Mathematics needn’t be fancy

Dr. Chris Brownell
Q.E.D.
Published in
3 min readOct 31, 2018

--

I have been thinking about a theory of the genesis of student errors in mathematical thinking. I want to call it something like “The disposition towards viewing mathematics as needing fancy ways to say things” As an instance of this consider this interchange related in a study on Prime number representations:

Students are told a number F is equal to 151 X 157 and asked, “Is F prime? Yes/No and explain your decision.” 14 out of the 74 students who correctly said No gave an answer like,

“The square root of 23707 (151X157) is approximately 153.9 so we check to see if any primes less than 153 divide 23707. We find that 23707 is divisible by 151 and also 157 so it is not prime.”

This is, strictly speaking not wrong. However, it seems to overcomplicate matters immensely. Here’s another:

“F is not prime because 151 and 157 are primes and the set of prime numbers is not closed under multiplication.”

Here the student almost sounds like they are channeling Sheldon from Big Bang Theory in the hopes of impressing the teacher with their encyclopedic knowledge of characteristic of multiplication of numbers in various sets.

Here’s what I mean…

I am pretty certain that these students understand the complementary nature of primes and composites, but are missing one of the most beautiful aspects of mathematics, SIMPLICITY. Furthermore, I don’t think that I can blame the students because, while trying to communicate mathematics teachers tend to overemphasize some ideas and fracture others, all in the hope of making things clearer.

Clearly, these students know a definition of primality and composite numbers that they have made useful for themselves. But they have been taught somewhere along the way that a simple answer is somehow “less than” one that sounds “fancy.” Hence they have developed the disposition towards complicated statements and round-about justifications. But wow!

The question of “Is F=151 X 157 Prime/explain?” is best answered by something akin to: “No, look you have two numbers multiplied together there, that means F is composite.” (I like the more British way of saying this, ‘full-stop’ here.) Nothing more need be said, and you have communicated both a well-reasoned explanation and your command of the facts.

As Albert Einstein is often paraphrased,

Things should be made as simple as possible, but no simpler…

Here’s my problem, please help…

How do I encourage my students to adopt the mathematical aesthetic of cleanliness and simplicity? How do we counteract this “Disposition toward fancy explanations”?

Mathematics, for me anyway, has been the ultimate pursuit of laziness. If I can say everything I need to say in a few symbols or words SCORE! If I need multiple lines to get the point across, perhaps I am either covering a lack of understanding or am confused.

While I have cited quotes from a research article, I have anecdotal experience with this myself. Having witnessed students dance about in their thoughts, and putting everything plus the kitchen sink into their explanations. Part of my job as a mathematics instructor is to encourage the love of this aesthetic and proficiency.

This disposition, while presented here with the intention of good humor, can be problematic as well. If you believe that your good thinking is somehow deficient because it is simple, then there we as mathematics teachers have done you a disservice, and your experiences ought to be re-examined. Cloudy thinking and needless convolutions do in fact lead to terribly misguided decisions (but I won’t get political in this piece).

Reference:

Zazkis, R., Liljedahl, P. (2004). Understanding Primes: The role of representation. Journal of research in mathematics education Vol. 35, No 3 p. 164–186. Published by NCTM.

--

--

Dr. Chris Brownell
Q.E.D.

Professor of Math & STEM Education, Co-Author with Sunil Singh of “Math Recess Playful Learning in an Age of Disruption.