SYMMETRY AND INTERCULTURALITY
Volume 2, Supplement 1, 2009SYMMETRY AND INTERCULTURALITYIuliana MarchisAbstract: Symmetry is one of the fundamental concepts in Geometry. It is a Mathematicalconcept, which can be very well connected with Art and Ethnography. The aim of thearticle is to show how to link the geometrical concept symmetry with interculturality. For thismosaics from different countries are used.Keywords: interculturality, symmetry, Mathematics Education1. IntroductionSymmetry is one of the fundamental concepts in Geometry. It is important to learn about symmetry byexploring things, which we use everyday but they are not necessary related to Mathematics (Knuchel,2004). We see symmetry in nature and in things, which are made by people (furniture, buildings, carsetc.). Symmetry has an important role in problem solving, as it connects various branches ofMathematics, as Geometry, Algebra, Probability and Mathematical Analysis (Leikin, Berman &Zaslavsky, 2000).Ethnomathematics is the study of the relation between Mathematics and culture. It is “theMathematics, which is practiced among identifiable cultural groups such as national-tribe societies,labor groups, children of certain age brackets and professional classes” (D’Ambrosio, 1985). Whileteaching Mathematics using ethnomathematics the students learn not only mathematical concepts butalso cultural elements.In literature there are some articles about teaching symmetry using cultural elements. Eglash (2001)has presented a way to teach symmetry using traditional Shoshone-Bannock beadwork. Abas (2001)have shown how to teach symmetry by Islamic patterns. According to Pumfrey and Beardon (2002),Mathematics and Arts have strong links. Using beadworks, mosaics or architecture pieces whileteaching, Mathematics could be very well connected with Arts or Ethnography. Dolinko (1996) hasused flags from different countries to present the concept of symmetry, which approach connectsMathematics with Social Sciences.There are projects on developing methodology for introducing interculturality and ethnomathematicsin Mathematics classes. For example, the IDMAMIM (Innovation in Mathematical Didactics inMulticultural contexts with pupils from Minorities and Immigrants) project (2000-2005) had the maintarget to appropriate Mathematics with the intercultural reality.The aim of this article is to show how to link the concepts of symmetry with interculturality.2. Teaching symmetry using mosaicsTo teach symmetry I have chosen photos of mosaics from different countries. In Figure 1 there aresome examples. By this activity pupils learn not only about symmetry, but also about interculturality,History and Geography. In the following the details of the activity are presented.The target group is 11-12 years old pupils.Aims/competencies:Related with symmetry, pupils have to be able toReceived 30 September 2009, accepted 15 October 2009, published 20 December 2009
58Iuliana Marchis-identify symmetric figures (symmetric to a line, symmetric to a point);find the number of symmetry axis of a symmetric figure.In the presented lesson we also want to develop intercultural competencies. Thus after the lessonpupils have to be able to-recognize that there are connections between different cultures;show respect to other cultures;recognize intercultural relations.Other competencies: after the lesson pupils have to be able to-search for information on the Internet;make a presentation on a given topic.Seville, SpainMonastir, TunisiaSeville, SpainMonastir, TunisiaFigure 1. Examples of mosaics usedTime: 100 minutes.Resources: photos with mosaics, laptop, videoprojector, computer presentation made by the teacher(presentation about the Royal Palace in Seville, Spain and the Mausoleum in Monastir, Tunisia), ahandout about symmetry (shapes symmetric to a line, shapes symmetric to a point, symmetry axis,etc), computer network with Internet connection.Steps of the lesson: In the following the main steps of this lesson are presented. Some of the activitiesare Mathematics orientated, other are related with intercultural education. In some of the activitiespupils are grouped in teams of two or three, in others they do individual work, and there are somesteps, where frontal work with the whole class is organized. Let see the main activities on the lesson:Acta Didactica Napocensia, ISSN 2065-1430
Symmetry and interculturality--59Students group in teams of two or three. Each team gets a set of ten photos of mosaics fromdifferent countries (without knowing from where they are). They have to guess from wherethe photos are.The teacher shows a short computer presentation about the Royal Palace in Seville, Spain andthe Mausoleum in Monastir, Tunisia from where the mosaics are.Every student chooses a mosaic, and then motivates to his/her team, why he/she has chosenthat one (for example, he/she likes the colors, the patters, etc).Every student identifies known geometrical shapes on the chosen mosaic. For example, inFigure 2 we see the hexagon identified in the third mosaic of Figure 1, and the squaresidentified in the forth mosaic. Then they show to the others in the group which geometricalshapes they have found on their image. This activity can be done with images in printedformat and pupils draw by hand on the photo, or, if they have enough knowledge in computergraphics, they can do it on the computer.Figure 2. Identified polygons-Every student gets and reads a short description about the symmetry: line symmetry and pointsymmetry.Every student tries to find symmetries on the chosen mosaic. As examples, see Figure 3.Students discuss in the group all the identified symmetries on all the chosen images.Figure 3. Drawing symmetry axis-Students discuss in the class, if there are similarities in the chosen images, and why thesesimilarities appeared. Teacher helps with historical facts, as Arabian influence in Spain. ThisVolume 2 Supplement 1, 2009
60Iuliana Marchis--discussion could lead to the conclusion, that the different cultures can be influenced by eachother.Every group gets the name of an ethnic group, and they have to discuss, what they can learnfrom the culture of that ethnic group. They make a short computer presentation, for whichthey can search pictures and information on the Internet.Each group presents its work in front of the class.Students discuss in the class about what kind of intercultural relations could be betweendifferent cultures.3. Learning symmetry by handcraftHandcraft can be used to develop Mathematical skills.The target group is 11-12 years old pupils.Aims/competencies:Related with symmetry, pupils have to be able to-identify symmetric figures (symmetric to a line, symmetric to a point);find the number of symmetry axis of a symmetric figure;complete a line symmetric figure with respect to a horizontal/vertical line;make patterns using symmetry.In the presented lesson we also want to develop intercultural competencies. Thus after the lessonpupils have to be able to-recognize that there are connections between different cultures;show respect to other cultures;recognize intercultural relations.Time: 100 minutes.Resources: snowflakes made from paper (see Figure 4), laptop, videoprojector, computer presentationmade by the teacher (presentation about beadworks in different cultures), a handout about symmetry(shapes symmetric to a line, shapes symmetric to a point, symmetry axis, etc), computer network withInternet connection, bead pattern designing software.Figure 4. SnowflakesSteps of the lesson: In the following the main steps of this lesson are presented. Some of the activitiesare Mathematics orientated, other are related with intercultural issues.-Each student gets one snowflake made from paper and he/she have to find he symmetry axisand the symmetry point of it.Each student designs a pattern for a bead bracelet using special software (see Figure 5).Pupils form group of four and study the patterns made by the group members to findsymmetries.Acta Didactica Napocensia, ISSN 2065-1430
Symmetry and interculturality-61The teacher shows a short computer presentation about bead bracelet patterns in differentcultures.Students discuss how cultures interact, how people of different cultures living in the samecommunity.Figure 5. Patterns for bead bracelets-The teacher shows the technique how to make a bead bracelet; pupils start to make theirbracelets based on the designed patter.Homework: Each student finishes the bracelet.ConclusionIn the above presented activities pupils learn Mathematics in an attractive way, but also they areconfronted by intercultural issues, they learn some Geography, Ethnography, History, and Arts too.Therefore these lessons are complex, uses the integrated teaching. The topic “symmetry” gives manyopportunities to introduce intercultural issues to Mathematics lessons.References[1] Abas, S. J. (2001). Islamic Geometrical patterns for the teaching of Mathematics of symmetry,Symmetry: Culture and Science, 12 (1-2), 53-65.[2] D’Ambrosio, U. (1985). Ethnomathematics and its place in the history and pedagogy ofmathematics, For the Learning of Mathematics, 5, 44-48.[3] Dolinko, L. (1996). Investigating flags: a multicultural approach, Teaching ChildrenMathematics, v3, 186-190.[4] Eglash, R. (2001). Rethinking symmetry in ethnomathematics, Symmetry: Culture and Science,12 (1-2), 159-166.[5] Knuchel, C. (2004). Teaching symmetry in the elementary curriculum, The MontanaMathematics Enthusiast, 1(1), 3-8.[6] Leikin, R., Berman, A., Zaslavsky, O. (2000). Applications of symmetry to problem solving,International Journal of Mathematical Education in Science and Technology, 31(6), 799-809.[7] Oliveras, M. L., Favilli, F., César, M. (2002). Teachers and Intercultural Education Based onEthnomathematics. In Ferreira, E. S. (Ed.), Proceedings of the II International Congress onEthnomathematics. Ouro Preto: Universidade Federal de Ouro Preto.[8] Pumfrey, E., Beardon, T. (2002). Art and mathematics – mutual enrichment, Micromath, v18/2,21-26.Volume 2 Supplement 1, 2009
62Iuliana MarchisAcknowledgementThe activities were developed and tested in the Comenius 2.1 project ICTime (ICT as a Tool forIntercultural and Media s julianna@yahoo.comActa Didactica Napocensia, ISSN 2065-1430University,Cluj-Napoca,Romania,e-mail:
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