DEVLIN’S ANGLE POSTS (2018–DATE)
The links take you to the current MAA site, MathValues. Posts there are listed most recent first. To navigate you may have to jump from one page of posts to the next, or use key-word search, so you may sometimes find this annotated index provides the quickest way to locate an old post. The 2018 posts were all duplicated from the old BlogSpot platform to provide for a smooth(er) transition.
2018
JAN Déjà vu, all over again My recent experience teaching a math class to a group of high school students at an elite private school in Silicon Valley in an elective vacation session brought back memories of my own high school experience in a working class town in the North of England in the 1960s. The demographics were vastly different, but the post-war “Boomer” generation I was part of had a thirst for knowledge that equaled that I saw in that self-selected group of students from elite Silicon Valley families. That realization led me to reflect on what it takes to produce students who can fully engage in challenging, open-ended, mathematical problem solving exercises. First post in a series of four.
FEB How today’s pros solve math problems: Part 1 A continuation of last month’s post about the problem solving courses I gave at an elective vacation course at an elite private school in Silicon Valley.
MAR How today’s pros solve math problems: Part 2 I take last month’s post a step further.
APR How today’s pros solve math problems: Part 3 (The Nueva School course) I describe the actual course problem referred to in the previous three posts.
MAY Calculation was the price we used to have to pay to do mathematics I consider the changes in mathematics education (at both school and university level) required now that there are widely available computer system that can carry out any mathematical procedure. What skills do today’s mathematics graduates need to operate in such a world?
JUN Cycling can be such a drag–and math can tell you exactly how much A chance encounter with a former aerospace engineer and cycling enthusiast who designed lightweight devices to add to racing bikes to reduces drag. I already knew that the mathematics of high performance racing bikes was heavy duty mathematics, and an exchange of email with this engineer confirmed that this was a fascinating new application of mathematics—and of mathematical thinking.
JUL 21st Century Math: The Movie In May, I participated in a global mathematics education summit in Geneva, Switzerland, organized to discuss the new way mathematics is being done and how best to prepare students to live and work in such a world. Both the United States Department of Education and the OECD’s (Organization for Economic Cooperation and Development’s) PISA educational testing organization were represented at the summit. The post recounts my experience.
AUG How a Fields Medal led to a mathematical roller-coaster journey When I headed off to Bristol University in the UK in 1968 to start on a PhD, I found myself at a place where research was being done into a hot new area of mathematics research. I quickly dropped the topic I had intended to work on, and jumped ship into Axiomatic Set Theory, an exciting roller coaster of new ideas that I rode for over a decade, moving on to something else (also exciting and new) only after the pace of change in set theory started to slow down. I was lucky to be in a right place at the right time, to be part of something new and exciting. Those periods are relatively few and far between.
SEP Is math really beautiful? The title is a question; the post provides an answer—of sorts.
OCT It’s high time to re-focus systemic mathematics education—and change the way we assess it The change I argue for is from assessing performance on the many individual topics in the Common Core State Standards, to assessing students’ achievements of the eight fundamental Mathematical Practices. Modern mathematics should not be approached as a large bag of tricks, but a holistic framework for understand the world and solving problems in the world for society to make progress. The Mathematical Practices should be more than just a “trailer” to the ”curriculum-ready content” that follows; the Practices are where the action should be.
NOV T-assessment: a bold suggestion modestly advanced A continuation of the previous month’s column, where I go into some specifics regarding assessment.
DEC To Boldly Go … 1996 was a year of major change at the MAA, with the launch of a new member service, MAA Online, and the start of a shift from the world of print to. The date in the first paragraph of this month’s column is wrong; the new Website was launched in December 1995, with the first posts coming out carrying the date January 1996, the new Devlin’s Angle being one of the regular features. Modulo that typo, the column recounts the shift-over activities that took place during 1996.
2019
JAN Teaching college-level math to students who grew up in a connected world I look at the new book published last December by Temple University social scientist Jordan Shapiro: The New Childhood: Raising Kids to Thrive in a Connected World. It’s much more than a “how to” manual for parenting (though he intended for it to be read that way). It’s also 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. Incidentally, the title refers to “college-level” math, but the book is equally applicable to K-12 teaching.
FEB What is mathematical creativity, how do we develop it, and should we try to measure it? Part 1 In an email exchange with an ex-teacher (not math) friend in the tech industry recently, we considered the question, “Can digital technologies, in particular digital mathematics learning games, help develop creativity, and can they measure it?” The first issue was to define what we meant by “mathematical creativity.” My post presents the definition we adopted (it’s not original to us), and recounts how we got there.
MAR What is mathematical creativity, how do we develop it, and should we try to measure it? Part 2 A continuation of the discussion in last month’s post, where I pull in insights from studies carried out by various scholars.
APR Summa de Arithmetica—a landmark in the development of the modern world The announcement by Christie’s that a rare first edition of Luca Pacioli’s landmark 1494 book on modern accounting methods prompted me to write this post. (I was familiar with Pacioli’s work from my research for my 2011 book The Man of Numbers, about Leonardo Fibonacci.)
MAY How double-entry bookkeeping changed the world A continuation of last month’s post where I look more closely as Pacioli’s work and the legacy he left behind.
JUN What topics should be covered in high school mathematics? What can we learn from advanced math? The first of a two-part series. I recount my own experience in doing mathematics, in particular what was involved in switching from very abstract, highly specialized, pure mathematics to working (in teams) in very applied areas. (Quick answer: Very little. And that’s the important message for K-12 education.)
JUL Mathematics is a Way of Thinking—How Can We Best Teach It? This is the meat that last month’s post prepared the way for. What won’t work is a menu-driven approach that sets out to equip the student with a collection of methods or procedures. Those are just tools of the trade. (Indeed, in today’s world, those tools all exist as pre-packaged apps, available freely and easily on the Web. The essence of doing/using mathematics is thinking. And as my first post made clear, to develop that capacity, it actually makes little difference what particular part of mathematics is the focus of the teaching. The tools are specific, but the method of thinking is universal.
AUG How Relevant is Cognitive Load Theory to Learning? The post explains what cognitive load theory is and addresses the question in the title. (Spoiler: I think it is veryrelevant.)
SEP What as a Mathematical Proof? I’ve written about this in Devlin’s Angle before, in December 1996 and June 2003. In this post I consider the definition: Proofs are stories that convince suitably qualified others that a certain statement is true.
OCT Student teaching evaluations are effective, but not in the way you think I comment on a large scale study. (That itself is unusual in the education world.) The relationship between good student evaluations and subsequent performance by those students in further courses raises questions, as you may suspect. Perhaps more surprising (though an explanation comes easily to hand), students who do worse in a course often end up doing better in subsequent courses than those who aced the first course. Cognitive scientists have much to say about these kinds of finding.
NOV Do Math and Chess Make You a Better Problem Solver? Teaching Math for Life in a Wicked World The majority of real world problems that mathematicians are called on to help solve are so-called “wicked problems”. In the 1980s, my university, Stanford, introduced a new interdisciplinary major to try to help students acquire the ability to solve such problems. Read the post to find out more.
DEC What Firefighting, Military Tactics, Cancer Treatment, and Teenage Parties Can Tell Us About Learning Math Using math to solve a novel real-world problem is hard. Cognitive science explains why. It also tells us what we need to do to overcome the feature of the human brain that causes the difficulty.
2020
JAN Reporting Numbers: Don’t Lie By Omission The message in this post is clear. Before you are swayed by a scary headline like “Doing X can double your chances of dying before you are 50”, check the actual data. If your risk increased from 0.001% to 0.002%, that “doubling” can safely be ignored.
FEB Giving a Math Course for Non-Stem Majors I’ve given many university math courses for non-stem majors over the years. In this post, I present some of the decisions I’ve made about how to give such a course. Student responses have been generally favorable, but that may be more a consequence of my enthusiasm—I do after all teach them because I like to—rather than the pedagogy. That can be summed up simply: I want them to get to the point where they can convince me they have engaged in some genuine mathematical thinking.
MAR Fibonacci-based music that Is not a Fibonacci Folly If you’ve read enough of my writing you will know I’m very dismissive of “Fibonacci-based” whatevers; they are almost always based on spurious claims about the Fibonacci numbers. This project is different. The only property of the Fibonacci numbers the project draws on is that it is a socially famous icon. The creativity comes not from any mathematical property of the sequence but the musical minds of the composers.
APR Making Sense of the Covid-19 Data A post I published on my profkeithdevlin.org blogsite, and some of the responses I got to it, led to this follow-up post about how to best communicate novel uses of mathematics to mathematically-lay-readers. (Even engineers had trouble understanding my argument; they were used to calculating probabilities, but not when such calculations were done to provide a model. My story about “God’s Book” was my attempt to convey what I was doing.)
MAY The Graph That Stopped The World As with last month’s post, the appearance of what became the iconic graph in the early days of the COVID-19 pandemic brought cries that it was a misuse of mathematics, since it was not a graph of data that had been collected, rather it was “made up”. Indeed it was, but it was “made up” using the known rules for the growth and spread of epidemics. As such, it provided a projection—a look into possible futures— to guide decision making. The goal was to take action to ensure that the graph was never aligned to reality! To many people, this was a new and unfamiliar use of graphs.
JUN Can We Really Understand Exponential Growth? The gist of this post is that the answer to the title question is “No.”
JUL An Unlikely Marriage of Mathematicians and Computers A look back to the late 1980s, when the AMS decided to be pro-active in encouraging and supporting mathematicians to make use of the new computing capabilities that were rapidly becoming available. (Mathematicians were much slower than scientists and engineers in embracing the new technology.)
AUG Of Course, 2 + 2 = 4 Is Cultural. That Doesn’t Mean The Sum Could Be Anything Else I wrote this post in response to an uninformed twitter storm trashing the self-evident truth that “2 + 2 = 4” is a cultural creation. The problem of course is that the trolls did not really understand the way mathematical formalisms work. They were essentially reading the standard interpretation into the formalism, thereby missing the way(s) mathematics (a social construct if ever there were one) connects to the world.
SEP Fibonacci in Pictures This is a link to an earlier post (September 2010). Reflecting on the way the MAA website had been changing, at the end of the article I provided a new link to the online Fibonacci photograph album since the one I provided in 2010 no longer worked. History repeats itself in that the new link I provided is itself now broken. The (currently, September 2023) working link for that album can be found HERE.
OCT Proving Unprovability In 1963, Stanford mathematician Paul Cohen developed a method for rigorously proving (with the framework of Zermelo-Fraenkel Set Theory) that certain formal statements cannot (themselves) be proved within that axiomatic framework. He used his new method to prove that Cantor’s Continuum Hypothesis (CH) cannot be proved. Since Kurt Gödel had previously shown that it could not be disproved, that meant that CH is formally undecidable. Cohen was awarded the Fields Medal for that result. I was one of many doctoral students in the decade that followed who chose the new field of undecidability results as my research focus.
NOV The 1,001 Ways To Have A Successful Career In Mathematics A tweet alerted me to the fact that the academic ranking service academicinfluence.com had declared me to be the mathematician in the world who had had the greatest societal impact during the past twenty years. It was a nice accolade, but I could not help but wonder why I ranked above many I would have put much higher. The crux if, of course, the criteria used by the algorithm the ranking organization clearly used. In my post, I tried to reverse-engineer their algorithm. It was a “Moneyball” ranking, and since the ranking was surely done on the Web, my online activities clearly played a big role. Hence my post’s title.
DEC Mathematics For The Million What should be the focus (and the pedagogy) for the K-10 mathematics education we provide to meet the life-needs of all our students, and what parts of mathematics should be provided as electives? It’s possible to have a thoughtful discussion about this question, but one thing is surely beyond debate: the current system we use in the US is wildly wrong. It’s beyond debate since it fails (miserably) the majority of our students, in many cases producing high-school graduates who not only have grossly inadequate math skills, they leave with a deep-felt hatred of the subject.
2021
JAN A Quarter Century And Counting Devlin’s Angle is 25-years old this month. This brief note reflects on the many structural changes that have taken place on the Web during that time. Since those changes have left previous posts of my column (as it was called in the early days), I provide a guide of where to find those archives. That guide no longer works, which is why I have created this new archive (with lost posts re-generated from the files I originally submitted). Sadly, the nature of the Web being what it is, there is no guarantee this new archive source will not one day become inaccessible.
FEB Covid-19 And Learning Loss As the world started to emerge from the COVID-19 pandemic, there was a flurry of articles discussing the “learning loss” that resulted from the societal restrictions that were required to keep the number of deaths well below what they would otherwise have been. The trouble is, no one had a good definition of “learning loss” or (hence) how to measure it. Indeed, once you try to formulate a useful definition, you find the ground sliding away beneath you. Understanding the relationship between teaching and learning figures largely in that morass.
MAR Covid-19 And The Math Of Life Expectancy A pandemic-related news story that caught my attention led me to wonder how the statistic “life expectancy” was calculated. What I learned when I dug into it, has important lessons for the way government agencies and other authorities communicate with the public at large. My post describes what I found.
APR How The Ancient Greeks Constructed The World’s First Mechanical Computer Researchers at University College London (UCL) have finally figured out the likely mechanism of the 2,000-year-old Antikythera device, arguably the world’s first mechanical computer. It’s a fascinating story.
MAY Email From A Former Student I’d Never Met This short post speaks for itself. Every educator who receives such a communication will know how it made me feel.
JUN Is Euler’s Identity Beautiful? And If So, How? I reflect on what we mathematicians mean when we say “mathematics is beautiful”.
JUL The Colorful History Of Binary Arithmetic This is a new take on my second ever Devlin’s Angle post, back in February 1996 (with illustrations, something not possible back in those days).
AUG When The Media Get The Math Wrong—Badly Wrong Sometimes the media can get a math story very badly wrong. I present some (worrying) examples. It’s in society’s interest to make sure people are able to detect when a sensational story just doesn’t ring true. The last five words of my post presents what I think we need to do.
SEP How Do We Help People Understand Statistics That Matter To Them? When major news organizations badly—and I mean badly—misunderstood (and hence misreported) a recent research finding about vaccine efficacy, you know we are failing to educate our citizens to understand statistical data. In examples like this, lives are quite literally at stake; millions of them. How can those of us in the math-ed biz help fix this? I present some specific pointers in my post.
OCT What We’ve Got Here Is Failure To Communicate–And Adequately Educate! This is a continuation (at some length) of the argument I presented in the previous month’s post. This problem is a major challenge to the mathematics education profession.
NOV Seeing The World Through Mathematical Eyes Arguably the biggest single life-skills-benefit that comes from experiencing mathematics is learning how to cope with being wrong. I present my reasoning.
DEC The “Wicked Problems” Problem, Recycled Even many experienced cyclists don’t realize that to turn left on a bicycle you have to turn the handlebars to the right. Yes, you turn the handlebars the opposite way to the direction you want to turn! Providing a scientific explanation of why that is the case is not easy; in fact, the mathematics of bicycle riding is extremely complex. Bicycle dynamics is not an example of a wicked problem. But it is where you end up when you (as a mathematician) ask the seemingly simple, wicked problem, “How do you ride a bicycle?” The mathematical analysis of bicycle dynamics is just one part of the answer.
2022
JAN Lights, Camera, Brainstorm! – Doing Science In Full View In A Global Pandemic I reflect on how leading scientists engaged with the general public in the course of doing the scientific research to understand, and learn how to cope with, the COVID-19 pandemic. Anyone with a Twitter account had a front-row seat to science in action. This provided not only up-to-date information of the consensus view of the experts, but more generally excellent insight into how science advances by a messy process of observation, experimentation, data collection, analysis, conjecture, debate, and peer-evaluation.
FEB How Many Words For Snow? I reprint a post I made 25-years earlier, in February 1997. Changes is societal norms required a change in the title, and that in itself is part of the message conveyed by this post.
MAR Pisa 2022 Gets It Right I take a look at the newly released PISA 2022 Mathematics Framework, which set out to answer the question, “What mathematical knowledge and skillsets does a person need for life in the 21st Century?” My title gives my view of what they produced, based on my own experience as a mathematician, in mathematics departments in research universities, at liberal arts colleges, and working with industry and for large departments of the US government.
APR Making The Mathematics Education Sausage In this follow-up to the previous month’s post, I look at the new California Mathematics Framework, finding that is very much aligned with PISA 2022, though people I know who were involved in the California initiative, they worked independently, so any similarities in the outcome reflect the fact that both groups had essentially the same goal and were operating with the same data.
MAY Mathematics And The Real World I elaborate on an aspect of mathematics I alluded to in last month’s post; namely, how mathematics fits into human activities; when it is used, why it is used, and how it is used. This is rarely made explicit in traditional mathematics education, but I think it should be.
JUN The Secret To Powerful Math Is Often A Simple Diagram A continuation of last month’s theme. I observe that even highly complex, abstract mathematics is often (in fact, almost always) based on a very simple (indeed simplistic) mental image. We math educators need to make our students (and our colleagues in other disciplines) more aware of this fact. It helps explain how we manage to produce the results we do. It’s not impenetrable “rocket science,” but then neither is “rocket science.”
JUL How Can We Picture Information? How Can We Model It Mathematically? Part I A (recent-) historical account of a 1980s project to try to develop a mathematically-grounded theory of information. It provides a contemporary illustration of some of the issues implicitly raised in the previous posts about K-12 mathematics education.
AUG How Can We Picture Information? How Can We Model It Mathematically? Part II This post begins where last month’s ends.
SEP Mathematics: Prescriptive Or Descriptive? Spurred by an email exchange with a musician who was familiar with my outreach work and was speculating about a possible collaboration, I reflected not only on the similarities between mathematics and music, but also on the fact that my childhood introduction to “music theory” was hampered by the (exclusively) “prescriptive” perception of mathematics I had at the time (a result, of course, of it being taught that way in K-12 education).
OCT Not A Walled Garden A well-known, highly respected mathematician starred in a television beer commercial recently. I explain why I think this was a good thing, both for mathematics and society. (I was not impartial, as I explain in the post.)
NOV Playful Math – Is There A “There” There? “Playful math” is serious stuff. I present a number of examples.
DEC What Use Is Math? The first part of my answer is to point readers to the new BBC-tv series The Secret Genius of Modern Life, presented by British mathematician Hannah Fry, and the new PBS Nova television documentary Zero to Infinity, hosted by American mathematician Talithia Williams. But there’s more; a lot more (including more videos). Do check out the post.
2022 MAA POSTS NOT FOR DEVLIN’S ANGLE
Mar 29, 2022 Writing About Mathematics For A General Audience Written, by request, for MAA members.
Nov 9, 2022 Coming To Grips With Mastodon Written, by request, for MAA members, at a time when Twitter seemed to be collapsing as a medium mathematicians could make good use of. (The MAA was a very early Twitter user.)
2023
JAN Understanding Mathematical Creativity Some (published) joint research I had been involved in suggests a novel way to explain mathematical creativity. I provide a blog-level summary.
FEB People Are Rational; But Are They Logical? [Creativity – Part 2] More detail on last month’s post.
MAR Might May Not Always Make Right; But It Often Wins The apparent effectiveness of the new tool ChatGPT led me to reflect on the work I did at Stanford in the 1980s that was at the time the cutting-edge approach to developing computers that could understand and process natural language. ChatGPT shows you can get linguistically fluent language processing if you ignore understanding. History will tell if this is a step forward or a distraction in terms of “natural language processing.”
APR Monuments To Algebra France has monuments to three giants in the development of algebra: Viete, Fermat, and Descartes. I take readers on a visually illustrated tour of those monuments.
MAY Does AI Pose A Threat To Mathematics Education? ChatGPT raised this question. On its own, it seems likely (indeed, almost certain) that ChatGTP, or indeed any Large Language Model, can threaten mathematics, unless you consider the downside of drawing researchers attention to a technology that has no “understanding” of the strings it processes. But if you combine a LLM with a mathematical resource, there will surely be ramifications to how mathematics is taught and done.
JUN When Patterns Mislead I continue the discussion of what effect LLMs like ChatGTP will have on mathematics. The new from the UK that ChatGTP enhanced with the Wolfram Alpha plug-in aced the school-levers college-entry exam is surely a weak-up call, but not necessarily a harbinger of doom.
JUL ChatGPT: For Mathematicians, A Tool In Search Of Good Applications I present some dramatic failures of ChatGTP in handling mathematical requests. Lack of a concept of “truth,” the program is not going to put mathematicians out of work. But it surely has the potential to produce major changes in mathematical praxis; just as tools like Mathematica and Wolfram Alpha did.
AUG California Passes Its New K-12 Mathematics Framework After a lengthy and at times contentious period of development and debate, California passed (unanimously) its new Mathematics Framework. I provide a summary.
SEP Attend To Precision. Why? One of the things the Common Core State Standards asks as a top-level goal of mathematics education is “Attend to precision.” Why did I ask why this goal is included? It came up when I was asked to provide expert testimony in a large class action lawsuit that hinged on a mathematical statement.
OCT New Devlin’s Angle archive goes live A post announcing the creation and publication of this new archive of all Devlin’s Angle posts since January 1996, explaining what led to the creation of this new archive.
NOV What’s the best way to teach calculus? A short post drawing attention to an important new, large-scale, randomized-control study of university-level calculus instruction, published recently in SCIENCE magazine.
DEC Fifty years of integer sequences A short post marking the 50th anniversary of the appearance of Neil Sloane’s A Handbook of Integer Sequences, which later morphed into the much-used Online Encyclopedia of Integer Sequences (OEIS).
2024
JAN Counting back in hundreds What were the major developments, if any, in each of the years 1924, 1824, etc.? (Why should we restrict anniversary articles to the decimal-number-base years?) I wrote the article only because I scored two big personal-research-interest hits with the 24s.
FEB How did human beings acquire the ability to do mathematics? I summarize the thesis I proposed in my book The Math Gene: How Mathematical Thinking Evolved And Why Numbers Are Like Gossip, published in 2001.
MAR How will the new AI impact mathematics research? I review the development of artificial intelligence and its use by mathematicians since the field of AI began in the 1950s, and consider the potential of the new, “Large Language Models” AI systems such as ChatGPT.
APR What’s it like to be a student in my class? I reflect on the perceptions (and resulting expectations) about mathematics that today’s students bring to their education, compared to those I had, back in the days when there were no computing devices and log tables and slide rules were the most useful calculation aids.
MAY Mathematical models – useful but often false I take a look at some well-known mathematical models that are in regular (and reliable) use, even though we know they are false, in some cases (e.g. the Bohr atom and electric current) badly so.
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