**Commenced**in January 2007

**Frequency:**Monthly

**Edition:**International

**Paper Count:**31903

##### The Fallacy around Inserting Brackets to Evaluate Expressions Involving Multiplication and Division

**Authors:**
Manduth Ramchander

**Abstract:**

Evaluating expressions involving multiplication and division can give rise to the fallacy that brackets can be arbitrarily inserted into expressions involving multiplication and division. The aim of this article was to draw upon mathematical theory to prove that brackets cannot be arbitrarily inserted into expressions involving multiplication and division and in particular in expressions where division precedes multiplication. In doing so, it demonstrates that the notion that two different answers are possible, when evaluating expressions involving multiplication and division, is indeed a false one. Searches conducted in a number of scholarly databases unearthed the rules to be applied when removing brackets from expressions, which revealed that consideration needs to be given to sign changes when brackets are removed. The rule pertaining to expressions involving multiplication and division was then extended upon, in its reverse format, to prove that brackets cannot be arbitrarily inserted into expressions involving multiplication and division. The application of the rule demonstrates that an expression involving multiplication and division can have only one correct answer. It is recommended that both the rule and its reverse be included in the curriculum, preferably at the juncture when manipulation with brackets is introduced.

**Keywords:**
Brackets,
multiplication,
division,
operations,
order.

**References:**

[1] G. Vivinetto, “Can you solve this math equation stumping the internet?” Today, 2 August 2019. Available: https://www.today.com/popculture/math-equation-stumping-folks-internet-t16008.

[2] G. Chrystal, Algebra: An Elementary Text-Book for the Higher Classes of Secondary Schools and for Colleges, Volume 1, Edinburg, 1904.

[3] S. Strogatz, “The equation that tried to stump the internet,” The New York Times, 2 August 2019. Available: https://www.nytimes.com/2019/08/02/science/math-equation-pedmas-bemdas-bedmas.html.

[4] S. LeConte, S, “Everyone’s arguing over this very simple math equation –here’s why it’s going viral,” BuzzFeed, 2 August 2019. Available: https://www.buzzfeed.com/stephenlaconte/viral-math-equation-controversial-pemdas.

[5] S. Strogatz, “That Vexing Math Equation? Here’s an Addition,” The New York Times, 5 August 2019. Available: https://www.nytimes.com/2019/08/05/science/math-equation-pemdas-bodmas.html.

[6] A. Daniels, “This Simple Math Problem Drove Our Entire Staff Insane. Can You Solve It?” Popular Mechanics, 31 July 2019. Available: https://www.popularmechanics.com/science/math/a28569610/viral-math-problem-2019-solved/.

[7] N. de Mestre, “Discovery with Neville de Mestre,” Australian Mathematics Teacher, vol. 65, no. 3, pp. 20- 21, 2009.

[8] C. Florian, A History of Mathematical Notations, Chicago: Open Court Pub. Co., 1928.

[9] R. Gunnarsson and A. Karlsson, Brackets and structure sense, School of Education and communication, Jonkoping University, 2014.

[10] A. Karlsson and R. Gunnarsson, “Students perceptions of brackets,” In Lindmeier, A.M. and A. Heinze. (Eds.), Proceedings of the 37th Conference of the International Group for the Psychology of Mathematics Education, vol. 40, no. 2, pp. 173-176, 2013.

[11] C. Kieran, C, “Children’s operational thinking within the context of bracketing and the order of operations,” In D. Tall (Ed.) Proceedings of the 3rd Conference of the International Group for the Psychology of Mathematics education, pp. 128-133, Warwick. UK: PME, 1979.

[12] R. Gunnarsson, B, Hernell and W.W. Sonnerhed, “Useless brackets in arithmetic expressions with mixed operations,” in Proceedings of the 36th Conference of the International Group for the Psychology of Mathematics Education, 2012, pp. 275-282, 2012.

[13] K.M. Robinson and J. LeFevre, “The inverse relationship between multiplication and division: Concepts, procedures, and a cognitive framework,” Educational Studies in Mathematics, vol. 79, no. 1, pp. 409-428, 2012.

[14] Robinson, K.M., Dube, A.K., and J. Beatch, “Children’s multiplication and division shortcuts: Increasing shortcut use depends on how shortcuts are evaluated,” Learning and Individual Differences, vol. 49, no. 1. pp. 97-304, 2016.

[15] K.N. Joseph, “College students’ misconceptions of the order of operations,” Unpublished master’s thesis, State University of New York, Fredonia, New York, 2014.

[16] D. Gordon, G, Achiman and D. Melman, “Rules in Mathematics,” Mathematics in School, vol.10, no, 3. pp. 2-4, 1981.