Using African Designs in Virtual Manipulatives for Geometrical Concept Development
Year: 2017 Authors: Philip Collett; Catherina Steyn
Core claim
Culturally familiar African design-based manipulatives can strengthen motivation and understanding of geometry, especially space and shape concepts.
Topics
geometry education, virtual manipulatives, ethnomathematics, tessellations, GeoGebra
Domains
geometry, space and shape, transformations, tessellations, Southern African indigenous art, Ndebele patterns, craft and beadwork, architectural decoration
Methods
GeoGebra applet design, rigid polygon puzzles, pilot testing, techno-blended learning
Media
GeoGebra, virtual manipulatives, TouchTutor™, PDFs, video
Paper text
The text below is the locally extracted OCR/Markdown version of the paper. Raw PDF files remain local and are not published here.
Bridges 2017 Conference Proceedings
Using African Designs in Virtual Manipulatives for Geometrical Concept Development
Philip Collett GMMDU, Nelson Mandela University Port Elizabeth, RSA Philip.collett@nmmu.ac.za
Catherina Steyn GMMDU, Nelson Mandela University Port Elizabeth, RSA Catherina.steyn@nmmu.ac.za
Abstract
Promoting interest and engagement in mathematics among learners in early high school grades is challenging in the South African educational scenario which is typified by traditional pedagogies, desolate classrooms, a lack of interesting resources and something of a rote learning culture. To enhance the attractiveness and the cultural congruence of learning materials for early high school grades we have used GeoGebra to design a set of virtual manipulatives, which incorporate Southern African indigenous art designs, for promoting concept formation in geometry. We explain the underlying pedagogy of the development and show an example. Plans for incorporating the material into existing programs and for researching the learning outcomes are also presented.
Introduction
The challenges faced by teachers and learners of mathematics in South African schools are well documented [1]. They include lack of adequate teacher training and professional development, lack of teaching and learning resources, desolate mathematics classrooms and reliance on outmoded pedagogies. In-school challenges are compounded in many communities by a socio-economic disadvantage which presents as low levels of parental literacy and numeracy, lack of parental and community support for children and a lack of role models who have succeeded as a result of education.
Learner achievement in mathematics is consequently poor with South Africa ranking low on international benchmarking measures such as TIMMS [2]. Annual National Assessments at various grades also yield low averages, and weaknesses are compounded from one grade to the next [3].
For those learners who persevere with mathematics, some areas of the CAPS curriculum present major conceptual difficulties. An area of pronounced weakness is geometry. Many senior high school learners have difficulty in constructing logically correct answers for geometrical questions as they lack the basic conceptual tools required [4]. The roots of these difficulties can be traced to problems with earlier development of concepts of shape and space as well as limited opportunities to develop logical arguments.
A Techno-Blended Approach to Mathematics Development in Schools
To engender enthusiasm and interest we have included recreational and conceptual manipulatives, developed in GeoGebra, in a curriculum-aligned, off-line, digital teaching and learning package. The TouchTutor™ package, developed over the past five years in South Africa, serves as the basis of a comprehensive techno-blended model [5] for the teaching and learning of mathematics.
The package includes a wide range of learning resources including video, PDFs, reference facilities and language support for mathematics, a self-testing system and a range of GeoGebra applets that are designed for learner investigations.
Collett and Steyn
The rationale for building and implementing such an integrated learning system rests on research reviews showing the potential enabling and liberating role of digital learning [6] and claims of positive learning outcomes associated with mobile learning [8, 9]. The GeoGebra applets that form part of the package were developed to be recreational but also educational. Through the project, we investigated the value of the applets and it was found that learners believed that using the applets improved their understanding of applicable mathematics concepts.
Southern African Indigenous Art Designs as Stimulus for Learning about Geometry
Many authors [10, 11] have documented the richness of incorporation of mathematics in African designs in the form of tessellations, sona, string figures, fractals and number patterns in games. Gray and Sarhangi [11] illustrate how beautiful cultural designs can easily be incorporated into the secondary school mathematics classroom. The inclusion of cultural designs narrows the gap between the learners’ world inside and outside the school environment [9, 10].
Gerdes [9] argues that it is increasingly necessary to “multi-culturalize” the mathematics curriculum as mathematics in Africa is often seen as useless and foreign. Including aspects of ethno-mathematics will improve learners’ confidence in the value of mathematics. Ethno-mathematics involves a valuing of indigenous mathematical knowledge, practices and applications in arts and crafts and the incorporation of these into the teaching and learning of mathematics.
The inspiration for the design in the applet described comes from the beautiful designs using basic geometrical shapes that the Ndebele woman use to decorate their homes. The Ndebele, a tribe in Southern Africa, has a culture rich in decorative geometrical patterns (Figures 1 - 3) applied to architecture, fabric and beadwork [12]. Other related South African cultures have their own decorative patterns.
Figure 1: Painted walls
Figure 2: Beadwork bracelet
Figure 3: Fabric design
An African Map Tessellation with Ndebele Patterns Created with GeoGebra
To enhance the attractiveness and the cultural congruence of learning materials for early high school grades we have used GeoGebra to design a set of virtual manipulatives for promoting concept formation in geometry. One such applet is illustrated in Figure 4 on the following page.
The puzzle pieces are created as rigid polygons in GeoGebra. We used polygons similar in shape and colour to those used in Ndebele designs and included a set that can be combined to form a stylized map of Africa. Each shape may be translated or rotated by the user of the applet to create their design (Figure 5).
Underlying Pedagogy
The learners involved in the project have typically experienced a learning environment poor in stimulus and challenge, particularly in space and shape. We believe attractive virtual manipulatives, which are
Using African Designs in Virtual Manipulatives for Geometrical Concept Development
culturally familiar in design, could be a catalyst to developing greater confidence and competence in geometry. Clements [13] contends that manipulatives can be used very successfully in the instruction of mathematics.
Figure 4: Elements of an Africa Map puzzle inspired by Ndebele designs.
Figure 5: Manipulating rigid puzzle pieces in GeoGebra.
Applets implementing simple tessellation type puzzles provide an opportunity to identify basic polygons (such as the triangles, rhombi, parallelograms, and hexagons in the puzzle described here) and to manipulate them by applying basic transformations such as translation and rotation. The degrees are shown during the rotation of the shapes, visually linking the degrees and the orientation of the rotated shape. There are many applications that can be demonstrated for e.g., the effects of rotation and another rotation returning the shape to its original position. The vertical and horizontal movement of the shape can be described to reinforce the effect of translation. Fitting pieces to the map teaches visual planning.
There are good opportunities for class discussion about ways of combining shapes to create bigger shapes, both in the form of larger regular polyhedral, quadrilaterals or more complex compound shapes. The design could also easily be reproduced on card allowing the learners to physically manipulate the objects, similarly to tangrams. In the digital form, applets can also assist learners to become familiar with the use of technology in a non-threatening manner.
Pilot testing of the example applet indicated that it was easy to use, enjoyable and motivating. It challenged learners’ design ability and required iterative techniques. The range of new composite figures which could were formed within the puzzle was interesting, as was the many different solutions which varied in the extent to which symmetry and colour patterns were used. The activity provided fruitful opportunities for discussion on classification of polygons and their properties, transformations, and tessellations. The cultural context of the design was clearly identified as African.
Collett and Steyn
Further Development and Research
The current indigenous art applets use simple shape identification, combinations and transformations for the junior grades. We already have various other applets that help learners in the senior grades to visualize the more complex theorems but none of them incorporates indigenous art. Therefore, we intend to extend the number of applets by identifying more indigenous patterns that lend themselves to dynamic investigation by senior learners.
The material will be made available via the TouchTutor™ system to groups in under-resourced rural schools in the Eastern Cape province of South Africa [9].
Pilot testing has provided a framework for studying learner interactions with the material to gauge effects on engagement with mathematics, motivation and, more specifically, the development of basic concepts in Space and Shape which form part of the current new curriculum.
Conclusion
Mathematics teaching and learning can be enhanced with a creative use of materials so that local cultural content resonates with and motivates learners. Future development, implementation and ideas will be guided by learner responses to these materials.
References
[1] N. Spaull, “South Africa’s Education Crisis: The quality of education in South Africa 1994-2011,” Johannesburg, 2013.
[2] I. V. S. Mullis, M. O. Martin, P. Foy, and A. Arora, Results in Mathematics, 1st ed. Chestnut Hill, MA: TIMSS & PIRLS International Study Center, 2012.
[3] Department of Basic Education, “Report on the Annual National Assessment of 2014 gr 1 to 6 & 9,” Pretoria, 2014.
[4] Department of Basic Education, “2015 National Senior Certificate Examination Diagnostic Report,” Pretoria, 2015.
[5] W. A. Olivier, “Maths Unit Boosts Pupils’ X Factor An Offline Tutoring Programme is Giving Talented but Sidelined Pupils a Fighting Chance,” Mail & Guardian, p. 4, 2015.
[6] D. Buckingham, Beyond Technology: Children’s Learning in the Age of Digital Culture. Cambridge: Polity Press, 2007.
[7] W. Daher, “Building Mathematical Knowledge in an Authentic Mobile Phone Environment,” Australas. J. Educ. Technol., vol. 26, no. 1, pp. 85–104, 2010.
[8] M. Sherman, “The Role of Technology in Supporting Students’ Mathematical Thinking: Extending the Metaphors of Amplifier and Reorganizer,” Contemp. Issues Technol. Teach. Educ., vol. 14, no. 3, pp. 220–246, 2014.
[9] P. Gerdes, “On Mathematics in the History of Sub-Saharan Africa,” Hist. Math., vol. 21, no. 3, pp. 345–376, 1994.
[10] M. Mosimege, “Indigenous Mathematical Knowledge at South African Cultural Villages: Opportunities for Integration in Mathematics Classrooms,” in Proceedings of the 14th Annual National Congress of AMESA, 2008, pp. 457–462.
[11] A. Gray and R. Sarhangi, “Study and Application of African Designs for Use in Secondary Education,” in Proceedings of the Bridges 2001 Conference, 2001, pp. 299–306.
[12] L. Rosenblum and M. Macedonia, “Ndebele painting in VR,” IEEE Comput. Graph. Appl., vol. 21, no. 2, pp. 10–13, 2001.
[13] D. H. Clements, “‘Concrete’ Manipulatives, Concrete Ideas,” Contemp. Issues Early Child., vol. 1, no. 1, pp. 45–60, 1999.