A HISTORICAL–EPISTEMOLOGICAL ANALYSIS OF THE ALGEBRAIC SYMBOLIC SYSTEM
Main Article Content
Abstract
This article presents a historical-epistemological analysis to clarify the formation and development of the algebraic symbolic system and identify viewpoints affecting the development and epistemological characteristics of the algebraic symbolic system. The analysis results show that the system developed in three intertwined stages: rhetorical or prose algebra, syncopated algebra, and symbolic algebra. In addition, three viewpoints greatly influenced the formation and development of the algebraic symbol system: geometry, arithmetic, and spreading mathematics. The results contribute to an epistemological analysis of the history of mathematics and serve as a basis for research on students' obstacles when accessing the algebraic symbol system.
Keywords
algebraic symbols, epistemological characteristics rhetorical algebra, syncopated algebra, symbolic algebra
Article Details
References
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Tall, D. (2008). The transition to formal thinking in mathematics. Mathematics Education Research Journal, 20(2), 5-24.
Tall, D., Gray, E., Bin Ali, M., Crowley, L., DeMarois, P., McGowen, M., Pitta, D., Pinto, M., Thomas, M., & Yusof, Y. (2001). Symbols and the bifurcation between procedural and conceptual thinking. Canadian Journal of Science, Mathematics and Technology Education, 1(1), 81-104. https://doi.org/10.1080/14926150109556452
Yetkin, E. (2003). Student Difficulties in Learning Elementary Mathematics. Bloomington, IN: ERIC Digest.
Bardini, C., & Pierce, R. (2015). Assumed mathematics knowledge: The challenge of symbols. International Journal of Innovation in Science and Mathematics Education, 23(1), 1-9.
Baumgart, J. K. (1969). The history of algebra: An overview. In Historical topics for the mathematics classroom. 31st National Council of Teachers of Mathematics Yearbook. Washington, DC: NCTM.
Blackhouse, J., Haggarty, L., Pirie, S., & Stratton, S. (1992). Improving the learning of mathematics. London: Cassell.
Boyer, C. B. (1991). A history of mathematics (2nd ed.). New York: John Wiley & Sons.
Britannica (2023). John Wallis. Encyclopedia Britannica. https://www.britannica.com/biography/John-Wallis
Britannica (2023). Mathesis Universalis. Encyclopedia Britannica. https://www.britannica.com/topic/Mathesis-Universalis
Cajori, F. (1993). A history of mathematical notations. Dover Publication, Inc. NewYork.
Cunningham, E. G. (1986). The language and development of algebra. In N. F. Ellerton (Ed.), Mathematics: Who needs what? (pp. 224-226).
Eves, H. (1983). An introduction to the history of mathematics. 5th ed. Philadelphia, Saunders College Pub.
Fauvel, J., & van Maanen, J. (Eds.). (2000). History in mathematics education: The ICMI Study. Dordrecht, The Netherlands: Kluwer Academic.
Godino, J. D., Batanero, C., & Font, V. (2007). The onto-semiotic approach to research in mathematics education. International Journal on Mathematics Education (ZDM), 39(1-2), 127-135.
Graham, A., & Thomas, M. O. J. (2000). Building a versatile understanding of algebraic variables with a graphic calculator. Educational Studies in Mathematics, 41, 265-282.
Güçler, B. (2014). The role of symbols in mathematical communication: the case of the limit notation. Research in Mathematics Education, 16(3), 251-268.
Gullberg, J. (1997). Mathematics from the birth of numbbers. New York: W.W. Norton.
Katz, V. J. (2009). A History of Mathematics: An Introduction, 3rd Edition. Pearson.
Katz, V. J., & Barton, B. (2007). Stages in the history of algebra with implications for teaching. Educational Studies in Mathematics, 66(2), 185-201.
Kieran, C. (1992). The learning and teaching of school algebra. In D. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 390-419).
Kızıltoprak, A., & Yavuzsoy Köse, N. (2017). Relational thinking: The bridge between arithmetic and algebra. International Electronic Journal of Elementary Education, 10(1), 131-145.
Küchemann, D. (1981). Algebra. In K. Hart (Ed.), Children’s Understanding of Mathematics: 11-16 (pp. 102-119). Murray, London.
Le, T. H. C. (2017). Sự cần thiết của phân tích tri thức luận đối với các nghiên cứu về hoạt động dạy học và đào tạo giáo viên [The necessity of epistemological analysis for research on teaching activities and teacher training]. In Proceedings of the sixth international conference on mathematics teaching (pp. 17-39). Ho Chi Minh City University of Education.
MacGregor, M., & Stacey, K. (1997). Students' understanding of algebraic notation: 11-15. Educational Studies in Mathematics, 33(1), 1-19.
Maharaj, A. (2008). Some insights from research literature for teaching and learning mathematics. South African Journal of Education, 28(3), 401-414.
Maracchia, S. (2013). The importance of symbolism in the development of algebra. Lettera Matematica, 1(3), 137-144.
Nataraj, M. S., & Thomas, M. O. J. (2012). The Concept of Generalised Number: Valuable Lessons from the History of Algebra. Mathematics Education: Expanding Horizons, Proceedings of the 35th Annual Conference of the Mathematics Education Research Group of Australasia, 562-569.
Nelson, D. (1993). Teaching mathematics from a multicultural standpoint. In D. Nelson, G. G. Joseph, & J. Williams, (Eds.). Multicultural mathematics. Oxford: Oxford University Press.
Neugebauer, O., & Sachs, A. (1945). Mathematical Cuneiform Texts, American Oriental Society, New Haven.
Schoenfeld, A. H., & Arcavi, A. (1988). On the Meaning of Variable. The Mathematics Teacher, 81(6), 420-427. http://www.jstor.org/stable/27965869
Sfard, A. (1995). The development of algebra: Confronting historical and psychological perspectives. Journal of Mathematical Behaviour, 14, 15-19.
Smartick. (n.d.). Mathematical Symbols to Represent Operations and Relations. https://www.smartick.com/blog/mathematics/algebra/mathematical-symbols/
Stallings, L. (2000). A Brief History of Algebraic Notation. School Science and Mathematics, 100(5), 230-235.
Stols, G. (2011). The importance of using and not using symbols in school mathematics. Proceedings: Towards Effective Teaching and Meaningful Learning in Mathematics, Science and Technology. ISTE International Conference on Mathematics, Science and Technology Education. 17-20 October 2011. Mopani Camp in Kruger National Park, Limpopo, South Africa, 255-264.
Struik, D. J. (1967). A concise history of mathematics. (3rd ed.). New York: Dover.
Sutori. (n.d.). The Evolution of Math and Algebraic Symbols. https://www.sutori.com/en/story/the-evolution-of-math-and-algebraic-symbols–3PuqpMcsdD1xnMBv5JWVNa2Q
Tall, D. (2008). The transition to formal thinking in mathematics. Mathematics Education Research Journal, 20(2), 5-24.
Tall, D., Gray, E., Bin Ali, M., Crowley, L., DeMarois, P., McGowen, M., Pitta, D., Pinto, M., Thomas, M., & Yusof, Y. (2001). Symbols and the bifurcation between procedural and conceptual thinking. Canadian Journal of Science, Mathematics and Technology Education, 1(1), 81-104. https://doi.org/10.1080/14926150109556452
Yetkin, E. (2003). Student Difficulties in Learning Elementary Mathematics. Bloomington, IN: ERIC Digest.