By Keith Devlin @KeithDevlin@fediscience.org, @profkeithdevlin.bsky.social

“What’s it like to be a student in my class?” I’ve periodically asked myself that question many times throughout my long career as a mathematics educator.

The longer I taught, the greater became the disconnect between how I had perceived mathematics when I was first learning it, and how mathematics must appear to my students. After all, those mostly young people were the same age every year, mostly between 18 and 21, but I was steadily getting older. And the world was changing. To do my job well, I needed to know how to motivate them.

For sure, their view of the role of mathematics in the world today—if indeed they had one—was surely different from the one I had when I was their age. If only for the simple reason that mathematics was constantly evolving and finding new uses. But that was not a source of a disconnect; we were all living in the same world after all.

No, the issue that bothered me was how they saw mathematics being done by people around them. Throughout my own education, and my entire career as a mathematician and educator, the introduction of more and more technological tools for mathematicians to use kept changing the landscape. Regardless of how much they knew about mathematics (from older friends, family members, the media, advertising, etc.), each generation of students I taught had access to more and more tools I never had at their age.

When the first hand-held electronic calculator appeared in 1970, I was approaching the completion of my Ph.D. For my entire school and university education, mental arithmetic and paper-and-pencil calculation were critical skills required for everyday life, and school subjects such as mathematics and physics required knowing how to use logarithm tables and slide rules. Not only did I need to know how to apply those methods and use those tools, they were part of my perception of what mathematics is and what it means to do math, including using mathematics in everyday life.

None of the students I ever taught had that perception of mathematics. For them, numerical calculation was something you do using a cheap, hand-held device. (That’s how I did it once those tools were available.) For my students, mastery of numerical calculation was no longer a self-evidently valuable skill. “Why do we have to learn this?” became a familiar call.

There are at least two very good answers to that particular question (“experience of algorithms” and “number sense”), though going by my own children’s school experience, it appeared that few teachers gave either. (The second term, “number sense”, came into use only in the 1980s.) As a result, all of a sudden we were in a world where many students started their mathematical education viewing it as a useless subject having little relevance to their lives.

Interestingly, I recall my arithmetically-proficient parents and grandparents say they could never do algebra and did not understood why they were taught it. (Most of my middle-school colleagues said the same.) Performing calculations in which there was an unknown number made no sense to them. (In retrospect, my finding it actually pretty cool was an early indicator of my subsequent career choice.) They were unable to see any reason to master the method used and developed (though not invented) by merchants and civil engineers in ninth century Baghdad to solve real world problems. But those kinds of problems only arose in certain professions, so my family’s perception was understandable. (I was the first person in my entire family to graduate high school, let alone go to university.)

In any event, given the way we are wired, if we don’t see why we should learn something, we are unlikely to put much effort into it. For a student of any age, passing a required course does provide a reason (of sorts), but most of us would invest the minimal effort to do so. And if, by the end of the course, we have not, somehow, come to realize the usefulness of what we are learning, our passing grade carries with it an enhanced sense that the subject is useless (even if it is an A-grade, as all of us competitive types always strive for).

This issue is by no means restricted to the schools. Almost all my teaching has been at the university level. For students majoring in a STEM subject, even if they find mathematics challenging (I do to this day) and are not good at it, they have a motivation to master the subject. A glance ahead in the textbook for their Major subjects will make it clear that they will need to be proficient in this stuff (even if they never really like it).

But every college and university I taught in had a mathematics requirement, usually completion of two courses, and as a result, the majority of students we mathematics faculty found ourselves teaching were non-STEM majors. Some institutions offer specific courses designed for non-STEM majors; at others, those students take regular first-year courses (where they can find themselves alongside STEM majors). It was when I was teaching non-STEM majors that I felt it was crucial that I have a good sense of how they would view the experience I would give them.

I say “experience I would give them” rather than “material I would present to them”, because that, surely, is what mathematics teaching is. Material can be gleaned from books and (these days) websites and videos. But mathematics is a first-person-shooter activity. It’s 5% knowing and 95% doing (which admittedly requires knowing how to do, but knowing-how is very different from knowing-that).

For the non-STEM majors in particular, I felt I had to do all I could to motivate them to make an effort. If they did not see how what I taught could benefit them, they were unlikely to do well in the course. Learning requires engagement, and engagement depends on motivation — which in turn requires a reason.

My starting point was to design the course on the (working-)assumption that the sole reason every student had to be in the class was to fulfill a graduation requirement, a goal that on its own is hardly a good grounding to survive an activity-course several months long in a subject that even the experts find challenging.

Early on in my career, I could try to meet this challenge by using mathematical examples that had demonstrable relevance to the various majors my students were taking. But after the introduction of the graphing calculator in the mid-1980s and even more so sophisticated computer systems such as Mathematica and Maple and freely available cloud resources such as Wolfram Alpha, which stormed onto the scene from 1987 onwards, relevance to their Major was no longer an adequate motivator.

Those tools can execute any and all mathematical procedures and solve any (solvable) equation, doing for all of university-level, procedural mathematics what the electronic calculator did for arithmetic back in the 1960s.

By the early 1990s, then, I had to face the fact that the answer to my question “What’s it like to be a student in my class?” included the reality that mathematics was something that (freely available and easy-to-use) consumer devices could do for them — much faster than anyone could do on their own, with essentially total accuracy, and for problems requiring volumes of data far beyond human capacity to handle.

Moreover, the non-STEM majors, who did not need mathematics for the subject they were pursuing, could see that their STEM-friends were all using those tools to do the math they were faced with in their classes! (Except, perhaps, the math classes.) As were we, the mathematical faculty, in our own research. (For those of us who needed to execute procedures.)

What to do?

For students who enjoy mathematics, there’s no problem. At any age. We are drawn to it by its inner beauty and the intellectual challenge. But we are a minority. For the (vast) majority of people alive today, mathematics is seen as something that computers can do for them; they don’t need to know much if anything about the inner working of those systems. Much as we view our automobiles — or any other technological aid (including computers themselves).

Where then, do we find the motivation to stimulate our students?

What we do know is that the many hours we (teachers) put into mastering the basic skills did more than give us procedural fluency with those skills. They were crucial to achieving conceptual understanding and learning to think mathematically. Moreover, I suspect I am not alone in thinking there is no other way to develop that discipline-thinking.

But there is no longer any need for a student to master those basics to a degree that they can execute them fluently at speed with a very low error rate, given that there are now digital tools that do it for us.

Learning and practicing basic skills to achieve understanding is very different from training to achieve industrial-strength, executional ability.

The answer to the motivation question, surely, is to take advantage of the perception of mathematics our students surely bring to their class, namely, it’s something we do by using machines.

So, why not make the procedural focus of the (non-STEM) course the use of the available tools to solve problems, either from their Major disciplines or from everyday life?

With those tools to hand, there is no longer any need to present students only with carefully crafted problems that can be solved in a few seconds, five minutes, or an hour (though hour-long problems rare, in my experience, even for homework problems). Those problems invariably look fake, and students can see they are.

But with Wolfram Alpha (say) and the information-seeking power of the Web to hand, we can safely and effectively present students with real problems.

Because you need a solid understanding of the mathematics those tools perform for you, as a teacher you have to help the students learn that mathematics, of course. But since you are teaching for understanding, not execution, you need only look at simple instances; for example, show them how to solve two simultaneous equations in two unknowns, let them try a few, and then simply point out that the same idea can be extended to more variables.

In fact, I used that specific example in a non-STEM-majors course I gave at Princeton in 2014, and again in a short course I gave at a local high school in 2018, where, working in small groups with my guidance, the students re-constructed the algorithm a big shipping company like UPS uses to route the packages it transports around the world. You can read my account in my February 2018 post.

Solving real world problems (which could be chosen from the students’ Major, be it a STEM discipline or some other subject) has several educational benefits:

1. It makes self-evident the fact that this math stuff is useful.

2. You can select a topic/problem that (at least some of) the students in the class see as directly relevant to them.

3. They experience first-hand the fact that the vast majority of applications of mathematics are in combination with several (sometimes many) other disciplines and techniques.

4. Most uses of mathematics in the world are in connection with so-called “wicked problems”, where the mathematics provides precise, accurate answers that contribute to the overall solution.

Regarding point 4 in the above list, in my student days, that fact was deliberately kept hidden from me so that I could focus on mastering a smorgasbord of specific techniques to the level where I could execute them efficiently and correctly at speed. But today, execution-level skill is no longer necessary.

For an example of a wicked problem: Do the benefits of recycling justify the costs? That problem alone can form the backdrop for an entire mathematics course. The instructor can guide the direction so that the students cover the various required curriculum items. At present, such courses can be found only at alternative schools (usually private) such as High Tech High in San Diego (that one is public), but they provide an example of what we could be moving towards.

Another good project topic that has been used (in various incarnations) at Stanford a number of times, is “Given a budget of X million dollars, what project in (say) a specific impoverished region of sub-Saharan Africa provides the greatest benefit to the population?” Such problems are rife with mathematical tasks within the capabilities of school and university students, given today’s mathematical toolkit and access to the Web.

To be sure, most teachers (particularly at the school level) won’t have the freedom to give such courses. But, at the very least, individual teachers could start to incorporate ideas from them into their own teaching. As the famous Chinese proverb says, “A journey of a thousand miles begins with a single step.”

Making that first step (then the second, etc.) would set us off on a transformation of mathematics education that would, I suspect (based on my own experiences as an educator), be better suited, and far more appealing, to the students who occupy our classrooms.

Further reading

I’ve written about wicked problems and their value in education a number of times in this column:

Do math and chess make you a better problem solver? Teaching math for life in a wicked world, November 2019.

What we've got here is failure to communicate – and adequately educate!  October 2021

The “wicked problems” problem, recycled, December 2021.