Skip to main content
SpringerLink
Log in

How Early Is Too Early for Thinking Algebraically?

  • Chapter
  • First Online:
Teaching and Learning Algebraic Thinking with 5- to 12-Year-Olds

Part of the book series: ICME-13 Monographs ((ICME13Mo))

Abstract

My answer is that it is never too early. In order to learn arithmetic it is necessary to think algebraically, although not necessarily using symbols. Some evidence for algebraic thinking amongst young children is given, followed by suggestions as to why such thinking has not always been promoted and developed. Specific pedagogic actions are outlined that focus on the expression of generality as the core of algebraic thinking, including examples of task-contexts that invoke reasoning both with and without using numbers. Finally, it is proposed that the critical feature for promoting algebraic thinking is not the tasks given to learners, but rather the opportunities noticed by teachers for calling upon learners’ powers to express and manipulate generalities, and that this is enriched when teachers engage in similar tasks at their own level, so as to sensitize themselves to pedagogic opportunities when working with learners.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

References

  • Andrews, N. (2016). Differences in how teachers make mathematical content available to learners over time. Unpublished DPhil. Thesis, University of Oxford.

    Google Scholar 

  • Blanton, M., & Kaput, J. (2011). Functional thinking as a route into algebra in the elementary grades. In J. Cai & E. Knuth (Eds.), Early algebraization: A global dialogue from multiple perspectives (pp. 5–23). Berlin: Springer.

    Google Scholar 

  • Boaler, J. (2010). The elephant in the classroom: Helping children learn and love maths. London: Souvenir Press.

    Google Scholar 

  • Bornstein, M., & Stiles-Davis, J. (1984). Discrimination and memory for symmetry in young children. Developmental Psychology, 20(4), 637–649.

    Google Scholar 

  • Bruner, J. (1966). Toward a theory of instruction, Cambridge, MA: Harvard University Press.

    Google Scholar 

  • Bruner, J. (1986). Actual minds, possible worlds, Cambridge, MA: Harvard University Press.

    Google Scholar 

  • Bruner, J. (1996). The culture of education. Cambridge, MA: Harvard University Press.

    Google Scholar 

  • Cai, J., Ng S. F., & Moyer, J. (2011). Developing students’ algebraic thinking in earlier grades: Lessons from China and Singapore. In J. Cai & E. Knuth (Eds.), Early algebraization: A global dialogue from multiple perspectives (pp. 25–41). Berlin: Springer.

    Google Scholar 

  • Chen, Ch.-S. (2015). Developing young children’s invention and identification of mathematical patterns. Poster presented at PME39, Hobart, AU.

    Google Scholar 

  • Chevallard, Y. (1985). La transposition didactique. Grenoble: La Pensée Sauvage.

    Google Scholar 

  • Chi, M., & Bassok, M. (1989). Learning from examples via self-explanation. In L. Resnick (Ed.), Knowing, learning, and instruction: Essays in honor of Robert Glaser. Hillsdale, NJ: Lawrence Erlbaum.

    Google Scholar 

  • Cooper, T., & Warren, E. (2011). Year 2 to year 6 students’ ability to generalize: Models, representations and theory for teaching and learning. In J. Cai & E. Knuth (Eds.), Early algebraization: A global dialogue from multiple perspectives (pp. 187–214). Berlin: Springer.

    Google Scholar 

  • Davis, B. (1996). Teaching mathematics: Towards a sound alternative. New York: Ablex.

    Google Scholar 

  • Davydov, V. (1990). Types of generalization in instruction (Soviet Studies in Mathematics Education, Vol. 2). Reston, VA: National Council of Teachers of Mathematics.

    Google Scholar 

  • Dougherty, B. (2008). Algebra in the early grades. Mahwah, NJ: Lawrence Erlbaum.

    Google Scholar 

  • Dweck, C. (2000). Self-theories: Their role in motivation, personality and development. Philadelphia, PA: Psychology Press.

    Google Scholar 

  • Fong, H. K. (1993). Challenging mathematical problems for primary school: The model approach. Singapore: Kingsford Educational Services.

    Google Scholar 

  • Gattegno, C. (1970). What we owe children: The subordination of teaching to learning. London: Routledge & Kegan Paul.

    Google Scholar 

  • Gattegno, C. (1973). In the beginning there were no words: The universe of babies. New York: Educational Solutions.

    Google Scholar 

  • Gattegno C. (1975/1988). The mind teaches the brain. New York: Educational Solutions.

    Google Scholar 

  • Gattegno, C. (1984). On infinity. Mathematics Teaching, 107, 19–20.

    Google Scholar 

  • Gattegno, C. (1987). The science of education. Part I: Theoretical considerations. New York: Educational Solutions.

    Google Scholar 

  • Gillings, R. (1972/1982). Mathematics in the time of the pharoahs. New York: Dover.

    Google Scholar 

  • Golomb, C. (1992). The child’s creation of a pictorial world. Berkeley: University of California Press.

    Google Scholar 

  • Goutard, M. (1958). Thoughts about problems. Mathematics Teaching, 7, 3–10.

    Google Scholar 

  • Henningsen, M., & Stein, M. (1997). Mathematical tasks and student cognition: Classroom-based factors that support and inhibit high-level mathematical thinking and reasoning. Journal for Research in Mathematics Education, 28(5), 524–549.

    Google Scholar 

  • Hewitt, D. (1998). Approaching arithmetic algebraically. Mathematics Teaching, 163, 19–29.

    Google Scholar 

  • Küchemann, D. (1981). Algebra. In K. Hart (Ed.), Children’s understanding of mathematics: 11–16 (pp. 102–119). London: John Murray.

    Google Scholar 

  • Lee, K., Khng, K. H., Ng, S. F., & Ng, L. K. J. (2013). Longer bars for bigger numbers? Children’s usage and understanding of graphical representations of algebraic problems. Frontline Learning Research, 1(1), 81–96.

    Google Scholar 

  • Love, E., & Mason, J. (1992). Teaching mathematics: Action and awareness. Milton Keynes, UK: Open University.

    Google Scholar 

  • Marton, F. (2015). Necessary conditions for learning. Abingdon, UK: Routledge.

    Google Scholar 

  • Maslow, A. (1971). The farther reaches of human nature. New York: Viking Press.

    Google Scholar 

  • Mason, J. (1988). Expressing generality. Milton Keynes, UK: Open University.

    Google Scholar 

  • Mason, J. (1990). Algebra (Supporting Primary Mathematics PM649). Milton Keynes, UK: Open University.

    Google Scholar 

  • Mason, J. (1998). Enabling teachers to be real teachers: Necessary levels of awareness and structure of attention. Journal of Mathematics Teacher Education, 1(3), 243–267.

    Google Scholar 

  • Mason, J. (1999). The expression of generality in early arithmetic texts. In I. Winchester & L. Lentz (Eds.), Toward scientific literacy: Proceedings of Fourth International Conference on History & Philosophy of Science Teaching (pp. 515–528). Alberta, CA: University of Calgary.

    Google Scholar 

  • Mason, J. (2002a). Researching your own practice: The discipline of noticing. London: RoutledgeFalmer.

    Google Scholar 

  • Mason, J. (2002b). Mathematics teaching practice: A guide for university and college lecturers. Chichester, UK: Horwood Publishing.

    Google Scholar 

  • Mason, J. (2003). Structure of attention in the learning of mathematics. In J. Novotná (Ed.) Proceedings of International Symposium on Elementary Mathematics Teaching (pp. 9–16). Prague, CZ: Charles University.

    Google Scholar 

  • Mason, J. (2008a). Making use of children’s powers to produce algebraic thinking. In J. Kaput, D. Carraher, & M. Blanton (Eds.), Algebra in the early grades (pp. 57–94). Mahwah, NJ: Lawrence Erlbaum.

    Google Scholar 

  • Mason, J. (2008b). Up and down sums, or why are we doing this Miss? Inner and outer aspects of tasks (Studies in algebraic thinking 1). Available from: www.PMTheta.com/studies.html Accessed: 21 February 2017.

  • Mason, J. (2014). Uniqueness of patterns generated by repetition. Mathematical Gazette, 98(541), 1–7.

    Google Scholar 

  • Mason, J. (2016). Overcoming the algebra barrier: Being particular about the general, and generally looking beyond the particular, in homage to Mary Boole. In S. Stewart (Ed.), And the rest is just algebra (pp. 97–117) New York: Springer.

    Google Scholar 

  • Mason, J., & Johnston-Wilder, S. (2004a). Fundamental constructs in mathematics education. London: RoutledgeFalmer.

    Google Scholar 

  • Mason, J., & Johnston-Wilder, S. (2004b). Designing and using mathematical tasks. Milton Keynes, UK: Open University (2006 reprint, St. Albans, UK: QED).

    Google Scholar 

  • Mason, J., & Metz, M. (2017). Digging beneath dual systems theory and the bicameral brain: Abductions about the human psyche from experience in mathematical problem solving. In U. Eligio (Ed.), Understanding emotions in mathematical thinking and learning. Cambridge, MA: Elsevier/Academic Press.

    Google Scholar 

  • Mason, J., Burton, L., & Stacey, K. (1982/2010). Thinking mathematically (second extended edition). Harlow, UK: Prentice Hall (Pearson).

    Google Scholar 

  • Mason, J., Graham, A., Pimm, D., & Gowar, N. (1985). Routes to, roots of, algebra. Milton Keynes, UK: Open University.

    Google Scholar 

  • Mason, J., Graham, A., & Johnston-Wilder, S. (2005). Developing thinking in algebra. London: Sage.

    Google Scholar 

  • Mason, J., Drury, H., & Bills, E. (2007). Explorations in the zone of proximal awareness. In J. Watson & K. Beswick (Eds.), Mathematics: Essential research, essential practice: Proceedings of the 30th annual conference of the Mathematics Education Research Group of Australasia (Vol. 1, pp. 42–58). Adelaide, AU: MERGA.

    Google Scholar 

  • Mason, J., Oliveira, H., & Boavida, A. M. (2012). Reasoning reasonably in mathematics. Quadrante, XXI(2), 165–195.

    Google Scholar 

  • Meyer, D. (2013). The three acts of a mathematical story. Available from: http://blog.mrmeyer.com/2011/the-three-acts-of-a-mathematical-story/. Accessed: 21 February 2017.

  • Moss, J., & Beatty, R. (2006). Knowledge building in mathematics: Supporting collaborative learning in pattern problems. Computer-Supported Collaborative Learning, 1, 441–465.

    Google Scholar 

  • Newton, I. (1707). Arithmetica universalis (Ed. William Whiston). Cambridge, UK: University of Cambridge.

    Google Scholar 

  • O’Brien, T. (2006). What is fifth grade? Phi Beta Kappan. January, 373–376.

    Google Scholar 

  • Open University. (1982). EM235: Developing mathematical thinking. A distance learning course. Milton Keynes, UK: Open University.

    Google Scholar 

  • Papic, M. (2007). Mathematical patterning in early childhood: An intervention study. Unpublished Ph.D. thesis. Sydney, AU: Macquarie University.

    Google Scholar 

  • Papic, M. (2013). Improving numeracy outcomes for young Australian indigenous children through the Patterns and Early Algebra Preschool (PEAP) Professional Development (PD) program. In L. English & J. Mulligan (Eds.), Reconceptualizing early mathematics learning (pp. 253–281). New York: Springer.

    Google Scholar 

  • Papic, M., & Mulligan, J. T. (2007). The growth of early mathematical patterning: An intervention study. In J. Watson & K. Beswick (Eds.), Mathematics: essential research, essential practice: Proceedings of the 30th annual conference of the Mathematics Education Research Group of Australasia (Vol. 2, pp. 591–600). Adelaide, AU: MERGA.

    Google Scholar 

  • Pólya, G. (1945). How to solve it: A new aspect of mathematical method. Princeton, NJ: Princeton University Press.

    Google Scholar 

  • Pólya, G. (1962) Mathematical discovery: On understanding, learning, and teaching problem solving (combined edition). New York: Wiley.

    Google Scholar 

  • Radhakrishnan S. (1953). The principal Upanishads. London: George Allen & Unwin.

    Google Scholar 

  • Renkl, A. (2002). Worked-out examples: Instructional explanations support learning by self-explanations. Learning and Instruction, 12, 529–556.

    Google Scholar 

  • Sangwin, C. (2006). Mathematical question spaces. In M. Danson (Ed.), Proceedings of 10th CAA International Computer Assisted Assessment Conference (pp. 377–386) Loughborough, UK: Loughborough University.

    Google Scholar 

  • Schmittau, J. (2004). Vygotskian theory and mathematics education: Resolving the conceptual-procedural dichotomy. European Journal of Psychology of Education, 19(1), 19–43.

    Google Scholar 

  • Shigematsu, S. (1981). A Zen forest. New York: Weatherhill.

    Google Scholar 

  • Skemp, R. (1976). Relational and instrumental understanding. Mathematics Teaching, 77, 20–26.

    Google Scholar 

  • Stein, M., Grover, B., & Henningsen, M. (1996). Building student capacity for mathematical thinking and reasoning: An analysis of mathematical tasks used in reform classrooms. American Educational Research Journal, 33(2), 455–488.

    Google Scholar 

  • Stein, S. (1987). Gresham’s law: Algorithm drives out thought. For the Learning of Mathematics, 7(2), 2–4.

    Google Scholar 

  • Tahta, D. (1972). A Boolean anthology: Selected writings of Mary Boole on mathematics education. Derby, UK: Association of Teachers of Mathematics.

    Google Scholar 

  • Thorndike, E. (1922). The psychology of arithmetic. New York: Macmillan.

    Google Scholar 

  • Ward, J. (1713). The young mathematician’s guide, being a plain and easie introduction to the mathematicks, in five parts …. London: Thomas Horne.

    Google Scholar 

  • Watson, A., & Mason, J. (2002). Extending example spaces as a learning/teaching strategy in mathematics. In A. Cockburn & E. Nardi (Eds.), Proceedings of 26 th Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 377–385). Norwich, UK: PME.

    Google Scholar 

  • Watson, A., & Mason, J. (2005). Mathematics as a constructive activity: Learners generating examples. Mahwah, NJ: Lawrence Erlbaum.

    Google Scholar 

  • Whiteside, D. (Ed.). (1972). The mathematical papers of Isaac Newton, Vol V 1683–1684. Cambridge, UK: Cambridge University Press.

    Google Scholar 

  • Young, R., & Messum, P. (2011). How we learn and how we should be taught: An introduction to the work of Caleb Gattegno. London: Duo Flamina.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Open University and University of Oxford, 27 Elms Road, Oxford, OX2 9JZ, UK

    John Mason

Authors
  1. John Mason
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to John Mason .

Editor information

Editors and Affiliations

  1. Département de Mathématiques, Université du Québec à Montréal, Montréal, Québec, Canada

    Carolyn Kieran

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Mason, J. (2018). How Early Is Too Early for Thinking Algebraically?. In: Kieran, C. (eds) Teaching and Learning Algebraic Thinking with 5- to 12-Year-Olds. ICME-13 Monographs. Springer, Cham. https://doi.org/10.1007/978-3-319-68351-5_14

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-68351-5_14

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-68350-8

  • Online ISBN: 978-3-319-68351-5

  • eBook Packages: EducationEducation (R0)

Publish with us

Policies and ethics

Search

Navigation