By Keith Devlin @profkeithdevlin

Fundamental misunderstandings early in our education can sometimes have long-term effects. Mathematics instructors in higher education regularly encounter students who harbor incorrect assumptions that can significantly hamper their progress. To give just one example, the false belief that multiplication is repeated addition is one I have written about a number of times. [See my January 2011 Devlin’s Angle post, though the links to early posts given there no longer work, so you will have to search the Archives by date.] Even when the claim is restricted to integer arithmetic, the belief that multiplication is an additive operation can lead to a plethora of conceptual misunderstandings.

On a more general level, the assumption that the core of mathematics is numerical calculation, or the successful manipulation of symbolic expressions (formulas and equations) and diagrams, can leave students unable to see and experience the rich and rewarding mental world that mathematicians know and love.

I often compare mathematics with music to try to make people aware that they are missing something. “Imagine that you had to master musical notation, both reading and writing, before being able to appreciate or create music,” I say. “If that were the case, few of us would like, or want to create, music.”

But our senses (in particular our hearing, for most of us) provide a direct pathway to the wonderful world of music. Musical notation is a useful tool, but its mastery is not the only entry portal, indeed not the main pathway. With mathematics, in contrast, for the most part, mastery of the notation — the language developed to do mathematics — is the only way in. [Modern technologies can bypass the symbols to provide some access, but only to limited, more elementary parts of mathematics. See Figure 1.]

For the mathematician, a page full of abstract symbols can convey — and bring alive — an entire mathematical domain, result, or problem. We see the symbols and experience a mathematical symphony.

Sadly, I never mastered musical notation to be able to do the same with music. And therein lies this month’s lament. For me, my early encounter with music instruction at high school (i.e., post age 11 in the UK in the 1960s) was a frustrating turnoff to music, as math class is for many others.

I was not turned off listening to and enjoying music. Heavens, I was growing up in England in the “Swinging Sixties”! But I had a deep-felt impression that, beyond the enjoyment my ears provided, I was lacking the “music gene” required to really understand it (and hence become a musician). A particular consequence was that I resisted all attempts to get me to learn an instrument other than the drums, which I took to with a passion, believing that my powerful sense for rhythm provided me with the only opportunity to participate. [In the end, when I went to London University to study mathematics in 1965, I abandoned music altogether (other than listening to it), while contemporary fellow London student Bryan May continued to create music while studying physics. That’s one reason why he became part of Queen and I didn’t — okay, there may be other factors!]

It was only many decades later that I began to understand how my first encounter with Music Theory in high school left me with a fundamental misunderstanding that influenced my entire life.

Since my attempts to understand it were all to no avail, I went through my life with a deep sense that I was missing something that most other people could get.

I also realized that my experience with music was in some ways parallel to the way the early encounters with math class lead many to develop similar feelings about mathematics.

In fact, I think there is even more to the similarity. Let me start with the way both mathematics and music were taught to me. (In retrospect, it’s pretty appalling, but it’s how things were in the UK back then.) Both subjects were taught by what is now referred to as direct instruction (DI), an approach which, while extremely efficient for students sufficiently far along the learning curve, can be disastrous for a beginning learner. My divergent experiences with math and music highlight the major problem with DI.

In both classes, we were presented with some initial vocabulary and grammatical rules, and given a series of routine, repetitive tasks to help us develop familiarity, interspersed with quizzes, homework assignments, and tests.

Being an intense kid who wanted to learn and, above all, understand, I asked questions. Why do we group in tens when we count? Why are there eight notes in an octave? [The fact that I remember this many decades later should indicate that this was a defining moment in my life. Momentous, in hindsight.]

This was for me the fork in the road. My math teacher explained that our ancestors counted on their fingers and when they ran out of fingers they recorded that group of ten and began again. The adult me can see all kinds of other questions I might have asked, and I probably did, but that first one set the tone: in math there are seemingly arbitrary rules, but there is a rationale. Perhaps not immediately understandable, but if you keep trying, you’ll eventually get it. Math is a subject where, if you put in the effort, you will eventually get it. It makes sense. Learning math (properly) is, at heart, sense making.

In response to my question about the octave, my music teacher said, “That’s just the way it is,” or words to that effect. “But why? Why eight?” I reacted. Of course, I didn’t give up right away. I put in some effort to try to find out why there are eight notes in an octave. In the process, I learned that an octave is the interval where the frequency doubles. So, there was a good reason for having a notion of an octave. But why split it into eight subunits? Surely, 4 or 16 would be just as rational?

I never got a satisfactory answer. It kept coming back to me as, “Because that’s what musicians do!” Eventually, I stopped asking. I felt I was on the outside of a cabal where everyone else could see something I could not. As a result, while they went on to acquire more and more knowledge of music and were able to conduct discussions filled with impressive sounding terminology, I just keep listening and enjoying, and simply endured classes where we all had to play the recorder, reinforcing my sense that music was a closed shop to me. I didn’t have the gene.

In fairness to that music teacher, he was up against a problem, in that my early encounters with math were several years before those with music. I had learned that, even when we moved on to some new math that initially did not make sense, provided I kept at it, and became fluent at the mechanics, I could solve all the problems, and eventually the understanding would come. Rapid, if not instant, gratification. I mastered the Direct Instruction game. Follow the instructions and first you can just do it and in time you’ll understand it as well. It all makes sense.

Moreover, the tasks we were given, the problems to be solved, all had unique answers. Everything was right or wrong. Clean and logical. No ambiguity. I bought the package. And I loved it. I still do.

Not surprisingly, when I got to Music Theory a few years later, which was taught in the same DI manner, I approached it in the same way. I expected it to be the same. Just learn the rules and apply them. But when some of the most basic rules seemed entirely arbitrary, and the only answer I got to me “Why?” questions was “Because that’s the way it is,” I started to flounder. I assumed that my problem with music was analogous to the problems some of my classmates had always had with math. They could not see the underlying logic in math that I did, and I could not see the underlying logic I thought they did in music. So, I did in music what many of them did in math: I gave up on Music Theory and focused on what I could understand and do well, namely math.

Fast forward many decades and the problem I had back then becomes crystal clear. School math back then was, and even today largely still is, prescriptive. You start with definitions and rules, and you work within that framework. Music Theory was, and is, descriptive. It sets out to provide a formal description of a human activity. I was fooled by the method of presentation. We were not being taught the rules that determine what music is (the way math was taught), we were being provided a framework to describe it.

How different my life might have been if I had not been put off music at an early age and assumed I didn’t have what it takes. [Even so, I doubt I would have been part of Queen; besides, I did play drums so in theory it could have been me sitting behind Bryan May instead of Roger Taylor.]

The irony is, most uses of mathematics by far are descriptive — not prescriptive. Yet, in my day, and to a large extent still today, mathematics is taught in schools (and well into the college years) in a prescriptive fashion.

There are benefits to that approach, to be sure. It is fast and focused, and it provides students with exposure to the vast, rich field we call Pure Mathematics. Many of us take to it like ducks to water and love it.

But when mathematics is presented to students only in a prescriptive fashion, it leaves the math-loving ducks largely ignorant of what mathematics is as a global, human activity, and for the much greater segment of society that are not “into math” it can create an impression that it is all a classroom game having little relevance to their lives. Neither group is served well. Time to change.

Which is the theme I have been pushing in Devlin’s Angle in every post this year starting with May (the month, not the rock star).