In an opinion piece for the New York Daily News, published in July 2012, mathematician Edward Frenkel and school superintendent Robert Ross write:
This Fourth of July will forever be remembered in the history of science as the day when the discovery of the Higgs boson was announced. The last remaining elementary particle among those predicted by the Standard Model of three forces of nature finally revealed itself through painstakingly assembled data of billions of collisions at the Large Hadron Collider, the most sophisticated machine ever built by humans.
But one important aspect of this great discovery has been largely hidden from the public view: the fact that this elusive particle was a mathematical prediction, and its discovery a triumph for an increasingly underappreciated discipline.
Unfortunately, we don’t hear much about the fascinating drama of ideas unfolding in modern mathematics — not even when they result in an epic discovery. This despite the fact that math is increasingly woven in the very fabric of our daily lives: every time we make a purchase online, send a text message, use a computer or GPS device, formulas and algorithms are launched to fulfill these tasks.
The language of math is as vital as ever, and the way forward is to rediscover its innate beauty and possibility. Replacing rigorous study of mathematics with “useful skills” is like teaching students how to paint a fence without ever showing them the paintings of Michelangelo and Picasso.
Educators and professional mathematicians should join forces to unlock the power and beauty of mathematics for students and inspire them to think big, so they can use this knowledge to create a freer and better world.
Poetic stuff, and I certainly agree with their basic point. I'm less sure about the details. It smacks a bit of the New Math, which was the short-lived, Sputnik-inspired idea that we should be teaching grade-schoolers about set theory and the axiomatic method. It's a good rallying cry to say we should get educators and mathematicians together to unlock the beauty of mathematics, but what does that mean in practice?
I would also note that the drill and drudgery of learning arithmetic does serve a purpose. Some amount of number sense and facility for manipulation is essential if you are ever going to have any success in learning algebra, geometry and calculus. It's hard to appreciate the beauty of mathematics when you are stumbling over the basic skills.
Somehow we need to find a balance between hammering the basic skills, while also making it clear that there is so much more to mathematics than arithmetic. I have no especially clever suggestions for how to do that, but more of my fellow mathematicians taking the attitudes of Frenkel and Ross would be a good start.
The current issue of the Notices of the American Mathematical Society turned up in my mailbox today. It features an editorial (PDF format) by mathematician Doron Zeilberger which is partly a reply to Frenkel and Ross. Zeilberger writes:
The reason so many mathematically talented students are so turned off from math is that once they go to university, even the science and engineering students are taught by professional mathematicians, whose rigid, pedantic, “rigor-or-nothing” philosophy is imposed on the courses, at least in part.
Communication in mathematics is, even at the “highest” level of conference talks, highly dysfunctional. Highly specialized specialists who attempt to communicate their their highly technical, usually very dry, preprepared laptop presentations, and (almost) no one has any clue. Indeed, pure math has gotten so splintered that very few people see the mathematical forest. Most can barely understand their own trees.
One example is the AMS Colloquium Lecture series at the Joint Mathematics Meetings. No doubt some of these three-hour lecture series have been very good. But too often they are delivered by talented mathematicians who do not even attempt to make the lectures accessible to a general mathematical audience. Rather, they give highly technical talks with completely unrealistic expectations about the background of the audience.
Yes, yes, yes! That's exactly right. It drives me crazy that so many of my fellow mathematicians are so inept at basic communication. Or worse, take positive delight in blowing their audience out of the water. This problem is especially acute in mathematics textbooks, which are so obsessed with rigor that they give no thought at all to how things look to the novice. Two consecutive sentences of exposition will be derided as excessively wordy. Sadly, my experience has been that too many of my colleagues think this is exactly as it should be!
On the subject of unrealistic expectations about your audience, I can't resist telling a story. The first time I ever spoke at a research conference was as a third-year graduate student. I was pretty nervous about it, and watching the talk just prior to mine did not help. The speaker gave a real barn-burner of a talk that was jargon-filled and utterly incomprehensible. To me it seemed like he must be the smartest man on the planet, and I had to speak after him! When it was my turn I got up and gave my rinky-dink little talk about what I was working on for my thesis. I couldn't get off the stage fast enough.
Well, that night was the big conference dinner. By sheer coincidence I found myself sitting next to the speaker who preceded me. At some point he leaned over and told me how impressed he was with my talk. He said that he only vaguely grasped the gist of what I was doing, since he felt it was very intricate and complex. At first I thought he was making fun of me, but it quickly became clear that he was not.
That was when I realized that the work I was doing seemed obvious and trivial to me only because I had been immersed in it for so long. I'm sure the other fellow thought his work was pretty straightforward, and would have been surprised that I found it so baffling.
Anyway, back to the subject of mathematical communication. Every four years the Fields Medal is awarded to recognize extraordinary work in mathematical research. It is often described as the mathematical equivalent of a Nobel Prize. The recipients of this award are the giants of my field, and their accomplishments are of fundamental importance to mathematics.
Now, I grant you that I am not a great mathematician. But I have been working in this business for close to twenty years (starting the count when I entered graduate school in 1995), and I do know a thing or two about mathematics. So you can imagine my dismay when I read the descriptions of the Fields-winning research in the Notices, and found that they were written in a manner suggestive of a research journal (as opposed to a journal directed at a general audience of mathematicians). There was tons of jargon and notation and no attempt to place the critical results in any sort of perspective. This is the most important work going on in my discipline, and I was not able to make heads or tails out of it. That's poor mathematical communication. How am I to convince my friends and colleagues from other disciplines that mathematics is terribly important and interesting, when so many of the writers in my field cannot even make it seem important and interesting to me?
All of which is my long-winded way of seconding Zeilberger's point. He has it exactly right.
There's more to Zeilberger's essay, and I'm less convinced of some of his other arguments. But since this post has already gotten a bit long, we shall save that for another time.
_Modern Mathematics in the Light of the Fields Medals_
I mostly agree. I'm not so sure about the "experimental math" part, but I largely agree with the rest.
When teaching math, I tried to develop the mathematical intuition of the students. There's a time for rigor, but building mathematical intuition is at least as important.
I once tried to persuade my colleagues to consider Sylvanus P. Thompson "Calculus Made Easy" as a calculus text. My colleagues were horrified at the idea.
Definitely right about conference talks. Half the time, the audience of experts is lost, and no one dares say so.
The most wonderful thing about my grad school advisor was that he wasn't afraid to point out that the Emperor had no clothes. He'd actually dare to raise his hand and ask the speaker to back up and explain things! A marvelous trait indeed.
The Op-Ed by Frenkel and Ross was published in July of LAST year, so it's old news. Frenkel has just published a book titled "Love & Math", see http://loveandmathbook.com/
Neil Rickert --
I share your skepticism about Zeilberger's remarks regarding experimental mathematics, as I shall discuss in a future post. I'm not familiar with Thompson's book, but it sounds like I should check it out.
Thanks for pointing out the date of the Frankel and Ross piece. I only learned about it from Zeilberger's essay, and found it with a bit of Googling, and failed to notice the date. I have corrected the opening of the post to indicate the date of the piece. As it happens, I downloaded Frenkel's book to my Kindle when I first became aware of it, but I have not read it yet.
As in any field, an incomprehensible, over-detailed, jargon-filled presentation is often the result of insecurity from the speaker - make it incomprehensible and you're less likely to be challenged on the content.
The two most clear and memorable colloquia I attended as a PhD student were from Steve Smale and V.I. Arnol'd, a Fields Medalist and a Crafoord Prize winner respectively, and both were absolute models of clarity - easily understandable even for a student working in an entirely different field. Also some of the clearest writings in mathematics I have come across recently have been Terence Tao's contributions to Tim Gowers' "Princeton Companion to Mathematics" - another two Fields' Medalists at work making advanced maths comprehensible to the mathematically-slightly-impaired.
The best mathematics communication that I ever saw was in an extracurricular lecture by the lecturer who taught subsidiary mathematics to us first-year undergraduate physics students. he was very popular: we physicists affectionately called him “Daddy” Clarke, for reasons too complicated to explain here, but which summed up the way we felt about him pretty well. Anyway, one evening he was due to give a talk on topology. The room was packed, with a fair crowd of physicists supplementing his regular mathematics students. Dr. Clarke fought his way to the lectern, opened a scruffy brown holdall, pulled out a huge tangle of rope, and announced “this is a knot”. After we had stopped laughing, he entertained us for a hour with what was then cutting-edge topology. More that fifty tears later, I can still recall some of the details, including the 19x19 matrix that described the knot. That is communication.
Re Neil Rickert: Thompson's book is an excellent introduction to calculus. Thompson was a physicist, a Fellow of the Royal Society no less, who obviously regarded calculus as a tool. Perhaps this approach does not endear him to professional mathematicians. His book can be downloaded free from the Project Gutenberg web site.
I'm sorry, I will not always remember the day the Higgs was announced or where I was on that momentous day - when was it again?
It must have been in the Seventies that Henry Pohlmann of my department, and a visitor, David Sánchez, proposed a Sylvanus P. Thompson Award, to be given each year to the author of the shortest Calculus text. Or was if the least massive? I think I may have gotten Thompson out of the library when I was in high school, and taught myself the beginnings.
In physics meetings and talks we often have the same incomprehensibility. Often, this is because the student or researcher feels a need to share everything THEY have personally contributed to the topic at hand. And since we spend most of our time furthering details and working out detailed calculations, that's what we want to share. It's hard to spend 20 minutes on background so everybody can understand 5 minutes of our contribution. We feel we're liable to get more useful feedback by spending no time on background and only talking to the specialists in our sub-field.
this is a test
Interesting test, without questions.
Did we pass?
Actually, it was a test of whether or not I could successfully leave a comment.
Anecdotally speaking, I was first taught math using the New Math (in elementary school). For most of the rest of my educational career, I was very good at math. Was there ever a good study of the success or failure of New Math or did it just fade away because parents didn't like it?