Advertisement

Innovative Uses of Digital Technology in Undergraduate Mathematics

  • Mike O. J. ThomasEmail author
  • Ye Yoon Hong
  • Greg Oates
Chapter
Part of the Mathematics Education in the Digital Era book series (MEDE, volume 9)

Abstract

The ways in which digital technology is often used in university teaching of mathematics can be quite different from how it is employed in schools. This has the potential to form a discontinuity between school and university, making the transition less than smooth for students. In this chapter we consider several examples of how digital technology has been used with first year mathematics students in both New Zealand and South Korea. The approaches employed include: intensive use of technology, including formative and summative assessment practice; lecturer modelling and privileging of technology use; a versatile approach to calculus concepts that encourages epistemic exploration of local properties of functions; and novel orchestration of mathematical thinking through smartphone communication technology. We analyse each of these approaches using the theory of instrumental orchestration and outline some innovative aspects and benefits of them. The student perspective is also considered, with some evidence of the influence on student engagement and attitudes. We conclude by suggesting that in order to teach with digital technology in the manner described here good pedagogical technology knowledge (PTK) is required.

Keywords

Instrumental orchestration Pedagogical technology knowledge Versatility Digital technology Undergraduate 

References

  1. Artigue, M. (2002). Learning mathematics in a CAS environment: The genesis of a reflection about instrumentalisation and the dialectics between technical and conceptual work. International Journal of Computers for Mathematical Learning, 7, 245–274.CrossRefGoogle Scholar
  2. Artigue, M. (2010). The school university interface. PowerPoint of a seminar given at Auckland University, April 2010.Google Scholar
  3. Bookman, J., & Friedman, C. P. (1999). The evaluation of Project Calc at Duke University 1989–1994. In B. Gold, S. Keith, & W. Marion (Eds.), Assessment practices in undergraduate mathematics (pp. 253–256). Washington DC: The Mathematical Association of America.Google Scholar
  4. Blyth, B., & Labovic, A. (2009). Using Maple to implement eLearning integrated with computer aided assessment. International Journal of Mathematical Education in Science and Technology, 40(7), 975–988.CrossRefGoogle Scholar
  5. Bressoud, D., Mesa, V., & Rasmussen, C. (Eds.). (2015). Insights and recommendations from the MAA national study of College Calculus. Washington DC: MAA Press.Google Scholar
  6. Burrill, G., Allison, J., Breaux, G., Kastberg, S., Leatham, K., & Sanchez, W. (2002). Handheld graphing technology in secondary mathematics: Research findings and implications for classroom practice, Michigan State University: Texas Instruments. Available from: http://education.ti.com/sites/UK/downloads/pdf/References/Done/Burrill,G.%2520%282002%29.pdf
  7. Cheung, A., & Slavin, R. E. (2011). The effectiveness of education technology for enhancing reading achievement: A meta-analysis. Retrieved on February, 8th, 2013, from http://www.bestevidence.org/reading/tech/tech.html
  8. Clark-Wilson, A., Sinclair, N., & Robutti, O. (Eds.). (2013). The mathematics teacher in the digital era. Dordrecht: Springer.Google Scholar
  9. Cobb, P., Confrey, J., diSessa, A., Lehrer, R., & Schauble, L. (2003). Design experiments in educational research. Educational Researcher, 32(1), 9–13.CrossRefGoogle Scholar
  10. Coffland, A., & Xie, Y. (2015). The 21st century mathematics curriculum: A Technology enhanced experience. In X. Ge et al. (Eds.), Emerging technologies for STEAM education, Educational communications and technology: Issues and innovations (pp. 311–329). Zurich: Springer. doi: 10.1007/978-3-319-02573-5_17
  11. 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(2), 213–234.CrossRefGoogle Scholar
  12. Drijvers, P., Tacoma, S., Besamusca, A., Doorman, M., & Boon, P. (2013). Digital resources inviting changes in mid-adopting teachers’ practices and orchestrations. ZDM: The International Journal on Mathematics Education, 45(7), 987–1001.Google Scholar
  13. Ellington, A. J. (2003). A meta-analysis of the effects of calculators on students’ achievement and attitude levels in precollege mathematics classes. Journal for Research in Mathematics Education, 34(5), 433–463.CrossRefGoogle Scholar
  14. Fernández-Plaza, J. A., Rico, L., & Ruiz-Hidalgo, J. F. (2013). Concept of finite limit of a function at a point: Meanings and specific terms. International Journal of Mathematical Education in Science & Technology, 44(5), 699–710.CrossRefGoogle Scholar
  15. Genossar, S.; Botzer, G., & Yerushalmy, M. (2008). Learning with mobile technology: A case study with students in mathematics education. Proceedings of the CHAIS conference, Open University. Available from: http://telem-pub.openu.ac.il/users/chais/2008/evening/3_2.pdf
  16. Goos, M., Galbraith, P., Renshaw, P., & Geiger, V. (2003). Perspectives on technology mediated learning in secondary school mathematics classrooms. Journal of Mathematical Behavior, 22(1), 73–89.CrossRefGoogle Scholar
  17. Graham, A. T., & Thomas, M. O. J. (2000). Building a versatile understanding of algebraic variables with a graphic calculator. Educational studies in mathematics, 41(3), 265–282.Google Scholar
  18. Hong, Y. Y., & Thomas, M. O. J. (2006). Factors influencing teacher integration of graphic calculators in teaching. Proceedings of the 11th Asian technology conference in mathematics (pp. 234–243). Hong Kong.Google Scholar
  19. Hong, Y. Y., & Thomas, M. O. J. (2015). Graphical construction of a local perspective on differentiation and integration. Mathematics Education Research Journal, 27, 183–200. doi: 10.1007/s13394-014-0135-6 CrossRefGoogle Scholar
  20. Hoyles, C., Kalas, I., Trouche, L., Hivon, L., Noss, R., & Wilensky, U. (2010). Connectivity and virtual networks for learning. In C. Hoyles & J.-B. Lagrange (Eds.), Mathematical education and digital technologies: Rethinking the terrain (pp. 439–462). New York: Springer.CrossRefGoogle Scholar
  21. Jaworski, B., Robinson, C., Matthews, J., & Croft, A. C. (2012). Issues in teaching mathematics to engineering students to promote conceptual understanding: A study of the use of GeoGebra and inquiry-based tasks. The International Journal for Technology in Mathematics Education, 19(4), 147–152.Google Scholar
  22. Kendal, M., & Stacey, K. (2001). The impact of teacher privileging on learning differentiation. International Journal of Computers for Mathematical Learning, 6(2), 143–165.CrossRefGoogle Scholar
  23. Kynigos, C. (2016). Constructionist mathematics with institutionalized infrastructures: The case of Dimitris and his students. In E. Faggiano, F. Ferrara, & A. Montone (Eds.), Innovation and technology enhancing mathematics education perspectives in the digital era. Dordrecht: Springer.Google Scholar
  24. 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 Academic Publishers.Google Scholar
  25. Larsen, S., Glover, E., & Melhuish, K. (2015). Beyond good teaching. In D. Bressoud, V. Mesa, & C. Rasmussen (Eds.), Insights and recommendations from the MAA national study of college calculus (pp. 93–106). Washington DC: MAA Press.Google Scholar
  26. Li, Q., & Ma, X. (2010). A meta-analysis of the effects of computer technology on school students’ mathematics learning. Educational Psychology Review, 22, 215–243.CrossRefGoogle Scholar
  27. Lin, C. T., & Thomas, M. O. J. (2011). Student understanding of Riemann integration: The role of the dynamic software GeoGebra. In J. Hannah, M. O. J. Thomas, & L. Sheryn (Eds.), Proceedings of Volcanic Delta 2011, The Eighth Southern Hemisphere conference on teaching and learning undergraduate mathematics and statistics, Rotorua, New Zealand, 27 November–2 December 2011 (pp. 216–227). Auckland, New Zealand: The University of Auckland and the University of Canterbury.Google Scholar
  28. McMullen, S., Oates, G., & Thomas, M. O. J. (2015). An integrated technology course at university: Orchestration and mediation. In K. Beswick, T. Muir, & J. Wells (Eds.), Proceedings of the 39th Conference of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 249–257). Hobart, Australia: PME.Google Scholar
  29. Meel, D. E. (1998). Honors students’ calculus understandings: Comparing calculus and Mathematica and traditional calculus students. CBMS Issues in Mathematics Education, Providence, Rhode Island: American Mathematical Society, 7, 163–215.CrossRefGoogle Scholar
  30. Mishra, P., & Koehler, M. (2006). Technological pedagogical content knowledge: A framework for teacher knowledge. Teachers college record, 108(6), 1017–1054. doi:  10.1111/j.1467-9620.2006.00684.x
  31. Mousley, J., Lambdin, D., & Koc, Y. (2003). Mathematics teacher education and technology. In A. J. Bishop, M. A. Clements, C. Keitel, J. Kilpatrick, & F. K. S. Leung (Eds.), Second international handbook of mathematics education (pp. 395–432). Dordrecht: Kluwer Academic Publishers.CrossRefGoogle Scholar
  32. Ng, W. L. (2011). Using an advanced graphing calculator in the teaching and learning of calculus. International Journal of Mathematical Education in Science & Technology, 42(7), 925–938.CrossRefGoogle Scholar
  33. Norris, C., Soloway, E., Tan, C., & Looi, C. (2013). Inquiry pedagogy and smartphones: Enabling a change in school culture. Educational Technology, 53(4), 33–40.Google Scholar
  34. Oates, G. N. (2011). Sustaining integrated technology in undergraduate mathematics. International Journal of Mathematical Education in Science and Technology, 42(6), 709–721.CrossRefGoogle Scholar
  35. Oates, G., Sheryn, L., & Thomas, M. O. J. (2014). Technology-active student engagement in an undergraduate mathematics course. In P. Liljedahl, S. Oesterle, C. Nicol, & D. Allan (Eds.), Proceedings of the 38th conference of the international group for the psychology of mathematics education (Vol. 4, pp. 330–337). Vancouver, Canada: PME.Google Scholar
  36. O’Malley, P., Jenkins, S., Wesley, B., Donehower, C., Rabuck, D., & Lewis, M. E. B. (2013). Effectiveness of using iPads to build math fluency. Paper presented at the 2013 Council for Exceptional Children Annual Meeting in San Antonio, Texas. Retrieved 1/9/2016 from http://files.eric.ed.gov/fulltext/ED541158.pdf
  37. Park, K., & Travers, K. J. (1996). A comparative study of a computer-based and a standard college first-year calculus course. CBMS Issues in Mathematics Education, Providence, Rhode Island: American Mathematical Society, 6, 155–176.CrossRefGoogle Scholar
  38. Paterson, J., Thomas, M. O. J., & Taylor, S. (2011). Decisions, decisions, decisions: What determines the path taken in lectures? International Journal of Mathematical Education in Science and Technology, 42(7), 985–996.CrossRefGoogle Scholar
  39. Pierce, R., & Stacey, K. (2011). Using dynamic geometry to bring the real world into the classroom. In L. Bu, & R. Schoen (Eds.), Model-centered learning modeling and simulations for learning and instruction (Vol. 6) (pp 41–55). Sense Publishers.Google Scholar
  40. Pierce, R., Stacey, K., & Wander, R. (2010). Examining the didactic contract when handheld technology is permitted in the mathematics classroom. ZDM International Journal of Mathematics Education, 42, 683–695. doi: 10.1007/s11858-010-0271-8 CrossRefGoogle Scholar
  41. Rasmussen, C., & Wawro, M. (2016, in Press). Post-calculus research in undergraduate mathematics education. Google Scholar
  42. Schoenfeld, A. H. (2010). How we think: A theory of goal-oriented decision making and its educational applications. New York: Routledge.Google Scholar
  43. Schoenfeld, A., Thomas, M. O. J., & Barton, B. (2016). On understanding and improving the teaching of university mathematics. International Journal of STEM Education, 3(4). doi: 10.1186/s40594-016-0038-z
  44. Schwingendorf, K. E. (1999). Assessing the effectiveness of innovative educational reform efforts. In B. Gold, S. Keith, & W. Marion (Eds.), Assessment practices in undergraduate mathematics (pp. 249–252). Washington DC: The Mathematical Association of America.Google Scholar
  45. Selinski, N. E., & Milbourne, H. (2015). The institutional context. In D. Bressoud, V. Mesa, & C. Rasmussen (Eds.), Insights and recommendations from the MAA national study of college calculus (pp. 31–44). Washington DC: MAA Press.Google Scholar
  46. Stacey, K. (2003). Using computer algebra systems in secondary school mathematics: Issues of curriculum, assessment and teaching. In S-C. Chu, W-C. Yang, T. de Alwis & M-G. Lee (Eds.), Technology connecting mathematics, Proceedings of the 8th Asian technology conference in mathematics, Taiwan R.O.C: ATCM.Google Scholar
  47. Stewart, S., Thomas, M. O. J., & Hannah, J. (2005). Towards student instrumentation of computer-based algebra systems in university courses. International Journal of Mathematical Education in Science and Technology, 36(7), 741–750.CrossRefGoogle Scholar
  48. Thomas, M. O. J. (2008). Developing versatility in mathematical thinking. Mediterranean Journal for Research in Mathematics Education, 7(2), 67–87.Google Scholar
  49. Thomas, M. O. J., De Freitas Druck, I., Huillet, D., Ju, M. -K., Nardi, E., Rasmussen, C., & Xie, J. (2015). Key mathematical concepts in the transition from secondary school to university. Proceedings of The 12th international congress on mathematical education (ICME-12) Survey Team 4 (pp. 265–284). Seoul, Korea.Google Scholar
  50. Thomas, M. O. J., & Holton, D. (2003). Technology as a tool for teaching undergraduate mathematics. In A. J. Bishop, M. A. Clements, C. Keitel, J. Kilpatrick, & F. K. S. Leung (Eds.), Second international handbook of mathematics education (Vol. 1, pp. 347–390). Dordrecht: Kluwer.Google Scholar
  51. Thomas, M. O. J., & Hong, Y. Y. (2005). Teacher factors in integration of graphic calculators into mathematics learning. In H. L. Chick & J. L. Vincent (Eds.), Proceedings of the 29th conference of the international group for the psychology of mathematics education (Vol. 4, pp. 257–264). Melbourne: University of Melbourne.Google Scholar
  52. 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
  53. Thomas, M. O. J., & Palmer, J. (2013). Teaching with digital technology: Obstacles and opportunities. In A. Clark-Wilson, N. Sinclair, & O. Robutti (Eds.), The mathematics teacher in the digital era (pp. 71–89). Dordrecht: Springer.Google Scholar
  54. Tobin, P., & Weiss, V. (2011). Teaching differential equations in undergraduate mathematics: Technology issues for service courses. In J. Hannah, M. O. J. Thomas, & L. Sheryn (Eds.), Proceedings of Volcanic Delta 2011, The Eighth Southern Hemisphere Conference on Teaching and Learning Undergraduate Mathematics and Statistics, Rotorua, New Zealand, 27 November–2 December 2011 (pp. 375–385). Auckland, New Zealand: The University of Auckland and the University of Canterbury.Google Scholar
  55. Trouche, L. (2004). Managing the complexity of human/machine interactions in computerized learning environments: Guiding students’ command process through instrumental orchestrations. Educational Studies in Mathematics, 9, 281–307.Google Scholar
  56. Trouche, L., & Drijvers, P. (2010). Handheld technology for mathematics education: Flashback into the future. ZDM, The International Journal on Mathematics Education, 42(7), 667–681. doi: 10.1007/s11858-010-0269-2 CrossRefGoogle Scholar
  57. Vandebrouck, F. (2011). Perspectives et domaines de travail pour l’étude des fonctions. Annales de Didactiques et de Sciences Cognitives, 16, 149–185.Google Scholar
  58. White, T., Booker, A., Carter Ching, C., & Martin, L. (2011). Integrating digital and mathematical practices across contexts: A manifesto for mobile learning. International Journal of Learning and Media, 3(3), 7–13. doi: 10.1162/ijlm_a_00076 CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.The University of AucklandAucklandNew Zealand
  2. 2.Ewha Womans UniversitySeoulSouth Korea

Personalised recommendations