Technology, Knowledge and Learning

, Volume 19, Issue 3, pp 255–286 | Cite as

Investigating the Potential of Computer Environments for the Teaching and Learning of Functions: A Double Analysis from Two Research Traditions

  • Jean-Baptiste Lagrange
  • Giorgos Psycharis


The general goal of this paper is to explore the potential of computer environments for the teaching and learning of functions. To address this, different theoretical frameworks and corresponding research traditions are available. In this study, we aim to network different frameworks by following a ‘double analysis’ method to analyse two empirical studies based on the use of computational environments offering integrated geometrical and algebraic representations. The studies took place in different national and didactic contexts and constitute cases of Constructionism and Theory of Didactical Situations. The analysis indicates that ‘double analysis’ resulted in a deepened and more balanced understanding about knowledge emerging from empirical studies as regards the nature of learning situations for functions with computers and the process of conceptualisation of functions by students. Main issues around the potential of computer environments for the teaching and learning of functions concern the use of integrated representations of functions linking geometry and algebra, the need to address epistemological and cognitive aspects of the constructed knowledge and the critical role of teachers in the design and evolution of students’ activity. We also reflect on how the networking of theories influences theoretical advancement and the followed research approaches.


Digital technology Research tradition Networking of theoretical frameworks Functions Integrated representations Double analysis 


  1. Artigue, M. (1997). Le logiciel DERIVE comme révélateur de phénomènes didactiques liés à l’utilisation d’environnements informatiques pour l’apprentissage. Educational Studies in Mathematics, 33, 133–169.CrossRefGoogle Scholar
  2. Artigue, M. (Ed.) (2009). Integrated theoretical framework (Version C). ‘ReMath’ (Representing Mathematics with Digital Media) FP6, IST-4 026751 (2005–2009). Del. 18. Retrieved September 21, 2012, from
  3. Artigue, M. & Bardini, C. (2010). New didactical phenomena prompted by TI-nspire specificities—the mathematical component of the instrumentation process. In V. Durand-Guerrier, S. Soury-Lavergne, F. Arzarello (Eds.), Proceedings of the 6th Congress of the European Society for Research in Mathematics Education (pp. 19–28). Lyon, France.Google Scholar
  4. Bikner-Ahsbahs, A., & Prediger, S. (2010). Networking of theories—an approach for exploiting the diversity of theoretical approaches. In B. Sriraman & L. English (Eds.), Theories of mathematics education: Seeking new frontiers (pp. 483–506). New York: Springer.CrossRefGoogle Scholar
  5. Bloch, I. (2003). Teaching functions in a graphic milieu: what forms of knowledge enable students to conjecture and prove? Educational Studies in Mathematics, 52, 3–28.CrossRefGoogle Scholar
  6. Breidenbach, D., Dubinsky, E., Hawks, J., & Nichols, D. (1992). Development of the process conception of function. Educational Studies in Mathematics, 23, 247–285.CrossRefGoogle Scholar
  7. Brousseau, G. (1997). The theory of didactic situations in mathematics. Dordrecht: Kluwer Academic Publishers.Google Scholar
  8. Carlson, M., Jacobs, S., Coe, E., Larsen, S., & Hsu, E. (2002). Applying covariational reasoning while modeling dynamic events: A framework and a study. Journal for Research in Mathematics Education, 33(5), 352–378.CrossRefGoogle Scholar
  9. Confrey, J., & Smith, E. (1995). Splitting, covariation, and their role in the development of exponential functions. Journal for Research in Mathematics Education, 26, 66–86.CrossRefGoogle Scholar
  10. Doorman, M., Drijvers, P., Gravemeijer, K., Boon, P., & Reed, H. (2012). Tool use and the development of function concept: From repeated calculations to functional thinking. International Journal of Science and Mathematics Education, 10, 1243–1276.CrossRefGoogle Scholar
  11. Drijvers, P., Godino, J. D., Font, V., & Trouche, L. (2013). One episode, two lenses. A reflective analysis of student learning with computer algebra from instrumental and onto-semiotic perspectives. Educational Studies in Mathematics, 82, 23–49.CrossRefGoogle Scholar
  12. Dubinsky, E. (1999). One theoretical perspective in undergraduate mathematics education research. In O. Zaslavsky (Ed.), Proceedings of the 23rd Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 65–73). Haifa, Israel.Google Scholar
  13. Duval, R. (2000). Basic issues for research in mathematics education. In T. Nakahara, M. Koyama (Eds.), Proceedings of the 24th Conference of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 55–69). Hiroshima University.Google Scholar
  14. Falcade, R., Laborde, C., & Mariotti, M. A. (2007). Approaching functions: Cabri tools as instruments of semiotic mediation. Educational Studies in Mathematics, 66(3), 317–333.CrossRefGoogle Scholar
  15. Harel, I., & Papert, S. (Eds.). (1991). Constructionism: Research reports and essays. Norwood, NJ: Ablex Publishing Corporation.Google Scholar
  16. Hazzan, O., & Goldenberg, E. P. (1997). Student’s understanding of the notion of function. International Journal of Computers for Mathematical Learning, 1(3), 263–290.CrossRefGoogle Scholar
  17. Hoffkamp, A. (2010). Enhancing functional thinking using the computer for representational transfer. In V. Durand-Guerrier, S. Soury-Lavergne, F. Arzarello (Eds.), Proceedings of the 6th Congress of the European Society for Research in Mathematics Education (pp. 1201–1210). Lyon, France.Google Scholar
  18. Hohenwarter, M., & Fuchs, K. (2005). Combination of dynamic geometry, algebra and calculus in the software system GeoGebra. In C. Sárvári (Ed.), Proceedings of Computer Algebra Systems and Dynamic Geometry Systems in Mathematics Teaching (pp. 128–133). Pecs, Hungary: University of Pecs.Google Scholar
  19. Hoyles, C., Lagrange, J.-B., & Noss, R. (2006). Developing and evaluating alternative technological infrastructures for learning mathematics. In J. Maasz & W. Schloeglmann (Eds.), New mathematics education research and practice (pp. 263–312). Rotterdam: Sense Publishers.Google Scholar
  20. 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). New York: Macmillan.Google Scholar
  21. Kieran, C. (2007). Learning and teaching algebra at the middle school through college levels. In F. K. Lester Jr. (Ed.), Second handbook of research on mathematics teaching and learning (pp. 707–762). Charlotte, NC: Information Age Publishing; Reston, VA: NCTM.Google Scholar
  22. Kieran, C., & Drijvers, P. (2006). The co-emergence of machine techniques paper-and pencil techniques, and theoretical reflection: a study of CAS use in secondary school algebra. International Journal of Computers for Mathematical Learning, 11, 205–263.CrossRefGoogle Scholar
  23. Kieran, C., & Yerushalmy, M. (2004). Research on the role of technological environments in algebra learning and teaching. In K. Stacey, H. Shick, & M. Kendal (Eds.), The future of the teaching and learning of algebra. The 12th ICMI (International Commission on Mathematical Instruction) Study. New ICMI Study Series (Vol. Vol. 8, pp. 99–152)., Kieran, C Dordrecht: Kluwer Academic Publishers.Google Scholar
  24. Kynigos, C. (2004). A ‘‘black-and-white box’’ approach to user empowerment with component computing. Interactive Learning Environments, 12(1–2), 27–71.CrossRefGoogle Scholar
  25. Kynigos, C. (2012). Constructionism: theory of learning or theory of design? Invited Regular Lecture to the 12th International Congress on Mathematical Education. COEX, Seoul, Korea.Google Scholar
  26. Lagrange, J.-B. (2005). Curriculum, classroom practices, and tool design in the learning of functions through technology-aided experimental approaches. International Journal of Computers for Mathematical Learning, 10, 143–189.CrossRefGoogle Scholar
  27. Lagrange, J.-B., Artigue, M. (2009). Students’ activities about functions at upper secondary level: a grid for designing a digital environment and analysing uses. In M. Tzekaki, M. Kaldrimidou, & C. Sakonidis (Eds.), Proceedings of the 33rd Conference of the International Group for the Psychology of Mathematics Education (pp. 465–472). Thessaloniki, Greece: PME.Google Scholar
  28. Mackrell, K. (2011). Introducing algebra with interactive geometry software. In W.-C. Yang, M. Majewski, T. de Alwis, E. Karakirk, (Eds.), Electronic proceedings of the 16th Asian Technology Conference in Mathematics, (pp. 211–220). Mathematics and Technology, LLC.Google Scholar
  29. Noss, R. (2004). Designing a learnable mathematics: a fundamental role for the computer? Regular Lecture to the 10th International Congress of Mathematics Education (ICMI 10). Roskilde, Denmark: Roskilde University.Google Scholar
  30. Noss, R., Healy, L., & Hoyles, C. (1997). The construction of mathematical meanings: Connecting the visual with the symbolic. Educational Studies in Mathematics, 33(2), 203–233.CrossRefGoogle Scholar
  31. Noss, R., & Hoyles, C. (1996). Windows on mathematical meanings. New York: Kluwer Academic Press.CrossRefGoogle Scholar
  32. Oehrtman, M., Carlson, M., & Thompson, P. (2008). Foundational reasoning abilities that promote coherence in students’ function understanding. In M. P. Carlson & C. Rasmussen (Eds.), Making the connection: Research and teaching in undergraduate mathematics (pp. 27–42). Washington DC: Mathematical Association of America.CrossRefGoogle Scholar
  33. Papert, S. (1980). Mindstorms. Children, computers and powerful ideas. N. Y.: Basic Books.Google Scholar
  34. Prediger, S., Bikner-Ahsbahs, A., & Arzarello, F. (2008). Networking strategies and methods for connecting theoretical approaches: first steps towards a conceptual framework. Zentralblatt für Didaktik der Mathematik, 40, 165–178.CrossRefGoogle Scholar
  35. Radford, L. (2005). The semiotics of the schema. Kant, Piaget, and the calculator. In M. H. G. Hoffmann, J. Lenhard, & F. Seeger (Eds.), Activity and sign—Grounding mathematics education. Festschrift for Michael Otte (pp. 137–152). New York: Springer.Google Scholar
  36. Radford, L. (2008). Connecting theories in mathematics education: Challenges and possibilities. Zentralblatt für Didaktik der Mathematik, 40, 317–327.CrossRefGoogle Scholar
  37. 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.CrossRefGoogle Scholar
  38. Slavit, D. (1997). An alternate route to the reification of function. Educational Studies in Mathematics, 332, 259–281.Google Scholar
  39. Tall, D. (1996). Functions and calculus. In A. Bishop, K. Clements, C. Keitel, J. Kilpatrick, & C. Laborde (Eds.), International handbook of mathematics education (pp. 289–325). Dordrecht: Kluwer Academic Publishers.Google Scholar
  40. Tall, D., & Vinner, S. (1981). Concept image and concept definition in mathematics, with special reference to limits and continuity. Educational Studies in Mathematics, 12, 151–169.CrossRefGoogle Scholar
  41. Thomas, M. O. J., Monaghan, J., & Pierce, R. (2004). Computer algebra systems and algebra: Curriculum, assessment, teaching, and learning. In K. Stacey, H. Chick, & M. Kendal (Eds.), The teaching and learning of algebra: The 12th ICMI study (pp. 155–186). Norwood, MA: Kluwer Academic Publishers.Google Scholar
  42. Thompson, P. W. (1994). Students, functions, and the undergraduate curriculum. In E. Dubinsky, A. H. Schoenfeld, & J. J. Kaput (Eds.), Research in collegiate mathematics education, I: Issues in mathematics education (Vol. 4, pp. 21–44). Providence, RI: American Mathematical Society.Google Scholar
  43. Van der Kooij, H. (2010). Mathematics at work. In M. M. F. Pinto & T. F. Kawasaki (Eds.), Proceedings of the 34th Conference of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 21–124). Belo Horizonte, Brazil: PME.Google Scholar
  44. Weigand, H. G., & Bichler, E. (2010). Towards a competence model for the use of symbolic calculators in mathematics lessons: the case of functions. Zentralblatt für Didaktik der Mathematik, 42, 697–713.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.University of Reims and LDAR University Paris DiderotParisFrance
  2. 2.Department of MathematicsNational and Kapodistrian University of AthensAthensGreece

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