Skip to main content
SpringerLink
Log in

Unknown, placeholder, or variable: what is x?

  • Original Article
  • Published:
Mathematics Education Research Journal Aims and scope Submit manuscript

Abstract

One of the most significant steps in learning algebra is understanding the change in the role of letters in mathematical expressions from unknowns to variables. We describe the historical development of this change in usage, starting with the ancient use of mathematical unknowns, detailing several important changes in practice that allowed for the idea of the placeholder, the birth of symbolic algebra, and the development of the variable. Focusing on these changes in practice, we interpret some classroom examples of 8th-grade students who interpret letters in terms of their experience with unknowns, rather than in terms of variables, to the confusion and dismay of the teachers. We also discuss how particular curricular and pedagogical treatments can support student learning by deliberately focusing on these changes in practice in the transition from unknowns to variables in the middle grades.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

Notes

  1. In 1637 Descartes introduced our current practice of denoting givens or placeholders with letters at the beginning of the alphabet and unknowns or variables with letters at the end of the alphabet.

  2. From W. Oughtred’s The Key of the Mathematicks New Forged and Filed, excerpted in Fauvel and Gray’s anthology.

  3. From Part I, Chapter 11 of N. Oresme’s De configurationibus, Clagett’s translation and commentary.

  4. Note that the Greeks used precisely this kind of ciphered system to represent their numbers: α = 1, β = 2, γ = 3, etc.

References

  • Achieve. (2010). State college- and career-ready high school graduation requirements. Retrieved from http://achieve.org/files/21CCRDiplomaTableJuly2010.pdf.

  • Asquith, P., Stephens, A. C., Knuth, E. J., & Alibali, M. W. (2007). Middle school mathematics teachers’ knowledge of students’ understanding of core algebraic concepts: Equal sign and variable. Mathematical Thinking and Learning, 9(3), 249–272.

    Article  Google Scholar 

  • Bardini, C., Radford, L., & Sabena, C. (2005). Struggling with variables, parameters, and indeterminate objects or how to go insane in mathematics. In H. L. Chick (Ed.), Proceedings of the 29th conference of the international group for the psychology of mathematics education (Vol. 2, pp. 129–136). Melbourne: PME.

    Google Scholar 

  • Billstein, R., & Williamson, J. (2008). Math thematics book 1. Evanston: McDougal Littell.

    Google Scholar 

  • Bombelli, R. (1966). Algebra. Milan: Feltrinelli.

    Google Scholar 

  • Boyer, C. B. (1968). A history of mathematics. New York: Wiley.

    Google Scholar 

  • Brousseau, G. (1997). Theory of didactical situations in mathematics: Didactique des mathématiques, 1970-1990. Dordrecht: Kluwer Academic Publishers.

    Google Scholar 

  • Carlson, M., Jacobs, S., Coe, E., Larsen, S., & Hsu, E. (2002). Applying covariational reasoning while modeling dynamic events: A framework and a study. Journal for Research in Mathematics Education, 33(5), 352–378.

    Article  Google Scholar 

  • Carpenter, T. P., Franke, M. L., & Levi, L. (2003). Thinking mathematically: Integrating arithmetic & algebra in elementary school. Portsmouth: Heinemann.

    Google Scholar 

  • Carraher, D., Schliemann, A., & Brizuela, B. (2006). Arithmetic and algebra in mathematics education. Journal for Research in Mathematics Education, 37(2), 87–115.

    Google Scholar 

  • Chazan, D. (2008). The shifting landscape of school algebra in the United States. In C. E. Greenes & A. Izsak (Eds.), Algebra and algebraic thinking in school mathematics (pp. 19–33). Reston: NCTM.

    Google Scholar 

  • Clagett, M. (Editor & Translator). (1968). Nicole Oresme and the medieval geometry of qualities and motions: A treatise on the uniformity and difformity of intensities known as Tractatus de configurationibus qualitatum et motuum. Madison, WI: University of Wisconsin Press.

  • Clement, J. (1982). Algebra word problem solutions: Thought processes underlying a common misconception. Journal for Research in Mathematics Education, 13(1), 16–30.

    Article  Google Scholar 

  • Collis, K. (1975). The development of formal reasoning. Newcastle, Australia: U. of Newcastle.

  • Datta, B. B., & Singh, A. N. (2001). History of Hindu mathematics: A source book. Parts I and II. New Delhi: Bharatiya Kala Prakashan.

  • Descartes, R. (1637). La geometrie. Translation in Hawking, S., (2005). God created the integers: The mathematical breakthroughs that changed history. Philadelphia: Running Press, 292–364.

  • Edwards, C. H. (1979). The Historical Development of the Calculus. New York: Springer-Verlag.

  • Fauvel, J., & Gray, J. (Eds.). (1987). The history of mathematics: A reader. Washington, D.C.: Open University.

    Google Scholar 

  • Filloy, E., Rojano, T., & Solares, A. (2010). Problems dealing with unknown quantities and two different levels of representing unknowns. Journal for Research in Mathematics Education, 41(1), 52–80.

    Google Scholar 

  • Fosnot, C., & Jacob, B. (2010). Young mathematicians at work: Constructing algebra. Portsmouth: Heinemann.

    Google Scholar 

  • Fujii, T., & Stephens, M. (2008). Using number sentences to introduce the idea of variable. In C. E. Greenes & R. Rubenstein (Eds.), Algebra and algebraic thinking in school mathematics (pp. 127–140). Reston: NCTM.

    Google Scholar 

  • Gay, A. S., & Jones, A. (2008). Uncovering variables in the context of modeling activities. In C. E. Greenes & R. Rubenstein (Eds.), Algebra and algebraic thinking in school mathematics (pp. 211–221). Reston: NCTM.

    Google Scholar 

  • Hake, S. (2007). Saxon math course 2. Orlando: Saxon.

    Google Scholar 

  • Høyrup, J. (2005). Tertium non datur: On reasoning styles in early mathematics. In P. Mancosu et al. (Eds.), Visualization, explanation and reasoning styles in mathematics (pp. 91–121). Dordrecht: Springer.

    Chapter  Google Scholar 

  • ICMI. (2001). Discussion document for the 12th ICMI study. The future of the teaching and learning of algebra. In H. Chick, K. Stacey, J. Vincent, & J. Vincent (Eds.), The future of the teaching and learning of algebra. Proceedings of the 12th ICMI study conference (Vol. 1). Australia: The University of Melbourne.

    Google Scholar 

  • Kaput, J. (1994). Democratizing access to calculus: New routes to old roots. In A. Schoenfeld (Ed.), Studies in mathematical thinking and learning: Mathematical thinking and problem solving (pp. 77–156). Hillsdale: Erlbaum.

    Google Scholar 

  • Kaput, J. (2008). What is algebra? What is algebraic reasoning? In J. J. Kaput, D. W. Carraher, & M. L. Blanton (Eds.), Algebra in the early grades, studies in mathematical thinking and learning (pp. 5–17). New York: Lawrence Erlbaum Associates.

    Google Scholar 

  • Katz, V. (Ed.). (2007). The mathematics of Egypt, Mesopotamia, China, India, and Islam: A sourcebook. Princeton: Princeton University Press.

    Google Scholar 

  • Kilpatrick, J., & Izsak, A. (2008). A history of algebra in the school curriculum. In C. Greenes & R. Rubenstein (Eds.), Algebra and algebraic thinking in school mathematics (pp. 3–18). Reston: NCTM.

    Google Scholar 

  • Klein, J. (1968). Greek mathematical thought and the origin of algebra. New York: Dover.

    Google Scholar 

  • Küchemann, D. (1978). Children’s understanding of numerical variables. Mathematics in School, 7(4), 23–26.

    Google 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 

  • Marcy, S. (1999). Prealgebra with pizzazz!: Practice in skills and concepts. Palo Alto: Creative Publications.

    Google Scholar 

  • Marcy, S. (2002). Elementary math with pizzazz!: Grade 3. New York: McGraw-Hill Education.

    Google Scholar 

  • Marcy, S., & Marcy, J. (2003). Mathimagination. Los Angeles: Marcy Mathworks.

    Google Scholar 

  • Marcy, S., & Marcy, J. (1973). Mathimagination. Palo Alto: Creative Publications.

    Google Scholar 

  • Philipp, R. (1992). The many uses of algebraic variables. The Mathematics Teacher, 85(7), 557–561.

    Google Scholar 

  • Piaget, J., & Garcia, R. (1989). Psychogenesis and the history of science. New York: Columbia University Press.

    Google Scholar 

  • Schoenfeld, A. H. (2002). Making mathematics work for all children: Issues of standards, testing, and equity. Educational Researcher, 31(1), 13–25. doi:10.3102/0013189X031001013.

    Article  Google Scholar 

  • Sfard, A. (1995). The development of algebra: Confronting historical and psychological perspectives. Journal of Mathematical Behavior, 14, 15–39.

    Google Scholar 

  • Stacey, K., & MacGregor, M. (1997). Ideas about symbolism that students bring to algebra. Mathematics Teacher, 90, 110–113.

    Google Scholar 

  • Tabach, M., & Friedlander, A. (2008). The role of context in learning beginning algebra. In C. Greenes & R. Rubenstein (Eds.), Algebra and algebraic thinking in school mathematics (pp. 223–232). Reston: NCTM.

    Google Scholar 

  • Usiskin, Z. (1999). Why is algebra important to learn? In B. Moses (Ed.), Algebraic thinking, grades K-12 (pp. 22–30). Reston: NCTM.

    Google Scholar 

  • Wagner, S. (1981). Conservation of equation and function under transformations of variable. Journal for Research in Mathematics Education, 12, 197–218.

    Article  Google Scholar 

  • Wagner, S., & Parker, S. (1999). Advancing algebra. In B. Moses (Ed.), Algebraic thinking, grades K-12 (pp. 328–340). Reston: NCTM.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Idaho, 314 Brink Hall, P.O. Box 441103, Moscow, ID, 83844, USA

    Robert Ely

  2. Department of Curriculum and Instruction, University of Idaho, 405C Education, P.O. Box 443082, Moscow, ID, 83844, USA

    Anne E. Adams

Authors
  1. Robert Ely
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Anne E. Adams
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Robert Ely.

Additional information

The research reported here was supported by the U.S. National Science Foundation, through grant REC 0529502—Coordinating Social and Individual Aspects of Generalizing Activity: A Multi-tiered ‘Focusing Phenomena’ Study. The PIs were Amy Ellis (University of Wisconsin-Madison) and Joanne Lobato (San Diego State University).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ely, R., Adams, A.E. Unknown, placeholder, or variable: what is x?. Math Ed Res J 24, 19–38 (2012). https://doi.org/10.1007/s13394-011-0029-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13394-011-0029-9

Keywords

Search

Navigation