Finding Meaning in Mathematical Mnemonics

By Miller, Geoffrey; Obara, Samuel | Australian Mathematics Teacher, Fall 2017 | Go to article overview

Finding Meaning in Mathematical Mnemonics


Miller, Geoffrey, Obara, Samuel, Australian Mathematics Teacher


Finding meaning in mathematical mnemonics

A mathematical mnemonic is a visual cue or verbal strategy used to aid initial memorisation and recall of a mathematical concept or procedure. Cross-multiplication, FOIL, and PEMDAS are common examples of mathematical mnemonics. Used wisely, mathematical mnemonics can increase performance and understanding, particularly within special education populations, about which the majority of research on mathematical mnemonics has been published (Greene, 1999; Kavale & Forness, 2000; Mastropieri, Scruggs, & Levin, 1987). In my experience, though, mathematical mnemonics have worked well in general student populations, too.

However, on occasion, mathematical mnemonics can be over-emphasised at the expense of conceptual fluency (Karp, Bush, & Dougherty, 2014; Kilpatrick, Swafford, & Finell, 2001). This is symptomatic of a difficult situation in some classrooms: "... too much focus is on learning procedures without any connection to meaning, understanding, or the applications that require these procedures" (National Council of Teachers of Mathematics, 2014, p. 3). Dr. Diane J. Briars (2014, July), the past president of the National Council of Teachers of Mathematics (NCTM), believes the overuse of mathematical mnemonics can impede some students' progress. When mathematical mnemonics are detached from conceptual understanding, they can become colourful tricks that usually get the right answer but distract from the meaning of mathematics.

Kalder (2012) and many others have seen first-hand how mathematical mnemonics can sometimes mislead students to view mathematics as a magical set of formulas to be memorised and applied without much thought to the matter. That is why educators such as Cardone (2013) have recently suggested eliminating mathematical mnemonics entirely from curricula. However, such a drastic approach is impractical and overlooks many of the positive findings of research (For example, Everett, Harsy, Hupp, & Jewell, 2014; Manalo, Bunnell, Stillman, 2000).

Indeed, students create and exchange mathematical mnemonics with one another quite naturally. Not only that, but encouraging students to explore how mathematical mnemonics work can provide them with opportunities to engage in proof and reasoning and, in fact, deepen their conceptual understanding.

Picture, for example, the following scenario culled from my various teaching and tutoring experiences over the years. This scenario is not a straightforward retelling of my interaction with a single student; rather, this scenario is an exercise in seeing how an instructor could use a mathematical mnemonic to initiate deeper discussions about mathematical meaning. It imagines a discussion with an introductory calculus student named John about his "butterfly method" for adding and subtracting fractions.

Dissecting the butterfly

John brings the butterfly method to my attention during a tutoring session. We are working on simplifying the rational expression:

[x.sup.2]/x - 2 + 3x + 1/[x.sup.2]

John draws curves and ellipses around the expression (see Figure 1) on a dry-erase board, and then writes in another place, without intermediate steps:

[x.sup.2]/x - 2 + 3x + 1/[x.sup.2] = (x - 2)(3x + 1) + ([x.sup.2])([x.sup.2])/(x - 2)([x.sup.2])

John is asked what he has done. "The butterfly method," John replies. "Did I get the right answer? I mean, it looks right. You said rational expressions work just like fractions, except you don't know what the x is. So, I thought you could just add them just like fractions, and you always add fractions using the butterfly method."

Ignorance about the butterfly method is expressed. The curves and ellipses drawn were taken as examples of fraction with no variables.

"Sure, no problem. It's easy."

He smiles and begins to write (see Figure 2).

"Say you want to add three quarters and two fifths. …

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