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Mathematics is a cultural construct The cultural dimension in (research into) mathematics teaching and learning P aul 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 19 th 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 Avoidance:Defines 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. Culture’s 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): 105–127).

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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 tenfifteentwenty-fiveeighty-sixninety-seven tienvijftienvijfentwintigzesentachtigzevenennegentig dixquinzevingt-cinqquatre-vingt-sixquatre-vingt-dix- sept zehnfünfzehnfünfundzwanzigsechsundachtzigsiebenundneunzig tiztizenöthuszonötnyolcvanhatkilencvenhét zececincisprezecedouăzeci şi cincicincizeci-sasenouăzeci-şapte diezquinceveinticincocincuenta y seisnoventa y siete

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The vocabulary of geometry EnglishDutchHungarian QuadrilateralVierhoekNégyszög ParallelogramParallellogramParalelogramma RectangleRechthoek Téglalap (brick shape) RhombusRuitRombusz SquareVierkantNégyzet

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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).

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