And the Rest is Just Algebra pp 173-201 | Cite as

# Rethinking Algebra: A Versatile Approach Integrating Digital Technology

## Abstract

Many have thought deeply about the construction of the school algebra curriculum, but the question remains as to why we teach the topics we do in the manner we do, stressing manipulations of symbols, and why some other avenues are ignored. In this chapter we consider the basic constructs in the school algebra curriculum and the procedural approach often taken to learning them and suggest some reasons why certain topics may be excluded. We examine how particular tasks, including some that integrate digital technology into student activity, could be used to rethink the algebra curriculum content with a view to motivating students and promoting versatile thinking. Some reasons why these topics have often not yet found their way into the curriculum are discussed.

## Keywords

Versatile thinking Algebra Tertiary Digital technology Representations## References

- Akkoc, H., & Tall, D. O. (2002). The simplicity, complexity and complication of the function concept. In A. D. Cockburn & E. Nardi (Eds.),
*Proceedings of the 26th Conference of the International Group for the Psychology of Mathematics Education*(Vol. 2, pp. 25–32). Norwich, UK.Google Scholar - Ball, D. L., Hill, H. C., & Bass, H. (2005). Knowing mathematics for teaching: Who knows mathematics well enough to teach third grade, and how can we decide?
*American Educator, 29*(1), 14–17, 20–22, 43–46.Google Scholar - Borowski, E. J., & Borwein, J. M. (1989).
*Dictionary of mathematics*. London: Collins.Google Scholar - Bruner, J. (1966).
*Toward a theory of instruction*. Cambridge, MA: Harvard University Press.Google Scholar - Chinnappan, M., & Thomas, M. O. J. (2003). Teachers’ function schemas and their role in modelling.
*Mathematics Education Research Journal, 15*(2), 151–170.CrossRefGoogle Scholar - Crowley, L., Thomas, M. O. J., & Tall, D. O. (1994). Algebra, symbols and translation of meaning. In J. P. da Ponte & J. F. Matos (Eds.),
*Proceedings of the 18th Conference of the International Group for the Psychology of Mathematics Education*(Vol. 2, pp. 240–247). Lisbon, Portugal: Program Committee.Google Scholar - de Alwis, A. (2012). Some curious properties and loci problems associated with cubics and other polynomials.
*International Journal of Mathematical Education in Science and Technology, 43*(7), 897–910.CrossRefGoogle Scholar - Dubinsky, E. (1991). Reflective abstraction in advanced mathematical thinking. In D. O. Tall (Ed.),
*Advanced mathematical thinking*(pp. 95–123). Dordrecht: Kluwer Academic.Google Scholar - Dubinsky, E., & McDonald, M. (2001). 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, The Netherlands: Kluwer Academic.Google Scholar - Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics.
*Educational Studies in Mathematics, 61*, 103–131.CrossRefGoogle Scholar - Ernest, P. (2006). A semiotic perspective of mathematical activity: The case of number.
*Educational Studies in Mathematics, 61*, 67–101.CrossRefGoogle Scholar - Filloy, E., & Rojano, T. (1984). From an arithmetical to an algebraic thought. In J. M. Moser (Ed.),
*Proceedings of the Sixth Annual Meeting of PME-NA*(pp. 51–56). Madison: University of Wisconsin.Google Scholar - Godfrey, D., & Thomas, M. O. J. (2008). Student perspectives on equation: The transition from school to university.
*Mathematics Education Research Journal, 20*(2), 71–92.CrossRefGoogle Scholar - Graham, A. T., Pfannkuch, M., & Thomas, M. O. J. (2009). Versatile thinking and the learning of statistical concepts.
*ZDM: The International Journal on Mathematics Education, 45*(2), 681–695.CrossRefGoogle Scholar - 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.CrossRefGoogle Scholar - Graham, A. T., & Thomas, M. O. J. (2005). Representational versatility in learning statistics.
*International Journal of Technology in Mathematical Education, 12*(1), 3–14.Google Scholar - Hansson, O., & Grevholm, B. (2003). Preservice teachers’ conceptions about
*y*=*x*+ 5: Do they see a function?*Proceedings of the 27th Conference of the International Group for the Psychology of Mathematics Education*(Vol. 3, pp. 25–32). Honolulu, Hawaii.Google Scholar - Heid, M. K., Thomas, M. O. J., & Zbiek, R. M. (2013). How might computer algebra systems change the role of algebra in the school curriculum? In A. J. Bishop, M. A. Clements, C. Keitel, J. Kilpatrick, & F. K. S. Leung (Eds.),
*Third international handbook of mathematics education*(pp. 597–642). Dordrecht: Springer.Google Scholar - Herscovics, N., & Linchevski, L. (1994). A cognitive gap between arithmetic and algebra.
*Educational Studies in Mathematics, 27*, 59–78.CrossRefGoogle Scholar - Hill, H., & Ball, D. L. (2004). Learning mathematics for teaching: Results from California’s mathematics professional development institutes.
*Journal for Research in Mathematics Education, 35*, 330–351. doi: 10.2307/30034819.CrossRefGoogle Scholar - Hodgen, J., Brown, M., Küchemann, D., & Coe, R. (2010).
*Mathematical attainment of English secondary school students: A 30-year comparison*. Paper presented at the British Educational Research Association (BERA) Annual Conference, University of Warwick.Google Scholar - Hodgen, J., Coe, R., Brown, M., & Küchemann, D. E. (2014). Improving students’ understanding of algebra and multiplicative reasoning: Did the ICCAMS intervention work? In S. Pope (Ed.),
*Proceedings of the Eighth British Congress of Mathematics Education*(BCME8) (pp. 1–8). University of Nottingham.Google Scholar - Hoehn, L. (1989, December). Solutions of
*x*^{n}+*y*^{n}=*z*^{n+1}.*Mathematics Magazine*, 342. doi: 10.2307/2689491. - Hong, Y. Y., & Thomas, M. O. J. (2006). Factors influencing teacher integration of graphic calculators in teaching. In
*Proceedings of the 11th Asian Technology Conference in Mathematics*(pp. 234–243). Hong Kong.Google Scholar - Hong, Y. Y., & Thomas, M. O. J. (2014). 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 - Hong, Y. Y., Thomas, M. O. J., & Kwon, O. (2000). Understanding linear algebraic equations via super-calculator representations. In T. Nakahara & M. Koyama (Eds.),
*Proceedings of the 24th Annual Conference of the International Group for the Psychology of Mathematics Education*(Vol. 3, pp. 57–64). Hiroshima, Japan: Programme Committee.Google Scholar - Kieran, C. (1981). Concepts associated with the equality symbol.
*Educational Studies in Mathematics, 12*(3), 317–326.CrossRefGoogle Scholar - Küchemann, D. E. (1981). Algebra. In K. M. Hart (Ed.),
*Children’s understanding of mathematics: 11-16*(pp. 102–119). London: John Murray.Google Scholar - Mason, J. (1987). What do symbols represent? In C. Janvier (Ed.)
*Problems of representation in the teaching and learning of mathematics.*Hillsdale, NJ: LEA.Google Scholar - Peirce, C. S. (1898). Logic as semiotic: The theory of signs. In J. Bucher (Ed.),
*Philosophical writings of Peirce*. New York: Dover.Google Scholar - Rosnick, P., & Clement, J. (1980). Learning without understanding: The effect of tutorial strategies on algebra misconceptions.
*Journal of Mathematical Behavior, 3*(1), 3–27.Google Scholar - Russell, B. (1903).
*The principles of mathematics*. Cambridge: Cambridge University Press.Google Scholar - Schoenfeld, A. H. (2011).
*How we think. A theory of goal-oriented decision making and its educational applications*. Routledge: New York.Google Scholar - Skemp, R. (1971).
*The psychology of learning mathematics*. Middlesex, UK: Penguin.Google Scholar - Skemp, R. R. (1976). Relational understanding and instrumental understanding.
*Mathematics Teaching, 77*, 20–26.Google Scholar - Stewart, S., & Thomas, M. O. J. (2010). Student learning of basis, span and linear independence in linear algebra.
*International Journal of Mathematical Education in Science and Technology, 41*(2), 173–188.CrossRefGoogle Scholar - Struik, D. J. (1969).
*A source book in mathematics, 1200-1800*. Cambridge, MA: Harvard University Press.Google Scholar - Tall, D. O. (2004). Building theories: The three worlds of mathematics.
*For the Learning of Mathematics, 24*(1), 29–32.Google Scholar - Tall, D. O. (2008). The transition to formal thinking in mathematics.
*Mathematics Education Research Journal, 20*(2), 5–24.CrossRefGoogle Scholar - Tall, D. O., & Thomas, M. O. J. (1991). Encouraging versatile thinking in algebra using the computer.
*Educational Studies in Mathematics, 22*, 125–147.CrossRefGoogle Scholar - Tall, D. O., Thomas, M. O. J., Davis, G., Gray, E., & Simpson, A. (2000). What is the object of the encapsulation of a process?
*Journal of Mathematical Behavior, 18*(2), 223–241.CrossRefGoogle Scholar - Thomas, M. O. J. (1988).
*A conceptual approach to the early learning of algebra using a computer*. Unpublished PhD thesis, University of Warwick.Google Scholar - Thomas, M. O. J. (1994). A process-oriented preference in the writing of algebraic equations. In G. Bell, B. Wright, N. Leeson, & J. Geake (Eds.),
*Challenges in mathematics education: Constraints on construction. Proceedings of the 17th Mathematics Education Research Group of Australasia Conference*(pp. 599–606). Lismore, Australia: MERGA.Google Scholar - Thomas, M. O. J. (2002). Versatile thinking in mathematics. In D. O. Tall & M. O. J. Thomas (Eds.),
*Intelligence, learning and understanding in mathematics*(pp. 179–204). Flaxton, Queensland, Australia: Post Pressed.Google Scholar - Thomas, M. O. J. (2003). The role of representation in teacher understanding of function. In N. A. Pateman, B. J. Dougherty, & J. Zilliox (Eds.),
*Proceedings of the 27th Conference of the International Group for the Psychology of Mathematics Education*(Vol. 4, pp. 291–298). Honolulu, Hawai’i: University of Hawai’i.Google Scholar - Thomas, M. O. J. (2008a). Conceptual representations and versatile mathematical thinking.
*Proceedings of ICME-10*(CD version of proceedings). Copenhagen, Denmark, 1–18. Retrieved from http://www.icme10.dk/proceedings/pages/regular_pdf/RL_Mike_Thomas.pdf - Thomas, M. O. J. (2008b). Developing versatility in mathematical thinking.
*Mediterranean Journal for Research in Mathematics Education, 7*(2), 67–87.Google Scholar - 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, Australia: University of Melbourne.Google Scholar - 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 - Thomas, M. O. J., & Stewart, S. (2011). Eigenvalues and eigenvectors: Embodied, symbolic and formal thinking.
*Mathematics Education Research Journal, 23*(3), 275–296. doi: 10.1007/s13394-011-0016-1.CrossRefGoogle Scholar - Thomas, M. O. J., & Tall, D. O. (2001). The long-term cognitive development of symbolic algebra. In
*Proceedings of the International Congress of Mathematical Instruction (ICMI) the Future of the Teaching and Learning of Algebra*(pp. 590–597). Melbourne.Google Scholar - Thompson, P. W. (1994). Students, functions, and the undergraduate curriculum. In E. Dubinsky, A. Schoenfeld, & 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 - Vandebrouck, F. (2011). Students’ conceptions of functions at the transition between secondary school and university. In M. Pytlak, T. Rowland, & E. Swoboda (Eds.),
*Proceedings of the 7th Conference of European Researchers in Mathematics Education*(pp. 2093–2102). Poland: Rzeszow.Google Scholar - Wagner, S. (1981). Conservation of equation and function under transformations of variable.
*Journal for Research in Mathematics Education, 12*, 118–197.CrossRefGoogle Scholar - Williams, C. G. (1998). Using concept maps to assess conceptual knowledge of function.
*Journal for Research in Mathematics Education, 29*(4), 414–421.CrossRefGoogle Scholar