By Keith Devlin

You are a math educator and, for whatever reason, you decide you want to write about mathematics for the general public. Power to you. Most people are woefully informed about mathematics—and are not able to solve a typical 8th Grade problem. So for the good of society, getting the word out is a laudable goal. What’s required? How do you do it?

And already I’m going to break one of the rules about writing for almost any audience. I’m going to begin with a negative. If you do that to a general audience, many will stop reading right then. But this article is not written for a general audience. It’s written for professional mathematicians and educators who want broaden their impact on society. And key to being successful at that is unlearning some of the lessons we have learned for, and in, our day jobs.

So here’s the skinny. As a general rule, good math teachers are terrible at communicating mathematics to a general audience. The reason is the very thing that makes them good teachers.

First, they are good at explaining things, particularly how to do math (since that’s how the effectiveness of math teaching is primarily assessed).

Second, they want to broaden their students’ knowledge and understand the math they are learning.

And third, while not all students like math or want to be in the class, they are a captive audience, and in most cases want to at least pass the course, so teachers become good at working with that kind of audience.

For a general audience, everything is different. If that’s the audience you want to reach, you should view yourself as providing diverting entertainment. After all, why else would they read an article about something they have no particular interest in or knowledge of — and in many cases have an intense fear or dislike of?

What was that? You’ve been very successful teaching your institution’s “Math for Poets” course. Doesn’t that prepare you to communicate with a general audience? Many Math for Poets students are in that class precisely because they don’t like math, right? True, but remember, they are there with a strong motivation: to pass the course.

Likewise, being successful writing for The Mathematical Monthly, Mathematics Magazine, Math Horizons, or even FOCUS or the NCTM’s Mathematics Teacher, and the like, doesn’t help. All those publications are for people who know some mathematics and are interested in math. To be sure, I listed those particular publications in descending requirements for mathematical background, but even if you end up having articles published by NCTM (I did, on two occasions), that does not even come close to writing for the general public. If anything, being good at writing for any of those outlets or the many like them gives you positive feedback for an approach that absolutely will not work for a general audience.

If you really want to communicate with laypersons, you have no choice: diverting entertainment is the only way to go.

Even then, only a minority of the “general public” will have any interest in an article on math. But I suspect you don’t actually want to reach everyone. Most likely, you are trying to connect with one or more of three groups:

• school teachers, to help guide their development, provide them with useful classroom ideas, or perhaps just “open their eyes” a bit more;

• parents of school students, many of whom want to help their children or at least understand something about what they are being taught (and not being taught!), and why;

• and finally, education administrators, politicians, and members of society who, for whatever reason, are interested in educational policy — all of whom, you feel, could benefit from acquiring greater knowledge of math, or having their beliefs about math (or math ed) supported or challenged (as the case may be).

For each of these groups, there are of course established channels through which they can be provided relevant information, but you probably think you have a better chance of reaching them during their off-duty hours when the pressure is off and they just want to relax. And I’d agree with you there.

(Many of the articles, columns, blog posts, and radio and television appearances I’ve done over several decades have found their way to members of all those groups, and had sufficient effect that I’ve heard back from people who took note of, posted, and made explicit use of, things I wrote, said, or showed.)

But to reach your reader when they are off duty, you face two challenges: getting their initial attention, and then keeping it long enough for you to get your point across.

To get their initial attention, you need some form of “hook”. These come in various forms.

For each of the three categories of potential readers I listed above, K-12 mathematics education issues can work as a “news hook”, particularly if there is a recent or current story doing the rounds in the mainstream news media.

Another good news hook is a current news story that affects, if not all then at least many of your desired readers. For example, all the time we are in the middle of a global pandemic, math relating to epidemiology can be “sold” with that hook. (I used the word “sold” deliberately. Just avoid adopting a bait-and-shift sales technique — your success depends on you having and maintaining credibility as a math communicator.)

Your news-related article does not have to focus on explaining or providing information about the math behind the news story. But there is one occasion where that might be necessary. Since many journalists have little or no math knowledge, and limited math skills, when working to deadlines they frequently make mistakes, even with simple arithmetic facts, and taking a mathematician’s look at their error can, if done well, make for a successful article. But you have to explain the error in the simplest way possible.

I gave an example of that in my September 2021 Devlin’s Angle post, which took a look at how some excellent math writers addressed different specific audiences. [For the point I want to make here, look for the paragraph that begins “Instead I would say this.”]

My one caution, however, would be to avoid pitching your article as a “take-down”. If it’s possible, adopt a constructive and sympathetic approach. That was the approach I adopted in one of my recent posts on my blog for the Stanford Mathematics Outreach Project. In that case, my initial desire to avoid going negative, which many commentators before me had done, led me to view the journalist’s error as a very understandable “there but for the Grace of God go I, in the heat of the moment” slip.

[BTW, the SUMOP site is aimed at math educators, so that post is not an example of an article for a general audience. The same is true for all the other posts on the site; they illustrate some features of writing for a general audience, but by no means all.]

You may be tempted to start your piece with a cute little puzzle. After all, puzzles provide a classic example of “diverting entertainment”, which I already said is how you should view your article. Such an opener could for sure get the attention of many people. Unfortunately, others in your potential audience will have no interest in puzzles, and maybe a dislike, so you’ll probably lose a substantial number of readers the moment they see it.

In any event, once you have found your hook, and a reader has cast their eyes at your first sentence, the challenge is to maintain their attention through to the end of your opening paragraph. In fact, you should be aware that they may well not go beyond that first paragraph. So, if possible, craft your initial paragraph with two goals in mind: first and foremost to get your main point across, and in addition interest them sufficiently to then keep going.

Which brings me nicely to another key factor in writing for a general audience. Write with the assumption that when your reader stops reading your article they will leave having assimilated just one point. In terms of your planning, no less, no more; exactly one point. You need to be very clear before you start, what that one point will be.

Regardless of the topic, for me the one point above all I want my readers to get from a general-audience article I write comes from the following short list:

1. Mathematics actually is relevant and important in today’s world.

2. Mathematics has relevance for that particular reader.

3. Mathematics is a powerful way of thinking, both for understanding, and for solving or helping to solve real-world problems.

4. Mathematics is created and done by people. It’s a human mental activity. The formulas on a page are just representations. Mathematics is to symbols on a page as music is to notes on paper.

5. If we don’t have sufficient knowledge of math, we are ill equipped to understand things that affect us and to make good decisions to our advantage.

6. A lot of the mathematics used in the world is beyond the capacity of all but the experts to do, but it can be crucial to have enough general knowledge of math to make sense of, and evaluate, the results of that mathematics. (For example, comparing reported risks from a novel virus to risks from a new vaccine or some other medical procedure.)

7. Mathematics includes numbers and arithmetic and calculation, but there is a whole lot more to it and it keeps growing.

8. This is much less common for general audience articles, but if the opportunity arises, there is the point that mathematics is a rich body of human knowledge, accumulated over thousands of years, having its own beauty, which many people pursue for pure pleasure.

These points are not all independent, and in all likelihood in hitting one you will meet others. But you should focus on just one and keep driving it home. Because the chances are high that having even one of them register won’t occur unless they keep encountering it.

Notice that these are not “math facts” they could repeat on a test. They are all general perceptions about the nature of math. But given the importance of math in today’s world, and the very poor knowledge of and appreciation for mathematics in the population at large, moving the needle on just one of those points is a huge win. If your article does that for some readers, it will have been a success.

I will note that in the process of achieving that one “move the needle” outcome, it’s certainly possible you will leave them remembering a significant, specific math fact. For example, “Google uses advanced math to find information for you”, or “Amazon uses advanced math to get your purchases to you in dazzlingly short time.”

But getting your reader to remember those specific facts is something for you to be proud of only if you work for Google or Amazon, respectively. The real win if they do remember, and the chances are high they will in this case, is they now have an actual example demonstrating general facts 1 and 2 in my above list — so you’ve hit two main (general) goals in one go — and moreover they are highly likely to remember those specific facts. So a teacher or parent faced with the familiar student question, “What is all this math good for?” now has a powerful, pithy response.

Of course, that teacher or parent will likely be unable to provide an explanation when asked the inevitable follow-up question, “How does Google/Amazon use math?” Even if your article did give some indication of how, that level of detail would most likely flow right past them. But now they can provide a good, practical response: “Ask Google that question”. (Check it out.) And now you have set the student on a path of discovery they may find themselves rapidly sucked into. A path that the teacher or parent may want to follow along with too. Big win on several levels.

The point is, going after just one, very general goal in your article can lead to a whole lot more. You just have to leave your reader with a general sense that stays with them, and for some reason may end up intriguing them.

To repeat, you need to get that one general idea across right away, in the opening paragraph, and then keep repeating it throughout the article. Not as a “fact to be learned” (and certainly not by rote learning!) but as a general idea picked up in reading your article. Keep it uppermost in your mind throughout the writing process. (This quickly becomes automatic as you write more and more general-audience articles, particularly if you get feedback from editors.)

Pick just one goal, remember. It’s possible you have a goal not on my list. It is just my list, though I’ve been using it for almost half a century now, and it covers almost everything I’ve written for that market, and in the early days I received some expert coaching from some award-winning, professional journalists. So it’s road-tested.

So now you have a hook, and a general goal that will be the reader’s takeaway. How do you write the article?

The simple answer is, you tell a story. And since you are telling it using words (what I say also applies to radio or podcasts) it should be a story that creates pictures and actions in the reader’s mind. (It sounds weird, but writing and radio broadcasting are visual media! The degree to which they succeed depends on how well the words create pictures and experiences in the reader’s mind.)

If at all possible, your story should be about people: the people who develop the math, the people who teach that math, the people who are learning the math, the people who use the math, the people who benefit from the math, maybe even the people who suffer as a result of things done with the math. (But be very careful how you frame that last one!)

And ideally, you should focus on just one person or a small group of people, as an exemplar. Our brains come hard-wired by evolution to be irresistibly interested in stories about other people. Particularly other individuals or small groups, when stories about them are frequently referred to as “gossip”.

(I actually made use of that scientific observation regarding the evolutionary development of gossip — and yes, it has been established scientifically, repeatedly — in providing an evolutionary account of how the human brain acquired the capacity to do math in The Math Gene, a book I published in 2000. The book’s subtitle includes the phrase “why numbers are like gossip”, supplied by the publisher’s editor, from which you might correctly infer that gossip is a key human activity, not to be dismissed as serving no evolutionary purpose.)

I note that, as mathematicians, we usually find the math or its applications of interest all on their own, of course. Discussion of the people involved is actually a distraction. But if you want to reach a general audience (even the restricted version of “general” I admitted earlier is the best we can hope for), you have to flip that around. Tell a story (necessarily a short story) about the people, and drop in the math as part of the furniture or the supporting cast.

If possible, start your story (this is that crucial first paragraph) with an intriguing, amusing, or surprising anecdote about a person or group. Definitely your main math point needs to be in there as a central feature, but the critical strategy now is to interest the reader sufficiently to read on to further paragraphs, where you will make sure the key point keeps cropping up one way or another.

As I noted once already, your story should if possible create a visual image or movie in the reader’s mind. You could maybe use an actual image to illustrate your story — a photograph or a very simple diagram — but in my view it’s usually better stick with words if you can, so the reader creates their own images. Try to reserve pictures and diagrams as wallpaper to break up the text and perhaps provide reinforcement for the story you tell with your words.

As a general rule (it’s almost a Golden Rule) don’t put any calculations in front of your reader, not even a simple contrived example to get a general idea across. Many readers will tune out at that point, and likely won’t tune back in later. If you feel you must include some symbolic math, put it in a Box, separate from the text, but you definitely run the risk the reader will see that before they read anything, which they then may well not do at all.

If you can, express your personality through your words. You might not be Terry Tao (and even if you were, that name would mean nothing to a general reader), but if you are a professional mathematician (of some variety), then to the reader you are a “math genius.” If you can get the reader to see you also as a regular person with a personality, you’ve already put some points on the scoreboard. (“Math geniuses are regular people.”)

And that’s it. Notice the things we are not doing in our article. We don’t teach. We don’t show how. We don’t explain how. We don’t lay out a logical argument. And we don’t present any calculation or math formula (let alone manipulate a formula). That stuff carries with it memories of the math class, which for many of your readers is a huge turnoff.

To reach a “general audience”, you need to stop being a teacher and become a storyteller.

One caution. Fellow mathematicians may critique you for your cursory, and inevitably incomplete coverage of the mathematical content in your article. (And there surely will be such, even though the long-term impact of your piece will likely be at most that one general idea you chose.) But remember, anyone who is in a position to comment that way does not need your article. You are writing for a very different audience, one much harder to reach.

As a final aside, I should perhaps remark that I think we mathematicians have it hardest of all when it comes to communicating with the general public. The readers we really want to reach likely had at best a boring school experience of math, and in many cases a traumatic one. Anything that reminds them of that experience will turn them off immediately. The appearance of a formula or a calculation is more than enough.

Good luck.