# Mapping Pedagogical Opportunities Provided by Mathematics Analysis Software

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## Abstract

This paper proposes a taxonomy of the pedagogical opportunities that are offered by mathematics analysis software such as computer algebra systems, graphics calculators, dynamic geometry or statistical packages. Mathematics analysis software is software for purposes such as calculating, drawing graphs and making accurate diagrams. However, its availability in classrooms also provides opportunities for positive changes to teaching and learning. Very many examples are documented in the professional and research literature, and in this paper we organize them into 10 types. These are displayed in the form of a ‘pedagogical map’, which further classifies them according to whether the opportunity arises from new opportunities for the mathematical tasks used, change to interpersonal aspects of the classroom or change to the point of view on mathematics as a subject. The map can be used in teacher professional development to draw attention to possibilities for lessons or as a catalyst for professional discussion. For research on teaching, it can be used to map current practice, or to track professional growth. The intention of the map is to summarise the potential benefits of teaching with technology in a form that may be useful for both teachers and researchers.

## Keywords

Technology Mathematics Pedagogical opportunities Computer algebra systems Dynamic geometry Spreadsheets Graphics calculators## Notes

### Acknowledgments

This work is based in part on research undertaken as part of projects C00002058 and LP 0453701 funded by the Australian Research Council with partners Texas Instruments, Casio and Hewlett-Packard and numerous schools. We thank all the teachers and students who participated.

## References

- Aldon, G. (Ed.). (2009). La function de l’ensegne.
*Mathématiques dynamiques.*(pp. 46–68). Paris, France: Hachette Livre.Google Scholar - Artigue, M. (2001). Learning mathematics in a CAS environment: The genesis of a reflection about instrumentation and the dialectics between technical and conceptual work. Paper presented at
*CAME 2001 Symposium. Communicating Mathematics Through Computer Algebra Systems*. Utrecht, The Netherlands. Retrieved 13th March 2008 from: http://Itsn.mathstore.ac.uk/came/events/freudenthal. - Artigue, M. (2005). The integration of symbolic calculators into secondary education: Some lessons from didactical engineering. In D. Guin, K. Ruthven, & L. Trouche (Eds.),
*The didactical challenge of symbolic calculators: Turning a computational device into a mathematical instrument*(pp. 231–294). Dordrecht, The Netherlands: Kluwer.CrossRefGoogle Scholar - Ball, L., & Pierce, R. (2004). CAS: Trainer wheels for algebra using the TI89. In B. Tadich, S. Tobias, C. Brew, B. Beatty, & P. Sullivan (Eds.), Towards excellence in mathematics.
*Proceedings of the 40th Annual Conference of the MAV*, (pp. 13–21). Melbourne: Mathematical Association of Victoria.Google Scholar - Ball, L., & Stacey, K. (2003). What should students record when solving problems with CAS? Reasons, information, the plan and some answers. In J. T. Fey, A. Cuoco, C. Kieran, L. Mullin, & R. M. Zbiek (Eds.),
*Computer algebra systems in secondary school mathematics education*(pp. 289–303). Reston, VA: The National Council of Teachers of Mathematics.Google Scholar - Ball, L., & Stacey, K. (2007). Using technology in high-stakes assessment: How teachers balance by-hand and automated techniques. In C.-S. Lim et al.
*Proceedings of EARCOME4 2007 4th East Asia Regional Conference on Mathematics Education*, (pp. 90–97) Penang, Malaysia: Universiti Sains Malaysia.Google Scholar - Black, T. (1999). Simulations on spreadsheets for complex concepts: Teaching statistical power as an example.
*International Journal of Mathematics Education in Science and Technology,**30*(4), 473–481.CrossRefGoogle Scholar - Brousseau, G. (1997).
*Theory of didactical situations in mathematics : didactique des mathématiques, 1970*–*1990*(Edited and translated by N. Balacheff, M. Cooper, R. Sutherland, V. Warfield). Dordrecht: Kluwer.Google Scholar - Brown, J. (2005). Affordances of a technology-rich teaching and learning environment. In P. Clarkson, A. Downton, D. Gronn, M. Horne, A. McDonough, R. Pierce, & A. Roche (Eds.),
*Building connections: Theory, research, and practice, Proceedings of the 28th Annual Conference of the Mathematics Education Research Group of Australasia (Vol. 1)*, (pp. 177–184). Sydney: MERGA.Google Scholar - Dickens, C. (1854).
*Hard Times.*Retrieved 5th September 2008 from: http://www.pagebypagebooks.com/. - Etlinger, L. (1974). The electronic calculator: A new trend in school mathematics.
*Educational Technology,**XIV*(12), 43–45.Google Scholar - Galbraith, P., Stillman, G., Brown, J., & Edwards, I. (2007). Facilitating middle secondary modelling competencies. In C. Haines, P. Galbraith, W. Blum, & S. Khan (Eds.),
*Mathematical modelling (ICTMA12): Education, engineering and economics*(pp. 130–140). Chichester, UK: Horwood Press.Google Scholar - Garner, S. (2004). The CAS classroom.
*Australian Senior Mathematics Journal,**18*(2), 28–42.Google Scholar - Garry, T. (2003). Computing, conjecturing and confirming with a CAS Tool. In J. T. Fey, A. Cuoco, C. Kieran, L. Mullin, & R. M. Zbiek (Eds.),
*Computer algebra systems in secondary school mathematics education*(pp. 289–303). Reston, VA: The National Council of Teachers of Mathematics.Google Scholar - Gibson, J. J. (1977). The theory of affordances. In R. Shaw & J. Bransford (Eds.),
*Perceiving, acting and knowing: Towards an ecological psychology*(pp. 67–82). Hillsdale, NJ: Erlbaum.Google Scholar - González-López, M. J. (2001). Using dynamic geometry software to simulate physical motion.
*International Journal of Computers for Mathematical Learning,**6*(2), 127–142.CrossRefGoogle Scholar - Heid, M. K. (1988). Resequencing skills and concepts in applied calculus using the computer as a tool.
*Journal for Research in Mathematics Education,**19*(1), 3–25.CrossRefGoogle Scholar - Heid, M. K., & Blume, G. W. (2008a).
*Research on technology and the teaching and learning of mathematics: Vol. 1: Research syntheses*. Charlotte, NC: Information Age Publishing.Google Scholar - Heid, M. K., & Blume, G. W. (2008b).
*Research on technology and the teaching and learning of mathematics: Vol. 2: Cases and perspectives*. Charlotte, NC: Information Age Publishing.Google Scholar - Heugl, H. (1997). Experimental and active learning with DERIVE.
*Zentralblatt für Didaktik der Mathematik,**29*(5), 142–148.CrossRefGoogle Scholar - Kendal, M., & Stacey, K. (2001). The impact of teacher privileging on learning differentiation with technology.
*International Journal of Computers for Mathematical Learning,**6*(2), 143–165.CrossRefGoogle Scholar - Kieran, C. (2007). Learning and teaching of algebra at the middle school through college levels: Building meaning for symbols and their manipulation. In F. Lester (Ed.),
*Second handbook of research on mathematics teaching and learning*(pp. 707–762). Charlotte, NC: Information Age Publishing.Google Scholar - Kieran, C. & Damboise, C. (2007). “How can we describe the relation between the factored form and the expanded form of these trinomials?-We don’t even know if our paper and pencil factorizations are right”: The case for computer algebra systems (CAS) with weaker algebra students. 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. 105–112)*.*Seoul: PME.Google Scholar - Kieran, C., & Yerushalmy, M. (2004). Computer algebra systems and algebra: Curriculum, assessment, teaching, and learning. In K. Stacey, H. Chick, & M. Kendal (Eds.),
*The future of the teaching and learning of algebra: The 12th ICMI study*(pp. 99–154). Norwood, MA: Kluwer.Google Scholar - Laborde, C. (2000). Dynamic geometry environments as a source of rich learning contexts for the complex activity of proving.
*Educational Studies in Mathematics,**44*(1/2), 151–161.CrossRefGoogle Scholar - Lagrange, J.-B. (2003). Learning techniques and concepts using CAS: A practical and theoretical reflection. In J. T. Fey, A. Cuoco, C. Kieran, L. McMullin, & R. M. Zbiek (Eds.),
*Computer algebra systems in secondary school mathematics education*(pp. 269–283). Reston, VA: NCTM.Google Scholar - 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. 237–270). Dordecht, Netherlands: Kluwer.Google Scholar - Oldknow, A. (1996). Micromaths: Tea cups, T Cubed, discharge and the elimination of drips.
*Teaching Mathematics and its Applications,**15*(4), 179–185.CrossRefGoogle Scholar - Pea, R. D. (1985). Beyond amplification: Using the computer to reorganise mental functioning.
*Educational Psychologist,**20*(4), 167–182.CrossRefGoogle Scholar - Pierce, R., & Stacey, K. (2001a). Reflections on the changing pedagogical use of computer algebra systems: Assistance for doing or learning mathematics.
*Journal of Computers in Mathematics and Science Teaching,**20*(1), 141–163.Google Scholar - Pierce, R., & Stacey, K. (2001b). Observations on students’ responses to learning in a CAS environment.
*Mathematical Education Research Journal,**3*(1), 28–46.Google Scholar - Pierce, R., & Stacey, K. (2004a). A framework for monitoring progress and planning teaching towards effective use of computer algebra systems.
*International Journal of Computers for Mathematical Learning,**9*(1), 59–93.CrossRefGoogle Scholar - Pierce, R., & Stacey, K. (2004b). Monitoring progress in algebra in a CAS active context: Symbol Sense, algebraic insight and algebraic expectation.
*International Journal for Technology in Mathematics Education,**11*(1), 3–11.Google Scholar - Pierce, R., & Stacey, K. (2006). Enhancing the image of mathematics by association with simple pleasures from real world contexts.
*Zentralblatt für Didaktik der Mathematik,**38*(2), 214–225.CrossRefGoogle Scholar - Pierce, R., & Stacey, K. (2008). Using pedagogical maps to show the opportunities afforded by CAS for improving the teaching of mathematics.
*Australian Senior Mathematics Journal,**22*(1), 6–12.Google Scholar - RITEMATHS. (n.d.)
*RITEMATHS project web site*. Retrieved August 10th, 2009, from http://extranet.edfac.unimelb.edu.au/DSME/RITEMATHS/. - Stacey, K. (2008). Pedagogical maps for describing teaching with technology. Retrieved October 31st, 2008, from http://www.sharinginspiration.org/info/contributions.php?lang=en&size=normal.
- 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*(pp. 351–394). Dordrecht, Netherlands: Kluwer.Google Scholar - 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 future of the teaching and learning of algebra: The 12th ICMI study*(pp. 155–186). Norwood, MA: Kluwer.Google Scholar - Vincent, J. (2003). Year 8 students’ reasoning in a Cabri environment. In L. Bragg, C. Campbell, G. Herbert, & J. Mousley (Eds.),
*Mathematics education researching: Innovation, networking, opportunity. Proceedings of the 26th Annual Conference of the Mathematics Education Research Group of Australasia.*(pp. 696–703). Sydney: MERGA.Google Scholar - Wander, R., & Pierce, R. (2009). Marina’s fish shop: A mathematically- and technologically-rich lesson.
*Australian Mathematics Teacher,**65*(2), 6–12.Google Scholar - Wilf, H. S. (1982). The disc with the college education.
*The American Mathematical Monthly,**89*, 4–8.CrossRefGoogle Scholar - Yearwood, J., & Glover, B. (1995). Computer algebra systems in teaching engineering mathematics.
*Australasian Journal of Engineering Education,**6*(1), 87–94.Google Scholar - Zbiek, R. M., Heid, M. K., Blume, G. W., & Dick, T. (2007). Research on technology in mathematics education: A perspective of constructs. In F. Lester (Ed.),
*Second handbook of research on mathematics teaching and learning*(pp. 1169–1207). Charlotte, NC: Information Age.Google Scholar