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Articles by Keith Devlin
In the world of mathematics outreach, it’s always prime time — MATH VALUES
In fact, that story was my first ever math piece (on any topic) in a national newspaper, and it’s what launched me onto a second career as “science writer”. My article was quickly picked up by other media outlets, and BBC Television reached out to me that very day for help doing a piece on their regular children’s news program John Craven’s Newsround (I always thought a children’s news program was a great idea) and later another segment for their popular tv series Record Breakers.
The mathematician who—incidentally—helped mathematicians to stop worrying and love the computer. — MATH VALUES
By Keith Devlin @KeithDevlin@fediscience.org, @profkeithdevlin.bsky.social Yes, this month’s title is a reference to the iconic 1964 movie Dr. Strangelove. Walk into any mathematics professor’s office and you’ll see a computer. Indeed, the chances are high that, even if said mathematician is not actively using it, it’s switched on. Back in 1985—the year AFTER the Macintosh launched—that computer would have been the exception to the norm. Huh? What?
The Shifting Sands of Algebra — MATH VALUES
By Keith Devlin @KeithDevlin@fediscience.org, @profkeithdevlin.bsky.social My August post about the growth (and changing nature) of algebra generated some feedback. For the most part, those who objected to one or more statements seemed to be motivated by an assumption of what the word “algebra” means (or should be taken to mean).
What is the Most Valuable Benefit You Got from Becoming a Mathematician? — MATH VALUES
Having a media presence, I often get asked for a quotable answer to that question. I usually answer, “It taught me never to be sure I’m right.” Mathematics is unforgiving. If you made an error in your solution or your proof, you have only two choices. Correct the error or (and this is by far the most common outcome) admit you were wrong. And then ask yourself what you can do about it. Giving up is usually to be avoided.
Algebra. It's powerful. But it's not what it was — MATH VALUES
To most non-mathematicians today, the word “algebra” conjures up an image of “calculation involving letters”. There is some truth in that, for school algebra at least. But it misses the essential distinction between algebra and arithmetic. Both can be used to solve numerical problems, and both involve calculation. The difference lies in the nature of the methods. In arithmetic, we calculate with numbers we are given to obtain a numerical answer.
Algebra. It’s powerful. But it’s not what it was
By Keith Devlin @KeithDevlin@fediscience.org, @profkeithdevlin.bsky.social What is algebra? That question arose in the Q&A session at a recent screening of the new documentary movie Counted Out that I wrote about in last month’s post. Here is the answer I alluded to in my response, but was way too much to launch into. Cover page of a manuscript of al-Khwārizmī’s Ninth Century book Algebra.
Math Skill is a Civil Right – The movie — MATH VALUES
By Keith Devlin @KeithDevlin@fediscience.org, @profkeithdevlin.bsky.social Access to basic math thinking skills is a key civil rights issue of our age. This is the top-level framing issue that shaped Counted Out, a cinema-released documentary film that was at least six years in the making, and had premiere screenings in San Francisco and Los Angeles earlier this year. Its overt focus is US mathematics education at both the K-12 level and adult education, but the issues are global.
What is the Most Important Skill Mathematicians Bring to Collaborative Work on a Wicked Problem? — MATH VALUES
During the 1990s, my main research project was an applied one on workplace communication, where I collaborated with a social scientist (wearing her ethnographer hat) who, in addition to her doctorate in sociolinguistics, had obtained a second PhD in computer science. (Much of our joint work was collected together into a 1996 research monograph, Language at Work.) When we presented our work at research conferences, we would typically each handle about half the presentation.
Mathematical Models – Useful But Often False — MATH VALUES
By Keith Devlin @KeithDevlin@fediscience.org, @profkeithdevlin.bsky.social According to the Ancient Greek philosopher Proclus (c.419–486) in his Commentary on Euclid, geometry (from geo metros, “measurement of land”) first arose in surveying practices among the ancient Egyptians, to assist them in redefining property boundaries following the annual the flooding of the Nile Valley.
What’s it like to be a student in my class? — MATH VALUES
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.
How Will the New AI Impact Mathematics Research? — MATH VALUES
In my case, after I graduated in mathematics from King’s College London in 1968 (during my time at which I had summer jobs at BP Chemicals programming mainframe computers in Algol-60 and assembly language), I went to the University of Bristol, intending to do a PhD in Computation Theory, a subject just beginning to emerge, of which AI was a part.
How did human beings acquire the ability to do mathematics? — MATH VALUES
How (and when) did human beings acquire the ability to do mathematics? Someone raised that question at the end of a workshop I participated in just before Christmas. It’s a question that has nagged me ever since Richard Dawkins’ book The Selfish Gene (1976) made me aware of just how evolution works (at least, to a first approximation).
Counting Back in Hundreds — MATH VALUES
The second notable result that year was Hilbert’s hotel, introduced by David Hilbert in a 1924 lecture “Über das Unendliche”. Hilbert proposed a thought experiment to illustrate a counterintuitive property of infinite sets. A simple argument shows that a fully occupied hotel with infinitely many rooms (i.e. rooms that can be numbered using all the natural numbers) may still accommodate additional guests. In fact, it can accommodate infinitely many additional guests.
Fifty Years of Integer Sequences — MATH VALUES
It would be remiss of me to let 2023 pass by without acknowledging that it’s the 50th Anniversary of the appearance of Neil Sloane’s A Handbook of Integer Sequences back in 1973. That was just two years after I completed my PhD, so his reference source has been part of the mathematical landscape throughout my career. That first edition had just 2,372 entries. In 1995, with Simon Plouffe as a co-author, it rebranded as The Encyclopedia of Integer Sequences, with 5,487 entries.
What's the Best Way to Teach Calculus? — MATH VALUES
Short post this week, to give you time to read an important (and how!) research paper on university-level calculus instruction, published recently in SCIENCE, Vol 381, Issue 6661 (31 August 2023), pp. 995-998. Yes, it’s yet another study comparing active student engagement with traditional, lecture-based instruction. But wait. The paper’s title is Establishing a new standard of care for calculus using trials with randomized student allocation. As that indicates, this one is different.
New "DEVLIN’S ANGLE" Archive Goes Live — MATH VALUES
In late August, I started getting emails from people who went to look up an old Devlin’s Angle post, only to get the dreaded “PAGE NOT FOUND” notice for anything posted prior to August 2011. During that period were the two most cited, referenced, linked-to, and argued-about threads in the entire 27 years of the Angle: the series of posts on multiplication (and why it isn’t repeated addition) and the posts on the Davydov Curriculum in the then Soviet Union.
Attend to Precision. Why? — MATH VALUES
The Common Core State Standards for Mathematics (CCSSM) document leads off with a list of eight basic practices for doing and using mathematics from which the rest of the Standards follow by increasingly narrowing the focus to meet the needs of classroom education.
California Passes its New K-12 Mathematics Framework — MATH VALUES
The Common Core, introduced in 2010, was one major national initiative to adjust the direction education should take to meet the needs of today’s world, but it went only so far. California (my home state) had a particularly vexing problem to address. Its hugely successful economy (large parts of which are based on STEM, including, these days, our vast agricultural industry) was a result of that Cold War educational push.
ChatGPT: for Mathematicians, a Tool in Search of Good Applications — MATH VALUES
I am sure we are going to see many such uses of LLM’s coupled with back-end systems built to perform specific tasks in various domains. As such, the LLMs will provide user-friendly front-ends to leverage the power of those tools without the need to master their controls.
When Patterns Mislead — MATH VALUES
One of the devices regular writers and bloggers turn to when an article is due and nothing quite right has cropped up, is to look for anniversaries that occur when the article is published. Which is what led to this month’s post.
When Patterns Mislead — MATH VALUES
One of the devices regular writers and bloggers turn to when an article is due and nothing quite right has cropped up, is to look for anniversaries that occur when the article is published. Which is what led to this month’s post.
Does AI Pose a Threat to Mathematics Education? — MATH VALUES
National Teacher Day in the US takes place on Tuesday May 2 this year. Many countries celebrate their Teachers' Day on 5 October each year, in conjunction with World Teachers' Day, which was established by UNESCO in 1994, but the US is not alone in setting a different date.
Monuments to Algebra — MATH VALUES
Most mathematicians, if asked to provide the five most significant figures in the chain of development that resulted in modern algebra would say Muḥammad al-Khwārizmī, (ca.780–ca.847), who lived at least most of his life in Baghdad, Leonardo “Fibonacci” (ca.1170–ca.1250), who was born and spent most of his life in Pisa, and three Frenchmen, Viète, Pierre de Fermat (1601–1665), and René Descartes (1596–1650). Diophantus (200–284) is ruled out by the way the question is asked.
Might May Not Always Make Right; but it Often Wins — MATH VALUES
The interest and debate started by the release into the wilds of the Large Language Model ChatGPT (the GPT stands for “Generative Pre-trained Transformer”, by the way) brought back memories of when I migrated across to the US from my native England back in 1987, to join a large and growing team of international researchers working at Stanford University’s Center for the Study of Language and Information (CSLI).
People are rational; but are they logical? [Creativity – Part 2] — MATH VALUES
Well, maybe I should amend my title to begin “People can be rational.” But my focus here is whether being rational is the same as being logical. If by the latter we mean “follow the rules of logic,” then I would say the answer is “No.” Even in mathematics, though we produce logical proofs to demonstrate mathematical facts to be so, we don’t use logical reasoning to find those proofs.
Understanding Mathematical Creativity — MATH VALUES
What is going on in our brain when we finally come up with the key idea to solve a mathematical problem we have been struggling with for some length of time (an hour, a few days, a month, a year, longer still)? The simple answer is, when the question is asked about the brain, no one has a clue.
What Use Is Math? — MATH VALUES
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.
Coming to grips with Mastodon — MATH VALUES
As Fig 2 indicates, what you will never see are toots from Mastodonians who have no connection to you or your instance. If you hear about them some other way, you can try to find them by entering, say, a URL in the Search facility (top left on my dashboard shown in Fig 1) and bring them into your network that way. But that’s it. Of course, as your network expands, so too will your accessible universe within the Federation.
Coming to grips with Mastodon — MATH VALUES
By Keith Devlin https://fediscience.org/@KeithDevlin I spent last weekend (November 5-6) in Mastodon. One of the first lessons I learned was, I was spelling it wrong; it’s not Mastadon as I instinctively wrote. It turned out I was not alone. In fact, most people I interacted with made the same mistake. But I digress. Why was I there?
Playful Math – Is there a "there" there? — MATH VALUES
Of course, graduate students – and professional research mathematicians (“graduate students who get paid more” when actually engaged in their research, and not teaching or engaged in administrative tasks) – already know a fair amount of mathematics and have made a decision to pursue the subject. So, they are able to keep the “play” from veering too far off course for too long.
Not a Walled Garden — MATH VALUES
The August/September issue of the MAA magazine FOCUS has a wonderful article by Jonathan Weisbrod about a Superbowl-week beer commercial starring renowned mathematics professor Ken Ono of the University of Virginia.
MAA Blog: Not a Walled Garden — MATH VALUES
The August/September issue of the MAA magazine FOCUS has a wonderful article by Jonathan Weisbrod about a Superbowl-week beer commercial starring renowned mathematics professor Ken Ono of the University of Virginia.
Mathematics: Prescriptive or Descriptive? — MATH VALUES
I was not turned off listening to and enjoying music. Heavens, I was growing up in England in the “Swinging Sixties”! But I had a deep-felt impression that, beyond the enjoyment my ears provided, I was lacking the “music gene” required to really understand it (and hence become a musician).
MAA Blog: Picturing and Modelling Information Part 2 — MATH VALUES
By Keith Devlin @profkeithdevlin This month’s essay completes a thread commenced with the May, June and July posts. Last month, I described the sequence of events that led me to join an interdisciplinary team of researchers at Stanford University’s Center for the Study of Language and Information (CSLI) in 1987 to try and develop a mathematically-based theory of information, that could support analysis of human–human, human-machine, and machine-machine communication.
MAA Blog: Picturing and Modelling Information Part I — MATH VALUES
By Keith Devlin @profkeithdevlin This month’s essay continues a thread commenced with the May and June posts. In 1958, Harold (Hal) Leavitt (1922–2007), a managerial psychologist, and Thomas Whisler (1920–2006), professor of industrial relations, both at the University of Chicago, published what would prove to be a highly influential article in the Harvard Business Review titled Management in the 1980s.
MAA Blog: The Secret to Powerful Math — MATH VALUES
By Keith Devlin @profkeithdevlin This month’s essay continues a thread commenced with last month’s post. I gave a guest lecture at a local retirement community recently, my first in-person speaking gig since the start of the pandemic. Located in the heart of Silicon Valley, my audience was, as I had initially assumed (and was confirmed during the initial exchange of emails with the organizer), a highly educated group; indeed, some of them were former professors from Stanford and elsewhere.
MAA Blog: Math in the Real World — MATH VALUES
I’ve been using the illustration in Figure 1 in courses and talks for several decades now. Not only to show how mathematics connects to the “outside world”, but also to illustrate how — and why — new mathematics is developed.
MAA Blog: Making the Math Education Sausage — MATH VALUES
By Keith Devlin @profkeithdevlin “People who like sausage should avoid watching it be made.” Old saying There’s an old Irish joke that goes like this. A motorist pulls up in front of a pub, winds down the window, and addresses an old man sitting at an outside table sipping a glass of Guinness: “Excuse me, what’s the best way to get to Killarney?” The old man puts down his glass, and responds: “Killarney, you say.
MAA Blog: Writing About Mathematics — MATH VALUES
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.
MAA Blog: PISA 2022 Gets It Right — MATH VALUES
The Organisation for Economic Co-operation and Development (OECD) Programme for International Student Assessment (PISA) provides international comparisons of national educational achievements What image comes to mind when you hear a reference to the website for the Common Core State Standards for Mathematics? In particular, do you think of a single Web page, or a document that would require almost 100 pages to print out? What are the most important parts of the document?
MAA Blog: How Many Words for Snow? — MATH VALUES
by Keith Devlin @profkeithdevlin This month’s post is a historical piece that looks back 25 years, and reprints an article I published in “Devlin’s Angle” in 1997. Society has moved on from then and, as is highlighted by the cartoon at the end, become much more aware of the problems that arise when one societal group views another through their own lens, especially if that lens assumes or conveys—or is perceived as conveying—a sense of superiority.
MAA Blog: Science in Public — MATH VALUES
I had been looking forward to attending the Joint Mathematics Meetings in Seattle early this month. I had attended every JMM since I moved to the US in 1987, and missed not being able to meet up with my many friends and colleagues last year, when the coronavirus pandemic led to the cancellation of the January 2021 meeting as an in-person event. But it was not to be. On December 22, the AMS announced that they had reluctantly decided to cancel this year’s in-person meeting as well.
MAA Blog: Revisiting "Wicked Problems" — MATH VALUES
I’ve written on this blog before about the false impression math educators can leave their graduates with, by focusing entirely on tightly constrained and carefully crafted problems, devoid of context, that yield a single right or wrong answer within a pre-determined length of time. For sure, such problems are ideal for developing specific math skills.
MAA Blog: A Mathematical Perspective — MATH VALUES
How does being a mathematician affect the way you approach the world? I was asked this question recently on a live interview, being streamed globally on a multi-national corporate network. The interviewer prefaced the question by noting that on more than one occasion I had gone down in print and on video saying that being a mathematician influenced almost everything I do. “Give me three specific examples,” the host said. My responses are below.
MAA Blog: Math and Science in Decision Making — MATH VALUES
A radio broadcaster I‘ve worked with in the past mentioned to me recently that she’d been reading some of my recent online posts on mathematical issues connected to the COVID-19 pandemic, in particular my posts on “Devlin’s Angle”. After we’d finished our brief exchange, I looked back to remind myself which specific topics I have covered. I was surprised to see that I’ve written eight coronavirus-related posts since the pandemic began (in the US) in March of last year.
MAA Blog: Helping People Understand Important Statistics — MATH VALUES
There was a flurry of reports and social media posts recently commenting on an Israeli study showing that nearly 60% of all patients hospitalized for COVID-19 (as of August 15, 2021) were fully vaccinated. A common take, by both journalists and commentators, was to conclude that the power of the vaccine must have waned dramatically since the early days. (Israel was way ahead of most nations in getting the vaccine out, and almost 80% of all Israelis over 12 years of age are now fully vaccinated.
MAA Blog: Data Science Literacy in Media — MATH VALUES
By Keith Devlin @profkeithdevlin As I was looking around for a topic for this month’s post, I wondered idly what I had written about in Devlin’s Angle exactly a quarter century ago, in August 1996. (The column had begun in January of that year, as had the MAA’s website.) Given the many major changes in the world since then (for example, Google would not be founded for over two more years!), would my words then have any relevance today? They do.
MAA Blog: The History of Binary Arithmetic — MATH VALUES
As the musical satirist (and mathematician) Tom Lehrer quipped in one of his songs (“New Math”), “Base-eight arithmetic is just like base-ten—if you are missing two fingers.” It was a funny line that always got a laugh, but it highlights a major feature of the Hindu-Arabic positional number system used around the world to perform integer arithmetic. We group in tens for the simple reason that we learn to count and perform elementary calculations using our fingers.
MAA Blog: Is Euler’s Identity Beautiful? — MATH VALUES
By Keith Devlin @profkeithdevlin Beauty, they say, is in the eye of the beholder. That’s certainly the case when it comes to beauty in mathematics. I long ago learned that not every mathematician agrees with me when I describe a particular mathematical concept or result as beautiful; they may see it as anything but.
MAA Blog: Email From a Former Student I’d Never Met — MATH VALUES
By Keith Devlin @profkeithdevlin For those of us who teach a subject so many find hard and frustrating (because it is), praise for our efforts from students comes relatively infrequently, so when it does it makes our day. I got this email from a former student of my online course on Coursera recently. Hi Keith, I just wanted to thank you for the role you played in inspiring me on my mathematical journey. About 7/8 years ago I took your course on Coursera "Introduction to mathematical thinking".
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