Scott Aaronson is Assistant Professor of Electrical Engineering and Computer Science at MIT. His research interests center around the limitations of quantum computers, and computational complexity theory more generally.
Yesterday DJ Strouse, a student in MIT’s quantum computing summer school, pointed me to A Mathematician’s Lament by Paul Lockhart, the most blistering indictment of K-12 “math” education I’ve ever encountered.
Lockhart says pretty much everything I’ve wanted to say about this subject since the age of twelve, and does so with the thunderous rage of an Old Testament prophet. If you like math, and more so if you think you don’t like math, I implore you to read his essay with every atom of my being.
Which is not to say I don’t have a few quibbles:
1. I think Lockhart gives too much credit to the school system when he portrays the bureaucratization, hollowing-out, and general doofusication of knowledge as unique to math. In my experience, science, literature, and other fields are often butchered with quite as much gusto. Not until grad school, for example, had I sufficiently recovered from eleventh-grade English to give Shakespeare another try (or from Phys Ed do push-ups).
2. Lockhart doesn’t discuss the many ways motivated students can and do end up learning what math is, despite the best efforts of the school system to prevent it. These side-channels include the web, the books of Martin Gardner, recreational programming, and math competitions and camps. Obviously it’s no defense of an execrable system to point out how some people learn in spite of it—but these omissions make the overall picture too depressing even for me (which is really saying something).
3. In describing math purely as a soul-uplifting pursuit of beautiful patterns, Lockhart leaves open the question of why, in that case, it’s been in bed with science and technology throughout its history—not merely for the education bureaucrats but for Archimedes, Newton, and Gauss. (Of course, like most relationships, this one is not without its sniping feuds.) Personally I have no problem with teachers who want to recognize and celebrate that aspect of math, provided the students respond to it. “So you say you want theorems that are not only beautiful, but also inspired by physics or economics or cryptography? Line up then, because here comes a heaping helping of them…”
4. Lockhart doesn’t address an interesting problem that’s arisen in my own teaching over the last few years. Namely, what happens when you try to teach as he advocates—with history and philosophy and challenging puzzles and arguments about the definitions and improvisation and digressions—but the students want more structure and drill and routine? Should you deny it to them? (For myself, I concluded that brains come in different types, and that it would be presumptuous to assume a teaching style that wouldn’t work for me can’t possibly work for anyone else. Still, before beginning a traditional rote drill session, it’s probably a good idea for all parties involved to agree on a safe-word.)
In the end, Lockhart’s lament is subversive, angry, and radical … but if you know anything about math and anything about K-12 “education” (at least in the United States), I defy you to read it and find a single sentence that isn’t permeated, suffused, soaked, and encrusted with truth.
I wonder how many readers have come across a book my father gave me when I was in high school, that gave me back a view of the world and the meaning of mathematical concepts - namely "Mathematics for the Million" by Lancelot Hogben. The companion book, "Science for the Citizen" is equally compelling, by the way. These books were Hogben's protest about math teaching back in the 1930's.
However, when I tried Mathematics for the Million out on my teenage son, he didn't even get though the first chapter! Never mind that he is highly intelligent, his mind-numbing exposure to US Middle and High School mathematics has already done the damage that Scott Aaronson and Paul Lockhardt lament.
Hogben's books need to be recast in more modern language but their essence is just what Scott is recommending in his paper. But schools have to have new mathematics testbooks every year or two, costing several decabucks, so that the rote-based concepts can be taught in the same mindless way, with new pretty pictures and multi-colored text to sugar-coat their bitter little pills.
In my first year of English high school, my math teacher, a Mr. Watkinson, told everyone to put away their math books and instead engage in some exercises with gummed paper strips, making Mobius loops or loops with multiple twists, and cutting them into interlocking circular loops; cutting out paper shapes and making the regular solids, then explaining why there can only ever be five of them in three dimensions -- then what about four dimensions? or five? That class was so much fun that it gave us all a completely different perspective on what math was all about. God forbid that any high school teacher here should resort to such an off-curriculum class, even for five minutes!
Watkinson was also my form master, and was involved in amateur opera, singing the role of Mephistopheles, among other things. Math and music often go together.
Those of us who learned that one can think in multiple dimensions in abstract forms can go on to invent, as I did, innovative ways to decode quadraphonic sound from two-channel recordings.
Today's high school students are crippled by the pseudo-mathematics they learned, and reduced to mind-numbing experiences like flipping hamburgers in a fast food restaurant, or worse.
I understand the argument in the essay, and agree with most of it. But let me present the case for the other side, only partly as devil's advocate.
In the first place, "mathematics" is not the only subject that is taught in grade school as a series of factoids rather than a coherent study (whether that study be art or science). Consider history, or what passes for "science" (physics, biology, chemistry). In each case, the subject is taught as names, dates, and assertions that must be memorized, or techniques that must be mastered. All pretty much arbitrary to the sophisticated eye -- that is, the eye that is properly aware of mathematics, history, or science.
Wait! This is a case for the other side? Well, yes. There is a collection of things that our society has decided every child should know, a basic part of the culture. In the case of history, it is a shared sense of who we are, what we believe, and how we got that way. In the case of math, it is a set of practical ways to deal with quantitative reality. Algebra, geometry, and trigonometry (as taught in grade school) is indeed something a knowledge worker might well be expected to use. (I use them all the time in golf club design, and in dealing with golf coaches and custom clubfitters.)
I suppose I could come up with a similar rationale for teaching physics, chemistry, and biology as they do in grade school.
Well, how about teaching both the practical factoids and the coherent art? What a concept!
If you do, I really believe you will find even fewer grade school students who are adept (or even simply not bored) by the coherent art than they are by the factoids. I fully agree that those students need to be exposed to it, but doubt that it would serve the system well to insist that all study it -- probably less so than the current system.
Then there's the practical issue of finding teachers capable of teaching math as an art... Or, for that matter, history or science they way they are taught in a good college. It is hard enough funding our tax-supported grade schools. It would be much harder if we had to attract large numbers of teachers who actually knew their subject on that level.
I know the last paragraph is cynical, and leads to even more cynical practical arguments along the same vein. (Students not used to working hard. Influence of teachers unions. Etc etc etc.) So I'll leave it at that.
Reading the comments here--and other recent Bloggs, I think I understand something painfully obvious, which everyone missed. I struggled with math for years, largely because it was forced on me, and I was never shown what it was good for. When I was a grade schooler, I was mocked by adults for not throwing myself at math (not that most of them were any good at helping me!), but I had no real use for the math. I mean, a small allowance did not require budgeting, I either had enough to buy a comic book or I did not. When I came to college and calculus, I asked my profs, 'what is this for?', and they could neither tell me, nor teach me worth a damn. But when I finally hit the science classes that required the math, I had a reason to learn it! A reason to WANT to learn it. So I minored in math. In the same way, when we stopped being a pioneer society, we stopped fighting the wilderness and started fighting each other. See how mean many bloggs are? And, we see major research into things like magnetic coupling to power communication appliances at a distance of a meter or so,,, instead of using a cheap and efficient cord a meter long. Sorry if that sounds mean. I live on a ranch, and the only regular utility I have is electricity. So, I put in back-up generators in case of emergency; I improved my well and storage tanks; I put in solar room and water heat, with more back-ups, and modern windows, cooling, etc. Now, I'm bored. I worked my way into comfort, and ran out of challenges. Our education system managed to do about the same, and now bores the students. NASA got out of the space business and into the political/job security business, and refuses to tackle the real dream of spaceflight--sorry if that is mean again. How is it that the prospect of loosing job or retirement today seems to scare us more than did the prospect of travesing a savage wilderness on foot two hundred years ago? We are not up to major challenges if we are focused on remaining 'comfortable'.
Massachusetts Institute of Technology
Comments
However, when I tried Mathematics for the Million out on my teenage son, he didn't even get though the first chapter! Never mind that he is highly intelligent, his mind-numbing exposure to US Middle and High School mathematics has already done the damage that Scott Aaronson and Paul Lockhardt lament.
Hogben's books need to be recast in more modern language but their essence is just what Scott is recommending in his paper. But schools have to have new mathematics testbooks every year or two, costing several decabucks, so that the rote-based concepts can be taught in the same mindless way, with new pretty pictures and multi-colored text to sugar-coat their bitter little pills.
In my first year of English high school, my math teacher, a Mr. Watkinson, told everyone to put away their math books and instead engage in some exercises with gummed paper strips, making Mobius loops or loops with multiple twists, and cutting them into interlocking circular loops; cutting out paper shapes and making the regular solids, then explaining why there can only ever be five of them in three dimensions -- then what about four dimensions? or five? That class was so much fun that it gave us all a completely different perspective on what math was all about. God forbid that any high school teacher here should resort to such an off-curriculum class, even for five minutes!
Watkinson was also my form master, and was involved in amateur opera, singing the role of Mephistopheles, among other things. Math and music often go together.
Those of us who learned that one can think in multiple dimensions in abstract forms can go on to invent, as I did, innovative ways to decode quadraphonic sound from two-channel recordings.
Today's high school students are crippled by the pseudo-mathematics they learned, and reduced to mind-numbing experiences like flipping hamburgers in a fast food restaurant, or worse.
martwill38
06/22/2009
Posts:8
In the first place, "mathematics" is not the only subject that is taught in grade school as a series of factoids rather than a coherent study (whether that study be art or science). Consider history, or what passes for "science" (physics, biology, chemistry). In each case, the subject is taught as names, dates, and assertions that must be memorized, or techniques that must be mastered. All pretty much arbitrary to the sophisticated eye -- that is, the eye that is properly aware of mathematics, history, or science.
Wait! This is a case for the other side? Well, yes. There is a collection of things that our society has decided every child should know, a basic part of the culture.
In the case of history, it is a shared sense of who we are, what we believe, and how we got that way.
In the case of math, it is a set of practical ways to deal with quantitative reality. Algebra, geometry, and trigonometry (as taught in grade school) is indeed something a knowledge worker might well be expected to use. (I use them all the time in golf club design, and in dealing with golf coaches and custom clubfitters.)
I suppose I could come up with a similar rationale for teaching physics, chemistry, and biology as they do in grade school.
Well, how about teaching both the practical factoids and the coherent art? What a concept!
If you do, I really believe you will find even fewer grade school students who are adept (or even simply not bored) by the coherent art than they are by the factoids. I fully agree that those students need to be exposed to it, but doubt that it would serve the system well to insist that all study it -- probably less so than the current system.
Then there's the practical issue of finding teachers capable of teaching math as an art... Or, for that matter, history or science they way they are taught in a good college. It is hard enough funding our tax-supported grade schools. It would be much harder if we had to attract large numbers of teachers who actually knew their subject on that level.
I know the last paragraph is cynical, and leads to even more cynical practical arguments along the same vein. (Students not used to working hard. Influence of teachers unions. Etc etc etc.) So I'll leave it at that.
dtutelman
06/22/2009
Posts:63
I struggled with math for years, largely because it was forced on me, and I was never shown what it was good for. When I was a grade schooler, I was mocked by adults for not throwing myself at math (not that most of them were any good at helping me!), but I had no real use for the math. I mean, a small allowance did not require budgeting, I either had enough to buy a comic book or I did not.
When I came to college and calculus, I asked my profs, 'what is this for?', and they could neither tell me, nor teach me worth a damn.
But when I finally hit the science classes that required the math, I had a reason to learn it! A reason to WANT to learn it. So I minored in math.
In the same way, when we stopped being a pioneer society, we stopped fighting the wilderness and started fighting each other. See how mean many bloggs are? And, we see major research into things like magnetic coupling to power communication appliances at a distance of a meter or so,,, instead of using a cheap and efficient cord a meter long. Sorry if that sounds mean.
I live on a ranch, and the only regular utility I have is electricity. So, I put in back-up generators in case of emergency; I improved my well and storage tanks; I put in solar room and water heat, with more back-ups, and modern windows, cooling, etc.
Now, I'm bored. I worked my way into comfort, and ran out of challenges. Our education system managed to do about the same, and now bores the students. NASA got out of the space business and into the political/job security business, and refuses to tackle the real dream of spaceflight--sorry if that is mean again.
How is it that the prospect of loosing job or retirement today seems to scare us more than did the prospect of travesing a savage wilderness on foot two hundred years ago? We are not up to major challenges if we are focused on remaining 'comfortable'.
kitk
06/23/2009
Posts:66