## 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.

## Access this chapter

Tax calculation will be finalised at checkout

Purchases are for personal use only

### Similar content being viewed by others

## Notes

- 1.
In this chapter we make some use of calculus differentiation techniques. While calculus is usually not studied in school in the USA , many countries do include it in the curriculum from age 16 or 17. Since the primary aim of school algebra is to lead to calculus some minimal use seems reasonable.

## 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. 25â€“32). Norwich, UK.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), 14â€“17, 20â€“22, 43â€“46.Borowski, E. J., & Borwein, J. M. (1989).

*Dictionary of mathematics*. London: Collins.Bruner, J. (1966).

*Toward a theory of instruction*. Cambridge, MA: Harvard University Press.Chinnappan, M., & Thomas, M. O. J. (2003). Teachersâ€™ function schemas and their role in modelling.

*Mathematics Education Research Journal, 15*(2), 151â€“170.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. 240â€“247). Lisbon, Portugal: Program Committee.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), 897â€“910.Dubinsky, E. (1991). Reflective abstraction in advanced mathematical thinking. In D. O. Tall (Ed.),

*Advanced mathematical thinking*(pp. 95â€“123). Dordrecht: Kluwer Academic.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. 275â€“282). Dordrecht, The Netherlands: Kluwer Academic.Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics.

*Educational Studies in Mathematics, 61*, 103â€“131.Ernest, P. (2006). A semiotic perspective of mathematical activity: The case of number.

*Educational Studies in Mathematics, 61*, 67â€“101.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. 51â€“56). Madison: University of Wisconsin.Godfrey, D., & Thomas, M. O. J. (2008). Student perspectives on equation: The transition from school to university.

*Mathematics Education Research Journal, 20*(2), 71â€“92.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), 681â€“695.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), 265â€“282.Graham, A. T., & Thomas, M. O. J. (2005). Representational versatility in learning statistics.

*International Journal of Technology in Mathematical Education, 12*(1), 3â€“14.Hansson, O., & Grevholm, B. (2003). Preservice teachersâ€™ conceptions about

*y*â€‰=â€‰*x*â€‰+â€‰5: Do they see a function?*Proceedings of the 27th Conference of the International Group for the Psychology of Mathematics Education*(Vol. 3, pp. 25â€“32). Honolulu, Hawaii.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. 597â€“642). Dordrecht: Springer.Herscovics, N., & Linchevski, L. (1994). A cognitive gap between arithmetic and algebra.

*Educational Studies in Mathematics, 27*, 59â€“78.Hill, H., & Ball, D. L. (2004). Learning mathematics for teaching: Results from Californiaâ€™s mathematics professional development institutes.

*Journal for Research in Mathematics Education, 35*, 330â€“351. doi:10.2307/30034819.Hodgen, J., Brown, M., KÃ¼chemann, 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.Hodgen, J., Coe, R., Brown, M., & KÃ¼chemann, D. E. (2014). Improving studentsâ€™ 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. 1â€“8). University of Nottingham.Hoehn, L. (1989, December). Solutions of

*x*^{n}â€‰+â€‰*y*^{n}â€‰=â€‰*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. 234â€“243). Hong Kong.Hong, Y. Y., & Thomas, M. O. J. (2014). Graphical construction of a local perspective on differentiation and integration.

*Mathematics Education Research Journal, 27*, 183â€“200. doi:10.1007/s13394-014-0135-6.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. 57â€“64). Hiroshima, Japan: Programme Committee.Kieran, C. (1981). Concepts associated with the equality symbol.

*Educational Studies in Mathematics, 12*(3), 317â€“326.KÃ¼chemann, D. E. (1981). Algebra. In K. M. Hart (Ed.),

*Childrenâ€™s understanding of mathematics: 11-16*(pp. 102â€“119). London: John Murray.Mason, J. (1987). What do symbols represent? In C. Janvier (Ed.)

*Problems of representation in the teaching and learning of mathematics.*Hillsdale, NJ: LEA.Peirce, C. S. (1898). Logic as semiotic: The theory of signs. In J. Bucher (Ed.),

*Philosophical writings of Peirce*. New York: Dover.Rosnick, P., & Clement, J. (1980). Learning without understanding: The effect of tutorial strategies on algebra misconceptions.

*Journal of Mathematical Behavior, 3*(1), 3â€“27.Russell, B. (1903).

*The principles of mathematics*. Cambridge: Cambridge University Press.Schoenfeld, A. H. (2011).

*How we think. A theory of goal-oriented decision making and its educational applications*. Routledge: New York.Skemp, R. (1971).

*The psychology of learning mathematics*. Middlesex, UK: Penguin.Skemp, R. R. (1976). Relational understanding and instrumental understanding.

*Mathematics Teaching, 77*, 20â€“26.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), 173â€“188.Struik, D. J. (1969).

*A source book in mathematics, 1200-1800*. Cambridge, MA: Harvard University Press.Tall, D. O. (2004). Building theories: The three worlds of mathematics.

*For the Learning of Mathematics, 24*(1), 29â€“32.Tall, D. O. (2008). The transition to formal thinking in mathematics.

*Mathematics Education Research Journal, 20*(2), 5â€“24.Tall, D. O., & Thomas, M. O. J. (1991). Encouraging versatile thinking in algebra using the computer.

*Educational Studies in Mathematics, 22*, 125â€“147.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), 223â€“241.Thomas, M. O. J. (1988).

*A conceptual approach to the early learning of algebra using a computer*. Unpublished PhD thesis, University of Warwick.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. 599â€“606). Lismore, Australia: MERGA.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. 179â€“204). Flaxton, Queensland, Australia: Post Pressed.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. 291â€“298). Honolulu, Hawaiâ€™i: University of Hawaiâ€™i.Thomas, M. O. J. (2008a). Conceptual representations and versatile mathematical thinking.

*Proceedings of ICME-10*(CD version of proceedings). Copenhagen, Denmark, 1â€“18. Retrieved from http://www.icme10.dk/proceedings/pages/regular_pdf/RL_Mike_Thomas.pdfThomas, M. O. J. (2008b). Developing versatility in mathematical thinking.

*Mediterranean Journal for Research in Mathematics Education, 7*(2), 67â€“87.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. 257â€“264). Melbourne, Australia: University of Melbourne.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. 71â€“89). Dordrecht: Springer.Thomas, M. O. J., & Stewart, S. (2011). Eigenvalues and eigenvectors: Embodied, symbolic and formal thinking.

*Mathematics Education Research Journal, 23*(3), 275â€“296. doi:10.1007/s13394-011-0016-1.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. 590â€“597). Melbourne.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. 21â€“44). Providence, RI: American Mathematical Society.Vandebrouck, F. (2011). Studentsâ€™ 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. 2093â€“2102). Poland: Rzeszow.Wagner, S. (1981). Conservation of equation and function under transformations of variable.

*Journal for Research in Mathematics Education, 12*, 118â€“197.Williams, C. G. (1998). Using concept maps to assess conceptual knowledge of function.

*Journal for Research in Mathematics Education, 29*(4), 414â€“421.

## Author information

### Authors and Affiliations

### Corresponding author

## Editor information

### Editors and Affiliations

## Rights and permissions

## Copyright information

Â© 2017 Springer International Publishing Switzerland

## About this chapter

### Cite this chapter

Thomas, M. (2017). Rethinking Algebra: A Versatile Approach Integrating Digital Technology. In: Stewart, S. (eds) And the Rest is Just Algebra. Springer, Cham. https://doi.org/10.1007/978-3-319-45053-7_10

### Download citation

DOI: https://doi.org/10.1007/978-3-319-45053-7_10

Published:

Publisher Name: Springer, Cham

Print ISBN: 978-3-319-45052-0

Online ISBN: 978-3-319-45053-7

eBook Packages: EducationEducation (R0)