Educational Studies in Mathematics

, Volume 54, Issue 1, pp 63–75 | Cite as

Focusing on informal strategies when linking arithmetic to early algebra

  • Barbara A. Van Amerom


In early algebra students often struggle with equation solving. Modeled on Streefland's studies of students' own productions a prototype pre-algebra learning strand was designed which takes students' informal (arithmetical) strategies as a starting point for solving equations. In order to make available the skills and tools needed for manipulating equations, the students are stimulated and guided to develop suitable algebraic language, notations and reasoning. One of the results of the study is that reasoning and symbolizing appear to develop as independent capabilities. For instance,students in grades 6 and 7 can solve equations at both a formal and an informal level, but formal symbolizing has been found to be a major obstacle.

arithmetic early algebra equations history of mathematics reasoning symbolizing 


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  1. Bednarz, N. and Janvier B.: 1996, 'Emergence and development of algebra as a poblemsolving tool: Continuities and discontinuities with arithmetic', in N. Bednarz. C. Kieran, and L. Lee(eds.), Approaches to Algebra, Kluwer Academic Publishers, Dordrecht, The Netherlands, pp. 115–136.Google Scholar
  2. Booth, L.R.: 1988, 'Children's difficulties in beginning algebra', in A.F. Coxford (ed.), The ideas of algebra, K-12 (1988 NCTM Yearbook), National Council of Teachers of Mathematics, Reston, VA, pp. 20–32.Google Scholar
  3. Da Rocha Falcão, J.: 1995, 'A case study of algebraic scaffolding: From balance scale to algebraic notation', in L. Meira and D. Carraher (eds.), Proceedings of the 19th International Conference for the Psychology of Mathematics Education, Vol. 2, Universidade Federal de Pernambuco, Recife, Brazil, pp. 66–73.Google Scholar
  4. Fauvel, J. and Van Maanen, J. (eds.): 2000, History in Mathematics Education: The ICMI Study, Kluwer Academic Publishers, Dordrecht, The Netherlands.Google Scholar
  5. Filloy, E. and Rojano, T.: 1989, 'Solving equations: The transition from arithmetic to algebra', For the Learning of Mathematics 9(2), 19–25.Google Scholar
  6. Freudenthal, H.: 1962, 'Logical analysis and critical study', in H. Freudenthal (ed.), Report on the Relations between Arithmetic and Algebra, Nederlandse Onderwijscommissie voor Wiskunde, Groningen, The Netherlands, pp. 20–41.Google Scholar
  7. Herscovics, N. and Linchevski, L.: 1994, 'A cognitive gap between arithmetic and algebra', Educational Studies in Mathematics 27(1), 59–78.CrossRefGoogle Scholar
  8. Kieran, C.: 1989, 'The early learning of algebra: A structural perspective', in S. Wagner and C. Kieran (eds.), Research Issues in the Learning and Teaching of Algebra, National Council of Teachers of Mathematics, Reston, VA, pp. 33–56.Google Scholar
  9. Kieran, C.: 1992, 'The learning and teaching of school algebra', in D. Grouws (ed.), Handbook of research on mathematics teaching and learning, MacMillan Publishing Company, New York, pp. 390–419.Google Scholar
  10. Linchevski, L. and 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, 39–65.CrossRefGoogle Scholar
  11. Mason, J.: 1996, 'Expressing Generality and Roots of Algebra', in N. Bednarz, C. Kieran and L. Lee (eds.), Approaches to Algebra, Kluwer Academic Publishers, Dordrecht, The Netherlands, pp. 65–111.Google Scholar
  12. Radford, L.: 1997, 'On psychology, historical epistomology, and the teaching of Mathematics: towards a socio-cultural history of mathematics', For the Learning of Mathematics 17(1), 26–33.Google Scholar
  13. 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 21, 1–36. FROM ARITHMETIC TO EARLY ALGEBRA 75CrossRefGoogle Scholar
  14. Sfard, A.: 1995, 'The development of algebra: confronting historical and psychological perspectives', Journal of Mathematical Behavior 14, 15–39.CrossRefGoogle Scholar
  15. Streefland, L.: 1995, 'Zelf algebra maken [Making algebra yourself]', Nieuwe Wiskrant 15(1), 33–37.Google Scholar
  16. Streefland, L.: 1996, Learning from History for Teaching in the Future, Freudenthal institute, Utrecht, The Netherlands. [see this Special Issue of Educational Studies in Mathematics]Google Scholar
  17. Streefland, L. and Van Amerom, A.: 1996, 'Didactical phenomenology of equations', in J. Giménez, R. Campos Lins and B. Gómez (eds.), Arithmetics and Algebra Education: Searching for the Future, Computer Engineering Department, Universitat Rovira i Virgili, Tarragona, pp. 120–131.Google Scholar
  18. Struik, D.J.: 1990, Geschiedenis van de wiskunde [History ofMathematics], Het Spectrum, Utrecht, The Netherlands.Google Scholar
  19. Usiskin, Z.: 1988, 'Conceptions of school algebra and uses of variables', in A. Coxford (ed.), The Ideas of Algebra, K-12, National Council of Teachers ofMathematics, Reston, VA, pp. 8–19.Google Scholar
  20. Van Amerom, B.A.: 2002, Reinvention of Early Algebra, CDβ-Press/Freudenthal Institute, Utrecht, The Netherlands. Freudenthal Institute, Utrecht University, Aïdadreef 12, 3561 GE Utrecht, The Netherlands Telephone +31 (0)302635555, Fax +31 (0)302660430, E-mail: b.vanamerom@fi.uu.nlGoogle Scholar

Copyright information

© Kluwer Academic Publishers 2003

Authors and Affiliations

  • Barbara A. Van Amerom
    • 1
  1. 1.Freudenthal InstituteUtrecht UniversityUtrechtThe Netherlands

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