk
These social revolutions occurred before most
countries even considered mass public education
as either a necessary or a desirable
provision. It is my conjecture that in every
country different societal norms, values, forms
of governance and industrialisation underpinned
the introduction of universal primary education
in the nineteenth century. In this respect
William Cummings has offered a helpful account of
the ways in which public educational systems
developed.
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Cummings' models of educational growth
Cummings, W. K. (1999). The InstitutionS of
education Compare, Compare, Compare! Comparative
Education Review, 43 (4) 413-437.
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Some explanatory perspectives

• Cummings argues that most of the world's
  educational systems are derivatives of the
  English, French, German, Japanese, Russian and
  the US.
• The English public school of the 19th century
  placed great emphasis on religion, innate talent,
  but almost nothing on science and mathematics A
  model that Holmes and McLean (1989) have
  described as essentialism.
• Post revolutionary France saw a broad rational
  curriculum which all, through effort, were
  expected to attain A model that Holmes and
  McLean have described as encyclopaedism.
• (Holmes, B., McLean, M. (1989). The Curriculum
  a comparative perspective. London Unwin Hyman).

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Hofstede's cultural dimensions

• Individualism Individualist cultures assume that
  any person looks primarily after /her own
  interest and the interest of his/her immediate
  family (husband, wife and children).
• Power Distance Defines the extent to which less
  powerful persons in a society accept inequality
  in power and consider it as normal
• Uncertainty AvoidanceDefines the extent to which
  people within a culture are made nervous by
  situations which they perceive as unstructured,
  unclear, or unpredictable.
• Masculinity Refers to the social roles
  associated with the biological fact of the
  existence of two sexes, and in particular in the
  social roles attributed to men.
• (Hofstede, Geert. 2001. Cultures consequences
  Comparing values, behaviors, institutions, and
  organizations across nations (2nd ed.). London
  Sage.)

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The UK Weak uncertainty avoidance and low power
distance. France Larger power distance and
greater uncertainty avoidance. Denmark...?
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Predicting classroom behaviour

• In such models we can explain, with some
  accuracy, how classroom roles play out. For
  example,
• English teachers are expected to emphasise
  personalised learning.
• Relationships between colleagues in English
  schools are informal and colleagues feel their
  opinions are valued.
• Teachers are free to apply for any job for which
  they believe they are qualified
• Relationships between teachers and students are
  formal, most schools insist that students wear
  uniforms, and frequently expect them to address
  teachers with titles like Sir.

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So how do such difference play out in the
classroom?

• In the following, I will show several short video
  clips highlighting how culturally determined
  norms are played out in classrooms.
• The first clips come from a grade 5 lesson on
  percentages.
• Flanders clip 1
• Flanders clip 2
• Flanders clip 3

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Interpreting the clips

• At no point did this teacher evaluate her
  expressions.
• Her goal was her students' understanding of the
  structural aspects, in particular,
  proportionality and the multiplicative properties
  of percentage calculations.
• This compares with the English teacher observed
  as part of the same project she spent one
  lesson encouraging her students to see
  calculating ten per cent as dividing by ten.
• The following lesson she asked her students how
  they would find twenty percent of something. They
  replied.....,
• Such differences are profound and reflect how
  mathematics is construed systemically in the two
  countries.

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A second lesson

• In this second lesson we see a Hungarian teacher
  working with her students on the solution of
  linear equations
• Hungary clip 1
• In contrast to this lesson's task. All the
  lessons in the comparable English sequence
  focused on solving equations with the unknown on
  one side.
• In the light of such differences it is not
  surprising to find a number of researchers
  asserting that mathematics and its teaching are
  culturally located.

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Differing classroom expectations

• Recently I worked with colleagues from five
  European countries to examine how mathematics is
  conceptualised and presented to students in the
  age range 10-14.
• Videotaped lessons were coded against a schedule
  developed by the team during the first year of
  the project.
• Unlike the TIMSS video studies which attempted a
  very close and highly specified classification of
  curriculum content we adopted seven generic
  categories of learning outcome and ten generic
  didactic strategies.
• (Andrews, P. (2007) Negotiating meaning in
  cross-national studies of mathematics teaching
  kissing frogs to find princes', Comparative
  Education, 43 (4) 489-509)

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Seven generic learning outcomes
(Andrews, P. (2009) Comparative studies of
mathematics teachers observable learning
objectives validating low inference codes,
Educational Studies in Mathematics 71 (2)
97-122).
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Ten generic didactic strategies
Andrews, P. (2009) Mathematics teachers didactic
strategies Examining the comparative potential
of low inference generic descriptors, Comparative
Education Review, 53 (4) (In Press)
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Looking at the same data differently
Suddenly the national script seems less secure
there are some interesting tendencies but each
cluster draws extensively on the episodes of
teachers from at least two countries
(Andrews, P. (2007) Mathematics teacher
typologies or nationally located patterns of
behaviour?, International Journal of Educational
Research, 46 306-318)
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Teacher beliefs

• Another influential, culturally informed, factor
  are the beliefs teachers' hold about mathematics
  and its teaching.
• Thompson (1984 105) asserts that teachers
  develop patterns of behavior that are
  characteristic of their instructional practice.
  In some cases, these patterns may be
  manifestations of consciously held notions,
  beliefs, and preferences that act as 'driving
  forces' in shaping the teacher's behavior. In
  other cases, the driving forces may be
  unconsciously held beliefs or intuitions that may
  have evolved out of the teacher's experience
• (Thompson, A.G (1984) The relationship of
  teachers conceptions of mathematics and
  mathematics teaching instructional practice,
  Educational Studies in Mathematics 15(2)
  105127).

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The cultural location of beliefs

• In an interview study of English and Hungarian
  teachers' beliefs the following emerged.
• Hungarian teachers saw mathematics as an
  intellectually challenging and problem-solving
  discipline. Learners' acquired the skills of
  logical reasoning through the collaborative
  solution of non-routine problems.
• English teachers were concerned with
  functionality, the teaching of applicable number,
  real-world preparation and differentiated
  curricula leading to pre-determined learner
  outcomes.
• (Andrews, P. (2007) The curricular importance of
  mathematics a comparison of English and
  Hungarian teachers espoused beliefs, Journal of
  Curriculum Studies, 39 (3) 317-338)

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Language in mathematics

• Lastly, one of the major elements of any
  mathematics classroom is language and the ways in
  which it can structure learning.
• It seems to me that some linguistic features
  facilitate learning while others do not. For
  example, what words would you use in your
  language to say the number 5.14?
• In English we would say five point one four. Many
  European languages would say five comma fourteen,
  others say five wholes fourteen,while a few would
  say something like five units fourteen
  hundredths.
• In Spain they say five comma fourteen but write,
  interchangeably, 5'14, 5.14 or 5,14. These are
  not inconsequential differences.

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Other number issues
ten fifteen twenty-five eighty-six ninety-seven
tien vijftien vijfentwintig zesentachtig zevenennegentig
dix quinze vingt-cinq quatre-vingt-six quatre-vingt-dix-sept
zehn fünfzehn fünfundzwanzig sechsundachtzig siebenundneunzig
tiz tizenöt huszonöt nyolcvanhat kilencvenhét
zece cincisprezece douazeci si cinci cincizeci-sase nouazeci-sapte
diez quince veinticinco cincuenta y seis noventa y siete
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The vocabulary of geometry
English Dutch Hungarian
Quadrilateral Vierhoek Négyszög
Parallelogram Parallellogram Paralelogramma
Rectangle Rechthoek Téglalap (brick shape)
Rhombus Ruit Rombusz
Square Vierkant Négyzet
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Closing thoughts

• In conclusion, there is much evidence that
  teachers in one country behave in ways that
  identify them more closely with compatriots than
  teachers elsewhere (Schmidt et al. 1996).
• This is because "teaching and learning are
  cultural activities (which)... often have a
  routineness about them that ensures a degree of
  consistency and predictability. Lessons are the
  daily routine of teaching and learning and are
  often organized in a certain way that is commonly
  accepted in each culture" (Kawanaka 1999, 91).
• This sense of routine predictability has been
  variously described as the traditions of
  classroom mathematics (Cobb et al. 1992), the
  cultural script (Stigler and Hiebert 1999),
  lesson signatures (Hiebert et al. 2003) and the
  characteristic pedagogical flow of a lesson
  (Schmidt et al. 1996).