“For all the time schools devote to the teaching of mathematics, very little (if any) is spent trying to convey just what the subject is about. Instead, the focus is on learning and applying various procedures to solve math problems. That's a bit like explaining soccer by saying it is executing a series of maneuvers to get the ball into the goal. Both accurately describe various key features, but they miss the what and the why of the big picture.”
― Keith Devlin, from "Introduction to Mathematical Thinking"
For the possible long weekend ahead, a real treat and long post with lots to chew on from Keith Devlin. How Keith finds the time/energy to do all he does I don't know, but he found time to answer some wordy questions from me! (on some things I was really curious about)
I've added bold to a few bits here and there for emphasis of notions I thought particularly interesting.
Anyway, Dr. Devlin ought need no introduction here, so without further adieu....
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1) These days, you're very active with the MOOC (massive open online course) movement, which will likely bring major changes to higher education in this country. Could you briefly paint us a picture of what you think college (or even secondary) education in the US may be like say 20-30 years from now, as contrasted with how it is today?
As the saying goes, prediction is difficult, particularly about the future. There are two parts to what we call "education" and I think the distinction between them will become clearer. First, for people who have essentially learned how to learn and to think critically, there is what I would call "training" -- learning some new skill or a variant of something already known. For learning of this kind, the textbook, the training manual, the classic instructional lecture, or the video instructional lecture are fine. A lot of vocational training is like this, as are many university level courses in computer science, mathematics, engineering, etc. This kind of teaching can easily be provided by MOOCs. Indeed, the first large scale MOOCs coming out of Stanford, then MIT, were all of this kind. With valued-brand universities offering this education online for free, possibly with payment required for certified accreditation, it's hard to see most middle ranked higher educational institutions surviving by focusing on such teaching. This kind of education is scalable.
Most of the media coverage of MOOCs has focused on this kind of education. But then there is the education that -- in theory, but I fear not always in practice -- the K-12 system is supposed to provide, and which most university courses in the Arts and Humanities, and many university courses in the Sciences and Engineering, do provide. These focus on learning how to learn in the lower grades, and on developing critical thinking and new ways of thinking at higher grades. The only way to provide that kind of education is with human-human interaction between learning and domain experts and the students, coupled with student-student interaction. This kind of education is not scalable.
The K-12 system will likely see little major change due to technology, and any change they do experience should be for the better, with teachers having more time to work individually with students. Higher education will focus much more on human interaction, with the classic lecture disappearing.
And relatedly, something you've often noted that I find interesting, is that college-level math and secondary school math involve very different skills (such that a person can sail through secondary math with high marks and yet hit a brick wall with college math) -- can digital resources help smooth out this transition, or will there likely always be a sharp inherent demarcation between the two levels of study?
The two kinds of mathematical education are almost separate disciplines. The demarcation lines are very similar to the ones I outlined above for education in general. Technology can help, but not very much, because, by definition, cognitive, conceptual mathematics is about the human brain.
2) You are consistently one of the clearest, most effective math writers/popularizers around for a general audience. To what do you attribute that knack? (have you always been a naturally good writer, or do you work especially hard at it, or just have super editors? ;-)
All of the above! Though all my mass market books have been written with great editors, and my early magazine articles and newspaper column were all professionally edited, these days hardly any of my articles -- and none of my blogs -- are edited by anyone except me.
3) I recently stumbled upon one of your older volumes (~1996), "Goodbye Descartes" -- another marvelous read. I was especially struck by the amount of material on linguistics and Noam Chomsky. My primary area of interest in grad school was actually psycholinguistics (though I had little interest in Chomsky's approach or work), so am curious how much you continue to follow work in that area, and has your opinion of Chomsky changed over the years?
Chomsky's early work on syntax has the same innate appeal as AI, and many mathematically-minded people such as I are initially seduced by the prospect, as were Chomsky himself for syntax and McCarthy for AI. But as soon as you delve deeper into the target domains, you realize that there are significant limits to the mathematical approach. Mathematics is a useful framework in both human domains, but it does not yield the same results that it does in the natural sciences. But in recent years I have continued to work in that general area, a lot of that work being for large corporations and for different government agencies related to national defense. In "Goodbye Descartes" I coined the term "soft mathematics" for such uses of the mathematical approach, where you blend mathematical thinking with other ways of working.
4) Much of what you write concerns the logic, history, and philosophy underlying mathematics. You don't often delve into recreational math, and I've not seen you reference Martin Gardner very much in your writings (though I've missed much of your prolific output). So I'm wondering what your view was of Gardner and if you ever had occasion to spend significant time with him? Is any lack of his mention in your writings just due to a lack of overlap in your interests, or does it possibly relate to basic disagreements over mathematics itself? -- I ask this especially because Martin was a very vocal and clear "Platonist" in his view of mathematics, while you have become a vocal NON-Platonist....
I corresponded occasionally with Martin, and re-read everything he had written when I started out on my popular writing, but I've never had much interest in recreational mathematics per se, except as a device to interest people in mathematics and as a pedagogic device. Mathematics has such power to do things in the world, I was never motivated to spend a lot of time on "recreational problems." It's not an issue of pure mathematics versus applied. My doctoral work was on infinitary set theory, which to date has no applications and may be one of the very few parts of mathematics that never finds (direct) applications. But it was a major piece of mathematics -- an abstract edifice -- developed over many decades, and that gave it purpose beyond any individual problem.
5) The whole Platonist/Non-Platonist debate actually fascinates me. Most working mathematicians I meet seem to assume a Platonist view is the norm, and that only a few 'fringe' elements out there hold the Non-Platonist perspective! Yet, over recent years, I've read more and more prominent math writers (like yourself), who started out as Platonists, but who have swung to the Non-Platonist side (and feel quite confident about it now). Do you have any sense of what percentage of professional mathematicians fall into each camp (in short, is it as heavily Platonist as some would have me believe, or not so one-sided in your view)?
My sense is the same as yours, that the non-Platonistic view has become more common, and maybe even dominant among successful professional mathematicians. But maybe I no longer mix with the Platonists!
And one follow-up to that: some I've talked to feel the whole P/Non-P debate is silly and unimportant; all that is significant is that we are able to successfully apply mathematics in life and science as well as we do; i.e. the Platonist debate is just mushy, mumbo-jumbo wordplay… what, if anything, might you say to that outlook? -- in short, does the debate even matter, beyond an intellectual exercise?
I think that the fact that doing mathematics seems to ENTAIL a Platonistic perception is of great interest. Why does it do that? I published a tentative explanation of that some years ago: "A mathematician reflects on the useful and reliable illusion of reality in mathematics" Proceedings of the workshop Towards a New Epistemology of Mathematics, held at the GAP.6 Conference in Berlin, September 14-16, 2006. Erkenntnis, Vol. 68, No. 3, May 2008, pp. 359-379.
-- This is a rich read (~20 pgs.) if you're interested in cognition, neuroscience, or even philosophy.
6) A second person I've not seen you discuss (completely apart from Gardner), and who I find very interesting, is the autistic savant Daniel Tammet. His introspective analysis of his own incredible math abilities seem fascinating. Have you read his writings on the subject of his own remarkable mathematical skills, and do you have any views regarding his talents or his personal analysis of them?
Actually, if you go back to my writing in the 1980s and 90s, you will find many references to Gardner -- but of course that was in the pre-Web era, so Google searches won't come up with what I wrote. I know nothing about Tammet and never heard of him. Prompted by your question, I'll take a look. Thanks.
[Wow, I was VERY surprised to learn of Dr. Devlin's unfamiliarity with Tammet who has received much publicity in recent years, and who writes frequently about his own keen mathematical cognition. I've finally received Tammet's latest book, "Thinking In Numbers," and will probably write some sort of blurb on it in the future... but this all seems to me to be a great example of just how broad and wide the mathematical landscape is -- that someone I simply presumed Dr. Devlin would be well-acquainted with and might have definite opinions about, is in fact, not even on his radar -- and I certainly don't mean that as a criticism, but rather as a tribute to how vast the sphere of mathematics is.]
7) Are you currently working on a new book, and if so, what about?
I probably am, but don't yet know it. I do have one finished in draft form that I have not yet figured out how to market -- a decision that will for sure entail a rewrite.
...something for us all to look forward to!
8) Speaking of books, an unusual one I recently finished is a self-published volume from Britain, entitled "The Mystery of the Prime Numbers" by Matthew Watkins [...I hope to have an interview here with Dr. Watkins soon]. It is one of the most fascinating/extraordinary math books for a mass audience I've ever read (entirely on the topic of prime numbers), and yet barely known because of weak distribution. I'm curious if you are familiar with it (and/or the author) and if so what you think of it?
No, I have not come across it or the author. Again, your question prompts me to take a look. I am very proud of having "discovered" Paul Lockhart and launched his large-market writing by publishing his "A Mathematician's Lament" in my MAA column. If I come across someone else with Paul's writing talent, I will try to do the same again.
...I do hope you access Watkins' book. I'd love to know if it is as exquisite as I find it, or are there flaws/failings I'm not competent to detect. Next time I contact Dr. Watkins I may even suggest he send you a complimentary copy.
9) To round yourself out a bit, when you're not involved in mathy things, what are some of your primary interests/hobbies/activities?
I am a physically active guy and like to spend a significant part ofd most days engaged in a physical pursuit. I used to be a distance runner (and before that a rock climber and skier), but when my knees started to complain in earnest about ten years ago I switched to cycling, and now own several different kinds of bikes, from upper end, all carbon road bikes to mountain and cross bikes. My website (http://profkeithdevlin.com/) and my Stanford homepage (http://www.stanford.edu/~kdevlin/) both have sections devoted to my biking.
10) What words of advice might you give to young people who are thinking they may want to pursue mathematics in college and professionally?
Just do it. The only prerequisite is a high tolerance for hard work, frequent frustration, and repeated failure. But those are what it takes to achieve the great highs that come with the successes.
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Several quite fascinating responses here on so many levels... I really can't thank Dr. Devlin enough for taking time out of his insane schedule to indulge me.
If you want more Devlin, he's all over the internet/blogosphere/YouTube etc., just google him, but I'll quickly cite 2 other references:
Another transcribed interview with Keith here:
http://www.guidetoonlineschools.com/interviews/how-to-become-a-mathematician
And to hear his British accent ;-) the very first podcast that "Wild About Math" blog ever did was fittingly with Dr. Devlin here:
http://wildaboutmath.com/2012/02/14/keith-devlin-inspired-by-math-1/
In the right-hand column, I also have several other past Math-Frolic posts 'tagged' for "Keith Devlin."
Hearing from Dr. Devlin is a fantastic way to close out the year, in the (possible) event that I don't get another post up before Jan. 1.....
