Teaching college-level math to students who grew up in a connected world

By Keith Devlin @profkeithdevlin


The primary purpose of K-16 education, we say, is to prepare the next generation for life in society.

CAUTION: The “we” here refers to those of us who are educators, and for an audience comprising only professional educators, my opening statement would stand on its own. But in an open, online forum such as Devlin’s Angle, I know from experience that my observation requires elaboration. Specifically, there is no shortage of people who think our job is to prepare our students for work. Of course, work is part of life, so if we prepare young people for life, that should definitely include preparation for work. But that should not be its sole, or even primary, focus. Equally (and arguably more) important, our task is to prepare our young charges to live full and rewarding lives as productively contributing members of society. SECOND CAUTION: Not the society we grew up in; rather, the one they will be part of. That’s a critical consideration.

That’s quite a challenge—particularly during a time of rapidly occurring, major societal changes, like today. Teaching well is hard at any time, doubly or triply so when the world our students will live in is not only different from the one we grew up in, but in all likelihood will have changed dramatically by the time they graduate.

One of the first nations to recognize that was Finland. Back in the 1970s, this tiny nation (population today just over 5.5M) realized that to prosper in the Information Age, they had to ensure maximum benefit from its most valuable natural resource: not the timber or the ships of previous ages, but its people. No, scratch that. Not its people, its society. They did not make the mistake of thinking it was about training people for work; rather, the trick was to create a cohesive, educated society where people can live and work together. They also understood that, as the ones tasked with producing the individuals who would make up that society, teachers were one of the most critical professions, alongside physicians, nurses, scientists, engineers, and business leaders. The result was that, thirty years later, in 2000, Finnish schoolchildren topped the international rankings in the OECD’s PISA (Programme for International Student Assessment) education tests.

To this day, the United States has, by and large, failed to meet the challenge, making up for the huge shortfall in adequately educated school graduates by massive immigration of talent educated elsewhere in the world. To be sure, that solution has many advantages in terms of a culturally more diverse society, but it consigns many native born Americans to less rewarding (and less remunerative) careers, often leading to resentment (and an antipathy to immigrants).


But, as I often do, I digress. My topic for this month’s post was occasioned by reading a remarkable new book from a social scientist at Temple University, who I have interacted with professionally on a few occasions: Jordan Shapiro. His book, The New Childhood: Raising Kids to Thrive in a Connected World, was published at the end of December. Its title suggests a “how to” manual for parenting, and indeed he has structured it that way, with each chapter ending with a summary that provides specific things parents can do to best prepare their children for life in today’s always-on, global society. But, on another level it’s much more than that. It’s a discussion of media that I found highly reminiscent both of Marshall McLuhan’s 1964 classic Understanding Media: The Extensions of Man, in which he coined the famous phrase “The medium is the message”, and of Alvin Toffler’s 1970 bestseller Future Shock. In fact, I’ll go further than that. I think it may well end up being viewed in the same way, as a seminal “taking stock of where we are as a society” study.

Divided into four sections—Self, Home, School, and Society—with three chapters in each (a symmetry sure to please mathematical readers), Shapiro’s book constantly asks the reader to consider the world from the child’s perspective.

As a father of two young sons, Shapiro has an in-house observational laboratory not available to many of us, but the way people (including children) use and react to media is also an area he has studied and written about professionally for many years, which is how our paths first crossed. He was one of the first ed tech commentators to write about my work on educational video games. [His book’s introduction is titled “Plato would have been a gamer”, which is very similar to my own oft-repeated remark “If video games had been around in 350 BCE, Euclid’s Elements would have been a video game.” It’s possible my meme inspired his header, but to anyone who really understands the nature of mathematics—particularly classroom geometry—and the nature of video games, as we both do, the sentiment is blindingly obvious.]

Though mathematics and math teaching are referred to throughout the book, they are there merely as examples of subjects that are taught and need to be mastered. The significance of Shapiro’s book for college math educators (or K-12 math teachers for that matter) is his discussion of how today’s globally-connecting technological infrastructure impacts what we need to teach and how best we can teach it.

Regular readers of this column have been exposed to my perspective on that topic for most of last year (beginning with the January post and continuing with just a couple of diversions through to the November post). Everything I wrote in those posts is entirely consistent with what Shapiro says, in large part because for several years we have followed each other’s work and consulted many of the same sources. But there is plenty in his book that was new to me, and my guess is it will be new to you as well. Since he writes superbly, I will for the most part leave it to you to check it out yourself.

I will, though, end by providing two BIG, and reassuring, takeaways, which come from Shapiro’s many years studying media—not just new media but different media stretching back thousands of years.

TAKEWAY 1: Nothing going on today is really new. It seems new to us, because we are in the middle of it. But if you put yourself in the position of people living when writing was introduced, when the printing press came along, when we acquired telephones, radio, film, television, and then all the generations of digital media we did live through, and finally the always-on, global network today’s kids take for granted, you will realize that each of those revolutions must have seemed very much the same to those living through them. Case in point. Already on page 5, Shapiro reminds us that Socrates thought much learning would be lost if ideas were written down. That did not deter his pupil Plato from doing just that, and the scholarly world rapidly adjusted to the radical new idea of learning being based on written texts. (Had video games been available at the time, Plato would have been able to stay closer to his teacher’s insistence that learning should involve active interaction of student and teacher by creating a video game rather than a book. Hence Shapiro’s introduction title. But Socrates would still have been unhappy. That brings us to the second takeaway.)

TAKEAWAY 2: It is an unavoidable consequence of being born and growing up at a certain time that we take our contextual environment as “the way things necessarily are.” That which we grow up with, we take for granted. We have no other choice. Society advances because each new generation eventually finds ways to go beyond what they encountered as children.

For the most part, the advances appear to be gradual and continuous, but every so often there is a kind of phase shift, where an accumulation of small changes has a dramatic effect.

Such was the case in mathematical praxis when, in the late 1980s, we acquired machines that could execute any mathematical procedure, rendering hand calculation unnecessary. (See my provocatively titled January 2017 article in the Huffington Post.) Since then, mathematicians spend their time very differently from that way their predecessors had operated for thousands of years. That major shift in how mathematics is done has been slow to percolate down to how it is taught in schools, but in due course the system will catch up. It has to.

An equally major rift occurred in mathematics education with the invention of the printing press. Prior to that, mathematical texts were written in words and numerals; no diagrams and no symbolic algebraic expressions. Part of learning mathematics back then was sketching diagrams and scribbling symbolic expressions in margins as part of the process of understanding what the prose argumentation meant. After the printing press came along, math textbooks were heavy on symbolic expressions and (in due course) diagrams, and students sketched diagrams and wrote prose comments in the margins as part of the process of understanding the symbolic expressions.

Neither of these specific examples is in Shapiro’s book, but there are a great many different ones of a less disciplinary-specialized nature. You may start out thinking, “Yes, but this particular point in history is different, because … ”, but eventually, Shapiro’s examples will overwhelm you, and you’ll cave. Scholarship does that.

So what is the change that tomorrow’s children will take for granted, but we will think is (a) impossible or (b) a disaster-in-the-making? According to Shapiro it is the abandonment of the fixed-period lesson, be it 30 minutes, 40 minutes, an hour, or whatever. It will, he argues, be replaced by “drip engagement.”

Shapiro introduces that term as “the process of turning one’s attention to small things as they arise… Think of academic content as if it were delivered like raindrops rather than a deluge.” You’ll need to read the book (and by now it should be clear I am urging you to do just that) to see what this amounts to, but the term itself is a good indicator. Alternatively, if you have young children, as Shapiro does, just watch how they study today.

Incidentally, this does not mean replacing the division of learning into one-hour classes, as we do now, by division into smaller chunks of something, say three minute videos—a change that has often been suggested by media technology folk who know virtually nothing about education. As Shapiro points out, the fixed-length class is a model we inherited from the monasteries in thirteenth century Europe. The introduction of devices that measure time accurately, combined with the necessity of bringing students into the same room as a teacher, resulted eventually in the establishment of the credit hour, that is the basic building block around which our entire current systemic educational system is built; from curriculum to finance and budgeting, to educational personnel workload, duties and compensation.

But a fixed unit of time has nothing to do with learning. From the perspective of learning, it is an imposed arbitrary constant around which educators must adjust everything to fit. The ideal unit of learning is not a unit of time, it is a … wait for it … unit of learning. Duh! Of course that’s what it should be! If it can be, that is—and in today’s world it can.

Time-to-reach-mastery should be a variable—because it is! It varies from subject to subject, topic to topic, student to student, and it depends of course on the availability of resources and on the degree of mastery required. Drips come in different sizes. (Like all analogies, you have to give the metaphor some latitude to be effective. In this case, ignore physics and think of idealized drips that can vary indefinitely in size.)

This may all seem strange to those of us who grew up in world dictated by the clock, with education delivered in credit-hour units, but to today’s students the only place they encounter that method of learning big-time is at school, and they make it clear they find it arbitrary and they don’t like it.

It’s not that they are not capable of spending long periods of time engrossed in one challenging task. Just watch them playing a difficult video game. In the video-game world, time is flexible; it is all about the challenge at hand. That’s why writer Gregg Toppo titled his excellent book about game-based learning “The Game Believes in You.”

In the case of math learning (and likely many other subjects as well), no one (not least Shapiro, who is both an academic and a parent of young children) is suggesting throwing the educational baby out with the bathwater. But we do need to recognize that the credit hour is part of the bathwater. Bathwater that has now cooled down over time to the point where we need to pull the plug and let it drain away.

I suspect that for math learning the situation will end up being very reminiscent of the way it changed with the printing press. Just as words and symbols swapped roles, so too I expect math learners will view drip engagement as the primary “delivery” medium and “extended periods of focused thought” as a secondary mode to adopt as and when required—the very opposite of the situation today.

[I’ll leave it to those with expertise in teaching math to younger children to determine how things will go there. In the early stages, education is as much as socialization and learning how to learn, as it is about any particular subject matter content. I do know from observation that elementary school classes today look very different from the way they did when I went through the system, so I suspect elementary school teachers are way ahead of the rest of us educators in adapting to today’s kids.]

I am sure this will all seem very strange to us. We may even think it could not possibly work. But the historical record—and Shapiro gives us a lot of it to reflect on—suggests otherwise. We are not in a unique historical moment.

In the meantime, if today’s math educators want to help prepare the way for tomorrow’s learners, we need to start stripping mathematics down to the individual components and reassembling it in a way that permits learning in a drip-engagement model. For, whether we like it or not, that is the future.