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ZDM

, Volume 49, Issue 5, pp 785–798 | Cite as

Learning extrema problems using a non-differential approach in a digital dynamic environment: the case of high-track yet low-achievers

  • Assaf Dvir
  • Michal Tabach
Original Article
First Online:

Abstract

High schools commonly use a differential approach to teach minima and maxima geometric problems. Although calculus serves as a systematic and powerful technique, this rigorous instrument might hinder students’ ability to understand the behavior and constraints of the objective function. The proliferation of digital environments allowed us to adopt a different approach involving geometry analysis combined with the use of the inequality of arithmetic and geometric means. The advantages of this approach are enhanced when it is integrated with dynamic e-resources tailored by the instructor. The current study adopts the abstraction in context framework to trace students’ knowledge construction processes while solving extremum problems in an e-resource GeoGebra-based environment using a non-differential approach. We closely monitored the learning of 5 pairs of high-track yet low achieving 17-year-old students for several lessons. We further assessed the students’ understanding at the end of the learning unit based on their explanations of extrema problems. Our findings allowed us to pinpoint the contributions (and pitfalls) of the e-resources for student learning at the micro level. In addition, the students demonstrated the ability to solve extrema problems and were able to explain their reasoning in ways that reflect the e-resources with which they worked.

Keywords

Knowledge construction Dynamic e-resource Non-differential math Creative reasoning Abstraction thinking 
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References

  1. Adler, J. (2000). Conceptualising resources as a theme for mathematics teacher education. Journal of Mathematics Teacher Education, 3(3), 205–224.CrossRefGoogle Scholar
  2. Arcavi, A. (2003). The role of visual representations in the learning of mathematics. Education Studies in Mathematics, 52(3), 215–241.CrossRefGoogle Scholar
  3. Berry, J., & Nyman, M. (2003). Promoting students’ graphical understanding of the calculus. Journal of Mathematical Behavior, 22(4), 481–497.CrossRefGoogle Scholar
  4. Birnbaum, I. (1982). Maxima and minima without calculus. Mathematics in School, 11(3), 8–9.Google Scholar
  5. Cullen, C., Hertel, J. T., & John, S. (2013). Investigating extrema with GeoGebra. Mathematics Teacher, 107(1), 68–72.CrossRefGoogle Scholar
  6. Erbas, A. K., & Yenmez, A. A. (2011). The effect of inquiry-based explorations in a dynamic geometry environment on sixth grade students’ achievements in polygons. Computers & Education, 57(4), 2462–2475.CrossRefGoogle Scholar
  7. Freedman, W. (2012). Minimizing areas and volumes and a generalized AM-GM inequality. Mathematics Magazine, 85(2), 116–123.CrossRefGoogle Scholar
  8. Granberg, C., & Olsson, J. (2015). ICT-supported problem solving and collaborative creative reasoning: Exploring linear functions using dynamic mathematics software. The Journal of Mathematical Behaviour, 37, 48–62.CrossRefGoogle Scholar
  9. Hershkowitz, R., Schwarz, B. B., & Dreyfus, T. (2001). Abstraction in context: epistemic actions. Journal for Research in Mathematics Education, 32(2), 195–222.CrossRefGoogle Scholar
  10. Hughes-Hallett, D., Gleason, A., Flath, D., Gordon, S., Lomen, D., Lovelock, D., McCallum, N., Osgood, B., Pasquale, A., Tecosky-Feldman, J., Thrash, J., & Thrash, K. (1994). Calculus. New York: Wiley.Google Scholar
  11. Israel Ministry of Education, High School Mathematics Curriculum for 5 units starting 2001.Google Scholar
  12. Kidron, I., & Dreyfus, T. (2010). Interacting parallel constructions of knowledge in a CAS context. International Journal of Computers for Mathematical Learning, 15(2), 129–149.CrossRefGoogle Scholar
  13. Klamkin, M. S. (1992). On two classes of extremum problems without calculus. Mathematics Magazine, 65(2), 113–117.CrossRefGoogle Scholar
  14. Kouropatov, A., & Dreyfus, T. (2014). Learning the integral concept by constructing knowledge about accumulation. ZDM–The International Journal of Mathematics Education, 46(4), 533–548.CrossRefGoogle Scholar
  15. Lithner, J. (2008). A research framework for creative and imitative reasoning. Educational Studies in Mathematics, 3(67), 255–276.CrossRefGoogle Scholar
  16. Metaxas, N. (2007). Difficulties in understanding the indefinite integral. In J. H. Woo, H. C. Lew, K. S. Park & D. Y. Seo (Eds.), Proceedings of the 31st conference of the international group for the psychology of mathematics education, (Vol. 3, pp. 265–272). Seoul.Google Scholar
  17. National Council of Teachers of Mathematics (NCTM) (2000). Principles and standards for school mathematics. Reston: NCTM.Google Scholar
  18. Niven, I. (1981). Maxima and minima without calculus. The Mathematical Association of America , the Dolciani mathematical expositions, No. 6.Google Scholar
  19. Pea, R. (1987). Cognitive technologies for mathematics education. In A. H. Schoenfeld (Ed.), Cognitive science and mathematics education (pp. 89–122). Hillsdale: Erlbaum.Google Scholar
  20. Schwarz, B. B., Dreyfus, T., & Hershkowitz, R. (2009). The nested epistemic actions model for abstraction in context. In B. B. Schwarz, T. Dreyfus & R. Hershkowitz (Eds.), Transformation of knowledge through classroom interaction (pp. 9–41). London: Routledge.Google Scholar
  21. Skemp, R. R. (1987). The psychology of learning mathematics. Hillsdale: Erlbaum.Google Scholar
  22. Stahl, G., Rosé, C. P., & Goggins, S. (2011). Analyzing the discourse of GeoGebra collaborations. Essays in Computer-Supported Collaborative Learning, 186, 33–40.Google Scholar
  23. Swidan, O., & Yerushalmy, M. (2014). Learning the indefinite integral in a dynamic and interactive technological environment. ZDM–The International Journal of Mathematics Education, 46(4), 517–531.CrossRefGoogle Scholar
  24. Thomas, M. O., Yoon, C., & Dreyfus, T. (2009). Multimodal use of semiotic resources in the construction of antiderivative. In R. Hunter, B. Bicknell & T. Burgess (Eds.), Crossing divides. Proceedings of the 32nd annual conference of the mathematics education research group of Australasia (Vol. 2). Palmerston North: MERGA.Google Scholar
  25. Thompson, P. W. (1994). Images of rate and operational understanding of the Fundamental Theorem of Calculus. Educational Studies in Mathematics, 26, 229–274.CrossRefGoogle Scholar
  26. Tikhomirov, V. (1991). Stories about maxima and minima. American Mathematical Society, Mathematical Association of America. Mathematical world. ISSN 1055–9426, 1.Google Scholar
  27. Trgalova, J., Clark-Wilson, A., & Weigand, H. G. (2017). Technology and resources in mathematics education. In T. Dreyfus, M. Artigue, D. Potari, S. Prediger, & K. Ruthven (Eds.), Developing research in mathematics education—twenty years of communication, cooperation and collaboration in Europe. London: Routledge.Google Scholar
  28. Yerushalmy, M. (2006). Slower algebra students meet faster tools: solving algebra word problems with graphic software. Journal for Research in Mathematics Education, 37(5), 356–387.Google Scholar
  29. Yerushalmy, M. (2009). Epistemological discontinuities and cognitive hierarchies in technology-based Algebra learning. 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 2017

Authors and Affiliations

  1. 1.Tel Aviv UniversityTel AvivIsrael

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