Teaching and learning of algebra has been one of the most debated areas in mathematics education. There are various ways to introduce algebra, but the most common practice has been to introduce algebra after students have undergone some amount of arithmetic instruction. In recent years, a fresh look is being taken at this approach. In this article, I wish to review a section of research literature which deals with identification of difficulties which students face in learning algebra, and their connection to arithmetic. Further, I will discuss various kinds of teaching experiments carried out so that students make more sense of the algebraic symbols and operations on them, thus explicating the arithmetic–algebra connection. Based on this review of literature, I will argue for a better and broader understanding of the arithmetic–algebra connection and will broadly describe the kind of arithmetic that is needed to be able to make a transition to algebra.
Balacheff, N. (2001). Symbolic arithmetic vs algebra: The core of a didactical dilemma. In R. Sutherland, T. Rojano, A. Bell & R. Lins (Eds), Perspectives on school algebra (pp. 249–260). Dordrecht, The Netherlands: Kluwer Academic Publishers.
2.
Banerjee, R. (2008). Developing a learning sequence for transiting from arithmetic to elementary algebra. Unpublished Ph.D. dissertation. Homi Bhabha Centre for Science Education, Tata Institute of Fundamental Research, Mumbai.
3.
Bednarz, N., & Janvier, B. (1996). Emergence and development of algebra as a problem-solving tool: Continuities and discontinuities with arithmetic. In Nadine Bednarz, Carolyn Kieran & Lesley Lee (Eds), Approaches to algebra: Perspectives for research and teaching (pp. 115–136). Dordrecht, The Netherlands: Kluwer Academic Publishers.
4.
Bell, A. (1995). Purpose in school algebra. Journal of Mathematical Behavior, 14(1), 41–73.
5.
Blanton, M., & Kaput, J.J. (2001). Algebrafying the elementary mathematics experience Part II: Transforming practice on a district-wide scale. In H. Chick, K. Stacey, & J. Vincent (Eds), Proceedings of the 12th ICMI study conference: The future of the teaching and learning of algebra (Vol. 1, pp. 87–95). Melbourne, Australia: The University of Melbourne.
Booth, L.R. (1988). Children’s difficulties in beginning algebra. In Arthur F. Coxford, & Albert P. Shulte (Eds), The ideas of algebra, K-12 (1988 yearbook of the NCTM) (pp. 20–32). Reston, VA: NCTM.
8.
Brizuela, B.M., & Schliemann, A.D. (2003). Fourth graders solving equations. In N.A. Pateman, B.J. Dougherty, & J. Zilliox (Eds), Proceedings of the 27th international conference for the psychology of mathematics education (Vol. 2, pp. 137–144). Honolulu, Hawaii: Psychology of Mathematics Education (PME).
9.
Carpenter, T.P., & Franke, M. (2001). Developing algebraic reasoning in the elementary school: Generalization and proof. In H. Chick, K. Stacey, & J. Vincent (Eds), Proceedings of the 12th ICMI study conference: The future of the teaching and learning of algebra (Vol. 1, pp. 155–162). Melbourne, Australia: The University of Melbourne.
10.
Carpenter, T.P., Franke, M.L., & Levi, L. (2003). Thinking mathematically: Arithmetic and algebra in elementary school. Portsmouth, NH: Heinemann.
11.
Carraher, D.W., Schliemann, A.D., & Brizuela, B.M. (2000). Early algebra, early arithmetic: Treating operations as functions. Plenary presentation at PME-NA XXII, Tucson, AZ.
12.
Cerulli, M., & Mariotti, M.A. (2001). Arithmetic and algebra, continuity or cognitive break? The case of Francesca. In Marja van-den Heuvel Pannhueizen (Ed.), Proceedings of the 25th conference of the international group for the psychology of mathematics education (Vol. 2, pp. 225–232). Utrecht, The Netherlands: PME.
13.
Chalough, L., & Herscovics, N. (1988). Teaching algebraic expressions in a meaningful way. In A.F. Coxford & A.P. Shulte (Eds), The ideas of algebra, K-12 (1988 yearbook) (pp. 33–42). Reston, VA: NCTM.
14.
Davis, R.B. (1975), Cognitive processes involved in solving simple algebraic equations. Journal of Children’s Mathematical Behaviour, 1(3), 7–35.
15.
Filloy, E., & Rojano, T. (1989). Solving equations: The transition from arithmetic to algebra. For the Learning of Mathematics, 9(2), 19–25.
16.
Fischbein, E., & Barash, A. (1993). Algorithmic models and their misuse in solving algebraic problems. In I. Hirabayashi, N. Nohda, K. Shigematsu & F. Lin (Eds), Proceedings of the 17th international group of the psychology of mathematics education (Vol. 1, pp. 162–72). Tsukuba, Ibaraki, Japan: PME.
17.
Fujii, T. (2003). Probing students’ understanding of variables through cognitive conflict problems: Is the concept of a variable so difficult to understand? In N.A. Pateman, B.J. Dougherty & J. Zilliox (Eds), Proceedings of the 27th international group of the psychology of mathematics education (Vol. 1, pp. 49–65). Honolulu, Hawaii: PME.
18.
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 12th ICMI study conference: The future of the teaching and learning of algebra (Vol. 1, pp. 258–264). Melbourne, Australia: The University of Melbourne.
19.
Herscovics, N., & Linchevski, L. (1994). A cognitive gap between arithmetic and algebra. Educational Studies in Mathematics, 27(1), 59–78.
20.
Kaput, J.J., & Blanton, M. (2001). Algebrafying the elementary Mathematics experience Part I: Transforming task structures. In H. Chick, K. Stacey & J. Vincent (Eds), Proceedings of the 12th ICMI study conference: The future of the teaching and learning of algebra (Vol. 2, pp. 344–352). Melbourne, Australia: The University of Melbourne.
21.
Kieran, C. (1981). Concepts associated with the equality symbol. Educational Studies in Mathematics, 12(3), 317–326.
22.
Kieran, C. (1992). Learning and teaching of school algebra. In D.A. Grows (Ed.), Handbook of research on mathematics teaching and learning (pp. 390–419). New York: Macmillan.
23.
Kieran, C. (2006). Research on the learning and teaching of algebra: A broad sources 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.
24.
Kirshner, D. (2001). The structural algebra option revisited. In R. Sutherland, T. Rojano, A. Bell, & R. Lins (Eds), Perspectives on school algebra (pp. 83–98). Dordrecht, The Netherlands: Kluwer Academic Publishers.
25.
Kuchemann, D.E. (1981). Algebra. In K.M. Hart, M.L. Brown, D.E. Kuchemann, D. Kerslake, G. Ruddock, & M. McCartney (Eds), Children’s understanding of mathematics: 11–16 (pp. 102–119). London: John Murray.
26.
Lee, L., & Wheeler, D. (1989). The arithmetic connection. Educational Studies in Mathematics, 20(1), 41–54.
27.
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 7th annual conference of the Southern African Association for Research in Mathematics and Science Education (SAARMSE) (pp. 249–256). Harare, Zimbabwe. Retrieved from http://academic.sun.ac.za/education/mathematics/MALATI/Files/Structure991.pdf
28.
Liebenberg, R., Sasman, M., & Olivier, A. (1999). From numerical equivalence to algebraic equivalence. Proceedings of the Fifth Annual Conference of the Association for Mathematics Education in South Africa (Vol. 2, pp. 173–83). Port Elizabeth: Port Elizabeth Technikon. Retrieved from http://academic.sun.ac.za/mathed/Malati/Files/Structure992.pdf
29.
Linchevski, L., & Herscovics, N. (1996). Crossing the cognitive gap between arithmetic and algebra: Operating on the unknown in the context of equations. Educational Studies in Mathematics, 30(1), 39–65.
30.
Linchevski, L., & Livneh, D. (1999). Structure sense: The relationship between algebraic and numerical contexts. Educational studies in Mathematics, 40(2), 173–96.
31.
Linchevski, L., & Livneh, D. (2002). The competition between numbers and structure: Why expressions with identical algebraic structures trigger different interpretations. Focus on Learning Problems in Mathematics, 24(2), 20–37.
32.
Lins, R.C. (1992). A framework for understanding what algebraic thinking is. Unpublished Ph.D. thesis, University of Nottingham, UK.
33.
Lins, R.C. (2001). The production of meaning for algebra: A perspective based on a theoretical model of semantic fields. In R. Sutherland, T. Rojano, A. Bell, & R. Lins (Eds), Perspectives on school algebra (pp. 13–36). Dordrecht, The Netherlands: Kluwer Academic Publishers.
34.
Lins, R., & Kaput, J. (2004). The early development of algebraic reasoning: The current state of the field. In K. Stacey, H. Chick & M. Kendal (Eds), Proceedings of the 12th ICMI study conference: The future of the teaching and learning of algebra (pp. 47–70). Dordecht, The Netherlands: Kluwer Academic Publishers.
35.
Livneh, D., & Linchevski, L. (2007). Algebrification of arithmetic: Developing algebraic structure sense in the context of arithmetic. In J. Woo, H. Lew, K. Park, & D. Seo (Eds), Proceedings of the 31st conference of the psychology of mathematics education (Vol. 3, pp. 217–225). Seoul, Korea: PME.
36.
MacGregor, M., & Stacey, K. (1997). Students’ understanding of algebraic notation: 11–15. Educational Studies in Mathematics, 33(1), 1–19.
37.
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.
38.
Mason, J., Graham, A., Pimm, D., & Gowar, N. (1985). Routes to/roots of algebra. Walton Hall, Milton Keynes: The Open University Press.
39.
Schliemann, A.D., Carraher, D.W., Brizuela, B., Earnest, D., Goodrow, A., Lara-Roth, S., & Peled, I. (2003). Algebra in elementary school. In N.A. Pateman, B.J. Dougherty & J. Zilliox (Eds), Proceedings of the 27th international conference for the psychology of mathematics education (Vol. 4, pp. 127–34). Honolulu, Hawaii: PME.
40.
Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin. Educational studies in Mathematics, 22(1), 1–36.
41.
Sfard, A., & Linchevski, L. (1994). Gains and pitfalls of reification. Educational Studies in Mathematics, 26(2–3), 191–228.
42.
Stacey, K., & MacGregor, M. (2000). Learning the algebraic method of solving problem. Journal of Mathematical Behavior, 18(2), 149–167.
43.
Tall, D., Thomas, M., Davis, G., Gray, E., & Simpson, A. (2000). What is the object of encapsulation of a process?Journal of Mathematical Behavior, 18(2), 223–241.
44.
Wagner, S., & Parker, S. (1999). Advancing algebra. In Barbara Moses (Ed), Algebraic thinking, grades K-12 (pp. 328–340). Reston, VA: NCTM.
