The Arithmetic-Algebra Connection: A Historical-Pedagogical Perspective
The problem of designing a teaching learning approach to symbolic algebra in the middle school that uses students’ knowledge of arithmetic as a starting point has not been adequately addressed in the recent revisions of the mathematics curriculum in India. India has a long historical tradition of mathematics with strong achievements in arithmetic and algebra. We review an explicit discussion of the relation between arithmetic and algebra in a historical text from the twelfth century, emphasizing that algebra is more a matter of insight and understanding than of using symbols. Algebra is seen as foundational to arithmetic rather than as a generalization of arithmetic. We draw implications from these remarks and present a framework that illuminates the arithmetic-algebra connection from a teaching-learning point of view. Finally, we offer brief sketches of an instructional approach developed through a design experiment with students of grade 6 that is informed by this framework, and discuss some student responses.
KeywordsOperational Composition Algebraic Expression Symbolic Expression Mathematics Textbook Primary Grade
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- ASER report (2010). Annual Status of Education Report (Rural) 2009 (Provisional). New Delhi: ASER Centre. Retrieved from http://asercentre.org/asersurvey/aser09/pdfdata/aser09.pdf.
- Banerjee, R. (2008a). Developing a learning sequence for transiting from arithmetic to elementary algebra. Unpublished doctoral dissertation. Mumbai: Homi Bhabha Centre for Science Education, Tata Institute of Fundamental Research. Google Scholar
- Banerjee, R. (2008b). Assessing the curriculum reforms in India: the case of integers and algebra for beginning middle school students. In International Conference on Mathematics Education (ICME-11), Mexico. Retrieved from http://dg.icme11.org/document/get/99.
- Banerjee, R., & Subramaniam, K. (2008). Bridging arithmetic and algebra: Evolution of a teaching sequence. In O. Figueras et al. (Eds.), International Group of the Psychology of Mathematics Education: Proceedings of the Joint Meeting of PME 32 and PME-NA XXX (PME29) (Vol. 2, pp. 121–128). Morelia, Mexico. Google Scholar
- Banerjee, R., & Subramaniam, K. (submitted). Evolution of a teaching approach for beginning algebra. Google Scholar
- Bose, A. (2009). Mathematical riddles among the Mushars: Linked to a historical tradition. In M. Tzekaki, M. Kaldrimidou, & H. Sakonidis (Eds.), Proceedings of the 33rd of the International Group for the Psychology of Mathematics Education (Vol. 5, p. 439). Thessaloniki, Greece. Google Scholar
- Cai, J., Lew, H. C., Morris, A., Moyer, J. C., Ng, S. F., & Schmittau, J. (2005). The development of students’ algebraic thinking in earlier grades: A cross-cultural comparative perspective. Zentralblatt fuer Didaktik der Mathematik (International Review on Mathematics Education), 37(1), 5–15. Google Scholar
- Chaiklin, S., & Lesgold, S. (1984). Prealgebra students’ knowledge of algebraic tasks with arithmetic expressions. Paper presented at the annual meeting of the American Research Association. Google Scholar
- Colebrooke, H. T. (1817). Algebra, with Arithmetic and Mensuration from the Sanscrit of Brahmegupta and Bhascara. London: John Murray. Google Scholar
- Datta, B., & Singh, A. N. (1938/2001). History of Hindu Mathematics, Vol. II, Edition 2001. Delhi: Bharatiya Kala Prakashan. Google Scholar
- Dewan, H. K. (2010). Pedagogy of mathematics. Learning Curve, XIV, 16–22. Google Scholar
- Educational Initiatives and Wipro (2006). Student learning in the Metros 2006: How well are our students learning? Report available online from http://www.ei-india.com/full-report.pdf. Accessed 1st Nov 2009.
- Fujii, T., & Stephens, M. (2001). Fostering an understanding of algebraic generalization through numerical expressions: The role of quasi-variables. In H. Chick, K. Stacey, & J. Vincent (Eds.), Proceedings of the 12 th ICMI Study Conference: The Future of the Teaching and Learning of Algebra (Vol. 1, pp. 258–264). Melbourne, Australia: The University of Melbourne. Google Scholar
- Fujii, T., & Stephens, M. (2008). Using number sentences to introduce the idea of a variable. In C. E. Greenes & R. Rubenstein (Eds.), Algebra and Algebraic Thinking in School Mathematics, Seventieth Yearbook (pp. 127–140). Reston, VA: NCTM. Google Scholar
- Fuson, K. C. (1992). Research on whole number addition and subtraction. In D. A. Grouws (Ed.), Handbook of Research in Mathematics Teaching and Learning. New York: MacMillan. Google Scholar
- Katz, V. (1998). A History of Mathematics: An Introduction (2nd ed.). Reading, Massachusetts: Addison Wesley. Google Scholar
- Katz, V. J. (2001). Using the history of algebra in teaching algebra. In H. Chick, K. Stacey, J. Vincent, & J. Vincent (Eds.), Proceedings of the 12th ICMI Study Conference: The Future of the Teaching and Learning of Algebra (Vol. 2, pp. 353–359). Melbourne: University of Melbourne. Google Scholar
- Kieran, C. (2006). Research on the learning and teaching of algebra: A broad source of meaning. In A. Gutierrez & P. Boero (Eds.), Handbook of Research on the Psychology of Mathematics Education: Past, Present and Future (pp. 11–49). Rotterdam, The Netherlands: Sense Publishers. Google Scholar
- Liebenberg, R., Linchevski, L., Sasman, M. C., & Olivier, A. (1999). Focusing on the structural aspects of numerical expressions. In J. Kuiper (Ed.), Proceedings of the 7 th Annual Conference of the Southern African Association for Research in Mathematics and Science Education (SAARMSE) (pp. 249–256). Harare, Zimbabwe. Google Scholar
- Malara, N., & Iaderosa, R. (1999). The interweaving of arithmetic and algebra: Some questions about syntactic and structural aspects and their teaching and learning. In I. Schwank (Ed.), Proceedings of the First Conference of the European Society for Research in Mathematics Education (Vol. 2, pp. 159–171). Osnabrueck: Forschungsinstitut fuer Mathematikdidaktik. Google Scholar
- Math-magic: Book 3 (2006). New Delhi: National Council of Educational Research and Training. Google Scholar
- Math-magic: Book 5 (2008). New Delhi: National Council of Educational Research and Training. Google Scholar
- Mathematics: Text book for class VI (2006). New Delhi: National Council of Educational Research and Training. Google Scholar
- Mukherjee, A. (2010). The nature of mathematics and its relation to school education. Learning curve, XIV, 16–22. Google Scholar
- Mumford, D. (2010). Review of Mathematics in India by Kim Plofker. Notices of the American Mathematical Society, 27(3), 385–390. Google Scholar
- National Centre for Educational Research and Training (2005). National Curriculum Framework. Retrieved from http://www.ncert.nic.in/html/pdf/schoolcurriculum/framework05/nf2005.pdf.
- National Centre for Educational Research and Training (2006). Position paper of the National Focus Group on the Teaching of Mathematics. Retrieved from www.ncert.nic.in/html/pdf/schoolcurriculum/position_papers/math.pdf.
- Plofker, K. (2007). Mathematics in India. In V. Katz (Ed.), The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook (pp. 385–513). Princeton, New Jersey: Princeton University Press. Google Scholar
- Plofker, K. (2009). Mathematics in India. Princeton, New Jersey: Princeton University Press. Google Scholar
- Pradhan, H. C., & Mavlankar, A. T. (1994). Compendium of Errors in Middle School Mathematics. Mumbai: Homi Bhabha Centre for Science Education, Tata Institute of Fundamental Research. Google Scholar
- Rajagopalan, S. (2010). Insights about student learning from an adaptive learning math program. Learning Curve, XIV, 93–99. Google Scholar
- Srinivas, M. D. (2008). Proofs in Indian mathematics, epilogue. In K. V. Sarma (Ed.), Ganita-Yukti-Bhasa of Jyesthadeva, Vol. 1—Mathematics (pp. 267–293). New Delhi: Hindustan Book Agency. Google Scholar
- Subramaniam, K. (2004). Naming practices that support reasoning about and with expressions. In Proceedings of the International Congress on Mathematics Education (ICME 10), Denmark. Available online at http://www.icme10.dk/proceedings/pages/regular~pdf/RL~K~Subramanian.pdf.
- Tripathi, P. (2007). Review of mathematics textbooks. Contemporary Education Dialogue, 4(1), 142–151. Google Scholar
- Van den Heuvel-Panhuizen, M. (1996). Assessment and Realistic Mathematics Education. Utrecht, The Netherlands: CD-ß Press Utrecht University. Google Scholar