DEVLIN’S ANGLE POSTS (AUG 2011–DEC 2018)
The links take you to the MAA’s live-archived BlogSpot site, where posts are listed most-recent first. The site has its own date-index in a sidebar, but the annotated index here provides the easiest way to locate particular posts. Note that the 2018 posts were all duplicated on the MAA’s current MATH VALUES site to provide for a smooth(er) transition.
2011
AUG The First Personal Computing Revolution Leonardo Fibonacci and his landmark book Liber abbaci
SEP The First Arithmetic Textbook in the Western World Liber abbaci and the more elementary books that followed
OCT Mathematics: A Recyclable Tool for the Modern Era Contemporary applications of mathematics
NOV How multiplication is really defined in Peano arithmetic This is usually regarded as college-level mathematics but it is pertinent to the MIRA discussion
DEC Christmas Trees from the Land of Santa Claus Infinitary trees, König’s Lemma, and the counter-intuitive Aronszajn trees
2012
JAN Patterns? What patterns? What exactly are mathematical patterns? The choral group Zambra sings Euler’s Identity
FEB If You Don’t Have a Web Presence, Are You Doing Your Job? A panel discussion at a math ed conference at Stanford urged math ed researchers to do a lot more to communicate their findings to teachers; an active Web presence is one good way.
MAR The difference between teaching and instruction Instruction is a part of teaching, but effective teaching needs to be much more.
APR When math and art meet If you put pi to music, can you copyright it? A court ruled “no.” The post continued to discuss the Zambra concert I worked on, which wat first mentioned in the January post.
MAY Math MOOC—Coming this fall. Let’s Teach the World The first math MOOC is about to be released; my Introduction to Mathematical Thinking. The birth of MOOCs certainly changed the math education landscape, do technically my opening sentence “Higher education as we know it just ended” but as it turned out MOOCs were just an addition to a system that continued much as it was. (In fact, the post went on to speculate that would be the outcome.)
JUN Telling stories with numbers A new book argues that the way numerical data is displayed on a page can make a big difference to how well it is understood, and hence the impact it has.
JUL “Can’t we all get along?” The rapid growth of the non-profit Khan Academy resulted in a firestorm of criticism met with an equally forceful partisan defense by KA supporters. I found myself having sympathies with both sides. In any event, the debate raised a number of important issues about the goal and nature of mathematics education.
AUG The future of textbook publishing is us I discuss my new MOOC and give my reasons for self-publishing the accompanying textbook using Amazon’s CreateSpace.
SEP What is mathematical thinking? It takes a fairly lengthy post to provide the answer, but the notion is of an important part of mathematics that is often overlooked.
OCT The Fibonacci dedication in Pisa Description of an important mathematical historical location in Italy.
NOV MOOC Lessons Based on my own recent experience, I describe the challenges of designing and giving a MOOC.
DEC The Darwinization of Higher Education This post about the future of MOOCs did not fair will with the passage of time, but it does reflect the feeling those of us involved in the original MOOC initiative had at the time.
2013
JAN R.I.P. Mathematics? Maybe. The post points to an answer I gave to the annual Edge question. I speculated that, given the trend of math students on my MOOC to spend a lot of time learning LaTeX to format their work for submission, presentation would take priority over deep thought, leading to a generation that did not become accomplished mathematicians. I did not, and do not, believe that, but it is a possibility.
FEB The Problem with Instructional Videos Also stimulated by my MOOC experience, I discuss the work of Derek Muller (of Veritasium) on what it takes for an educational video to be effective, a topic that was the focus of his PhD. I find his work interesting and for the most part convincing. I also suggest his work has implications for the design of effective MOOCs.
MAR Can we make constructive use of machine-graded, multiple-choice questions in university mathematics education? I discuss the evolution of my MOOC, pointing to my decision to lean even more heavily on my original design model of one-on-one tutoring and student peer-interaction. I point to my blog MOOCtalk.org for more details.
APR Only in Silicon Valley This is an April Fools spoof that satirizes Silicon Valley.
MAY The Mother of All NCTM Addresses I discuss mathematics education professor Uri Treisman’s dynamite keynote address at the NCTM Annual Conference in Denver, Colorado, titled “Keeping Our Eyes on the Prize”.
JUN Will Cantor’s Paradise Ever Be of Practical Use? My PhD and the early years of my research career were in infinitary, axiomatic set theory. I can imagine a way that Set Theory would find practical applications, but in the end I think it will not happen.
JUL “It Only Takes About 42 Minutes To Learn Algebra With Video Games” I discuss the new mathematics learning video game DragonBox, which I like, and which has much in common with my own game Wuzzit Trouble.
AUG “Will this (mathematics) be of any use?” With the research I engaged with after I essentially left Set Theory to work on a new Theory of Information in the 1980s, the post 9/11 world saw me use my newer research experience (which had seemed very esoteric and decidedly non-practical, at the time) working on an intelligence analysis project for the NSA. I was forced to face up to the fact that if you do not want to engage in research that can have military applications, you have to avoid going into mathematics.
SEP Two Startups in One Week From book writing and publication to online courses to educational technology products, modern technology tools, and the Web as a distribution platform, have made it possible for anyone in education to create an educational product and make it available, either for fee use or for sale.
OCT Math Ed? Sometimes It Takes a Team A follow-up to the previous month’s post, where I observe that once you take the step to creating an educational product (other than a textbook), you almost certainly need to assemble a team to bring in the requisite areas of expertise required.
NOV The Educational Power of Elementary Arithmetic I present the take-home message I get from reading Liping Ma’s work on mathematics education. In a nutshell, you don’t need a broad curriculum for successful K-10 education. It’s not what you teach, it’s how you teach it.
DEC MOQR, Anyone? Learning by Evaluating I describe my attempts to modify my MOOC to fulfil the requirement of a quantitative reasoning (QR) course. On feature is my requirement that students learn to evaluate proofs rather than construct them, and are assessed on that skill.
2014
JAN 23 and Me. Play it again, Sam I provide links to all the Devlin’s Angle posts I get most requests for, on multiplication and repeated addition, mathematical thinking, and MOOCs. (Many of those links no longer work; this archive now provides that information.)
FEB Want to learn how to prove a theorem? Go for a mountain bike ride A somewhat whimsical, but accurate description of my attempts to master a notoriously tricky mountain bike trail ascent on near Stanford University and how closely it resembles attempts to prove a theorem in mathematics. First of a two-part series. Readers could substitute their own pursuits for my mountain biking; this is really about human problem solving.
MAR How Mountain Biking Can Provide the Key to the Eureka Moment Completion of the essay started in last month’s column. I speculate as to the nature of mathematical creativity in finding proofs to theorems.
APR What good is math and why do we teach it? A link to the 30-minute narrated video-stream of a conference presentation I gave at a middle school mathematics education conference. (The link in the post no longer works. The video is available here.)
MAY Déjà vu all over again: Fibonacci and Steve Jobs A link to the first of a two-video recording of a public presentation I gave at Princeton University. The embedded video link in the column no longer works. Here is a working link.
JUN Déjà vu all over again: Fibonacci and Steve Jobs — Part 2 The second-video of the two-part series started last month. The embedded video link in the column no longer works. Here is a working link.
JUL The Power of Dots I comment on the news story that a parent took her daughter out of school to teach her math “the old fashioned way she had been taught.” In fact, the school had been exposing the student to the kind of mathematical thinking that is important in today’s world. I look more generally at the power of simple diagrams in doing mathematics.
AUG Most Math Problems Do Not Have a Unique Right Answer After ruminating at some length on the myth that math problems have unique right answers (a result of feeding students a diet consisting only of such problems), the column concludes by looking at some excellent new resources to bring to the school classroom.
SEP Will the Real Geometry of Nature Please Stand Up? A wide ranging discussion of how mathematical theories relate to the world, including the two identities 0.999… = 1 and 1 + 2 + 3 + … = –1/12. Both are equally correct, and depend upon assumptions as to how we handle infinity.
OCT The Straw Teacher The “teachers “ the front-line troops in the Math Wars often argue over—the ones who focus exclusively on repetitive drill and the ones who talk about conceptual understanding but don’t care about getting a right answer—surely don’t exist. All the ones I’ve met lie somewhere in the middle, positioned somewhere on a spectrum. In reality, it comes down to relative emphasis, and area where productive debate can take place. The links in the column don’t work, but the dates are there and this index can get you to the right articles.
NOV Against Answer Getting Math educator Phil Daro’s excellent video, linked early in the article warns of the problems that arise in K-12 education when students come to view mathematics as “answer getting.” My column looks at the problems college instructors face when their students arrive on campus with that misleading perception of math firmly entrenched.
DEC How do you find good math learning apps? With hindsight, I see the title of the article could be misleading. It’s not a checklist of features to look for in apps; rather a discussion of the process of creating apps and a suggestion that the most reliable guide to selecting an educationally-good app is to check out the folks who designed and produced it. I also provide a cautionary tale against assuming that people who use math all the time, such as engineers or even math professors, can provide reliable advice on K-10 math learning; in general, they cannot. Knowing how to do or use math does not automatically make anyone able to teach it or advise on K-10 math education. (Almost the opposite is true.)
2015
JAN Your Father’s Mathematics Teaching No Longer Works Today’s K-12 math education system was developed to meet the societal needs of the 19th Century, when computers were people (aided by mechanical devices as the 20th Century wore on). The needs today are different; mathematics rather than computing (which is now done entirely by machines). The Common Core was developed to direct focus onto today’s needs. I provide links to talks by Sugata Mitra, Sir Ken Robinson Eric Mazur, and other sources to highlight the educational approach we need today, at both school and college level.
FEB The Greatest Math Teacher Ever? Following on last month’s column, I combine earlier posts from May and June 1999 into a single jumbo essay. The Moore Method had its focus on producing research mathematicians, but math instructors at all levels, for any students, can make productive use of “Discovery Learning.” It’s not easy to pull off (I tried several times), but for the students’ sake its worth persisting. [It is not “letting the students engage in free exploration”, as opponents frequently make out!]
MAR Pi Day, Cyclical Motion, and a Great Video Explanation of Multiplication Pi Day is special this year: 3.14.15, so we get four decimal places (and if you timed it right, you caught 3.14.15 9:26:53). The column this month has links to a lot of fun (and highly informative) pi-stuff that teachers can make use of.
APR The Importance of Mathematics Courses in Computer Science Education A discussion of the different perspectives mathematicians and computer scientists often bring to those two “formal systems”. It can provide (minor) hurdles to computer scientists learning math and vice versa, but understanding the differences can help develop expertise of both disciplines.
MAY Time to re-read (or read) What’s Math Got To Do With It? The article makes it clear why I recommend you do this. There is much to be learned from Boaler’s research.
JUN PIACC – PISA for grown-ups The OECD’s PISA program is well known. But the organization also examines the nation-based adult skillsets that are most significant to national prosperity in a modern society. Whereas the PISA surveys focus on specific age-groups of school students, PIAAC studies adults across the entire age range 16 to 65.
JUL Is Math Important? The post is essentially just a link to a video of two discussions (back-to-back) hosted in Aspen, Colorado, by New York Times journalist David Leonhardt. The topic is the question that’s the title for this post. What makes this particularly worth watching is the selection of speakers and the views they express.
AUG Hard fun – video games creep into the math classroom The publication of Greg Toppo’s book The Game Believes in You, about educational video games prompted this column on the potential of video games to greatly enhance mathematics learning. The majority of math video games concentrate on repetitive-practice of basic skills, which has some value, but the medium offers so much more.
SEP A Brilliant Young Mind: The IMO goes to the movies It’s a good movie, that follows closely an earlier television documentary on the same story of young mathematicians at the International Mathematical Olympiad. Start by watching the two-minute problem-solving clip I link to in the article. I temper my enthusiasm for the movie by stating some concerns I have about such film portrayals of mathematics.
OCT Letter to a calculus student – The Sequel In August 2015, I received an email from a math student in Washington State, a much belated response to my July 2006 column “Letter to a calculus student”. This post presents (with the author’s permission) that entire email. Receiving it made my day.
NOV Today is George Boole’s 200th Birthday A short post pointing to a number of tributes to George Boole on the 200th anniversary of his birth.
DEC Life inside an impossible Escher figure I was a fan of the new video-game Monument Valley from the moment I first opened it and entered its magical recreation of a Escher impossible figure. The column it prompted me to write examines the features of video games that make creations like this one such effective learning environments. (Enjoyment and engagement are part of it. That’s why good video games are sometimes called “hard fun.”)
2016
JAN Do your kids find learning math hard? There may be an app for that! A new math app got rave reviews. Can we believe the claims? In this case, yes; the study was carried out by a team of researchers at the University of Chicago. Studies conducted by companies are always suspect (though may nonetheless be perfectly legitimate). But academics are not going to risk their careers faking a study, especially if the stakes are the distribution figures for a free (or low cost) educational app.
FEB Theorem: You are exceptional I present one of my favorites classroom examples of mathematical modeling. It’s very simple, it demonstrates how today’s powerful online computational tools can facilitate rapid problem exploration, and the result is initially surprising.
MAR The Math Myth that permeates “The Math Myth” A much-hyped new book claiming to show why teaching algebra should be dropped from the school curriculum does not stand up to even a cursory analysis by anyone who knows what algebra is (me in the case of the commentary I wrote); the book’s author does not. His argument does amount to a critique, which I would support, that there is a lot wrong with the way algebra is often taught in our schools. But the message there is change the teaching, don’t throw out the baby with the bathwater. (Or do I mean “mathwater”?)
APR Algebraic roots – Part 1 A look back at the origins of algebra in 9th Century Baghdad, and how it developed into the subject we know today.
MAY Algebraic Roots – Part 2 A follow up to the previous month’s look at the history of algebra (beginning as “al-jabr”) where I look at a recent educational video-game that requires you solve algebraic equations. The twist is that the familiar symbols of algebra are replaced by game-like artwork objects. That purely presentational change turns out to be enough to get schoolkids to being able to solve equations after a little over an hour with the game.
JUN Infinity and Intuition Prompted by an article in the New York Times, I look at some of the ways our intuitions can initially mislead us when we start to think about (mathematical) infinity—which mathematicians were forced to do when calculus came along.
JUL What Does the UK Brexit Math Tell Us? The recent Brexit vote in the UK was one of the worst misuses of mathematics in my lifetime. Basing a decision on a single number (total vote count) is irresponsibility on a massive scale, since is ignored the rich, informative information available in the voting data. I present a few examples of the kinds of factors business managers consider when trying to reach a decision based on numerical data.
AUG Mathematics and the End of Days I reflect on the recent documentary movie Zero Days, about the very real threat to today’s digitally-based world posed by malware that can potentially be numerically far more deadly than nuclear weapons.
SEP Then and Now: Devlin’s Angle Turned Twenty This Year Devlin’s Angle turned twenty this year (on January 1, to be precise). I reflect on why I was asked to write a regular column on the newly established MAA website. We were all learning as we took our first steps onto the then-new World Wide Web. (Back then I would probably have said “in the Web, rather than “on” it.)
OCT It was Twenty Years Ago Today A follow up to the previous post where I look back at what I wrote about in that first year of the column. The links are all broken, of course, and the original files seemingly lost, which is what prompted me to create this archive.
NOV Mathematical Milk and the U.S. Presidential Election With mail-in voting in the upcoming US election, I reflect on Down’s Paradox: why does each one of us who votes choose to do so, despite the fact that the probability of our individual vote having significance in vanishingly small? What benefit do we get? In addition to providing the (standard) resolution to the paradox, I compare the solution with the issues that arise when we develop a mathematical model of some aspect of reality.
DEC You can find the secret to doing mathematics in a tubeless bicycle tire You might now realize it from the title, but this month’s column was an argument against the direct instruction approach to K-12 mathematics education. If the goal is to develop effective mathematical thinkers, the approach doesn’t work. (Though it can play a role in an approach that is effective.)
2017
JAN So THAT’s what it means? Visualizing the Riemann Hypothesis I provide a refresher overview of the Riemann Hypothesis as a prelude to sending you off to view a remarkable new video by Grant Sanderson (of 3Blue1Brown). If, like me, you are not an analytic number theorist, this may provide the crucial internal “picture” of the hypothesis you were never able to create from the symbolic mathematics.
FEB Hans Rosling, July 27, 1948 – February 7, 2017 The post is a link to one of his powerful video presentations about how you can get valuable, persuasive information from data if you present it well. His presentation speaks for itself, so I restricted my commentary to the “Comments” section to my post.
MAR Finding Fibonacci On the occasion of the publication of my book Finding Fibonacci, I reflect on my own unplanned entry into the world of “mathematics outreach” (in large part by writing popular books) and how that led to me fascination with Leonardo Fibonacci.
APR Fibonacci and Golden Ratio Madness How did all those urban myths about the ubiquity of the Golden Ratio come about? This post provides the answer.
MAY The Math Gift Myth I push back against the myth that some people “have a gift” for mathematics (and hence, by extension, others do not). All the evidence points to that perception being false.
JUN Classroom Clickers Are Good; Except When They Are Not There’s been a lot of articles written of late about the use of clickers in classes. But research indicates that they can be detrimental to good learning.
JUL The Power of Simple Representations I present examples of simple representations or diagrams that facilitate understanding and advancement of the field (mathematics), ending with James Tanton’s wonderful “exploding dots”.
AUG What are universities for and how do they work? Every so often, governments decide they need to “make universities more efficient.” That generally means they see higher education as a production line for producing future employees with workplace skills they believe the nation’s industries require. Such action is based on ignorance of what universities do, and the result is invariably a national disaster that can take decades to recover from.
SEP The Legacy of Jonathan Borwein My good friend, colleague, and book co-author Jonathan Borwein passed away last August. In this post, I look back on some of our professional interactions.
OCT Monty Hall may now rest in peace, but his problem will continue to frustrate The American television personality Monty Hall passed away recently. To mathematicians, he will always be remembered for the “Monty Hall Problem,” the counter-intuitive probability puzzler named after him because of its use in a quiz show he hosted.
NOV Mathematics and the Supreme Court With partisan state legislatures using increasingly sophisticated computer programs to jerrymander electoral maps, democracy-supportive mathematicians have been developing methods to identify jerrymandered maps. A case currently before the Supreme Court addresses this issue.
DEC Clash of representations Numerical data presented graphically can be very powerful. But unless the presentation is constructed carefully, it can convey entirely the wrong message.
2018
[THESE ARE ALL DUPLICATED ON MATH VALUES]
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 summer 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 summer 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 apart.
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.
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