International Electronic Journal of Mathematics Education

Grade 11 Students’ Proof Construction Ability in Relation to Classroom Resources
AMA 10th edition
In-text citation: (1), (2), (3), etc.
Reference: Shongwe B. Grade 11 Students’ Proof Construction Ability in Relation to Classroom Resources. INT ELECT J MATH ED. 2020;15(2), em0571. https://doi.org/10.29333/iejme/6278
APA 6th edition
In-text citation: (Shongwe, 2020)
Reference: Shongwe, B. (2020). Grade 11 Students’ Proof Construction Ability in Relation to Classroom Resources. International Electronic Journal of Mathematics Education, 15(2), em0571. https://doi.org/10.29333/iejme/6278
Chicago
In-text citation: (Shongwe, 2020)
Reference: Shongwe, Benjamin. "Grade 11 Students’ Proof Construction Ability in Relation to Classroom Resources". International Electronic Journal of Mathematics Education 2020 15 no. 2 (2020): em0571. https://doi.org/10.29333/iejme/6278
Harvard
In-text citation: (Shongwe, 2020)
Reference: Shongwe, B. (2020). Grade 11 Students’ Proof Construction Ability in Relation to Classroom Resources. International Electronic Journal of Mathematics Education, 15(2), em0571. https://doi.org/10.29333/iejme/6278
MLA
In-text citation: (Shongwe, 2020)
Reference: Shongwe, Benjamin "Grade 11 Students’ Proof Construction Ability in Relation to Classroom Resources". International Electronic Journal of Mathematics Education, vol. 15, no. 2, 2020, em0571. https://doi.org/10.29333/iejme/6278
Vancouver
In-text citation: (1), (2), (3), etc.
Reference: Shongwe B. Grade 11 Students’ Proof Construction Ability in Relation to Classroom Resources. INT ELECT J MATH ED. 2020;15(2):em0571. https://doi.org/10.29333/iejme/6278

Abstract

Despite proof being fundamental to the mathematics discipline and its role as a means to convey mathematical content, little is known about the effect of resources on influencing students’ proof construction ability. The purpose of this study was to compare two didactic environments, one regarded as resourced (favored) and the other under-resourced (disadvantaged), in relation to the construction of a mathematical proof. Motivated by the discrepancies in the literature on the influence of school resources on students’ performance and the unfortunately prevalent view that the sole function of proof in mathematics is verification by using confirmatory cases, this study sought to examine the differences (if any) between resourced and under-resourced classrooms in relation to students’ proof construction ability. To this end, data were drawn from a proof-related task performed by 78 Grade 11 students in the Ethekwini Metropolitan area, South Africa. A modified version of the Proof Construction Assessment tool showed that students in resourced schools significantly performed better than those in under-resourced schools in relation to proof construction. In addition, there was an observable and noticeable effect of this. Specifically, at an alpha = .05, the t-test for independent means revealed a significant difference between the two groups, t(76) = 2.749, p < .01, d = .624 SD. The practical significance of the results emphasizes the importance of taking into account the role of resources when investigating the learning and teaching of proofs. Further, preliminary results also suggested that most students struggled to even begin to prove the proposition. Recommendations and implications for the students’ careers and future research are raised and discussed.

References

  • Anderson, K. A. (2015). Introduction to optimal resource theory: A framework for enhancing student achievement. The Journal of Negro Education, 84(1), 25-39. https://doi.org/10.7709/jnegroeducation.84.1.0025
  • Arzarello, F., Bairral, M., Danie, C., & Yasuyuki, I. (2013). Ways of manipulation touchscreen in one geometrical dynamic software. In E. Faggiano, & A. Montone (Eds.), Proceedings of the 11th International Conference on Technology in Mathematics Teaching (pp. 59-64). Bari: University of Bari.
  • Bertram, C., & Hugo, W. (2008). Social justice through epistemological access: A tale of two classrooms. In A. Muthukrishna (Ed.), Educating for social justice and inclusion in an African context: Pathways and transitions (pp. 133-143). New York: Nova.
  • Bloch, G. (2009). The toxic mix. What’s wrong with South Africa’s schools and how to fix it. Cape Town: Tafelberg. https://doi.org/10.1016/j.tox.2009.04.005
  • Brodie, K. (2006). Teaching mathematics for equity: Learner contributions and lesson structure. African Journal of Research in SMT Education, 10(1), 13-24. https://doi.org/10.1080/10288457.2006.10740590
  • Burtless, G. (Ed.). (1996). Does money matter? The effect of school resources on student achievement and adult success. Washington, D.C.: Brookings Institution Press.
  • CadwalladerOlsker, T. (2011). What do we mean by mathematical proof? Journal of Humanistic Mathematics, 1(1), 33-60. https://doi.org/10.5642/jhummath.201101.04
  • Chazan, D. (1993). High school geometry students’ justification for their views of empirical evidence and mathematical proof. Educational Studies in Mathematics, 24(4), 359-387. https://doi.org/10.1007/BF01273371
  • Christou, C., Mousoulides, N., Pittalis, M., & Pitta-Pantazi, D. (2004). Proofs through exploration in dynamic geometry environments. Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education, 2, 215-222. https://doi.org/10.1007/s10763-004-6785-1
  • Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Hillsdale, NJ: Lawrence Erlbaum Associates.
  • Creswell, J. W. (2012). Educational research: Planning, conducting, and evaluating quantitative and qualitative research (4th ed.). Boston, MA: Pearson Education.
  • de Villiers, M. D. (1990). The role and function of proof in mathematics. Pythagoras, 24, 17-24.
  • de Villiers, M. D. (1997). The role of proof in investigative, computer-based geometry: Some personal relections. In J. R. King, & D. Schattschneider (Eds.), Geometry turned on! (pp. 15-24). Washington: MAA. Retrieved from http://frink.michighway.com/~dynamicm/joint-AMS-MAA.pdf
  • de Villiers, M. D. (1998). An alternative approach to proof in dynamic geometry. In R. Lehrer, & D. Chazan (Eds.), Designing learning environments for developing understanding of geometry and space (pp. 369-393). Mahwah, NJ: Lawrence Erlbaum.
  • de Villiers, M. D. (2004). 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. D. (2012). Rethinking proof with the Geometer’s Sketchpad (Vol. 5). Emeryville, CA: Key Curriculum Press.
  • de Villiers, M. D., & Heideman, N. (2014). Conjecturing, refuting and proving within the context of dynamic geometry. Learning and Teaching Mathematics, 17, 20-26.
  • Denton, J. (2017). Transforming mathematics: Using dynamic geometry software to strengthen understanding of enlargement and similarity. Warwick Journal of Education, 1, 69-84.
  • Douek, N. (2009). Approaching proof in school: From guided conjecturing and proving to a story of proof construction. In F.-L. Lin, F.-J. Hsieh, G. Hanna, & M. de Villiers (Eds.), Proceedings of the ICMI Study 19 Conference: Proof and proving in Mathematics Education (pp. 142-147). Taipei, Taiwan: National Taiwan Normal University.
  • Dreyfus, T., & Hadas, N. (1987). Euclid may stay—and even be taught. In M. M. Lindquist, & A. P. Shulte (Eds.), Learning and teaching geometry, K-12 (pp. 47-58). Reston, VA: National Council of Teachers of Mathematics.
  • Easdown, D. (2012). The role of proof in mathematics teaching and the Plateau Principle. Proceedings of The Australian Conference on Science and Mathematics Education (formerly UniServe Science Conference), (pp. 28-33).
  • Geary, D. C. (1998). Male, female: The evolution of human sex differences. Washington, DC: American Psychology Association. https://doi.org/10.1037/10370-000
  • Geary, D. C. (1999). Sex differences in mathematical abilities: Commentary on the math-fact retrieval hypothesis. Contemporary Educational Psychology, 24, 267-274. https://doi.org/10.1006/ceps.1999.1007
  • Grant, C. C. (2014). Leading for social justice in South African schools: Where have all the activists gone? In I. Bogotch, & C. M. Shields (Eds.), International handbook of educational leadership and social (in)justice (pp. 521-541). Springer. https://doi.org/10.1007/978-94-007-6555-9_29
  • Greenwald, R., Hedges, L. V., & Laine, R. D. (1996). The effect of school resources on student achievement. Review of Educational Research, 66(3), 361-396. https://doi.org/10.3102/00346543066003361
  • Gustafsson, J.-E. (2003). What do we know about effects of school resources on educational results? Swedish Economic Policy Review, 10, 77-110.
  • Hanna, G. (2000). Proof, explanation and exploration: An overview. Educational Studies in Mathematics, 44, 5-23. https://doi.org/10.1023/A:1012737223465
  • Hanna, G. (2007). The ongoing value of proof. In P. Boero (Ed.), Theorems in school: From history, epistemology and cognition to classroom practice (pp. 3-18). Rotterdam: Sense. https://doi.org/10.1163/9789087901691_002
  • Hanna, G., de Villiers, M. D., Arzare, F., Dreyfus, T., Durand-Guerrier, V., Jahnke, H. N., . . . Yevdokimov, O. (2009). Proof and proving in mathematics education: Discussion document. In F.-L. Lin, F.-J. Hsieh, & M. de Villiers (Eds.), Proceedings of the ICMI Study 19 Conference: Proof and proving in mathematics education. National Taiwan Normal University in Taipei, Taiwan, May 10 to May 15 (pp. xix-xxx).
  • Hanushek, E. A. (1986). The economics of schooling: Production and efficiency in public schools. Journal of Economic Literature, 24, 1141-1117.
  • Hanushek, E. A. (1997). Assessing the effects of school resources on student performance: An update. Education Evaluation and Policy Analysis, 19(2), 141-164. https://doi.org/10.3102/01623737019002141
  • Harel, G. (2013). Intellectual need. In K. R. Leatham (Ed.), Vital directions for mathematics education research (pp. 119-151). New York, NY: Springer. https://doi.org/10.1007/978-1-4614-6977-3_6
  • Harel, G., & Sowder, L. (1998). Students’ proof schemes: Results from exploratory studies. Issues in Mathematics Education, 7, 234-283. https://doi.org/10.1090/cbmath/007/07
  • Healy, L., & Hoyles, C. (2000). A study of proof conceptions in algebra. Journal for Research in Mathematics Education, 31(4), 396-428. https://doi.org/10.2307/749651
  • Herbst, P. G., & Miyakawa, T. (2008). When, how, and why prove theorems? A methodology for studying the perspective of geometry teachers. ZDM Mathematics Education, 40, 469-486. https://doi.org/10.1007/s11858-008-0082-3
  • Hofstede, G. (1986). Cultural differences in teaching and learning. International Journal of Intercultural Relations, 10, 301-320. https://doi.org/10.1016/0147-1767(86)90015-5
  • IBM Corp. (Released 2016). IBM SPSS Statistics for Windows, Version 24.0. Armonk, NY: IBM Corp.
  • Inglis, M., Mejia-Ramos, J. P., Weber, K., & Alcock, L. (2013). On mathematicians’ different standards when evaluating elementary proofs. Topics in Cognitive Science, 5, 270-282. https://doi.org/10.1111/tops.12019
  • Jones, K. (2011). The value of learning geometry with ICT: Lessons from innovative educational research. In A. Oldknow, & C. Knights (Eds.), Mathematics education with digital technology (pp. 39-45). London: Continuum.
  • Kane-Berman, J. (2017). Achievement and enterprise in school education. Johannesburg, South Africa: South African Institute of Race Relations.
  • Knuth, E. J. (2002). Teachers’ conceptions of proof in the context of secondary school mathematics. Journal of Mathematics Teacher Education, 61-88. https://doi.org/10.1023/A:1013838713648
  • Lee, V. E., & Zuze, T. L. (2011). School resources and academic performance in Sub-Saharan Africa. Comparative Education Review, 55(3), 369-397. https://doi.org/10.1086/660157
  • Lubben, F., Sadeck, M., Scholtz, Z., & Braund, M. (2010). Gauging students’ untutored ability in argumentation about experimental data: A South African case study. International Journal of Science Education, 32(16), 2143-2166. https://doi.org/10.1080/09500690903331886
  • Martin, T. S. (1997). Calculus students abilities to solve geometric related rate problems and their understanding of related geometric growth factors (Unpublished doctoral dissertation), Boston University, Boston, MA.
  • McCrone, S. M., & Martin, T. S. (2004). Assessing high school students’ understanding of geometric proof. Canadian Journal of Science, Mathematics and Technology Education, 4(2), 223-242. https://doi.org/10.1080/14926150409556607
  • Mogodi, T. K. (2013). The use of ICT for learning at a Dinaledi school in the Limpopo province (Unpublished masters dissertation), University of Johannseburg, South Africa.
  • Pandiscio, E. A. (2002). Exploring the link between preservice teachers’ conception of proof and the use of dynamic geometry software. School Science and Mathematics, 102(5), 216-221. https://doi.org/10.1111/j.1949-8594.2002.tb18144.x
  • Reddy, V., Prinsloo, C., Visser, M., Arends, F., Winnaar, L., Rogers, S., . . . Mthethwa, M. (2012). Mathematics and science achievement of South African schools in TIMSS 2011. South Africa: HSRC Press.
  • Ruthven, K. (2012). The didactical tetrahedron as a heuristic for analysing the incorporation of digital technologies into classroom practice in support of investigative approaches to teaching mathematics. ZDM, 44(5), 627-640. https://doi.org/10.1007/s11858-011-0376-8
  • Sarracco, L. (2005). The effects of using dynamic geometry software in the middle school classroom. EDT 896 Research Report, Iona College, NY. Retrieved May 21, 2019
  • Sedibe, M. (2011). Inequality of access to resources in previously disadvantaged South African high schools. Journal of Social Sciences, 28(2), 129-135. https://doi.org/10.1080/09718923.2011.11892937
  • Smith, K. (2014). How teacher beliefs about mathematics affect student beliefs about mathematics. Honors Theses and Capstones, University of New Hampshire.
  • Soudien, C. (2007). The “A” factor: Coming to terms with the question of legacy in South African education. International Journal of Educational Development, 27, 182-193. https://doi.org/10.1016/j.ijedudev.2006.07.006
  • South African Government News Agency. (2016). Motshekga moots scrapping of quintile system. Pretoria: South African Government News Agency. Retrieved on January 23, 2017, from http://www.sanews.gov.za/south-africa/motshekga-moots-scrapping-quintile-system
  • Thompson, D. R., Senk, S. L., & Johnson, G. J. (2012). Opportunities to learn reasoning and proof in high school mathematics textbooks. Journal for Research in Mathematics Education, 3, 253-295. https://doi.org/10.5951/jresematheduc.43.3.0253
  • Toulmin, S. E. (2003). The uses of argument (Updated Edition). Cambridge: Cambridge University Press. https://doi.org/10.1017/CBO9780511840005
  • Uploaders. (2013). Siyavula: Mathematics (Grade 11). Retrieved on January 23, 2016, from http://www.everythingmaths.co.za/maths/grade-11/siyavula-mathematics-grade-11-caps
  • Willingham, W. W., & Cole, N. S. (1997). Gender and fair assessment. Mahwah, NJ: Erlbaum.
  • Wilson, S., & MacLean, R. (2011). Research methods and data analysis for psychology. Berkshire: McGaw-Hill.

License

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.