Advertisement

ZDM

, Volume 45, Issue 3, pp 377–391 | Cite as

Learning beginning algebra in a computer-intensive environment

Original Article

Abstract

We present a design research on learning beginning algebra in an environment where spreadsheets were available at all times but the decision about using them or not, and how, in any particular situation was left to the students. Students’ activity is analyzed in Kieran’s framework of generational, transformational and global/meta-level activity, and compared to the designers’ intentions. We do this by focusing on the activity of one student in four sessions spread over several months and discussing the activity of 51 additional students in view of the analysis of the focus student. We show that the environment enables a number of different entries into algebra and as such supports students in becoming autonomous learners of algebra, and in making the shift from arithmetic to algebra via generational and global/meta-level activity before dealing with the more technical transformational activities.

Keywords

Beginning algebra Computer intensive environment Algebraic generational activity Algebraic transformational activity Global/meta-level activity Instrumental genesis and instrumental orchestration 

References

  1. Ainley, J. (1996). Purposeful contexts for formal notation in a spreadsheet environment. Journal of Mathematical Behavior, 15, 405–422.CrossRefGoogle Scholar
  2. Arcavi, A. (1994). Symbol sense: Informal sense-making in formal mathematics. For the Learning of Mathematics, 14, 24–35.Google Scholar
  3. Artigue, M. (2002). Learning mathematics in a CAS environment: The genesis of a reflection about instrumentation and the dialectics between technical and conceptual work. International Journal of Computers for Mathematical Learning, 7, 245–274.CrossRefGoogle Scholar
  4. Bednarz, N., Kieran, C., & Lee, L. (Eds.). (1996). Approaches to algebra. Perspectives for research and teaching. Dordrecht: Kluwer.Google Scholar
  5. Cobb, P., Stephan, M., McClain, K., & Gravemeijer, K. (2001). Participating in classroom mathematical practices. The Journal of the Learning Sciences, 10, 113–163.CrossRefGoogle Scholar
  6. 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: Kluwer.Google Scholar
  7. Dörfler, W. (2008). En route from patterns to algebra: Comments and reflections. ZDM—The International Journal on Mathematics Education, 40, 143–160.CrossRefGoogle Scholar
  8. Dreyfus, T., Hershkowitz, R., & Schwarz, B. B. (2001). Abstraction in context: The case of peer interaction. Cognitive Science Quarterly—An International Journal of Basic and Applied Research, 1, 307–358.Google Scholar
  9. Drijvers, P., Doorman, M., Boon, P., Reed, H., & Gravemeijer, K. (2010). The teacher and the tool: Instrumental orchestrations in the technology-rich mathematics classroom. Educational Studies in Mathematics, 75, 213–234.CrossRefGoogle Scholar
  10. Drouhard, J. P., & Teppo, A. R. (2004). Symbols and language. In K. Stacey, H. Chick, & M. Kendal (Eds.), The future of the teaching and learning of algebra: The 12th ICMI study (pp. 227–266). Dordrecht: Kluwer.Google Scholar
  11. Eisenmann, T., & Even, R. (2010). Enacted types of algebraic activity in different classes taught by the same teacher. International Journal of Science and Mathematics Education, 9, 867–891.CrossRefGoogle Scholar
  12. Friedlander, A., Hershkowitz, R., & Arcavi, A. (1989). Incipient “algebraic” thinking in pre-algebra students. In Proceedings of the 13th conference of the international group for the psychology of mathematics education (Vol. 1, pp. 283–290). Paris: PME.Google Scholar
  13. Friedlander, A., & Tabach, M. (2001). Developing a curriculum of beginning algebra in a spreadsheet environment. In H. Chick, K. Stacey, & J. Vincent (Eds.), The future of teaching and learning of algebra. Proceedings of the 12th ICMI study conference (Vol. 2, pp. 252–257). Australia: The University of Melbourne.Google Scholar
  14. Gardner, J., Morrison, H., & Jarman, R. (1993). The impact of high access to computers on learning. Journal of Computer Assisted Learning, 9, 2–16.CrossRefGoogle Scholar
  15. Goos, M. (2012). Digital technologies in the Australian curriculum: Mathematics—a lost opportunity? In B. Atweh, M. Goos, R. Jorgensen, & D. Siemon (Eds.), Engaging the Australian National Curriculum: Mathematics—Perspectives from the Field (pp. 135–152). Mathematics Education Research Group of Australasia: Online Publication. http://www.merga.net.au/sites/default/files/editor/books/1/Chapter%207%20Goos.pdf. Accessed 27 Aug 2012.
  16. Guin, D., Ruthven, K., & Trouche, L. (Eds.). (2005). The didactical challenge of symbolic calculators: Turning a computational device into a mathematical instrument. New York: Springer.Google Scholar
  17. Haspekian, M. (2005). An “instrumental approach” to study the integration of a computer tool into mathematics teaching: The case of spreadsheets. International Journal of Computers for Mathematics Learning, 10, 109–141.CrossRefGoogle Scholar
  18. Heid, M. K. (1995). Curriculum and evaluation standards for school mathematics, Addenda series: Algebra in a technological world. Reston, VA: National Council of Teachers of Mathematics.Google Scholar
  19. Hershkowitz, R., & Arcavi, A. (1990). The interplay between student behaviors and the mathematical structure of problem situations—issues and examples. In Proceedings of the 14th conference of the international group for the psychology of mathematics education (Vol. 2, pp. 193–200). Mexico: Oaxtepec.Google Scholar
  20. Hershkowitz, R., Dreyfus, T., Ben-Zvi, D., Friedlander, A., Hadas, N., Resnick, T., et al. (2002). Mathematics curriculum development for computerized environments: A designer–researcher–teacher–learner activity. In L. D. English (Ed.), Handbook of international research in mathematics education (pp. 657–694). Mahwah, NJ: Lawrence Erlbaum.Google Scholar
  21. Hoyles, C., Noss, R., & Kent, P. (2004). On the integration of digital technologies into mathematics classrooms. International Journal of Computers for Mathematical Learning, 9, 309–326.CrossRefGoogle Scholar
  22. Kieran, C. (1992). The learning and teaching of school algebra. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 390–419). Reston, VA: National Council of Teachers of Mathematics.Google Scholar
  23. Kieran, C. (2004). The core of algebra: Reflections on its main activities. In K. Stacey, H. Chick, & M. Kendal (Eds.), The future of teaching and learning of algebra: The 12th ICMI study (pp. 21–34). Dordrecht: Kluwer.CrossRefGoogle Scholar
  24. Knuth, E. J., Stephens, A. C., McNeil, N. M., & Alibali, M. W. (2006). Does understanding the equal sign matter? Evidence from solving equations. Journal for Research in Mathematics Education, 37, 297–311.Google Scholar
  25. Laborde, C. (2003). Technology used as a tool for mediating knowledge in the teaching of mathematics: The case of Cabri-geometry. In W. C. Yang, S. C. Chu, T. de Alwis, & M. G. Lee (Eds.), Proceedings of the 8th Asian technology conference in mathematics (Vol. 1, pp. 23–38). Hsinchu: Chung Hua University.Google Scholar
  26. Lagrange, J. B., Artigue, M., Laborde, C., & Trouche, L. (2003). Technology and mathematics education: A multidimensional study of the evolution of research and innovation. In A. J. Bishop, M. A. Clements, C. Keitel, J. Kilpatrick, & F. K. S. Leung (Eds.), Second international handbook of mathematics education (pp. 239–271). Dordrecht: Kluwer.Google Scholar
  27. Malle, G. (1993). Didaktische Probleme der elementaren Algebra. Wiesbaden: Vieweg.Google Scholar
  28. Mariotti, M. A. (2002). Technological advances in mathematics learning. In L. D. English (Ed.), Handbook of international research in mathematics education (pp. 695–723). Mahwah, NJ: Lawrence Erlbaum.Google Scholar
  29. Mason, J., & Pimm, D. (1984). Generic examples: Seeing the general in the particular. Educational Studies in Mathematics, 15, 277–289.CrossRefGoogle Scholar
  30. Radford, L. (2000). Signs and meanings in students’ emergent algebraic thinking: A semiotic analysis. Educational Studies in Mathematics, 42, 237–268.CrossRefGoogle Scholar
  31. Rockman et al. (1999). A more complex picture: Laptop use and impact in the context of changing home and school access. http://www.rockman.com/projects/projectDetail.php?id=126. Accessed 27 Aug 2012.
  32. Rojano, T. (2002). Mathematics learning in the junior secondary school: Students’ access to significant mathematical ideas. In L. D. English (Ed.), Handbook of international research in mathematics education (pp. 143–164). Mahwah, NJ: Lawrence Erlbaum.Google Scholar
  33. Shternberg, B., & Yerushalmy, M. (2003). Models of functions and models of situations: On the design of modeling-based learning environments. In H. M. Doerr & R. Lesh (Eds.), Beyond constructivism: A model and modeling perspective on teaching, learning, and problem solving in mathematics education (pp. 479–498). Mahwah, NJ: Lawrence Erlbaum.Google Scholar
  34. Simon, M. (1995). Reconstructing mathematics pedagogy from a constructivist perspective. Journal for Research in Mathematics Education, 26, 114–145.CrossRefGoogle Scholar
  35. Stacey, K., & MacGregor, M. (2001). Curriculum reform and approaches to algebra. In R. Sutherland, T. Rojano, A. Bell, & R. Lins (Eds.), Perspectives on school algebra (pp. 141–153). Dordrecht: Kluwer.Google Scholar
  36. Star, J. R. (2007). Foregrounding procedural knowledge. Journal for Research in Mathematics Education, 38, 132–135.Google Scholar
  37. Sutherland, R., & Balacheff, N. (1999). Didactical complexity of computational environments for the learning of mathematics. International Journal of Computers for Mathematics Learning, 4, 1–26.CrossRefGoogle Scholar
  38. Sutherland, R., & Rojano, T. (1993). A spreadsheet approach to solving algebra problems. Journal of Mathematical Behavior, 12, 353–383.Google Scholar
  39. Tabach, M. (2011a). The dual role of researcher and teacher: A case study. For the Learning of Mathematics, 31, 32–34.Google Scholar
  40. Tabach, M. (2011b). Symbolic generalization in a computer intensive environment: The case of Amy. In CERME 7—seventh conference of european research in mathematics education. http://www.cerme7.univ.rzeszow.pl/WG/15b/CERME7-WG15B-Paper01_Tabach.pdf. Accessed 27 Aug 2012.
  41. Tabach, M. (2011c). A mathematics teacher’s practice in a technological environment: A case study analysis using two complementary theories. Technology, Knowledge and Learning, 16, 247–265.Google Scholar
  42. Tabach, M., Arcavi, A., & Hershkowitz, R. (2008a). Transitions among different symbolic generalizations by algebra beginners in a computer intensive environment. Educational Studies in Mathematics, 69, 53–71.CrossRefGoogle Scholar
  43. Tabach, M., & Friedlander, A. (2004). Levels of student responses in a spreadsheet-based environment. In M. J. Høines, & A. B. Fuglestad (Eds.), Proceedings of the 28th international conference for the psychology of mathematics education (Vol. 2, pp. 423–430). Bergen: PME.Google Scholar
  44. Tabach, M., & Friedlander, A. (2008). Understanding equivalence of algebraic expressions in a spreadsheet-based environment. International Journal of Computers in Mathematics Education, 13, 27–46.Google Scholar
  45. Tabach, M., Hershkowitz, R., Arcavi, A., & Dreyfus, T. (2008b). Computerized environments in mathematics classrooms: A research-design view. In L. D. English, M. Bartolini Bussi, G. A. Jones, R. A. Lesh, B. Sriraman, & D. Tirosh (Eds.), Handbook of international research in mathematics education (2nd ed., pp. 784–805). New York: Routledge.Google Scholar
  46. Tabach, M., Hershkowitz, R., & Schwarz, B. B. (2006). Constructing and consolidating of algebraic knowledge within dyadic processes: A case study. Educational Studies in Mathematics, 63, 235–258.CrossRefGoogle Scholar
  47. Trouche, L. (2004). Managing complexity of human/machine interactions in computerized learning environments: Guiding students’ command process through instrumental orchestrations. International Journal of Computers for Mathematical Learning, 9, 281–307.CrossRefGoogle Scholar
  48. Vérillon, P., & Rabardel, P. (1995). Cognition and artifact: A contribution to the study of thought in relation to instrumented activity. European Journal of Psychology in Education, 9, 77–101.CrossRefGoogle Scholar
  49. Wilson, K., Ainley, J., & Bills, L. (2005). Naming a column on a spreadsheet: Is it more algebraic? In D. Hewitt, & A. Noyes (Eds.), Proceedings of the sixth British congress of mathematics education (pp. 184–191). Warwick, UK.Google Scholar
  50. Yackel, E., & Cobb, P. (1996). Socio-mathematical norms, argumentation, and autonomy in mathematics. Journal for Research in Mathematics Education, 27, 458–477.CrossRefGoogle Scholar
  51. Yerushalmy, M. (2005). Challenging known transitions: Learning and teaching algebra with technology. For the Learning of Mathematics, 2, 37–42.Google Scholar
  52. Yerushalmy, M. (2009). Technology-based algebra learning, epistemological discontinuities and curricular implications. In B. B. Schwarz, T. Dreyfus, & R. Hershkowitz (Eds.), Transformation of knowledge through classroom interaction (pp. 56–64). London: Routledge.Google Scholar

Copyright information

© FIZ Karlsruhe 2012

Authors and Affiliations

  • Michal Tabach
    • 1
  • Rina Hershkowitz
    • 2
  • Tommy Dreyfus
    • 1
  1. 1.Tel Aviv UniversityTel AvivIsrael
  2. 2.Weizmann Institute of ScienceRehovotIsrael

Personalised recommendations