Mathematics for the million

By Keith Devlin @profkeithdevlin

Lancelot Hogben’s classic general-audience expository mathematics book, published in 1936, and still in print.

Older readers will doubtless remember the popular mathematics book Mathematics for the Million, by Lancelot Hogben, first published in 1936. There were few general audience books about advanced mathematics available when I was a teenager becoming interested in mathematics, so we all read Hogben’s classic. It was my experience reading such books that helped propel me into majoring in mathematics and eventually becoming a mathematician. They also provided my inspiration for writing such books myself.

The title is a good one, but the reality is that, no matter how accessibly written, expository mathematics books don’t reach a million readers. Those of us who see the beauty in mathematics and thrive on its intellectual challenge may not like to admit it, but the mathematics we love, for all that it’s one of humankind’s greatest cultural creations, is a niche discipline that should be taught as an elective for those who want it, alongside most other disciplines.

In making mathematics compulsory, we do significant damage in that a large number of students—and now we really are talking about millions—are put off, often for life, by anything remotely like mathematics.

The standard counter-argument to making mathematics an elective is that a basic knowledge of some mathematics is an important skill for anyone to have, and somewhat more knowledge is an essential requirement for many careers. But before we nod our heads in agreement, we in the math ed biz should step back and define our terms.

There is one other discipline that education systems around the world universally view as non-elective: the natural language of the society. But in that case an obvious distinction can be (and often is) made between teaching grammar and language use on the one hand, and offering courses on literature and creative writing on the other. (There is of course some overlap between the two in terms of content, though not intent.) The first is an essential life skill, and moreover, even students who struggle with language will recognize its importance to their lives. The second, in contrast, while a gateway to a great deal of pleasure (as consumer or creator), is not an essential life skill.

With mathematics, we do not have universally accepted different names to denote the basic mathematical life skills that really are essential for life in the Twenty-first Century, and the cultural creation we call the academic field of (predominantly pure) mathematics. There was an attempt in the 1990s to promote the term “quantitative literacy” for the former, that got some traction but never managed to take a firm hold. “Practical mathematics” is another name that has been used to refer to the basic-life-skills side to mathematics, but it too did not come to dominate. 

In my own case, taking expository advantage of the effective elimination of manually executing formal procedures as an essential component of doing and using mathematics, I (and others) started using the term “mathematical thinking” to refer to the non-procedural part of doing and using mathematics, leaving “mathematics” to refer to the whole shebang (as it always has). That actually puts the dividing line, insofar as there is a divide, in a different place, but I think it does offer us an opportunity to try again to split mathematics education into two (connected) threads.

[I first discussed mathematical thinking in an MAA post in September 2012, when I was just about to launch my online course Introduction to Mathematical Thinking on the then-new MOOC platform Coursera. My thinking on this has evolved since then, as discussed in the three Blue Notepad Videos on sumop.org. This more recent view is better aligned with this essay.]

Regardless of terminology, however, we do need to start talking (again) about separating “quantitative stuff for the millions” from mathematics-for-the-interested. The cost of not doing so, to individuals, to society, and to the discipline of mathematics itself, is simply too great.

The reason we should reignite the discussion now is that two powerful drivers are present that are today far more significant than they were in the Nineties, when “QL courses” were being enthusiastically touted. 

First, widely available, and highly effective, powerful digital tools are now available both for using mathematics and for learning mathematics. (Wolfram Alpha and Desmos are examples, respectively.) Our education system has lagged way behind in adjusting to these changes in mathematical praxis. When it finally catches up, the stage will be set for splitting mathematics education into two separate threads.

This graph was the focus of much attention in March of this year as the US prepared for the coronavirus to hit our shores. Being able to correctly interpret this theoretical graph, and many subsequent graphs based on real data, showing far more details, was key to formulating policy.

The second driver, which is a consequence of the widespread availability and use of digital quantitative tools, is that data science is now an essential requirement in the education of the next generation.

Nothing could illustrate more dramatically the need for good data-science education than the crucial need to be able to read the plethora of graphs and other quantitative representations produced regarding the growth of the coronavirus pandemic. I have seen claims made publicly by influential people that are based on hopelessly naïve—and hence hopelessly wrong—interpretations of simple trend lines produced to help decision makers select optimal courses of action. 

Focusing on the height or slope of a curve rather than the second derivative is perhaps the most dangerous error. With a highly contagious pandemic, we are facing exponential growth, which will rapidly dominate the current height or slope, so the second derivative is the key indicator. [A good data science education can also raise awareness of the counter-intuitive nature of exponential growth. See my June post earlier this year for a short discussion.]

Incidentally,I looked at the use of prediction graphs for pandemic planning in my May post. US decision makers did a poor job of reading the data, and the result was a death count an order of magnitude larger than it could have been. Good data science skills are crucial in a modern society.

Absent a widely accepted term, for the purpose of this essay I’ll use “mathematical thinking” to refer to the mathematical-life-skills education that should be obligatory to all, and “mathematics” for the classical discipline. (You can mentally replace the former with “quantitative literacy” if you prefer. “Thinking” and “literacy” both convey the essence that the education is aimed at improving people’s thinking process, not mastering the highly stilted form of thinking that is required in advanced mathematics.)

When would the two threads of mathematical education diverge? After middle school surely, and most likely during the final two years of high school, so around age 16, though I could mount an argument for it being a couple of years before then. In any event, education before the split should take account of the fact that such a branching is going to occur.

(I’ll be the first to admit that it’s easy to write opinion pieces, by the way. Making things happen in the real world tends to get messy and difficult very quickly. Fortunately, my days doing the latter are now over, and I’m free to be just a sage-with-age.)

For the discipline of mathematics (viewed and presented as a part of human culture), we surely have to provide all students with a taste. But there is no reason for it to be compulsory beyond an initial exposure. It should be an elective, just like music, literature, drama, painting, sports, and all the other (eventually) optional disciplines.

Mathematics-as-a-discipline courses should focus on the classical discipline, developed so all can experience the beauty, the elegance, the rigor, the challenge, and the power and reach of the subject. Some will love it and want more; others will not. After the initial compulsory courses, teachers of mathematics courses can assume all students are there because they have interest. That will benefit everyone.

What goes into the mathematics for the millions curriculum?

The design of the mathematical-life-skills education that I am current calling mathematical thinking, is where there is real work to be done—and likely battles to be fought.

I have argued elsewhere on a number of occasions that it does not really matter what mathematical topics are chosen, since the essence is the kind of thinking required. Recent “Devlin’s Angle” posts on this issue were in July 2019, November 2019, and December 2019. In particular, the widespread, cost-free availability of digital tools to perform all mathematical procedures means there is no need to develop a high level of fluency and accuracy with any formal procedures. Education should focus on the students achieving sufficient understanding. That surely requires spending considerable time hand executing a range of procedures, but the requirements of speed and accuracy that were important for earlier generations (including mine) evaporated in the 1990s.

The mathematics educator Liping Ma made a good case some years ago for arithmetic (integer and fractions) being an adequate base for students to master all the key cognitive skills for mathematical thinking. Ma based her case on teaching done in China, where she began her teaching career before moving to the United States and obtaining a doctorate at Stanford University.

I see her point, but my vote would be arithmetic, algebra, and geometry, since they are all entry-level subjects, and all are relevant in many careers and various walks of life. Moreover, all of today’s teachers are familiar with those subjects.

To that list, we should surely add data science (which includes the topic of algorithms), since that subject plays such a major role in today’s world. My Stanford colleague Jo Boaler has of late been arguing for a focus on data science. 

The importance of data science. This figure is taken from a government planning document used by the UK government in early March to explore various mitigation strategies for the coronavirus epidemic that was just beginning. The actual figures are estimates produced by a mathematical model based on existing techniques of mathematical epidemiology. Effective use of such representations requires the ability to correctly interpret the relative shapes of the different scenario-curves, understanding what they each signify, in order to make better decisions. Knowing what to ignore is as important as knowing what is critical information, in order to make good use of predictive graphs such as these.

What specific topics in those subjects would we need to cover, and to what extent? I think this should be driven in part by the need for a course designed for the millions for everything to be self-evidently relevant to the lives of the students in the course. We should provide a sufficiently broad perspective to facilitate a wide variety of real-world examples, all self-evidently highly relevant to everyday life, which can help motivate students. With the range of technological aids available today, students can work on real-world problems of relevance to their lives, free from the constraint that governed mathematics education in past centuries that classroom problems had to be solvable using hand methods that were within the capabilities of the students.

For instance, in terms of algebra, linear equations and inequalities in two or more unknowns are hugely important in today’s world, which provides an argument for building an algebra course around that topic. The main use of that algebra is in formulating and solving optimization problems. Google, shipping companies like UPS and FedEx, major airlines, and large online retailers all make heavy use of optimization using linear algebra. 

To be sure, the optimization problems those companies depend on typically involve thousands or even millions of unknowns, which is well beyond human capabilities. Computer packages are used to solve them. No human could ever cope with that. But for linear optimization, working on examples with just two or three unknowns provides a good understanding of the method. (This is not true for all mathematical topics, but it is here.)

So, given the frequency with which solutions of linear equations and inequalities crop up in a great many of today’s real-world problems that affect our lives, there is a good case to make for teaching methods to solve, by hand, one, two, and three variables cases, with the learner being able to get correct answers in the one- and two-variable cases and to solve three-variable examples without worrying too much if they make arithmetic slips. 

In contrast to the subjects I have suggested, calculus is most definitely not necessary at the school level. For one thing, it is only important for students who wish to pursue science or engineering, or mathematics itself. (In contrast, data science is important for everyone.) But also, calculus cannot be taught well at the high-school level, since it is considerably more sophisticated than anything else in the school curriculum. Students who do take calculus at school frequently have trouble with university calculus courses later on, since their school experience leaves them with a superficial, and essentially procedural, understanding that gives them a false sense of security during the first weeks, which eventually gives way to an unpleasant, and occasionally disastrous, train-wreck when they find their weak understanding is inadequate for the more advanced part of the course. Calculus is a must in the elective, disciplinary mathematics courses.

The point is, however, that while what we teach is subject to some debate, what should not be up for discussion is the purpose for which we teach it, and how we teach it. The answer to the second question (“how”) depends on the answer to the first (“why”). 

And the reason there should be no debate about the “why” or the “how” is that the goal of mathematical thinking education should be to ensure that future generations are able to make effective use of mathematics in the world they will inhabit, and in that world, mathematical thinking is the key skillset. (Computers calculate and execute procedures; people think.) Number sense is a part of mathematical thinking, by the way. So too is the capability to reason logically from assumptions to conclusions.

I think it is worth noting that, with the availability of digital tools to execute procedures, mathematical thinking is much closer to the humanities or the creative arts than it was in the days when you could not get far into any kind of mathematics without first acquiring fluent, accurate procedural skills. When it comes to education for the millions, this, I believe, is a huge advantage.

How do we teach mathematics for the millions?

As to the “how” do we teach it, this is where society will sooner or later have to face the stark reality that developing mathematical thinking ability requires a very different conception of teaching than the one most teachers are familiar with.

Except that teachers are familiar with it. Just not in the mathematics classroom. 

In the early stages of learning, many school disciplines involve the acquisition of a substantial amount of factual information. To progress in the subject, the student first has to know quite a lot. It involves a lot of teacher instruction, reading, and (these days, but not when I was a student) video watching. Tests and exams typically ask the student to demonstrate that they have assimilated enough of that information. 

Mathematics is a glaring exception. Yes, there are some facts to be learned, but mostly, learning mathematics involves solving math problems. Take a look at any mathematics textbook. After a short passage presenting some information, the student is presented with a much longer section that provides examples of the thinking required to solve problems using that new information, followed by a list of problems for the student to attempt. Compare that with a textbook on introductory biology or physics, say, or history, social studies, geography, or literature.

Learning mathematics is primarily about doing; not about knowing. In that regard, it is more akin to sports or music or handicrafts (or the laboratory sections of science classes, which typically follow, and depend upon, fact-learning classes).  What is the role the teacher plays in those disciplines? (Actually, they are so different, they are usually called “pursuits,” rather than “disciplines.”) 

As a teacher (and whoever else would be reading this column), think back to when you took lessons in, say, driving, playing a musical instrument, tennis, golf, skiing, chess, creative writing, or speaking a foreign language. What role did the instructor play? 

“Coach” is surely the best way to describe it. You learned how to perform that activity by doing it under the watchful eye of an expert who was able to guide you towards improvement. That’s how we best learn how to do something. Yes, we may be able to make progress on our own, by attending lectures, reading books, or watching videos. But it generally goes much faster, and we achieve way better levels of performance, when our learning is guided by an expert coach.

As a doing subject, that is how we best learn mathematics. One of the reasons I advocate using the term mathematical thinking is to emphasize the fact that it is primarily a doing subject, and the doing involved is active, and frequently creative, thinking.

At present, the coaching-style approach to mathematics learning is standard only at the university doctoral education level. A few universities also provide it systemically at the undergraduate level; for example Oxford and Cambridge in the UK, with their Tutor system, and some US liberal arts colleges do the same. (The “tutor” is a coach.)

Knowing the value of the coaching approach, many college and university faculty try to carve out time to provide coaching sessions, as do some school mathematics teachers. But with systemic school education structured the way it is, it is hard to do this. When a teacher focuses on one particular student, as good coaching requires, they cannot be attending to the needs of the others in the class. Splitting a class into small groups can go some way towards the desired goal, as can the use of teaching assistants (a rare circumstance few teachers have had available). 

Technology can help, though the “AI-driven, personal digital tutor” systems that are much touted do not live up to the hype, and indeed can result in disastrous results. (Google “Benny’s Rules” to get a sense of the educational nightmare that can result when the coaching is not provided by a human expert.)

And this brings me to one of the main points I want to make. The school systems adopted throughout most of the world take a “production-line” approach, modeled on, and designed to prepare future citizens for, the industrial age of the Nineteenth Century, when they were developed and deployed. In the case of mathematics, curricula and textbooks were designed to support that model. The main societal need was for an arithmetically able workforce to support, in particular, the mechanized, production-line manufacturing that drove the Industrial Revolution. There was little need for individual creativity. Fast, efficient, accurate rule-following, with everyone doing things the same way, was the order of the day.

Today, things are very different. Anything that can be done by routine rule-following is now done by machines. The primary need today is for creative thinkers and problem solvers to do the things the machines cannot. And the reality is, the very automation that made redundant the human skills of the Nineteenth Century, has given rise to a society where there is great demand for such (human) skills. Most people involved in education know this, of course. What makes it difficult to change education to meet the need is that it requires major restructuring of the entire education system, including the structure of the teaching profession. 

Some smaller nations, such as Finland and Estonia, have made significant strides in that direction, but larger countries such as England and the United States face far greater challenges. (I am somewhat familiar with all four of those examples.) But there is still a long way to go for everyone.

My own view is that school mathematics classes need (at least) two instructors, with the requirement for each being different. One will be the classroom teacher, who is in charge of the entire class. That teacher would, ideally, have a bachelors degree in mathematics and an additional credential in mathematics pedagogy. 

The other instructor (and maybe it requires more than one) would be a “tutor” who, on a rotating schedule, spends some time each week with each student in a one-on-one basis. (Maybe twice a week, with additional sessions on an as-needed basis if the teacher thinks it is required.) The tutor could very likely operate remotely, over a video link with a shared workspace.

The tutor has to be someone with a broad knowledge of mathematics who has used mathematics extensively in their career, possibly a retired scientist or engineer. While they would certainly require some training in pedagogy and how people learn mathematics, the main skill they bring is a deep, broad knowledge and experience of mathematics, as it is practiced in the world today. Few teachers have such a background—teaching is a demanding career on its own. (Equally, the tutor would be unlikely to bring the pedagogic, social-organization, and developmental skills necessary to provide a good education.)

In major conurbations, there is surely a good supply of individuals who could be tutors. In more sparsely populated regions, video-links would surely be the way to go. In either case, the role of the tutor might best be served by adopting an “Uber-like,” gig-economy system that connects classroom teachers to tutors with particular areas of expertise.

I will note that middle-class parents of sufficient financial means frequently ensure that their children do have regular coaching from a tutor, by hiring freelance tutors to come to their homes once or twice a week and work with those children. As a graduate student, I supplemented my state maintenance grant with income from private tutoring. But regular coaching should be available to all.

In any event, I put these suggestions out there for reflection and discussion. As always, I stress that I write from the perspective of a professional mathematician who worked in pure mathematics in the first half of my fifty-years career, followed by extensive consulting work for various industries and US government agencies in the second. My teaching experience is largely, though not exclusively, in colleges and universities, with some brief excursions into high schools, and occasional one-off classes in middle schools. My suggestion is thus driven by what might best serve society (which of course is made up of the people in it), based on my experience, and a strong belief that there are really not many options open to us if we are to provide future generations with the mathematics education they, and hence society, need.

I leave it to those with years of experience in the school classroom and those who have labored long and hard to design and run systemic education to provide input as to how to meet such a need.