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The Future of the Teaching and Learning of Algebra The 12th ICMI Study pp 45–70Cite as

The Early Development of Algebraic Reasoning: The Current State of the Field

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Part of the book series: New ICMI Study Series ((NISS,volume 8))

Abstract

The main aim of this chapter is to argue that an early start to algebra education is possible and of great relevance for mathematics education because it provides a special opportunity to foster a particular kind of generality in our students’ thinking. To argue this, we map the various views on algebra education found historically, and trace how the perceptions that mathematics educators hold about children’s thinking and learning have changed. Overall, a great realisation that children can do more in mathematics than was previously believed leads to the adoption of more ambitious objectives for the initial years of school, and to the development of new classroom approaches to algebra education in the early grades. That does not mean teaching the same old school algebra in the same usual way to younger children, but rather to introduce them to new algebraic ways of thinking and immersing them in the culture of algebra. The chapter ends with a research agenda to further developments in this particular sub-field of mathematics education.

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References

  • Arzarello, F., Bazzini, L., & Chiappini, G. (2001). A model for analysing algebraic processes of thinking. In R. Sutherland, T. Rojano, A. Bell, & R. Lins (Eds.), Perspectives on school algebra (pp. 61–82). Dordrecht, The Netherlands: Kluwer Academic.

    Google Scholar 

  • 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.

    Google Scholar 

  • Balacheff, N., & Sutherland, R. (1994). Epistemological domain of validity of Microworlds: The case of LOGO and Cabri-géometre. In R. Lewis & P. Mendelson (Eds.), Lessons from learning (pp. 137–150). North Holland: Elsevier Science B.V.

    Google Scholar 

  • Becker, G. (1988). A classification of students’ errors in secondary level algebra. In A. Borbás (Ed.), Proceedings of the 12thannual conference of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 131–138). Veszprém, Hungary: Program Committee.

    Google Scholar 

  • Biggs, J., & Collis, K. (1982). Evaluating the quality of learning: The SOLO Taxonomy. New York: Academic Press.

    Google Scholar 

  • Blanton, M., & Kaput, J. (2001). Algebrafying the elementary mathematics experience. Part II: Transforming practice on a district-wide scale. In H. Chick, K. Stacey, J. Vincent, & J. Vincent (Eds.), The future of the teaching and learning of algebra (Proceedings of the 12th ICMI Study Conference, pp. 87–95). Melbourne, Australia: The University of Melbourne.

    Google Scholar 

  • Blanton, M., & Kaput, J. (2002). Design principles for tasks that support algebraic thinking in elementary school classrooms. In A. D. Cockburn & E. Nardi (Eds.), Proceedings of the 26thConference of the International Group for the Psychology of Mathematics Education (Vol. 2, 105–112). University of East Anglia: Norwich, UK.

    Google Scholar 

  • Blanton, M., & Kaput, J. (in press). Design principles for instructional contexts that support students’ transition from arithmetic to algebraic reasoning. Elements of task and culture. In R. Nemirovsky, B. Warren, A. Rosebery, & J. Soloman (Eds.), Learning environments: The encounter of everyday and disciplinary experiences. Mahwah, NJ: Lawrence Erlbaum.

    Google Scholar 

  • Boero, P. (2001). Transformation and anticipation as key processes in algebraic problem solving. In R. Sutherland, T. Rojano, A. Bell, & R. Lins (Eds.), Perspectives on school algebra (pp. 99–120). Dordrecht, The Netherlands: Kluwer Academic.

    Google Scholar 

  • Booth, L. (1984). Algebra: Children’s strategies and errors. Windsor, UK: NFER-NELSON.

    Google Scholar 

  • Booth, L. (1989). Grade 8 students’ understanding of structural properties in mathematics. In G. Vergnaud, J. Rogalski, & M. Artigue (Eds.), Proceedings of the 13thannual conference of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 141–148). Paris, France: Program Committee.

    Google Scholar 

  • Brizuela, B., & Lara-Roth, S. (2001a). Additive relations and function tables. In H. Chick, K. Stacey, J. Vincent, & J. Vincent (Eds.), The future of the teaching and learning of algebra (Proceedings of the 12th ICMI Study Conference, pp. 110–119). Melbourne, Australia: The University of Melbourne.

    Google Scholar 

  • Brizuela, B. M., & Lara-Roth, S. (2001b). Additive relations and function tables. Journal of Mathematical Behavior, 20(3), 309–319.

    Article  Google Scholar 

  • Brousseau, G. (1983). Les Obstacles epistemologiques et les problemes en mathematiques. Revista Reserches en Didactique des Mathématiques, 4(2), 165–180.

    Google Scholar 

  • Camp, J., & Marchionini, G. (1984). Programming and learning: Implications for mathematics education. In V. P. Hanson & M. J. Zweng (Eds.), Computers in mathematics education, 1984 Yearbook (pp. 118–126). Reston, VA: National Council of Teachers of Mathematics.

    Google Scholar 

  • Carpenter, T., Ansell, E., Franke, M., Fennema, E., & Weisbeck, L. (1993). Models of problem-solving: A study of kindergarten children’s problem-solving processes. Journal for Research in Mathematics Education, 24, 429–441.

    Google Scholar 

  • Carpenter, T., & Franke, M. (2001). Developing algebraic reasoning in the elementary school: Generalization and proof. In H. Chick, K. Stacey, J. Vincent, & J. Vincent (Eds.), The future of the teaching and learning of algebra (Proceedings of the 12th ICMI Study Conference, pp. 155–162). Melbourne, Australia: The University of Melbourne.

    Google Scholar 

  • Carpenter, T. P., & Levi, L. (1999). Developing conceptions of algebraic reasoning in the primary grades. Paper presented at the Annual Meeting of the American Educational Research Association, Montreal, Canada (Available from the first author at the University of Wisconsin-Madison).

    Google Scholar 

  • Carraher, D., Brizuela, B., & Earnest, D. (2001). The reification of additive differences in early algebra: Viva La Différence! In H. Chick, K. Stacey, J. Vincent, & J. Vincent (Eds.), The future of the teaching and learning of algebra (Proceedings of the 12th ICMI Study Conference, pp. 163–170). Melbourne, Australia: The University of Melbourne.

    Google Scholar 

  • Carraher, D., Schliemann, A.D., & Brizuela, B.M. (in press). Treating operations as functions. In D. W. Carraher, R. Nemirovsky, & C. DiMattia, C. (Eds.), Media and Meaning. CD-ROM issue of Monographs for the Journal of Research in Mathematics Education.

    Google Scholar 

  • Confrey, J. (1991). Steering a course between Vygotsky and Piaget. Educational Researcher, 20(8), 28–32.

    Google Scholar 

  • Davis, E. (2003). Untangling dimensions of middle school students’ beliefs about scientific knowledge and science learning. International Journal for Science Education, 25(4), 439–468.

    Google Scholar 

  • Davis, R. B. (1975). Cognitive processes involved in solving simple algebraic equations. Journal of Children’s Mathematical Behavior, 1(3), 7–35.

    Google Scholar 

  • Davis, R. B. (1984). ICME-5 report: Algebraic thinking in early grades. The Journal of Mathematical Behavior, 4, 195–208.

    Google Scholar 

  • Davydov, V. (1962). An experiment in introducing elements of algebra in elementary school. Soviet Education, 8, 27–37.

    Google Scholar 

  • Davydov, V. (1975). The psychological characteristics of the “prenumerical” period of mathematics instruction. In L. P. Steffe (Ed.), Soviet studies in the psychology of learning and teaching mathematics (Vol. 7, pp. 109–206). Chicago, IL: The University of Chicago Press.

    Google Scholar 

  • Davydov, V. (1982). The psychological characteristics of the formation of elementary mathematical operations in children. In T. Carpenter, J. Noser, & T. Romberg (Eds.), Addition and subtraction: A cognitive perspective (pp. 224–238). Hillsdale, NJ: Lawrence Erlbaum Associates.

    Google Scholar 

  • Davydov, V. (1983). A concept of educational activity for school children. Soviet Psychology, 11(2), 50–76.

    Google Scholar 

  • Dettori, G., Garuti, R., & Lemut, E. (2001). From arithmetic to algebraic thinking by using a spreadsheet. In R. Sutherland, T. Rojano, A. Bell, & R. Lins (Eds.), Perspectives on school algebra (pp. 191–208). Dordrecht, The Netherlands: Kluwer Academic.

    Google Scholar 

  • Dienes, Z. (1973). The six stages in the process of learning mathematics (P. L. Seaborne, Trans.). New York: Humanities Press.

    Google Scholar 

  • Dougherty, B. J., & Zilliox, J. (2003). Voyaging from theory to practice in teaching and learning: A view from Hawai’i. In N. Pateman, B. Dougherty, & J. Zilliox (Eds.), Proceedings of the 27thannual conference of the International Group for the Psychology of Mathematics Education (Vol. 1, 13–31). Honolulu, Hawaii: Program Committee.

    Google Scholar 

  • Feurzeig, W., Lucas, G., Grant, R., & Faflick, P. (1969). Programming languages as a conceptual framework for teaching mathematics. (Technical Report No. 1899). Cambridge, MA: Bolt, Beranek, & Neuman.

    Google Scholar 

  • Fey, J. T. (1984). Computing and mathematics: The impact on secondary school curricula. Reston, VA: National Council of Teachers of Mathematics.

    Google Scholar 

  • Fey, J. (1989). Technology and mathematics education: A survey of recent developments and important problems. Educational Studies in Mathematics, 20, 237–272.

    Article  Google Scholar 

  • Filloy, Y. (1987). Modelling and the teaching of algebra. In J. Bergeron, N. Herscovics, & C. Kieran (Eds.), Proceedings of the 11thannual conference of the International Group for the Psychology of Mathematics Education (Vol. I, pp. 295–300). Montreal, Canada: Program Committee.

    Google Scholar 

  • Franke, M., & Carey, D. (1997). Young children’s perceptions of mathematics in problem-solving environments. Journal for Research in Mathematics Education, 28(1), 8–25.

    Google Scholar 

  • Freudenthal, H. (1974). Soviet research on teaching algebra at the lower grades of elementary school. Educational Studies in Mathematics, 5, 391–412.

    Google Scholar 

  • Fujii, T. (2003) Probing students’ understanding of variables through cognitive conflict problems: Is the concept of a variable so difficult for students to understand? In N. Pateman, B. Dougherty, & J. Zilliox (Eds.), Proceedings of the 27thannual conference of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 49–66). Honolulu, Hawaii: Program Committee.

    Google Scholar 

  • Fujii, T., & Stephens, M. (2001). Fostering an understanding of algebraic generalisation through numerical expressions: The role of quasi-variables. In H. Chick, K. Stacey, J. Vincent, & J. Vincent (Eds.), The future of the teaching and learning of algebra (Proceedings of the 12th ICMI Study Conference, pp. 258–264). Melbourne, Australia: The University of Melbourne.

    Google Scholar 

  • Gallardo, A. (1990). Avoidance and acknowledgement of negative numbers in the context of linear equations. In G. Booker, P. Cobb, & T. de Mendicuti (Eds.), Proceedings of the 14thannual conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 43–50). Oaxtepex, Mexico: Program Committee.

    Google Scholar 

  • Gallardo, A., & Rojano, T. (1987). Common difficulties in the learning of algebra among children displaying low and medium pre-algebraic proficiency levels. In J. Bergeron, N. Herscovics, & C. Kieran (Eds.), Proceedings of the 11thannual conference of the International Group for the Psychology of Mathematics Education (Vol. I, pp. 301–307). Montreal, Canada: Program Committee.

    Google Scholar 

  • Garcia, R., & Piaget, J. (1984). Psicogénesis e História de la Ciencia. Mexico City, Mexico: Siglo Veintuno Editores.

    Google Scholar 

  • Harper, E. (1987). Ghosts of Diophantus. Educational Studies in Mathematics, 18, 75–90.

    Article  Google Scholar 

  • Hart, K. (1984). Children’s understanding of mathematics: 11–16. London: John Murray.

    Google Scholar 

  • Hativa, N., & Cohen, D. (1995). Self learning of negative number concepts by lower division elementary students through solving computer-provided numerical problems. Educational Studies in Mathematics, 28, 401–431.

    Article  Google Scholar 

  • Heath, T. (1964). Diophantus of Alexandria: A study in the history of Greek algebra. New York: Dover.

    Google Scholar 

  • Herscovics, N. (1989). Cognitive obstacles encountered in the learning of algebra. In C. Kieran & S. Wagner (Eds.), Research issues in the learning and teaching of algebra (pp. 60–86). Reston, VA: National Council of Teachers of Mathematics, Hillsdale, NJ: Lawrence Erlbaum Associates.

    Google Scholar 

  • Kaput, J. (1979). Mathematics and learning: Roots of epistemological status. In J. Clement & J. Lochhead (Eds.), Cognitive process instruction (pp. 289–303). Philadelphia, PA: Franklin Institute Press.

    Google Scholar 

  • Kaput, J. (1986). Information technology and mathematics: Opening new representational windows. The Journal of Mathematical Behavior, 5(2), 187–207.

    Google Scholar 

  • Kaput, J. (1992), Technology and mathematics education. In D. Grouws (Ed.), Handbook on research in mathematics teaching and learning (pp. 515–556). New York: Macmillan.

    Google Scholar 

  • Kaput, J. (1994). The representational roles of technology in connecting mathematics with authentic experience. In R. Biehler, R. W. Scholz, R. Strasser, & B. Winkelman (Eds.), Mathematics didactics as a scientific discipline (pp. 379–397). Dordrecht, The Netherlands: Kluwer Academic.

    Google Scholar 

  • Kaput, J. (1998). Transforming algebra from an engine of inequity to an engine of mathematical power by “algebrafying” the K-12 curriculum. In National Council of Teachers of Mathematics and Mathematical Sciences Education Board (Eds.), The nature and role of algebra in the K-14 curriculum: Proceedings of a national symposium (pp. 25–26). Washington, DC: National Research Council, National Academy Press.

    Google Scholar 

  • Kaput, J. (1999). Teaching and learning a new algebra. In E. Fennema & T. Romberg (Eds.), Mathematics classrooms that promote understanding (pp. 133–155). Mahwah, NJ: Lawrence Erlbaum Associates.

    Google Scholar 

  • Kaput, J., & Blanton M. (2001). Algebrafying the elementary mathematics experience. Part I: Transforming task structures. In H. Chick, K. Stacey, J. Vincent, & J. Vincent (Eds.), The future of the teaching and learning of algebra (Proceedings of the 12th ICMI Study Conference, pp. 344–351). Melbourne, Australia: The University of Melbourne.

    Google Scholar 

  • Kaput, J., Noss, R., & Hoyles, C. (2001). Developing new notations for learnable mathematics in the computational era. In L. D. English (Ed.), The handbook of international research in mathematics (pp. 51–73). London: Kluwer Academic.

    Google Scholar 

  • Kieran, C. (1981). Pre-algebraic notions among 12-and 13-year-olds. In Equipe de Recherche Pedagogique Laboratoire (Eds.), Proceedings of the 5thannual conference of the International Group for the Psychology of Mathematics Education (pp. 158–164). Grenoble, France: Program Committee.

    Google Scholar 

  • Kieran, C., Boileau, A., & Garançon, M. (1996). Introducing algebra by means of a technology-supported functional approach. In N. Bednarz, C. Kieran, & L. Lee (Eds.), Approaches to algebra (pp. 257–293). Dordrecht, The Netherlands: Kluwer Academic.

    Google Scholar 

  • Kirshner, D. (1987). The myth about binary representation in algebra. In J. Bergeron, N. Herscovics, & C. Kieran (Eds.), Proceedings of the 11thannual conference of the International Group for the Psychology of Mathematics Education (Vol. I, pp. 308–315). Montreal, Canada: Program Committee.

    Google Scholar 

  • Kirshner, D. (1990). Acquisition of algebraic grammar. In G. Booker, P. Cobb, & T. de Mendicuti (Eds.), Proceedings of the 14thannual conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 75–84). Oaxtepex, Mexico: Program Committee.

    Google Scholar 

  • 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.

    Google Scholar 

  • Klein, J. (1968). Greek mathematical thought and the origins of algebra. Cambridge, MA: The MIT Press.

    Google Scholar 

  • Küchemann, D. (1978). Children’s understanding of numerical variables. Mathematics in School, September, 23–26.

    Google Scholar 

  • Küchemann, D. (1984). Stages in understanding algebra. Journal of Structured Learning, 8, 113–124.

    Google Scholar 

  • Lacampagne, C., Blair, W., & Kaput, J. (Eds.). (1995). The algebra initiative colloquium: Vols.1 and 2. Washington, DC: U.S. Department of Education.

    Google Scholar 

  • Lee, L. (1997). Algebraic understanding: The search for a model in the mathematics education community. Unpublished doctoral thesis. Université du Québec à Montréal, Canada.

    Google Scholar 

  • Lee, L. (2001), Early algebra-But which algebra? In H. Chick, K. Stacey, J. Vincent, & J. Vincent (Eds.), The future of the teaching and learning of algebra (Proceedings of the 12th ICMI Study Conference, pp. 392–399). Melbourne, Australia: University of Melbourne.

    Google Scholar 

  • Lewis, C. (1981). Skill in algebra. In J. R. Anderson (Ed.), Cognitive skills and their acquisition (pp. 85–110). Hillsdale, NJ: Lawrence Erlbaum Associates.

    Google Scholar 

  • Lins, R. (1992.) A framework for understanding what algebraic thinking is. Unpublished Ph.D. Thesis. Shell Centre for Mathematical Education/Nottingham University, UK.

    Google Scholar 

  • Lins, R. (1994). Eliciting the meanings for algebra produced by students: Knowledge, justification and semantic fields. In J. P. da Ponte, & J. F. Matos (Eds.), Proceedings of the 18thannual conference of the International Group for the Psychology of Mathematics Education (Vol. III, pp. 184–191). Lisbon, Portugal: Program Committee.

    Google Scholar 

  • Lins, R. (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. 37–60). Dordrecht, The Netherlands: Kluwer.

    Google Scholar 

  • Marjanovic, M., & Kadijevich, D. (2001). Linking arithmetic to algebra. In H. Chick, K. Stacey, J. Vincent, & J. Vincent (Eds.), The future of the teaching and learning of algebra (Proceedings of the 12th ICMI Study Conference, pp. 425–429). Melbourne, Australia: The University of Melbourne.

    Google Scholar 

  • Mason, J. (1989). Mathematical abstraction as the result of a delicate shift of attention. For the Learning of Mathematics, 9(2), 2–8.

    Google Scholar 

  • Mason, J. (1991). Supporting primary mathematics: Algebra. Milton Keynes, UK: Open University Press.

    Google Scholar 

  • Mason, J. (1996). Expressing generality and roots of algebra. In N. Bednarz, C. Kieran, & L. Lee (Eds.), Approaches to algebra: Perspectives for research and teaching (pp. 65–86). Dordrecht, The Netherlands: Kluwer.

    Google Scholar 

  • Mason, J., Graham, A., Gower, N., & Pimm, D. (1985). Routes to / roots of algebra. Milton Keynes, UK: Open University Press.

    Google Scholar 

  • Matz, M. (1980). Towards a computational theory of algebraic competence. Journal of Mathematical Behavior, 3, 93–166.

    Google Scholar 

  • Matz, M. (1982). Towards a process model for high school algebra errors. In D. H. Sleeman & J. S. Brown (Eds.), Intelligent tutoring systems (pp. 25–50). New York: Academic Press.

    Google Scholar 

  • McArthur, D., Stasz, C., & Zmunidzinas, M. (1990). Tutoring techniques in algebra. Cognition and Instruction, 7, 197–244.

    Google Scholar 

  • Moses, R. (1995). Algebra, the new civil right. In C. Lacampagne, W. Blair, & J. Kaput (Eds.), The algebra colloquium: Vol.2. Working group papers (pp. 53–67). Washington, DC: U.S. Department of Education.

    Google Scholar 

  • Mulligan, J., & Mitchelmore, M. (1997). Young children’s intuitive models of multiplication and division. Journal for Research in Mathematics Education, 28, 309–330.

    Google Scholar 

  • National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author.

    Google Scholar 

  • National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.

    Google Scholar 

  • National Council of Teachers of Mathematics, & Mathematical Sciences Education Board, National Research Council (Eds.). (1998). The nature and role of algebra in the K-14 curriculum: Proceedings of a national symposium. Washington: National Academy Press.

    Google Scholar 

  • Nemirovsky, R. (1996). A fundamental approach to algebra: Two issues that emerge. In N. Bednarz, C. Kieran, & L. Lee (Eds.), Approaches to algebra (pp. 295–313). Dordrecht, The Netherlands: Kluwer Academic.

    Google Scholar 

  • Pereira-Mendoza, L. (1987). Error patterns and strategies in algebraic simplification. In J. Bergeron, N. Herscovics, & C. Kieran (Eds.), Proceedings of the 11thannual conference of the International Group for the Psychology of Mathematics Education (Vol. I, pp. 331–337). Montreal, Canada: Program Committee.

    Google Scholar 

  • Petitto, A. (1979). The role of formal and non-formal thinking in doing algebra. Journal of Children’s Mathematical Behavior, 2(2), 69–82.

    Google Scholar 

  • Pimm, D. (1987). Speaking mathematically. London: Hodder & Stoughton.

    Google Scholar 

  • Radford, L. (1994). Moving through systems of mathematical knowledge: From algebra with one unknown to algebra with two unknowns. In J. P. da Ponte & J. F. Matos (Eds.), Proceedings of the 18thannual conference of the International Group for the Psychology of Mathematics Education (Vol. IV, pp. 73–80). Lisbon, Portugal: Program Committee.

    Google Scholar 

  • Radford, L. (1995). Before the other unknowns were invented: Didactic inquiries in the methods and problems of medieval Italian algebra. For the Learning of Mathematics, 15(3), 28–38.

    Google Scholar 

  • Schifter, D. (Ed.). (1999). Voicing the new pedagogy: Teachers interpret the rhetoric of mathematics education reform. New York: Teachers College Press.

    Google Scholar 

  • Schliemann, A., Lara-Roth, S., & Goodrow, A. (2001). Tables as multiplicative function tables. In H. Chick, K. Stacey, J. Vincent, & J. Vincent (Eds.), The future of the teaching and learning of algebra (Proceedings of the 12th ICMI Study Conference, pp. 531–540). Melbourne, Australia: The University of Melbourne.

    Google Scholar 

  • Sfard, A. (1989). Transition from operational to structural conception: The notion of function revisited. In G. Vergnaud, J. Rogalski, & M. Artigue (Eds.), Proceedings of the 13thannual conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 151–158). Paris, France: Program Committee.

    Google Scholar 

  • 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, 1–36.

    Article  Google Scholar 

  • Sfard, A. (1995). The development of algebra: Confronting historical and psychological perspectives. In New perspectives on school algebra: Papers and discussions of the ICME-7 algebra working group. Journal of Mathematical Behavior (Special Issue), 14, 15–39.

    Google Scholar 

  • Sleeman, D. (1986). Introductory algebra: A case study of misconceptions. Journal of Mathematical Behavior, 5(1), 25–52.

    Google Scholar 

  • Soloway, E., Lochhead, J., & Clement, J. (1982). Does computer programming enhance problem solving ability? Some positive evidence on algebra word problems. In R. J. Seidel, R. E. Anderson, & B. Hunter (Eds.), Computer literacy (pp. 171–185). New York: Academic Press.

    Google Scholar 

  • Steffe, L. P. (2001). What is algebraic about children’s numerical operating? In H. Chick, K. Stacey, J. Vincent, & J. Vincent (Eds.), The future of the teaching and learning of algebra (Proceedings of the 12th ICMI Study Conference, pp. 556–563). Melbourne, Australia: The University of Melbourne.

    Google Scholar 

  • Sutherland, R. (1989). Providing a computer-based framework for algebraic thinking. Educational Studies in Mathematics, 20, 317–344.

    Article  Google Scholar 

  • Sutherland, R. (1993). Connecting theory and practice: Results from the teaching of LOGO. Educational Studies in Mathematics, 24, 95–113.

    Article  Google Scholar 

  • Sutherland, R., & Rojano, T. (1993). Spreadsheet approach to solving algebra problems. The Journal of Mathematical Behavior, 12(3), 353–383.

    Google Scholar 

  • Sutherland, R., Rojano, T., Bell, A., & Lins, R. (2001). Perspectives on school algebra. Dordrecht, The Netherlands: Kluwer Academic

    Google Scholar 

  • Thompson, P. W. (1993). Quantitative reasoning, complexity, and additive structures. Educational Studies in Mathematics, 25(3), 165–208.

    Article  Google Scholar 

  • Urbanska, A. (1993). On the numerical competence of six-year-old children. Educational Studies in Mathematics, 24, 265–275.

    Google Scholar 

  • Wagner, S. (1981). Conservation of equation and function under transformations of variable. Journal for Research in Mathematics Education, 12, 107–118.

    Google Scholar 

  • Warren, E. (2001). Algebraic understanding: The importance of learning in the early years. In H. Chick, K. Stacey, J. Vincent, & J. Vincent (Eds.), The future of the teaching and learning of algebra (Proceedings of the 12th ICMI Study Conference, pp. 633–640). Melbourne, Australia: The University of Melbourne.

    Google Scholar 

  • Wenger, R. (1987). Some cognitive science perspectives concerning algebra and elementary functions. In A. Schoenfeld (Ed.), Cognitive science and mathematics education (pp. 217–252). Hillsdale, NJ: Lawrence Erlbaum Associates.

    Google Scholar 

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Authors and Affiliations

  1. Mathematics Dept/PGEM, UNESP-Rio Claro, Brazil

    Romulo Lins & James Kaput

  2. Department of Mathematics, UMass-Dartmouth, MA, USA

    Romulo Lins & James Kaput

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  1. Romulo Lins
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  2. James Kaput
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  1. The University of Melbourne, Australia

    Kaye Stacey , Helen Chick  & Margaret Kendal ,  & 

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Lins, R., Kaput, J. (2004). The Early Development of Algebraic Reasoning: The Current State of the Field. In: Stacey, K., Chick, H., Kendal, M. (eds) The Future of the Teaching and Learning of Algebra The 12th ICMI Study. New ICMI Study Series, vol 8. Springer, Dordrecht. https://doi.org/10.1007/1-4020-8131-6_4

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