The State of South African Mathematics Education: Situating the Hidden Promise of Multiple-solution Tasks

Sfiso Cebolenkosi Mahlaba

doi:10.29333/ejmste/9279

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Sfiso Cebolenkosi Mahlaba ¹ ^*

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¹ North-West University, SOUTH AFRICA^* Corresponding Author

Abstract

We live in the challenging times of the 21st-century with the increased need for humans to possess specific skills that will help them to be successful in this era. This means that education should in learners, develop these skills effectively. Different global countries have begun to recognize the significance of multiple solutions tasks in the teaching and learning of mathematics in the 21st-century. However, this practice is not visible in South Africa. Hence, the current study explores and synthetize the sparsely available literature on MSTs to answer the question: What is the significance of multiple-solution tasks (MSTs) in mathematics education and why is it relevant for South African mathematics education to make the exercise of producing multiple solutions accessible to learners? The literature that is being synthetized here is viewed through the optic lens of the social constructivism theory as proposed by Vygotsky and explicated in Jean Lave and Etienne Wenger`s Situated learning: Legitimate peripheral participation. In the conclusion I engage in an argumentation that illuminates the significance of MSTs in mathematics education and provide reasons why it would be beneficial for the South African mathematics curriculum to incorporate MSTs.

Keywords

multiple solution tasks
problem solving
creativity
geometry

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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Article Type: Review Article

EURASIA J Math Sci Tech Ed, Volume 16, Issue 12, December 2020, Article No: em1921

https://doi.org/10.29333/ejmste/9279

Publication date: 28 Nov 2020

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References

Alex, J. K. (2019). The preparation of secondary school mathematics teachers in South Africa: Prospective teachers’ student level disciplinary content knowledge. Eurasia Journal of Mathematics, Science and Technology Education, 15(12), em1791. https://doi.org/10.29333/ejmste/105782
Alex, J. K., & Mammen, K. J. (2014). An Assessment of the Readiness of Grade 10 Learners for Geometry in the Context of Curriculum and Assessment Policy Statement (CAPS) Expectation. International Journal of Educational Sciences, 7(1), 29-39. https://doi.org/10.1080/09751122.2014.11890167
Anderson, H. H. (1960). The Nature of Creativity. Studies in Art Education, 1(2), 10-17. https://doi.org/10.2307/1319842
Apino, E., & Retnawati, H. (2017). Developing Instructional Design to Improve Mathematical Higher Order Thinking Skills of Students. Journal of Physics: Conference Series, 812, 012100. https://doi.org/10.1088/1742-6596/812/1/012100
Arends, F., Winnaar, L., & Mosimege, M. (2017). Teacher classroom practices and mathematics performance in South African schools: A reflection on TIMSS 2011. South African Journal of Education, 37(3), 1-11.
Bayaga, A., Mthethwa, M. M., Bosse, M. J., & Williams, D. (2019). Impacts of implementing GeoGebra on eleventh grade student`s learning of Euclidean geometry. South African Journal of Higher Education, 33(6).
Bingolnali, E. (2011). Multiple Solutions to Problems in Mathematics Teaching: Do Teachers Really Value Them? Australian Journal of Teacher Education, 36(1), 18-31.
Chapman, O. (2013). Mathematical-task knowledge for teaching. Journal of Mathematics Teacher Education, 16(1), 1-6. https://doi.org/10.1007/s10857-013-9234-7
Chen, E. (2016). Euclidean geometry in mathematical olympiads. United States of America: The Mathematical Association of America (Incorporated).
Christison, A. (2019). A look back : the International Mathematics Olympiad, 1959. Learning and Teaching Mathematics, 2019(27), 26-29. Retrieved from https://journals.co.za/content/journal/10520/EJC-19eb16bb03
Cropley, A. J. (1997). More ways than one: Fostering creativity. Norwood, New Jersey: Ablex Publishing Corporation.
Daher, W., Tabaja-Kidan, A., & Gierdien, F. (2017). Educating Grade 6 students for higher-order thinking and its influence on creativity. Pythagoras, 38(1), 350 2017. https://doi.org/10.4102/pythagoras.v38i1.350
De Villiers, M. (2004a). The role and function of quasi‐empirical methods in mathematics. Canadian Journal of Science, Mathematics and Technology Education, 4(3), 397-418. https://doi.org/10.1080/14926150409556621
De Villiers, M. (2004b). Using dynamic geometry to expand mathematics teachers’ understanding of proof. International Journal of Mathematical Education in Science and Technology, 35(5), 703-724. https://doi.org/10.1080/0020739042000232556
De Villiers, M. (2016). A multiple solution task: A SA mathematics olympiad problem. Learning & Teaching Mathematics, (20), 18-20.
De Villiers, M. (2017). A multiple solution task : another SA Mathematics Olympiad problem. Learning and Teaching Mathematics, 2017(22), 42-46. Retrieved from https://journals.co.za/content/journal/10520/EJC-7c1634cc0
De Villiers, M., & Heideman, N. (2014). Conjecturing, refuting and proving within the context of dynamic geometry. Learning and Teaching Mathematics, 2014(17), 20-26. Retrieved from https://journals.co.za/content/amesal/2014/17/EJC164266
Department of Basic Education. (2011). Curriculum and Assessment Policy Statement: Mathematics (Grade 10-12). Pretoria: Government Printers
Department of Basic Education. (2017). The SACMEQ IV project in South Africa: A study of the conditions of schooling and the quality of education. South Africa
Department of Basic Education. (2019). Mathematics teaching and learning framework in South Africa: Teaching mathematics for understanding. Pretoria: Government Printers
Dewey, J. (1933). How we think: A restatement of the relation of reflective thinking to the educative process. Boston: D. C. Heath and Company.
Dhombres, J. (1993). Is one proof enough? Travels with a mathematician of the baroque period. Educational Studies in Mathematics, 24(4), 401-419. https://doi.org/10.1007/BF01273373
Donohue, D., & Borman, J. (2014). The challenges of realising inclusive education in South Africa. South African Journal of Education, 34(2), 1-14.
Erbas, A. K., & Okur, S. (2012). Researching students’ strategies, episodes, and metacognitions in mathematical problem solving. Quality & Quantity, 46(1), 89-102. https://doi.org/10.1007/s11135-010-9329-5
Ervynck, G. (1991). Mathematical Creativity. In D. Tall (Ed.), Advanced Mathematical Thinking (pp. 42-53). Dordrecht: Springer Netherlands.
Flavell, J. H. (1976). Metacognitive aspects of problem solving. In L. Resnick (Ed.), The nature of intelligence (pp. 231-236). Hillside, NJ: Erlbaum.
Flavell, J. H. (1979). Metacognition and cognitive monitoring: A new are of cognitive-developmental inquiry. American Psychologist, 34(10), 906-911.
Gleason, N. W. (2018). Singapore’s Higher Education Systems in the Era of the Fourth Industrial Revolution: Preparing Lifelong Learners. In N. W. Gleason (Ed.), Higher Education in the Era of the Fourth Industrial Revolution (pp. 145-169). Singapore: Springer Singapore.
Große, C. S. (2014). Mathematics learning with multiple solution methods: effects of types of solutions and learners’ activity. Instructional Science, 42(5), 715-745. Retrieved from www.jstor.org/stable/43575255
Guberman, R., & Leikin, R. (2013). Interesting and difficult mathematical problems: changing teachers’ views by employing multiple-solution tasks. Journal of Mathematics Teacher Education, 16(1), 33-56. https://doi.org/10.1007/s10857-012-9210-7
Guilford, J. P. (1967). Creativity: Yesterday, Today and Tomorrow. The Journal of Creative Behavior, 1(1), 3-14. https://doi.org/10.1002/j.2162-6057.1967.tb00002.x
Hadamard, J. (1945). The psychology of invention in the mathematical field: Princeton University Press.
Hamming, R. W. (1980). The Unreasonable Effectiveness of Mathematics. The American Mathematical Monthly, 87(2), 81-90. https://doi.org/10.2307/2321982
Hashimoto, Y. (1997). The methods of fostering creativity through mathematical problem solving. ZDM, 29(3), 86-87. https://doi.org/10.1007/s11858-997-0005-8
Hoth, J., Kaiser, G., Busse, A., Döhrmann, M., König, J., & Blömeke, S. (2017). Professional competences of teachers for fostering creativity and supporting high-achieving students. ZDM, 49(1), 107-120. https://doi.org/10.1007/s11858-016-0817-5
Howie, S. J. (2003). Language and other background factors affecting secondary pupils’ performance in Mathematics in South Africa. African Journal of Research in Mathematics, Science and Technology Education, 7(1), 1-20. https://doi.org/10.1080/10288457.2003.10740545
Julie, C., & Gierdien, F. (2020). Reflections on Realistic Mathematics Education from a South African Perspective. In M. van den Heuvel-Panhuizen (Ed.), International Reflections on the Netherlands Didactics of Mathematics: Visions on and Experiences with Realistic Mathematics Education (pp. 71-82). Cham: Springer International Publishing.
Kilpatrick, J. (1981). The Reasonable Ineffectiveness of Research in Mathematics Education. For the Learning of Mathematics, 2(2), 22-29. Retrieved from www.jstor.org/stable/40247734
Kilpatrick, J. (1985). Reflection and recursion. Educational Studies in Mathematics, 16(1), 1-26. https://doi.org/10.1007/BF00354880
Knowles, M. S. (1975). Self-directed learning: A guide for learners and teachers. Englewood Cliffs, NJ: Cambridge Adult Education.
Leikin, R. (2007). Habits of the mind associated with advanced mathematical thinking and solution spaces of mathematical tasks. Paper presented at the Proceedings of the fifth conference of the European Society fo Research in Mathematics Education -CERME-5.
Leikin, R. (2010). Teaching the Mathematically Gifted. Gifted Education International, 27(2), 161-175. https://doi.org/10.1177/026142941002700206
Leikin, R. (2011). Multiple-solution tasks: from a teacher education course to teacher practice. ZDM, 43(6), 993-1006. https://doi.org/10.1007/s11858-011-0342-5
Leikin, R. (2014). Challenging Mathematics with Multiple Solution Tasks and Mathematical Investigations in Geometry. In Y. Li, E. A. Silver, & S. Li (Eds.), Transforming Mathematics Instruction: Multiple Approaches and Practices (pp. 59-80). Cham: Springer International Publishing.
Leikin, R., & Lev, M. (2013). Mathematical creativity in generally gifted and mathematically excelling adolescents: what makes the difference? ZDM, 45(2), 183-197. https://doi.org/10.1007/s11858-012-0460-8
Leikin, R., & Levav-Waynberg, A. (2008). Solution Spaces of Multiple-Solution Connecting Tasks as a Mirror of the Development of Mathematics Teachers’ Knowledge. Canadian Journal of Science, Mathematics and Technology Education, 8(3), 233-251. https://doi.org/10.1080/14926150802304464
Leikin, R., & Pitta-Pantazi, D. (2013). Creativity and mathematics education: the state of the art. ZDM, 45(2), 159-166. https://doi.org/10.1007/s11858-012-0459-1
Levav-Waynberg, A., & Leikin, R. (2012a). The role of multiple solution tasks in developing knowledge and creativity in geometry. The Journal of Mathematical Behavior, 31(1), 73-90. https://doi.org/10.1016/j.jmathb.2011.11.001
Levav-Waynberg, A., & Leikin, R. (2012b). Using Multiple Solution Tasks for the Evaluation of Students’ Problem-Solving Performance in Geometry. Canadian Journal of Science, Mathematics and Technology Education, 12(4), 311-333. https://doi.org/10.1080/14926156.2012.732191
Lockhart, P. (2009). A mathematician`s lament. New York: Bellevue Literary Press.
Lynch, K., & Star, J. R. (2014). Teachers’ Views About Multiple Strategies in Middle and High School Mathematics. Mathematical Thinking and Learning, 16(2), 85-108. https://doi.org/10.1080/10986065.2014.889501
Mann, E. L. (2006). Creativity: The Essence of Mathematics. Journal for the Education of the Gifted, 30(2), 236-260. https://doi.org/10.4219/jeg-2006-264
Mhlolo, M. K. (2017). Regular classroom teachers’ recognition and support of the creative potential of mildly gifted mathematics learners. ZDM, 49(1), 81-94. https://doi.org/10.1007/s11858-016-0824-6
Miri, B., David, B.-C., & Uri, Z. (2007). Purposely Teaching for the Promotion of Higher-order Thinking Skills: A Case of Critical Thinking. Research in Science Education, 37(4), 353-369. https://doi.org/10.1007/s11165-006-9029-2
Naidoo, J., & Kapofu, W. (2020). Exploring female learners’ perceptions of learning geometry in mathematics. South African Journal of Education, 40, 1-11. Retrieved from http://www.scielo.org.za/scielo.php?script=sci_arttext&pid=S0256-01002020000100014&nrm=iso
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.
Organization for Economic Co-operation and Development. (2018). PISA 2021 mathematics framework (draft). Retrieved from https://pisa2021-maths.oecd.org/#Formulate
Oswald, M., & de Villiers, J.-M. (2013). Including the gifted learner: perceptions of South African teachers and principals. South African Journal of Education, 33, 1-21. Retrieved from http://www.scielo.org.za/scielo.php?script=sci_arttext&pid=S0256-01002013000100004&nrm=iso
Pehkonen, E. (1997). The state-of-art in mathematical creativity. ZDM, 29(3), 63-67. https://doi.org/10.1007/s11858-997-0001-z
Pfeiffer, S. I., & Blei, S. (2008). Gifted Identification Beyond the IQ Test: Rating Scales and Other Assessment Procedures. In S. I. Pfeiffer (Ed.), Handbook of Giftedness in Children: Psychoeducational Theory, Research, and Best Practices (pp. 177-198). Boston, MA: Springer US.
Phakeng, M. S., & Moschkovich, J., N. (2013). Mathematics Education and Language Diversity: A Dialogue across Settings. Journal for Research in Mathematics Education, 44(1), 119-128. https://doi.org/10.5951/jresematheduc.44.1.0119
Pillay, P. (2017). A multiple solution problem. Learning & Teaching Mathematics, 23, 32-37.
Poincare, H. (1948). Science and method. New York: Dover.
Polya, G. (1954). Mathematics and plausible reasoning: Induction and analogy in mathematics (Vol. 2). Princeton: Princeton University Press.
Polya, G. (1973). How to solve it: A new aspect of mathematics method. Princeton: Princeton University Press.
Pournara, C., Hodgen, J., Adler, J., & Pillay, V. (2015). Can improving teachers’ knowledge of mathematics lead to gains in learners’ attainment in Mathematics? South African Journal of Education, 35, 1-10. Retrieved from http://www.scielo.org.za/scielo.php?script=sci_arttext&pid=S0256-01002015000300007&nrm=iso
Reddy, V., Visser, M., Winnaar, L., Arends, F., Juan, A., Pronsloo, C., & Isdale, K. (2016). TIMSS 2015: Highlights of mathematics and science achievement of grade 9 South African learners.
Renzulli, J. S. (2011). What Makes Giftedness?: Reexamining a Definition. Phi Delta Kappan, 92(8), 81-88. https://doi.org/10.1177/003172171109200821
Resnick, L. (1987). Education and learning to think. Washington, DC: National Academy.
Rittle-Johnson, B., Star, J. R., & Durkin, K. (2012). Developing procedural flexibility: Are novices prepared to learn from comparing procedures? British Journal of Educational Psychology, 82(3), 436-455. https://doi.org/10.1111/j.2044-8279.2011.02037.x
Runco, M. A. (1993). Creativity as an educational objective for disadvantaged students. Storrs: University of Connecticut: The National Research Center on the Gifted and Talented.
Runco, M. A., & Acar, S. (2012). Divergent Thinking as an Indicator of Creative Potential. Creativity Research Journal, 24(1), 66-75. https://doi.org/10.1080/10400419.2012.652929
Samson, D. (2015). Devising explorative Euclidean Geometry questions. Learning and Teaching Mathematics, 2015(19), 13-16. Retrieved from https://journals.co.za/content/amesal/2015/19/EJC181296
Samson, D. (2017). Euclidean geometry - Nurturing multiple solutions. Learning & Teaching Mathematics, 23, 15-18.
Samson, D., & Kroon, S. (2019). A multiple solution task. Learning and Teaching Mathematics, 2019(26), 7-11. Retrieved from https://journals.co.za/content/journal/10520/EJC-16d84dd913
Santos-Trigo, M., & Reyes-Rodriguez, A. (2016). The use of digital technology in finding multiple paths to solve and extend an equilateral triangle task. International Journal of Mathematical Education in Science and Technology, 47(1), 58-81. https://doi.org/10.1080/0020739X.2015.1049228
Schindler, M., Joklitschke, J., & Rott, B. (2018). Mathematical Creativity and Its Subdomain-Specificity. Investigating the Appropriateness of Solutions in Multiple Solution Tasks. In F. M. Singer (Ed.), Mathematical Creativity and Mathematical Giftedness: Enhancing Creative Capacities in Mathematically Promising Students (pp. 115-142). Cham: Springer International Publishing.
Schoenfeld, A. H. (1983). Problem solving in the mathematics curriculum: A report, recommendations` and an annotated bibliography. Washington, DC: The Mathematical Association of America.
Schoenfeld, A. H. (1985). Mathematical problem sovling. Orlando: Academic Press, Inc.
Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving, metacognition, and sense making in mathematics. In D. Grouws (Ed.), Handbook for research on mathematics teaching and learning (pp. 334-370). New York: Macmillan.
Schon, D. A. (1983). The reflective practitioner: How professionals think in action. United States of America: Basic Books.
Schukajlow, S., Krug, A., & Rakoczy, K. (2015). Effects of prompting multiple solutions for modelling problems on students’ performance. Educational Studies in Mathematics, 89(3), 393-417. https://doi.org/10.1007/s10649-015-9608-0
Segal, R., Stupel, M., & Flores, A. (2017). Examples of multiple proofs in geometry: Part 1, tasks and hints. Ohio Journal of School Mathematics, 35-43.
Sigler, A., Segal, R., & Stupel, M. (2016). The standard proof, the elegant proof, and the proof without words of tasks in geometry, and their dynamic investigation. International Journal of Mathematical Education in Science and Technology, 47(8), 1226-1243. https://doi.org/10.1080/0020739X.2016.1164347
Silver, E. A. (1997). Fostering creativity through instruction rich in mathematical problem solving and problem posing. ZDM, 29(3), 75-80. https://doi.org/10.1007/s11858-997-0003-x
Sriraman, B. (2005). Are Giftedness and Creativity Synonyms in Mathematics? Journal of Secondary Gifted Education, 17(1), 20-36. https://doi.org/10.4219/jsge-2005-389
Sriraman, B. (2009). The characteristics of mathematical creativity. ZDM, 41(1), 13. https://doi.org/10.1007/s11858-008-0114-z
Sriraman, B., & Dickman, B. (2017). Mathematical pathologies as pathways into creativity. ZDM, 49(1), 137-145. https://doi.org/10.1007/s11858-016-0822-8
Sriraman, B., Haavold, P., & Lee, K. (2013). Mathematical creativity and giftedness: a commentary on and review of theory, new operational views, and ways forward. ZDM, 45(2), 215-225. https://doi.org/10.1007/s11858-013-0494-6
Stanislaw, S., & Krug, A. (2014). Do Multiple Solutions Matter? Prompting Multiple Solutions, Interest, Competence, and Autonomy. Journal for Research in Mathematics Education, 45(4), 497-533. https://doi.org/10.5951/jresematheduc.45.4.0497
Stols, G., Ferreira, R., Pelser, A., Olivier, W. A., Van der Merwe, A., De Villiers, C., & Venter, S. (2015). Perceptions and needs of South African Mathematics teachers concerning their use of technology for instruction. South African Journal of Education, 35, 01-13. Retrieved from http://www.scielo.org.za/scielo.php?script=sci_arttext&pid=S0256-01002015000400004&nrm=iso
Stols, G., Van Putten, S., & Howie, S. (2010). Making Euclidean geometry compulsory: are we prepared? Perspectives in Education, 28(4), 22-31. Retrieved from https://www.ingentaconnect.com/content/sabinet/persed/2010/00000028/00000004/art00004
Stupel, M., & Ben-Chaim, D. (2017). Using multiple solutions to mathematical problems to develop pedagogical and mathematical thinking: A case study in a teacher education program. Investigations in Mathematics Learning, 9(2), 86-108. https://doi.org/10.1080/19477503.2017.1283179
Torrance, E. P. (1966). Torrance tests of creative thinking. Norms-technical manual. Research edition. Verbal tests, forms A and B. Figural tests, forms A and B. Princeton: Personnel Press.
Ubah, I., & Bansilal, S. (2019). The use of semiotic representations in reasoning about similar triangles in Euclidean geometry. Pythagoras, 40(1), 480 2019. https://doi.org/10.4102/pythagoras.v40i1.480
Umugiraneza, O., Bansilal, S., & North, D. (2018). Exploring teachers’ use of technology in teaching and learning mathematics in KwaZulu-Natal schools. Pythagoras, 39(1), 342 2018. https://doi.org/10.4102/pythagoras.v39i1.342
Visser, M., Juan, A., & Feza, N. (2015). Home and school resources as predictors of mathematics performance in South Africa. South African Journal of Education, 35(1), 1-10.
Weisberg, R. W. (1988). Problem solving and creativity. In R. J. Sternberg (Ed.), The nature of creativity (pp. 148-176). New York: Cambridge University Press.
Wessels, H. M. (2014). Levels of mathematical creativity in model-eliciting activities. Journal of Mathematical Modelling and Application, 1(9), 22-40.
Winner, E. (2000). The origins and ends of giftedness. American Psychologist, 55(1), 159-169.
World Economic Forum. (2015). The global competitiveness repot 2015-2016. Retrieved from http://www3.weforum.org/docs/gcr/2015-2016/Global_Competitiveness_Report_2015-2016.pdf

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APA

Mahlaba, S. C. (2020). The State of South African Mathematics Education: Situating the Hidden Promise of Multiple-solution Tasks. Eurasia Journal of Mathematics, Science and Technology Education, 16(12), em1921. https://doi.org/10.29333/ejmste/9279

Vancouver

Mahlaba SC. The State of South African Mathematics Education: Situating the Hidden Promise of Multiple-solution Tasks. EURASIA J Math Sci Tech Ed. 2020;16(12):em1921. https://doi.org/10.29333/ejmste/9279

AMA

Mahlaba SC. The State of South African Mathematics Education: Situating the Hidden Promise of Multiple-solution Tasks. EURASIA J Math Sci Tech Ed. 2020;16(12), em1921. https://doi.org/10.29333/ejmste/9279

Chicago

Mahlaba, Sfiso Cebolenkosi. "The State of South African Mathematics Education: Situating the Hidden Promise of Multiple-solution Tasks". Eurasia Journal of Mathematics, Science and Technology Education 2020 16 no. 12 (2020): em1921. https://doi.org/10.29333/ejmste/9279

Harvard

Mahlaba, S. C. (2020). The State of South African Mathematics Education: Situating the Hidden Promise of Multiple-solution Tasks. Eurasia Journal of Mathematics, Science and Technology Education, 16(12), em1921. https://doi.org/10.29333/ejmste/9279

MLA

Mahlaba, Sfiso Cebolenkosi "The State of South African Mathematics Education: Situating the Hidden Promise of Multiple-solution Tasks". Eurasia Journal of Mathematics, Science and Technology Education, vol. 16, no. 12, 2020, em1921. https://doi.org/10.29333/ejmste/9279