# Order of operations: On convention and met-before acronyms

## Abstract

In our exploration of the order of operations we focus on the following claim: “In the conventional order of operations, division should be performed before multiplication.” This initially surprising claim is based on the acronym BEDMAS, a popular mnemonic used in Canada to assist students in remembering the order of operations. The claim was voiced by a teacher and then presented for consideration to a class of prospective elementary school teachers. We investigate the participants’ understanding of the order of operations, focusing on the operations of multiplication and division. We report on participants’ ways of resolving a cognitive conflict faced as a result of relying on memorized mnemonics.

## Keywords

Order of operations Cognitive conflict Relearning Met-before## References

- Ameis, J. A. (2011). The truth about PEMDAS.
*Mathematics Teaching in the Middle School*,*16*(7), 414–420.Google Scholar - Bachelard, G. (1938/1986). The formation of the scientific mind: A contribution to a psychoanalysis of objective knowledge. Boston: Beacon Press.Google Scholar
- Bachelard, G. (2002).
*The formation of the scientific mind*. Manchester: Clinamen Press.Google Scholar - Bay-Williams, J. M., & Martinie, S. L. (2015). Order of operations: The myth and the math.
*Teaching Children Mathematics*,*22*(1), 20–27.CrossRefGoogle Scholar - Blando, J. A., Kelly, A. E., Schneider, B. R., & Sleeman, D. (1989). Analyzing and modeling arithmetic errors.
*Journal of Research in Mathematics Education*,*20*(3), 301–308.CrossRefGoogle Scholar - Chernoff, E., & Zazkis, R. (2011). From personal to conventional probabilities: From sample set to sample space.
*Educational Studies in Mathematics*,*77*(1), 15–33.CrossRefGoogle Scholar - Dupree, K. M. (2016). Questioning the order of operations.
*Mathematics Teaching in the Middle School*,*22*(3), 152–159.CrossRefGoogle Scholar - Erlwanger, S. (1973). Benny’s conception of rules and answers in IPI Mathematics.
*Journal of Children’s Mathematical Behavior*,*1*, 7–26.Google Scholar - Flavell, J. H. (1977).
*Cognitive development*. Englewood Cliffs, NJ: Prentice-Hall.Google Scholar - Glidden, P. L. (2008). Prospective elementary teachers’ understanding of order of operations.
*School Science and Mathematics*,*108*(4), 130–136.CrossRefGoogle Scholar - Herscovics, N. (1989). Cognitive obstacles encountered in the learning of algebra. In C. Kieran & S. Wagner (Eds.),
*Research issues in the learning and teaching of algebra*(pp. 60–86). Hillsdale, NJ: Lawrence Erlbaum Associates.Google Scholar - Hewitt, D. (1999). Arbitrary and necessary: A way of viewing the mathematics curriculum.
*For the Learning of Mathematics*,*19*(3), 2–9.Google Scholar - Hewitt, D. (2012). Young students learning formal algebraic notation and solving linear equations: Are commonly experienced difficulties avoidable?
*Educational Studies in Mathematics*,*81*(2), 139–159.CrossRefGoogle Scholar - Kalder, R. S. (2012). Are we contributing to our students’ mistakes?
*Mathematics Teacher*,*106*(2), 90–91.CrossRefGoogle Scholar - Kontorovich, I., & Zazkis, R. (2017). Mathematical conventions: Revisiting arbitrary and necessary.
*For the Learning of Mathematics, 37*(1).Google Scholar - Leikin, R., & Zazkis, R. (2010). Teachers’ opportunities to learn mathematics through teaching. In R. Leikin & R. Zazkis (Eds.),
*Learning through Teaching Mathematics: Developing teachers’ knowledge and expertise in practice*(pp. 3–22). Dordrecht, Netherlands: Springer.CrossRefGoogle Scholar - Linchevski, L., & Livneh, D. (1999). Structure sense: The relationship between algebraic and numerical contexts.
*Educational Studies in Mathematics*,*40*, 173–196.CrossRefGoogle Scholar - Martin, L. C., & Towers, J. (2016). Folding back, thickening and mathematical met-befores.
*Journal of Mathematical Behavior*,*47*, 89–97.CrossRefGoogle Scholar - McGowen, M., & Tall, D. (2010). Metaphor or met-before? The effects of previous experience on the practice and theory of learning mathematics.
*Journal of Mathematical Behavior*,*29*(3), 169–179.CrossRefGoogle Scholar - Musser, G. L., Burger, W. F., & Paterson, B. E. (2006).
*Mathematics for elementary teachers: A contemporary approach*. New York: Wiley.Google Scholar - Pappanastos, E., Hall, M. A., & Honan, A. S. (2002). Order of operations: Do business students understand the correct order?
*Journal of Education for Business*,*78*(2), 81–84.CrossRefGoogle Scholar - Piaget, J. (1977).
*The development of thought: Equilibration of cognitive structures*. New York: Viking.Google Scholar - Sierpinska, A. (1994).
*Understanding in mathematics*. London: Falmer Press.Google Scholar - Shahbari, J. A., & Peled, I. (2015). Resolving cognitive conflict in a realistic situation with modeling characteristics: Coping with a changing reference in fraction.
*International Journal of Science and Mathematics Education*,*13*(4), 891–907.CrossRefGoogle Scholar - Tall, D. (2013).
*How humans learn to think mathematically: Exploring the three worlds of Mathematics*. New York, NY: Cambridge University Press.CrossRefGoogle Scholar - Tirosh, D., & Graeber, A. O. (1990). Evoking cognitive conflict to explore preservice teachers’ thinking about division.
*Journal for Research in Mathematics Education*,*21*(2), 98–108.CrossRefGoogle Scholar - Van de Walle, J. A., Folk, S., Karp, K. S. Bay-Williams, J. M. (2011).
*Elementary and middle school mathematics: Teaching developmentally. 3rd Canadian Edition.*Toronto: Pearson Canada.Google Scholar - Watson, J. (2007). The role of cognitive conflict in developing students’ understanding of average.
*Educational Studies in Mathematics*,*65*(1), 21–47.CrossRefGoogle Scholar - Waxter, M., & Morton, J. B. (2011). Cognitive conflict and learning. In N. M. Seel (Ed.),
*Encyclopedia of the sciences of learning*(pp. 585–587). New York: Springer.Google Scholar - Zaslavsky, O. (1997). Conceptual obstacles in the learning of quadratic functions.
*Focus on Learning Problems in Mathematics*,*19*(1), 20–44.Google Scholar - Zazkis, R. (2011).
*Relearning mathematics: A challenge for prospective elementary school teachers*. Charlotte, NC: Information Age Publishing.Google Scholar - Zazkis, R., & Kontorovich, I. (2016). A curious case of superscript (−1): Prospective secondary mathematics teachers explain.
*Journal of Mathematical Behavior*,*43*, 98–110.Google Scholar - Zazkis, R., Liljedahl, P., & Gadowsky, K. (2003). Students’ conceptions of function translation: Obstacles, intuitions and rerouting.
*Journal of Mathematical Behavior*,*22*(4), 437–450.CrossRefGoogle Scholar

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