The Mathematics Teacher in the Digital Era pp 163-185 | Cite as

# Teaching Roles in a Technology Intensive Core Undergraduate Mathematics Course

## Abstract

We discuss the dual teaching roles of university mathematics tutors, as teachers and policy makers, in relation to the classroom implementation of technology while guided by departmental policies. The main contribution of this chapter is the exemplification of these roles in an undergraduate mathematics programme, called *Mathematics Integrated with Computers and Applications* (MICA), with systemic technology integration. The current classroom practices of tutors in one of the MICA core courses for mathematics majors and future teachers of mathematics are examined. The role of the tutors in this course is to carefully guide the students’ instrumental genesis of programming technology for the investigation of both mathematics concepts and conjectures, and real-world applications. Acting as a mentor, the tutor encourages students’ mathematical creativity as they design, program, and use their own interactive mathematics *Exploratory Objects*.

## Keywords

University mathematics education Technology integration Tutors’ roles as teacher and policy maker Mathematics department Programming Exploratory objects/microworlds Instrumental integration Creativity## References

- Abrahamson, D., Berland, M., Shapiro, B., Unterman, J., & Wilensky, U. (2006). Leveraging epistemological diversity through computer-based argumentation in the domain of probability.
*For the Learning of Mathematics, 26*(3), 19–45.Google Scholar - 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 Mathematics Learning, 7*(3), 245–274.CrossRefGoogle Scholar - Assude, T. (2007). Teachers’ practices and degree of ICT integration. In D. Pitta- Pantazi & G. N. Philippou (Eds.),
*Proceedings of the fifth congress of the European Society for Research in Mathematics Education*(pp. 1339–1348). Larnaka: Department of Education, University of Cyprus.Google Scholar - Ben-El-Mechaiekh, H., Buteau, C., & Ralph, W. (2007). MICA: A novel direction in undergraduate mathematics teaching.
*Canadian Mathematics Society Notes, 39*(6), 9–11.Google Scholar - Burtch, M. (2003). The evolution of conjecturing in a differential equations course. Retrieved from http://mcli.maricopa.edu/book/export/html/1024
- Buteau, C., & Muller, E. (2006, December 3-8). Evolving technologies integrated into undergraduate mathematics education. In L. H. Son, N. Sinclair, J. B. Lagrange, & C. Hoyles (Eds.),
*Proceedings for the seventeenth ICMI study conference: Digital technologies and mathematics teaching and learning: Revisiting the terrain*(8 pp.). Hanoi: Hanoi University of Technology (c42)[CD-ROM].Google Scholar - Buteau, C., & Muller, E. (2010). Student development process of designing and implementing exploratory and learning objects.
*Proceedings of the sixth conference of European Research in Mathematics Education*(pp. 1111–1120). Lyon. Retrieved from http://www.inrp.fr/editions/editions-electroniques/cerme6/working-group-7 - Buteau, C., Marshall, N., Jarvis, D., & Lavicza, Z. (2010b). Integrating computer algebra systems in post-secondary mathematics education: Preliminary results of a literature review.
*International Journal for Technology in Mathematics Education, 17*(2), 57–68.Google Scholar - Buteau, C., Jarvis, D., & Lavicza, Z. (forthcoming). On the integration of computer algebra systems (CAS) by Canadian mathematicians: Results of a national survey. Accepted for publication in
*Canadian Journal of Science, Mathematics and Technology Education.*Google Scholar - Chae, S., & Tall, D. (2001). Construction of conceptual knowledge: The case of computer-aided exploration of period doubling.
*Research in Mathematics Education, 3*(1), 199–209.CrossRefGoogle Scholar - Dubinsky, E., & McDonald, M. (2002). APOS: A constructivist theory of learning. In D. Holton (Ed.),
*The teaching and learning of mathematics at university level: An ICMI study*(pp. 275–282). Dordrecht: Kluwer.CrossRefGoogle Scholar - Dubinsky, E., & Tall, D. (1991). Advanced mathematical thinking and the computer. In D. Tall (Ed.),
*Advanced mathematical thinking*(pp. 231–248). Dordrecht: Kluwer.Google Scholar - Elliott, P. (1976). Programming – an integral part of an elementary mathematics methods course.
*International Journal of Mathematical Education in Science and Technology, 7*(4), 447–454.CrossRefGoogle Scholar - Ervynck, G. (1991). Mathematical creativity. In D. Tall (Ed.),
*Advanced mathematical thinking*(pp. 42–53). Dordrecht: Kluwer.Google Scholar - Goos, M., & Cretchley, P. (2004). Teaching and learning mathematics with computers, the internet and multimedia. In B. Perry, G. Anthony, & C. Diezmann (Eds.),
*Research in mathematics education in Australasia 2000–2003*(pp. 151–174). Flaxton, Queensland, Australia: Post Pressed.Google Scholar - Goos, M., & Soury-Lavergne, S. (2010). Teachers and teaching: Theoretical perspectives and classroom implementation. In C. Hoyles & J.-B. Lagrange (Eds.),
*ICMI Study 17, technology revisited, ICMI study series*(pp. 311–328). New York: Springer.Google Scholar - Gravemeijer, K. (1999). How emergent models may foster the constitution of formal mathematics.
*Mathematical Thinking and Learning, 1*(2), 155–177.CrossRefGoogle Scholar - Halvorson, M. (2010).
*Microsoft Visual Basic 2010 step by step*(p. 579). Washington, USA: Microsoft Press.Google Scholar - Healy, L., & Kynigos, C. (2010). Charting the microworld territory over time: Design and construction in mathematics education.
*ZDM, 42*(1), 63–76. doi: 10.1007/s11858-009-0193-5.CrossRefGoogle Scholar - Holton, D. (Ed.). (2001).
*The teaching and learning of mathematics at university level*(ICMI study series: New ICMI study series, Vol. 7, p. 560). Dordrecht/Boston/London: Kluwer Academic Publishers.Google Scholar - Holton, D. (2005). Tertiary mathematics education for 2024.
*International Journal of Mathematical Education in Science and Technology, 36*(2–3), 305–316.Google Scholar - Howson, A. G., & Kahane, J. P. (Eds.). (1986).
*The influence of computers and informatics on mathematics and its teaching*(ICMI study series, Vol. 1, p. 155). Cambridge, UK: Cambridge University Press.Google Scholar - Hoyles, C., & Lagrange, J.-B. (Eds.). (2010).
*Mathematics education and technology – rethinking the terrain: The 17th ICMI study*(p. 494). Springer: New York.Google Scholar - Jonassen, D. H. (1996).
*Computers in the classroom: Mindtools for critical thinking*. Englewood Cliffs, NJ: Prentice-Hall.Google Scholar - Jonassen, D. H. (2006).
*Modeling with technology: Mindtools for conceptual change*(3rd ed.). Upper Saddle River, NJ: Merrill.Google Scholar - Karian, Z. A. (Ed.). (1992).
*Symbolic computation in undergraduate mathematics education*(MAA Notes, Vol. 24, p. 200). Washington, DC: Mathematical Association of America.Google Scholar - Keynes, H., & Olson, A. (2001). Professional development for changing undergraduate mathematics instruction. In D. Holton (Ed.),
*The teaching and learning of mathematics at the university level: An ICMI study*(pp. 113–126). Dordrecht: Kluwer.Google Scholar - King, K., Hillel, J., & Artigue, M. (2001). Technology – a working group report. In D. Holton (Ed.),
*The teaching and learning of mathematics at university level: An ICMI study*(pp. 349–356). Dordrecht: Kluwer.Google Scholar - Kynigos, C. (2012). Constructionism: Theory learning or theory of design?
*Proceedings of the 12th International Congress on Mathematical Education*(ICME 12), 8*–*15 July 2012, Seoul (Korea). 24 pp.Google Scholar - Lagrange, J. B., Artigue, M., Laborde, C., & Trouche, L. (2003). Technology and mathematics education: Multidimensional over- view of recent research and innovation. In F. K. S. Leung (Ed.),
*Second international handbook of mathematics education*(Vol. 1, pp. 237–270). Dordrecht: Kluwer.CrossRefGoogle Scholar - Lavicza, Z. (2010). Integrating technology into mathematics teaching at the university level.
*ZDM, 42*, 105–119.CrossRefGoogle Scholar - Maplesoft. (n.d.). http://www.maplesoft.com/
- Marshall, N. (2012a). Simulation and Brock University’s MICA Program – reflections of a graduate. In E. R. Muller, J-P. Villeneuve & P. Etchecopar (Eds.), Using simulation to develop students’ mathematical competencies – post secondary and teacher education,
*Proceedings of Canadian Mathematics Education Study Group/GCEDM 2011 meeting*(pp. 59–76).Google Scholar - Marshall, N. (2012b).
*Contextualizing the learning activity of designing and experimenting with interactive, dynamic mathematics exploratory objects.*Unpublished M.Sc. thesis, Brock University, St.Catharines.Google Scholar - Martinovic, D., Muller, E., & Buteau, C. (2013). Intelligent partnership with technology: Moving from a mathematics school curriculum to an undergraduate program.
*Computers in the Schools, 30*(1–2), 76–101.CrossRefGoogle Scholar - Mason, J. H. (2002).
*Mathematics teaching practice – a guide for university and college lecturers*. Chichester: Horwood Publishing.Google Scholar - Mason, J., Burton, L., & Stacey, K. (1985).
*Thinking mathematically*. Wokingham: Addison-Wesley.Google Scholar - Mathematical Association of America. (2004). In W. Barker et al. (Eds.),
*Undergraduate programs and courses in the mathematical sciences: CUPM curriculum guide*. Washington, DC: Mathematical Association of America.Google Scholar - Mgombelo, J. R., Orzech, M., Poole, D., & René de Cotret, S. (2006). Report of the CMESG working group: Secondary mathematics teacher development (La formation des enseignants de mathématiques du secondaire). In L. Peter (Ed.),
*Proceedings of the annual meeting of Canadian Mathematics Education Study Group (CMESG)*, University of Calgary, CalgaryGoogle Scholar - Microsoft Visual Studio. (n.d.). http://www.microsoft.com/visualstudio/fr-ca
- Morselli, F. (2006). Use of examples in conjecturing and proving: An exploratory study. In Novotnà et al. (Eds.),
*Proceedings of the 30th conference of the international group for the Psychology of Mathematics Education*(Vol. 4, pp. 185–192), Prague.Google Scholar - Muller, E., Buteau, C., Ralph, B., & Mgombelo, J. (2009). Learning mathematics through the design and implementation of Exploratory and Learning Objects.
*International Journal for Technology in Mathematics Education, 16*(2), 63–73.Google Scholar - National Science Foundation. (1996).
*Shaping the future: New expectations for undergraduate education in science, mathematics, engineering, and technology*. Arlington: National Science Foundation.Google Scholar - Noss, R. (1999). Learning by design: Undergraduate scientists learning mathematics.
*International Journal of Mathematical Education in Science and Technology, 30*(3), 373–388.CrossRefGoogle Scholar - Pea, R. D. (1987). Cognitive technologies for mathematics education. In A. H. Schoenfeld (Ed.),
*Cognitive science and mathematics education*(pp. 89–122). Hillsdale: Lawrence Erlbaum.Google Scholar - Profetto, A. (2005).
*Mandelbrot Set*Exploratory Object. http://www.brocku.ca/mathematics/resources/learningtools/learningobjects/mathobjects/mandelbrot/executable/mainpage.swf - Ralph, B. (1999).
*Journey through Calculus*. Pacific Grove: Brooks/Cole/Thomson Learning. (CD)Google Scholar - Ralph, B. (2001). Mathematics takes an exciting new direction with MICA program.
*Brock Teaching*,*1*(1), 1. Retrieved October 30, 2011 from http://www.brocku.ca/webfm_send/18483 - Schurrer, A., & Mitchell, D. (1994). Technology and the mature department.
*Electronic Proceedings of the 7th international conference on Technology in Collegiate Mathematics*. Retrieved June 29, 2012 from http://archives.math.utk.edu/ICTCM/VOL07/C002/paper.txt - Tall, D. (1991). Reflections. In D. Tall (Ed.),
*Advanced mathematical thinking*(pp. 251–259). Dordrecht: Kluwer.CrossRefGoogle Scholar - Wilensky, U. (1995). Paradox, programming, and learning probability: A case study in a connected mathematics framework.
*The Journal of Mathematical Behavior, 14*(2), 253–280.CrossRefGoogle Scholar - Willis, S., & Kissane, B. (1989). Computer technology and teacher education in mathematics. In
*Department of employment, education and training, discipline review of teacher education in mathematics and science*(Vol. 3, pp. 57–92). Canberra, Australia: Australian Government Publishing Service.Google Scholar - Wolfram Mathematica. (n.d.). http://www.wolfram.com/mathematica/