What Use Is Math?
By Keith Devlin @KeithDevlin@fediscience.org
To those of us in MathWorld, the familiar student question “What good is this stuff?” is so far out, we are tempted to reply “What is it not good for?” In fact, I suspect that answer has been given frequently, if only as a segue. But a more productive response is to give some specific examples.
If the question is really about basic arithmetic, responses that talk about dealing with money when shopping, or measuring areas of rooms to purchase a carpet, don’t work. Examples like that used to be just fine before shops had checkout registers and we all had calculating devices (these days most commonly on our smartphones). But in today’s world, not so much. Indeed, I don’t think there are any simple examples you can give that will convincingly show a young child why knowing, say, how to multiply two natural numbers, will be of any practical use to them. As a boomer, I was in the last generation of beginning school students for whom that worked.
The best way to handle the “Why are we learning this?” question for early-grade math, in my opinion, is to teach the basics in a way that is sufficiently interesting and stimulating that the question doesn’t arise. (Lots of games and puzzles require basic arithmetic, for example.) And there is no shortage of teachers who teach that way every day.
With more advanced mathematics, in contrast, there are examples galore. The question is, how do you present them when the question comes from someone who has not yet met much, if any, math beyond basic arithmetic? How, for example, do you demonstrate the value of school algebra?
The answer (of course?) is you describe important applications they can relate to, but without going into the mathematical details. Unfortunately, many school teachers are not familiar with the multitude of examples mathematicians and other math-users are aware of; their training and expertise are focused elsewhere, on mathematics pedagogy, a whole subject in itself. (Most mathematicians are not familiar with that domain, which I mention only because many seem totally unaware of that lack of knowledge, which can be problematic when they become involved in K-12 education issues; see Devlin’s Angle, April 2022, for example.)
This is where television and social media can step in. Recent years have seen a significant growth in television programs and web-posted videos that describe mathematics, and in particular everyday applications of mathematics, in a highly accessible manner.
I was prompted to write about this topic by the release last month of the new BBC television series The Secret Genius of Modern Life, presented by University College London mathematics professor Hannah Fry, which US-based viewers can access on YouTube.
[ADDED AFTER PUBLICATION: The YouTube videos were taken down shortly after this article went live. Presumably due to a distribution rights issue, possibly triggered by the publication of this article. Readers in the US may have to take my word for it that the first three episodes at least (which I was able to view on YouTube) are superb, and wait until the series becomes available in the US.]
Scenes from the BBC television series The Secret Genius of Modern Life, 2022, presented by Dr. Hannah Fry.
In some ways, the new BBC series is an up-to-date version of the PBS series Life by the Numbers that I worked on back in the late 1990s. But with one important difference that I think is significant. To overcome the “math fear factor” that would keep many views from tuning in, the LBTN producers hired movie star Danny Glover as on-screen host and narrator. (They originally wanted Oprah, but she did not want to dilute her educational focus on literacy.) Hannah Fry, besides being an honest-to-goodness mathematician with a Ph.D., is a television natural. (I’ve never met her, but I guarantee she has worked hard on coming across that way on a screen.) That means that the programs don’t have to keep using the word “mathematics” (or “maths”). [I’ll mention for US readers, that she was known to British television viewers for some years before this most recent series. They know she’s a mathematician and an engaging presenter.]
The result is a series that’s highly entertaining, informative, and a superb resource to draw on to answer that “What’s this useful for?” question. [If you’re that teacher and the student responds, “But where’s the math in what Dr. Fry just showed me?”, then you have an excellent, Google-assisted discovery project to assign. Moreover, one that you, the teacher, will surely enjoy!] Do check it out.
Scene from the PBS Nova program Zero to Infinity (2022), presented by Dr. Talithia Williams.
Hannah Fry’s series shows some of the many practical applications of mathematics. But if providing an answer to the “Why are we learning this?” question (I’ve changed the phrasing deliberately) consists entirely of practical applications, even if they are cool applications relevant to the students’ everyday and future lives, then we’ve also done those students a disservice.
Mathematics is a major part of our culture, a product of human creativity over thousands of years, having intrinsic interest and beauty. We need to show our students that side to mathematics as well.
There too, television and video can help provide that skeptical student with an answer.
Last month also saw a new PBS television Nova documentary about one of the most abstract parts of pure mathematics: infinity. Titled Zero to Infinity, the program is hosted by Talithia Williams, Professor of Mathematics at Harvey Mudd College.
Like Fry, Dr. Williams is no newcomer to television, and once again, as a professional mathematician, she speaks with authority. Unlike the BBC series, however, the focus in this PBS special is entirely pure mathematics, occasionally veering into philosophy. Besides (and because of) the more esoteric topic, the PBS documentary does not have such a wide-cast as the BBC series. But it is still engaging and highly accessible.
Williams, like Fry, makes use of the affordances of modern television and a good budget to bring the story to life for the lay viewer. In particular, she talks with a variety of mathematicians, math educators, and other experts, allowing their enthusiasm to convey its own message(s), beyond the mathematics itself.
Do check this one out too. [Disclosure: I was an adviser to the program, but my involvement was fairly minimal.]
Two excellent sources, then, that show the wide scope of mathematics without miring the viewer in any of the details.
But wait! There’s more!
The Fourier Transformation. From the 3Blue1Brown animated video by Grant Sanderson (see text).
Last month I saw a third excellent new video documentary on mathematics. This one however, is not aimed at a general audience. Its topic is Fast Fourier Transforms, and it does not hide all the details. (If you hide the details, there’s not that much to say!) But much of the program is history, and those parts are entirely accessible to a lay viewer. They can fast-forward the technical parts. And watching it that way would definitely be worthwhile, since the story it tells is a compelling one about one of the most important uses of mathematics in the modern world. That story was completely new to me, and I suspect to many MAA MATH VALUES readers as well. And it’s a zinger.
Titled The Algorithm That Transformed the World, the video is the product of Veritasium, the (essentially one-person) science-video communication operation of Derek Muller. With a Ph.D. on the use of media in science education research from the University of Sydney, Muller has amassed an impressive repertoire of science-related videos.
When I saw a reference to his new video, my initial thought was that he was focusing on linear programming, and in particular the Simplex Algorithm, which plays a crucial role in much of present-day commerce. I assumed there must have been a major new advance in that area. It never occurred to me his video would be about Fourier Transforms.
Fourier transforms are, for sure, a part of mathematics for which there are many important applications, but there is no shortage of excellent math videos that explain them. My favorites are those of Grant Sanderson, published on his 3Blue1Brown Web-platform. But such is the nature of Fourier’s beast, even his introductory video on the topic (But What Is the Fourier Transform?) makes it clear that this is not a topic for laypersons. Even the kind of laypersons who find their way to Veritasium.
Dr Derek Muller describing the role played by the Fast Fourier Transform during the Cold War, in the Veritasium video The Algorithm That Transformed the World (2022)
What I did not know was the historical story Muller has brought to the table; and does so in brilliant fashion.
Though he initially misspeaks in saying that the Fast Fourier Transform (FFT) is “The most important algorithm of all time” (at the very least, the classical algorithm for integer multiplication has a far better claim on being that), it is, as mathematician Gilbert Strang observed (and Muller quotes toward the end of his video) “The most important numerical algorithm of our lifetime.” [My emphasis.]
Even if the very sight of the Fourier integral scares you (it would surely scare most K-10 students), there is value to watching this video. If you are a teacher, show your skeptical students the video in class, fast-forwarding the math bits as you follow the historical story.
Just be sure they have chance to glimpse what you are hiding from them. If this story doesn’t convince them that all that abstract math can have major consequences in all our lives, I don’t know what would.
POSTSCRIPT: All the cool uses of mathematics presented in that late-1990s PBS series Life By The Numbers I mentioned had been superseded within ten years, leaving the series mostly of historical interest. The same will be true for the Hannah Fry series. New mathematics is being developed all the time, and people are always finding new uses of mathematics. Things change fast in MathWorld.