Thinking Mathematically About Mathematics

By Robert Berkman,
MCS Math Coordinator

If you’ve been reading the Sunday Times over the summer, you may have come across an article by Andrew Hacker entitled “Is Algebra Necessary?”

In his piece, Hacker argued that for many high school and college students, algebra was an “onerous stumbling block” that prevented students from pursuing more useful forms of mathematics, particularly statistics and finances. Hacker concluded the article by likening algebra to a huge boulder that we all had to pull, with little payback beyond high dropout rates and negative attitudes towards studying mathematics as a profession.

I have taught algebra for most of my nearly three decades in education, and I have returned to this question again and again: “Is algebra necessary.” After reading Hacker’s essay, I would answer his question with some of my own: Are sonnets “necessary?” Is cubism “necessary?” Is bebop “necessary?” Algebra is no more “necessary” than any of those; however, that doesn’t make it (or them) unimportant.

In my response to this article, I made the case that we should be careful to distinguish between what is “algebra” and what is “algebraic thinking.” Since this is my first blog post for Manhattan Country School, I’d like to expand on this dichotomy and write briefly about what distinguishes “mathematics” from “mathematical thinking.” This point was brilliantly brought across in an essay by Keith Devlin, in which he postulated the following: “Mathematical thinking is a whole way of looking at things, of stripping them down to their numerical, structural, or logical essentials, and of analyzing the underlying patterns.” Note that there is no mention of doing long division, factoring polynomials or expanding a Taylor Series. To Devlin, mathematical thinking is not so much about content as it is about methodology. That is, mathematics is a style of thinking which transcends the language of numbers, operations and equations.

What Devlin is careful to state is that it is entirely possible to be adept at thinking mathematically, yet have little proficiency with actual mathematics; that is, we can use underlying mathematical tools to strip away the superficial features of the world and analyze the patterns we discern without having to know anything how to divide fractions. Devlin deserves the gratitude of mathematics educators everywhere for sharing this essential truth, for it relieves us of our continuous burden to justify a profession in which the subject matter instills so much anxiety and anger in its participants.

Perhaps it’s time for you, a concerned parent, to ratchet up your skills as a mathematical thinker. Okay, maybe you’re not ready to go back in the classroom with a new notebook and sharpened #2 pencil, but Keith Devlin is offering a free “massively open online course” through Coursera called “Introduction to Mathematical Thinking.” This is an opportunity for all of us to improve our understanding of this thing called mathematics from the comfort a computer screen. The course starts on Monday, September 17th, and yes, it is truly free. If your last experience with mathematics was the dreaded algebra described by Andrew Hacker, maybe it’s time to try again with an expert like Keith Devlin and see if it will change your mind.

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