What topics should be covered in high school mathematics? What can we learn from advanced math?

By Keith Devlin @profkeithdevlin

This month’s column is about what topics to cover in high school mathematics. I say that up front because that’s not how I am going to begin. But bear with me. Everything I say will be relevant.

Earlier this month I attended an international mathematics research conference at the Fields Institute in Toronto, Canada. The occasion was the fiftieth anniversary of the (University of) Toronto Set Theory Seminar. I spent three lengthy periods as a visiting faculty member at the University of Toronto in the late 1970s and early 1980s, participating actively in the seminar. Though I left the field in the mid-1980s, the moment I heard about the anniversary celebration, I could not resist registering my interest in attending. Though I had left the field, most of the mathematicians who were part of my mathematical world back then had not, and a fair number of them were going to be present. It would be wonderful to meet up with them again. In the event, the organizers graciously sent me an invitation, together with a request to give both a seminar talk (I had a topic that was relevant, though far from central) and an associated RCI - Fields Institute Public Lecture.

Though an absence from the field of thirty-five years left me unable to follow any of the talks except in a very superficial way, I greatly enjoyed being back in that particular intense, research domain for a few days. I had, after all, been immersed in it for almost a quarter of a century of my life.

A number of talks aroused my interest in particular, being on specific research questions I had worked on. Among them was a talk titled Analytic Quasi-Orders and Two Forms of Diamond, given by Prof Assaf Rinot of Bar-Ilan University in Israel. I did not know Prof Rinot (he is much younger than me, from a later generation of set theorists), but I did know his research advisor.

Slide 16 from Prof Rinot’s seminar talk

Slide 16 from Prof Rinot’s seminar talk

I struggled (mostly in vain) to follow the details in a talk on a topic I once had at my fingertips. That I had anticipated. What I did not expect was a slide Prof Rinot put up at the very end of his talk (slide 16 of 18), shown here. I had absolutely no recollection either of formulating that rather complex looking definition or proving that theorem. And I had only a very vague sense of what it all meant. (I could not remember exactly what constituted the “non-ineffable case”.) I was also taken aback (though pleased) when Rinot went on to say that the result had recently been observed to be useful in more recent work. For those interested, a video of Prof Rinot’s talk is available on the conference archive. 

So why do I recount this personal story? The point of relevance is that the concepts and specific methods in that branch of mathematics are almost entirely separate from anything I studied and mastered either at school or as a mathematics undergraduate, and have only minimal overlap with courses I took as a graduate student. Rather, I acquired all that specialized knowledge and knowhow “on the job,” as I followed various research leads.

After I left set theory in the mid-1980s, I focused on very different mathematical problems in the real world, first in industry, then, after the September 11, 2001 attack on the World Trade Center, in a series of Defense Department projects for the CIA, the US Navy, and the US Army. Again, I acquired the specialist knowledge and knowhow to work in those domains “on the job”. [That was the research I talked about in my conference presentation. See here for an abstract of my talk; at the time of writing, the video of my talk has not yet been processed and published on the conference archives. As you will notice from the talk’s title, my DoD work was inspired by my earlier work in set theory, but was very different in many essential ways.]

The point is, although I have been professionally engaged in mathematics since I began work on my doctoral degree in 1968, essentially none of the concepts, definitions, or methods I learned (and practiced) either in school or in my undergraduate mathematics degree played any role in any of that professional work. (That, by the way, is typical.)

You may, therefore, be tempted to think that my entire mathematical education was a waste of time. Not at all. Without it, I could not possibly have done any of that subsequent research in set theory. For what I took away from all those years of struggle in high school and university was the ability to think mathematically.

Likewise, the very detailed, expert knowledge of set theory I acquired in my many years research in that field proved to be invaluable in my work for the DoD, even though the two domains are about as far apart as you could imagine. (My work in set theory was about the properties of sets of higher orders of infinity, my work for the DoD focused on improving intelligence analysis.)

[In fact, it was some earlier research I had done based on set theory that led to me being asked to work on those post-9/11 projects. Someone at the DoD had seen the potential relevance to intelligence analysis—actually, far more likely someone at a Defense contractor organization.]

So what does all this have to do with high school mathematics education?

First, what is taught is not, in itself, of any significance. The chances of anyone who finds they need to make use of mathematics at some point in their life or career being able to use any specific high school curriculum topic is close to zero. In fact, by the time a student today graduates from university, the mathematics they may find themselves having to use may well have not been developed when they were at school. Such is the pace of change today.

Second, what is crucial to effective math learning is what is sometimes called “deep learning”; the ability to think fluidly and creatively, adapting definitions and techniques already mastered, reasoning by analogy with reasoning that has worked (or not worked) on similar problems in the past), and combining (in a creative fashion) known approaches to a novel situation. 

But here’s the rub. The mass of evidence from research in cognitive science tells us that the only way to acquire that all important “deep learning” is by prolonged engagement with some specific mathematics topics. So, to be effective, any mathematics curriculum has to focus on a small number of specific topics.

But according to my first remark, there is no set of “most appropriate topics,” at least in terms of subsequent  “applicational utility”.

So what to do? How should we determine a curriculum? That’s my topic in next month’s post. (Hint: There are other educationally important criteria besides utility.)