This article was downloaded by: [Heriot-Watt University]On: 06 October 2014, At: 13:38Publisher: RoutledgeInforma Ltd Registered in England and Wales Registered Number: 1072954Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH,UK

Intercultural EducationPublication details, including instructions for authorsand subscription information:http://www.tandfonline.com/loi/ceji20

Mathematics learning is alsointercultural learningSusanne PredigerPublished online: 01 Jul 2010.

To cite this article: Susanne Prediger (2001) Mathematics learning is also interculturallearning, Intercultural Education, 12:2, 163-171, DOI: 10.1080/14675980120064809

To link to this article: http://dx.doi.org/10.1080/14675980120064809

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all theinformation (the “Content”) contained in the publications on our platform.However, Taylor & Francis, our agents, and our licensors make norepresentations or warranties whatsoever as to the accuracy, completeness, orsuitability for any purpose of the Content. Any opinions and views expressedin this publication are the opinions and views of the authors, and are not theviews of or endorsed by Taylor & Francis. The accuracy of the Content shouldnot be relied upon and should be independently verified with primary sourcesof information. Taylor and Francis shall not be liable for any losses, actions,claims, proceedings, demands, costs, expenses, damages, and other liabilitieswhatsoever or howsoever caused arising directly or indirectly in connectionwith, in relation to or arising out of the use of the Content.

This article may be used for research, teaching, and private study purposes.Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly

forbidden. Terms & Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions

Dow

nloa

ded

by [

Her

iot-

Wat

t Uni

vers

ity]

at 1

3:38

06

Oct

ober

201

4

Intercultural Education, Vol. 12, No. 2, 2001

Mathematics learning is also interculturallearningSUSANNE PREDIGER

ABSTRACT This article represents a plea to widen our perspective. Intercultural learningdoes not only take place when people of different origins come together, but also within oneculture. This also applies to the various (scienti� c) subcultures that exist in society. If weview mathematics as a culture of formal thinking, then the learning of mathematics can beunderstood as intercultural learning. In order to explain this view, mathematics is presentedas a separate culture. The characteristics of mathematics learning (as intercultural learn-ing) are discussed and we re� ect on what the � eld of intercultural education might gainfrom this perspective.

Introduction

This is a plea to widen our perspective. Intercultural learning does not only takeplace when people of different origins meet, but also when interaction takes placewithin one culture. This also applies to the various (scienti� c) subcultures that existin society. If we view mathematics as a culture of formal thinking, then mathematicseducation research can pro� t enormously from understanding mathematics learningas a type of intercultural learning. In this paper we will elaborate on this perspectiveand we will re� ect on what the � eld of intercultural education might gain from thisdiscussion.

Let me start by explaining what the title of this paper does not mean: thepaper does not attempt to point out how intercultural issues can be brought upin a mathematics classroom (such as explaining that there were importantmathematicians in the rest of the world, too). These are important issues, butenough discussion has taken place in mathematics education around these issues(especially in the � eld of ethnomathematics—see, for example, Bishop, 1991and Barton, 1996—and also in the � eld of educational science—see Schroder,1994).

The main premise of this article is that mathematics is a culture, and whenever weexpect students to learn mathematics they are confronted with an interculturallearning situation. In order to clarify this idea further, we shall commence byexplaining what we mean by “mathematics is a culture”.

ISSN 1467-5986 print; 1469-8489 online/01/020163-09 Ó 2001 Taylor & Francis LtdDOI: 10.1080/14675980120064809

Dow

nloa

ded

by [

Her

iot-

Wat

t Uni

vers

ity]

at 1

3:38

06

Oct

ober

201

4

164 S. Prediger

Mathematics as a Culture

The � rst scientist who described mathematics as a culture was the anthropologistLeslie White. Starting his philosophical quest by focusing on the locus of mathemat-ical reality, he tried to solve an old problem: do mathematical truths reside in theexternal world, there to be discovered by human beings, or are they human-con-structed inventions? Does mathematical reality have an existence and a validityindependent of the human species, or is it merely a function of the human nervoussystem? He answered these queries by locating mathematics in the intersubjectiveworld of culture:

Mathematics does have objective reality. And this reality […] is not thereality of the physical world. But there is no mystery about it. Its reality iscultural: the sort of reality possessed by a code of etiquette, traf� c regula-tions, the rules of baseball, the English language or rules of grammar.(White, 1947, p. 302f.)

Thus White considered mathematics to be a part of human culture and, by doingthis, the apparent contradiction was clari� ed. Mathematical formulas, like otheraspects of culture, do have—in a sense—an “independent existence and intelligenceof their own” (White, 1947, p. 295). Nevertheless, they are not independent of thehuman species. They are only independent of the individual (a more detaileddiscussion about the relationship of the individual to mathematics as a culture canbe found in Prediger, 2001b).

In his early re� ections, White did not posit an elaborate concept of culture. Heunderstood culture as the “mode of life of any people, no matter how primitive oradvanced” (White, 1947, p. 292). For our further discussion, we feel it is morefruitful to rely on a more re� ned concept of culture. Very instructive for the analysisof mathematics as a culture is the de� nition provided by Alexander Thomas:

In general, culture can be understood as a universal, but (for a society,nation, organisation or group) speci� c system of orientation. This orien-tation-system is passed down from one generation to the next within asociety, nation, or group. It in� uences perceptions, thoughts, values andthe actions of all members […]. This orientation-system provides a speci� cway of managing life and the environment for all members of the group.(Thomas, 1988, pp. 82–83, my translation)

This concept of culture includes all norms, values and attitudes that in� uence theperceiving, thinking, valuing and acting of humans. Viewing mathematics as such anorientation-system has become more popular recently. Furthermore, the � eld ofethnomathematics has developed an understanding of mathematics as a “system ofcodi� cation which allows describing, dealing, understanding and managing reality”(D’Ambrosio, cited in Barton, 1999, p. 54). Empirical evidence for this has beenfound in numerous ethnological investigations in different civilizations (see es-pecially Bishop, 1991). These studies reveal that throughout the centuries each

Dow

nloa

ded

by [

Her

iot-

Wat

t Uni

vers

ity]

at 1

3:38

06

Oct

ober

201

4

Mathematics Learning is also Intercultural Learning 165

civilization has created a culture of formal thinking which serves to cope with thespeci� c challenges presented by their respective environments.

There is also an increasing number of individuals within the philosophy andhistory of mathematics who argue in favour of a culture-based view of mathematics(Wilder, 1981; Hersh, 1997; Tymoczko, 1985; Ernest, 1998). Embracing a socialconstructivist perspective, they emphasize that mathematics must be understood asa socio-cultural historical phenomenon:

From the viewpoint of philosophy, mathematics must be understood as ahuman activity, a social phenomenon, part of human culture, historicallyevolved, and intelligible only in a social context. (Hersh, 1997, p. 11)

It should be emphasized that this perspective runs counter to the classical approachto mathematics, which views it as objective science, in which ultimate truths arediscovered and indubitable knowledge is collected in a cumulative fashion. Ernestcontrasts the “absolutist view” with the “fallibilist view”. The latter view acknowl-edges that, even in mathematics, results are sometimes falsi� ed, and the maincriterion for the correctness of mathematical proof is social acceptance within thecommunity (cf. Ernest, 1998; for empirical evidence see also Heintz, 2000).

Even if one is attached to the absolutist view of mathematics, one might perhapsbe willing to consider mathematics as a speci� c cognitive approach to the world bywhich we can gain particular insights into the world, whereas certain other aspectselude mathematical comprehension. In this context, the metaphor of using“mathematical glasses”, through which the world is viewed, is a very instructive one.The speci� c cognitive approach to the world is linked to particular ways of thinking,speci� c concepts and strategies. All of these have to be learned if mathematicalinsight is to be acquired, for example in school.

From the culture-based point of view, the “speci� c cognitive approach” is only apart of an overall orientation-system. Thus, non-cognitive aspects such as values andmeaning must be added. If we understand mathematics as a culture then this � eldof enquiry comprises knowledge about the origin of concepts, the aims of mathemat-ical activities, the unconscious and conscious purposes of theory development andmuch more. Such a wider understanding of mathematics is embraced by themovement of “general mathematics” (Wille, 1995).

Viewing mathematics as a culture is not completely uncommon in mathematicseducation. For example, in the current OECD study (PISA), which comparesmathematical competencies in different countries, the need to develop“mathematical glasses” is considered to be an essential part of mathematical com-petence:

Mathematics Literacy is an individual’s capacity to identify and understandthe role that mathematics plays in the world, to make well-founded mathemat-ical judgements and to engage in mathematics, in ways that meet the needsof that individual’s current and future life as a constructive, concerned andre� ected citizen. (PISA-Framework, cited in Baumert et al., 1999, myitalics)

Nevertheless, this approach is not very well developed yet, either in theory or in real

Dow

nloa

ded

by [

Her

iot-

Wat

t Uni

vers

ity]

at 1

3:38

06

Oct

ober

201

4

166 S. Prediger

classroom practice (Heymann, 1996). If we desire to formulate more precisely thedifferent levels of aims associated with a culture-based mathematics education, thecharacteristics of mathematics learning as intercultural learning need to be elabo-rated.

Characteristics of Mathematics Learning as Intercultural Learning

Traditionally, mathematics teaching was understood as a transition of concepts andtechniques on a purely cognitive level. This � ts with an absolutist attitude regardingthe nature of mathematics and a very narrow understanding of learning as a processin which information in received and skills are acquired (for a detailed critique seeErnest, 1998).

A more convincing didactical approach, which is based on a culture-based view ofmathematics, is represented by Bishop’s conceptualization of “mathematical encul-turation”, which refers to a cultural learning process. He explains this process asbeing a re-creative act that every individual has to go through:

Each young person and every new generation of young people re-createsthe cultural symbols and values of their culture, “lives” and validates themwithin their lifetime, and then engages with the next generation who intheir turn re-create, rede� ne, and therefore “re-live” them. […] Culturallearning is thus no simple one-way process from teacher to learner. Encul-turation, as it is more formally called, is a creative, interactive processengaging those living the culture with those born into it, which results inideas, norms and values which are similar from one generation to the nextbut which inevitably must be different in some way due to the re-creationrole of the next generation. (Bishop, 1991, p. 89)

The goal of mathematical enculturation is the induction of children into thesymbols, concepts and values of mathematical culture. The relation between chil-dren and the culture is described by Bishop (1991) as follows:

It clearly involves both “process” and “object”, and we shall therefore needto examine both. It cannot be just process-oriented because of the culture’sframe of knowledge, but nor should it just attend to that knowledge, sinceeducation is more than mere transmission. Enculturation, equally, has aresponsibility to both child and culture, respecting the individuality andpersonality of children as well as the characteristics of the culture. Toignore the � rst would lead to indoctrination, while to ignore the secondwould lead to anarchy. Mathematical enculturation needs to be conceptu-alized as a social interactive process carried out within a certain knowledgeframe but with the goal of recreating and rede� ning that frame. (p. 89)

There are many critical aspects of this conceptualization that need to be considered,but Bishop neglects one important dimension: the fact that various con� ictualsituations can result from the interface of the different cultural environments inwhich the students live. When students are � rst confronted with mathematical

Dow

nloa

ded

by [

Her

iot-

Wat

t Uni

vers

ity]

at 1

3:38

06

Oct

ober

201

4

Mathematics Learning is also Intercultural Learning 167

culture, and when they are expected to enter into this culture, they have alreadybeen socialized into their everyday culture with all its knowledge, beliefs, values andways of thinking. Thus, in addition to respecting mathematical culture and a child’sindividuality, as Bishop claims, we must also respect and include their everydaycultural environments. It is especially due to the need for such sensitivities thatintercultural learning is very instructive. In order to explain this further, we mustspecify in what way mathematics learning can be viewed as intercultural learning.

Starting from the preliminary de� nition that intercultural learning is “learning ina situation of overlapping cultural environments” (e.g. Thomas, 1988), we can focuson the different cultures that are involved in mathematics learning. In a mathematicsclass the culture of formal mathematical thinking (which is to be acquired) alwaysoverlaps with the culture of common sense that the students bring with them intothe classroom. Whenever students are exposed to the culture of mathematics, their“basic” culture is involved and it is advisable to respect this.

Proponents of intercultural learning argue that whenever intercultural learningtakes place, the educational process should not focus on the cognitive dimensionalone (i.e. the acquisition of knowledge about different cultural standards and theirbackground), but it should also include addressing (and hopefully changing) atti-tudes and behaviour (see, for example, Breitenbach, 1979). This should also be animportant aspect of mathematics teaching in the classroom, since various studieshave demonstrated how students’ attitudes in� uence the learning process (see, forexample Lerman, 1994).

Winter (1988) has distinguished four levels of intercultural learning aims:

1. intercultural learning as an act of acquiring knowledge so that an individual canlearn to “get-along” with those from another culture (for example basic knowl-edge about manners, morals, customs, acts of courtesy, etc.);

2. intercultural learning as an act of understanding collective norms, values, atti-tudes, convictions which underlie the behavior of the majority of people belong-ing to the foreign culture in question;

3. intercultural learning as a process of coordinating culturally different schemes ofacting: understanding and interpreting codes and scripts in the course of aninteraction and translating the insight gained into practical actions;

4. intercultural learning as a process of generalized culture-learning. An evaluationof different intercultural communication situations that allows the developmentof general rules, strategies and techniques that help individuals to orient them-selves and assimilate in various foreign cultures. (p. 167ff.)

How can we make these insights relevant to mathematics learning? The � rst levelcorresponds to 90% of what usually takes place in mathematics learning: studentsacquire concepts and skills in order to cope with the common tasks in a mathematicsclass. The second level of learning is often not achieved: an understanding of generalmathematical strategies, underlying norms and aims. To gain such understandingstudents need to understand the concept of “mathematical glasses”. They mustlearn what speci� c characteristics are associated with the mathematical approach tothe world. This can be accomplished by using mathematical modelling processes in

Dow

nloa

ded

by [

Her

iot-

Wat

t Uni

vers

ity]

at 1

3:38

06

Oct

ober

201

4

168 S. Prediger

mathematics classes (for a review of present developments in this � eld see, forexample, Galbraith et al., 1998; de Lange, 1996).

The third level of intercultural learning leads to the transfer of mathematical waysof thinking to other areas, especially applying mathematical strategies and conceptsto everyday thinking. A long-debated issue in mathematics education is why thistransfer does not happen automatically (cf. Lenne, 1969; Bauersfeld, 1983). This isthe reason why more explicit methods must be found to support this transfer (see,for example, Lengnink & Prediger, 2000).

The fourth level, translated into the realm of mathematics learning, relates to thephilosophy of science. Here, students are supposed to develop a general understand-ing of the differences between cultures (subjects) and strategies to cope with variouscultures (subjects). It involves moving away from one’s concrete cultural reality andmoving towards a more abstract level of thinking (here, mathematics). Formulatedas a goal for mathematics learning, generalized cultural learning implies that stu-dents will not only understand and use mathematics through these speci� c glasses,but that they will also recognize that all sciences need to be seen through speci� cglasses. They must learn to compare and evaluate the different approaches that areavailable and decide which one is appropriate in which situation. Key questions arethen, for instance: when can mathematics help to describe and understand aproblem, when should we choose biology, when psychology? What can each sciencecontribute to the problem being addressed? This competence is a very importantgoal in education, but it is not easy to reach this goal in most schools (compareHeymann’s idea regarding cultural coherence as an aim of mathematical education;Heymann, 1996).

In a similar vein, Winter emphasizes that in most situations involving interculturallearning, the learning process does not progress beyond the second level of learning.He argues that we should at least strive to reach the third level, the synthesis ofdifferent systems of orientation (Winter, 1988). It would be useful to concentrate onthe second and the third levels of learning in mathematics education as well. Thefourth level is a long-term aim and is dif� cult to attain, but it should be kept inmind.

Using Winter’s levels of intercultural learning as a frame of reference, we havebeen able to formulate the goals of mathematics learning (viewed as interculturallearning). In another paper (Prediger, 2000), the methodology of intercultural learn-ing was described and its contributions to the improvement of mathematics edu-cation were discussed. Let me brie� y sketch the main ideas of this other paper. Theproposed methodology of intercultural learning, based on constructivist ideas oflearning, focuses on:

· the design of adequate learning environments, i.e. the development of a classroomculture that allows lively and open confrontations with reality-oriented mathemat-ics in an open and pleasant social climate (see, for example, Henn, 1999;Wittmann, 1991);

· the explicit teaching of transfer, especially of mathematical ways of thinking and

Dow

nloa

ded

by [

Her

iot-

Wat

t Uni

vers

ity]

at 1

3:38

06

Oct

ober

201

4

Mathematics Learning is also Intercultural Learning 169

their connection to everyday thinking (cf. Lengnink & Prediger, 2000; Lengnink& Peschek, 2001);

· the important role of organized re� ection on the experiences that students have(cf. Skovsmose, 1994);

· the important role of addressing intercultural con� icts.

Contributions of Mathematics Learning to Intercultural Learning in Gen-eral

What can mathematics learning, if it is conceptualized as intercultural learning,contribute to intercultural learning in general? Our thesis is that mathematicslearning can help to construct a general re� ective cultural identity which is animportant precondition and aim of all intercultural learning. We can only develop awell-de� ned, re� ective position towards other cultures if we have a heightenedconsciousness about our own culture (Thomas, 1988, pp. 77–99).

Due to its impact on the natural sciences and technology, mathematics has atremendous in� uence on our civilization and our culture. Hence, a better under-standing of mathematics implies that we will be able to better understand our world,and it will allow us to re� ect more effectively on our cultural identity. H.W.Heymann has elaborated upon this aspect of mathematics in his book on generaleducation in mathematics (Heymann, 1996). He subsumes the idea of a re� ectivecultural identity under two tasks: “construction of a world view” and “culturalcoherence”. Cultural coherence comprises, on the one hand, the maintenance ofcultural continuity (i.e. the traditions of a speci� c culture from one generation to thenext—the diachronous aspect). On the other hand, it comprises the mediationbetween different subcultures (the synchronous aspect). These tasks can only beperformed in the mathematics classroom if students experience and critically re� ecton the role of mathematics as an orientation-system, which can help them to copewith the world around them.

Beyond the general contribution that a culture-based mathematics education canoffer to the development of a re� ective cultural identity, such education can help todevelop relativistic points of view that correspond with an attitude of tolerance, andalso the will to understand other perspectives. How can this be achieved? Whenstudents realize that mathematical solutions to problems are often shaped by certain(and quite speci� c) mathematical points of view, facility in using “mathematicalglasses” can help students place mathematical results into a more relative frame-work. In such circumstances they will no longer uncritically accept a mathematicallydetermined result (Skovsmose has emphasized this critical aspect of mathematicseducation; see Skovsmose, 1994). If the metaphor of the speci� c glasses developedfor mathematics can be transferred to other domains, it may help students toquestion implicit presuppositions. It can also help them to understand that differentbasic conditions, norms, values, etc. can lead to cultural differences.

For all the reasons given above, I believe that scholars and practitioners inintercultural education should also be interested in mathematics learning. Just aspolitical education has long been an essential part of intercultural learning, science

Dow

nloa

ded

by [

Her

iot-

Wat

t Uni

vers

ity]

at 1

3:38

06

Oct

ober

201

4

170 S. Prediger

and mathematics should also be included in future discussions (since they shape ourworld in profound ways).

On a � nal note, if this paper can trigger a lively discussion about the possible anddesirable contributions of mathematics learning to intercultural education in gen-eral, then this article will have had its desired effect.

Address for correspondence: Susanne Prediger, Technische Universitat Darmstadt, Fach-bereich Mathematik, Schlossgartenstraße 7, D-64289 Darmstadt, Germany; e-mail: predi-ger@mathematik.tu-darmstadt.de

References

BARTON, B. (1996) Making sense of ethnomathematics: ethnomathematics is making sense.Educational Studies in Mathematics, 31, 1–2, pp. 201–233.

BARTON, B. (1999) Ethnomathematics and philosophy. ZDM, 2, pp. 54–58.BAUERSFELD, H. (1983) Subjektive Erfahrungsbereiche als Grundlage einer Interaktionstheorie

des Mathematiklernens und -lehrens. In: BAUERSFELD, H. et al. (Eds) Lernen und Lehren vonMathematik, pp. 1–56, IDM 6. Koln: Aulis.

BAUMERT, J. et al. (1999) Internationales und nationales Rahmenkonzept fur die Erfassung vonmathematischer Kompetenz in PISA. Paper, Max-Planck-Institut fur Bildungsforschung,Berlin.

BISHOP, A.J. (1991) Mathematical Enculturation. A Cultural Perspective on Mathematics Education.Dordrecht: Kluwer.

BREITENBACH, D. (Ed.) (1979) Kommunikationsbarrieren in der Internationalen Jugendarbeit.Forschungsprojekt im Auftrag des Bundesministers fur Jugend, Familien und Gesundheit, Band 5.Saarbrucken: Fort Lauderdale.

ERNEST, P. (1998) Social Constructivism as a Philosophy of Mathematics. New York: SUNY Press.GALBRAITH, P. et al. (1998) Mathematical Modelling. Teaching and Assessment in a Technology-rich

world. Chichester: Horwood Publishing.HEINTZ, B. (2000) Die Innenwelt der Mathematik. Zur Kultur und Praxis einer beweisenden Disziplin.

Vienna and New York: Springer.HENN, H.W. (Ed.) (1999) Mathematikunterricht im Aufbruch. Zum Projekt “Weiterentwicklung der

Unterrichtskultur im Fach Mathematik” des Kultusministeriums Baden-Wurttemberg, Landesin-stitut fur Erziehung und Unterricht Stuttgart. Schroedel: Hannover.

HERSH, R. (1997) What is Mathematics, Really? London: Jonathan Cape.HEYMANN, H.W. (1996) Allgemeinbildung und Mathematik. Weinheim: Beltz.LANGE, J. DE (1996) Using and applying mathematics in education. In: BISHOP, A.J. et al. (Eds)

International Handbook of Mathematics Education, pp. 49–97. Dordrecht: Kluwer.LENGNINK, K. & PESCHEK, W. (2001) Das Verhaltnis von Alltagsdenken und mathematischem

Denken als Inhalt mathematischer Bildung. In: LENGNINK, K., PREDIGER, S. & SIEBEL, F.(Eds) Mathematik und Mensch. Sichtweisen der Alljemeinen Mathematik. Darmstadler Schriftenzur Alljemeinen Wissenschaft. Band 2. Darmstadt: Verlag Alljemeinen Wissenschaft.

LENGNINK, K. & PREDIGER, S. (2000) Mathematisches Denken in der Linearen Algebra. Zentral-blatt fur Didaktik der Mathematik, 32, 4, pp. 111–122.

LENNE, H. (1969) Analyse der Mathematikdidaktik in Deutschland. Stuttgart: Klett.LERMAN, S. (Ed.) (1994) Cultural Perspectives on the Mathematics Classroom. Dordrecht: Kluwer.PREDIGER, S. (2000) Mathematiklernen als interkulturelles Lernen. Unpublished manuscript, TU

Darmstadt.

Dow

nloa

ded

by [

Her

iot-

Wat

t Uni

vers

ity]

at 1

3:38

06

Oct

ober

201

4

Mathematics Learning is also Intercultural Learning 171

PREDIGER, S. (2001) Mathematik als kulturelles Produkt menschlicher Denktatigkeit und ihrBezug zum Individuum. In: HEYMANN, H.W., LENGNINK, K., PREDIGER, S. & SIEBEL, F.(Eds) Mathematik und Mensch. Sichtweisen der Alljemeinen Mathematik. Darmstadler Schriftenzur Alljemeinen Wissenschaft. Band 2. Darmstadt: Verlag Alljemeinen Wissenschaft.

SCHRODER, J.Z.W. (1994) Zahlen-Welten Bausteine fur einen interkulturellen Mathematikunterricht.Langenau-Ulm: Vaas.

SKOVSMOSE, O. (1994) Towards a Philosophy of Critical Mathematics Education. Dordrecht: Kluwer.THOMAS, A. (Ed.) (1988) Interkulturelles Lernen im Schuleraustausch. Saarbrucken: Fort Laud-

erdale.TYMOCZKO, T. (Ed.) (1985) New Directions in the Philosophy of Mathematics: An Anthology. Boston

and Basel: Birkhauser.WHITE, L.A. (1947) The locus of mathematical reality: an anthropological footnote. Philosophy of

Science, 14, pp. 289–303.WILDER, R.L. (1981) Mathematics as a Cultural System. Oxford: Pergamon Press.WILLE, R. (1995) Allgemeine Mathematik als Bildungskonzept fur die Schule. In: BIELE, R.,

HEYMANN, H.W. & WINKELMANN, B. (Eds) Mathematik allgemeinbildend unterrichten, pp. 41–55, IDM 21. Cologne: Aulis.

WINTER, G. (1988) Konzepte und Stadien interkulturellen Lernens. In: ALEXANDER, T. (Ed.)Interkulturelles Lernen im Schuleraustausch, pp. 151–177. Saarbrucken: Fort Lauderdale.

WITTMANN, E.C. (1991) Mathematikunterricht zwischen Skylla und Charybdis. Mitteilungen dermathematischen Gesellschaft in Hamburg, Festschrift zum 300 jahrigen Bestehen, XII, 3,pp. 663–679.

Dow

nloa

ded

by [

Her

iot-

Wat

t Uni

vers

ity]

at 1

3:38

06

Oct

ober

201

4