A Mathematician's Lament — an Indictment of US Math Education 677
Scott Aaronson recently had "A Mathematician's Lament" [PDF], Paul Lockhardt's indictment of K-12 math education in the US, pointed out to him and takes some time to examine the finer points. "Lockhardt 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 [...] In the end, Lockhardt'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 and find a single sentence that isn't permeated, suffused, soaked, and encrusted with truth."
The way math is structured is disconnected from... (Score:4, Interesting)
... interesting things kids want to do.
Lets face it a minority of people will like math, but matehmaticians have done a lot to make mathematics overly complicated.
I struggled with the symbolic format math was presented in highschool because it was so disconnected from the world, only as I got older did I realize how arbitrary and how that was only one way to present mathematics. To really teach math one must learn how to observe first before one even gets into symbolic computation, math at it's most basic is about observing relationships, patterns of : Size, ratio, proportion, etc. It's really a language invented to systematize structure and relationships of the real world, therefore how math is represented and structured and is taught matters a hell of a lot.
I've learned over the years that many mathematical systems are totally arbitrary are are more obtuse then they need to be, math comes from the simplest observations. Math has built up a lot of cruft and wasteful jargon disconnecting math from the world.
For instance I had no idea for a long time that the way math is structured could be restructured when I was young and it was one group of peoples perspective on mathematical principles, I came across debates and alernative systems like:
http://www.symmetryperfect.com/ [symmetryperfect.com]
And it showed me how arbitrary mathematical systems and their structures really are and they are built to suit particular kinds of minds or cultures.
For instance the ancient mayans used shapes for numbers, instead of 1, 2, 3
See here:
http://en.wikipedia.org/wiki/Maya_numerals [wikipedia.org]
Math is a very rich subject which unfortunately has a lot of cultish like people who think themselves the gatekeepers of mathematics.
I've thought about writing a book in my spare time about how badly mathematicians and the academia has blinded themselves to simplifying mathematics by focusing too much on symbolic jargon and not teaching children how 'mathematical' relationships are related to our simplest observations of the world: Size, shape, form, color, motion, etc.
Could be worse... (Score:1, Interesting)
Eh. (Score:5, Interesting)
The whole idea behind his essay is that he liked playing with numbers and shapes as if it's an art, but he doesn't seem to realize most people don't share this love for math, like pretty much 90% of any student population. This is me speaking as a just-graduated senior: the things he suggests is beyond the ability of most math students in high school.
US K-12 MATH = Real world fail. (Score:1, Interesting)
Just the other day, I was watching "Who wants to be a Millionare?" And a 24 year long high school teacher didn't know what the sign for factorial means. Choices where along the lines of : ! & %
An allegedly true story from a professor (Score:5, Interesting)
While I was in university, a computer science professor in the faculty of mathematics told me (and the rest of the class) a cute and funny story about what happens "when the children of math professors get together". He and a colleague, who each had a young daughter at that time, were walking together in a park with their daughters. The children were old enough to have picked up some math-related words and phrases from their fathers, but young enough to have no idea what they really meant - six or seven years old, maybe? The daughters went off to play and their fathers overheard them arguing about who had seen the most flowers in the park.
My professor's daughter said, "I saw five flowers!"
"And I saw... six!", the other girl replied.
Not to be outdone, my professor's daughter said, "I saw a million flowers."
"Oh yeah? I saw infinity flowers."
This, according to my professor, caused his daughter to pause - she had never heard of "infinity" before. How could she top "infinity flowers", especially since she didn't know what it meant?
But after thinking for a few seconds, she said, "Well, I saw all the flowers."
Re:tl;dr (Score:5, Interesting)
Bingo, and that's one of the big problems with trying to do anything about the issues the paper raises: there are only so many people with the 1) ability, 2) knowledge, and 3) inclination, to do the kind of real mathematics he's talking about.
We'd have to re-vamp our teacher training along the lines of what's talked about in the paper to try to increase the number of people who could do it, and hope Lockhart's right about this being an art with universal appeal so that enough of the teacher candidates "get" it. Even if elementary schools began using dedicated math teachers (some already do, but many don't) we'd still need a shitload of people trained in this "math as an art/math as play" style, and we currently have approximately zero in elementary education.
Re:I Sympathize With Him But Too Idyllic (Score:4, Interesting)
Passions come to people unexpectedly. We should deal with the fact that more people are passionate about topics like Art and Humanities than Math and Computer Science. It's just the reality of academia right now.
Isn't his point that we don't really know if that's true, since math isn't taught in a way to inspire passion? That if more people were able to glimpse some of the beauty and creativity in it, there might be more interest in it?
Well if you're not asking for teachers needing to be professional published mathematicians, what was that paragraph about?
I agree we can't expect every teacher to be awe-inspiring; even getting (and retaining) enough marginally competent teachers is a challenge. However, you needn't be a university-level mathematics professor to know some of what he's suggested. For example, public school teachers are supposed to have Master's degrees, right? Now, isn't there something funny about the fact that teachers will go and get their BS in the subject they will teach, but get their Master's degree in "education"? Cue the "Philosophy of science is about as useful to scientists as ornithology is to birds" quotes. I'd think that teachers might be better served by a decent master's degree in their field of teaching, rather than "education". That would allow them the opportunity to study the history and philosophy of their subject, get a grasp of recent developments (maybe not in all subjects, but they could at least be able to pick up journals), etc. The really good ones could even get published (I just got my Master's degree, and was able to get a paper published, so yes, it's possible).
Re:Cue the other subjects (Score:3, Interesting)
Personally, I put the blame less on fundamentalists and more on decreasing importance of education in the home.
There are dozens of examples (single mothers with multiple jobs and multiple kids who just don't have time to parent, illegal immigrants raising kids that accept no-skill jobs as manual labor as sufficient for a lifetime instead of working to get an education and work in a skilled field), but the basic problem is that kids don't believe that they need a real education to live.
Re:US School System compared to Europes School Sys (Score:3, Interesting)
I saw the issue the OP stated in all of the K-12 schools I went to. As my father was in the military, I got to go to 3 elementary schools, 1 middle school, and 2 high schools. Some teachers were able to handle students at different reading/math levels in elementary school, but once I hit middle/high school, everything except math was lowest common denominator. In Seventh grade, the English class was using the reader I used in Fifth grade. And people in the class were having a hard time with it! The only way I could get away from the morons was to get into an AP class. Of course, I couldn't get into the AP English classes as my grades were too low in Eighth grade (should have actually done my homework.) Math was something I was good at. I had excellent Math teachers in HS. Sadly, I went to college. By my second quarter, I had enough of the stupid rote memorization of proofs that had to be regurgitated on exams to just stop attending classes. Feh.
Re:Cue the other subjects (Score:4, Interesting)
American education was designed to fail. Read the book (it's free online) The Underground History of American Education by John Taylor Gatto. He is a former New York State and New York City Teacher of the Year
http://www.johntaylorgatto.com/underground/
Re:Cue the other subjects (Score:2, Interesting)
Read The Fundamentals of Extremism: The Christian Right in America
They've staged a long and protracted anti-education war in government for 3 decades.
http://www.amazon.com/Fundamentals-Extremism-Christian-Right-America/dp/0972549609/ref=sr_1_1?ie=UTF8&s=books&qid=1245437556&sr=8-1 [amazon.com]
U.S. Public Education (Score:3, Interesting)
Discussing "US Public Education" is about as specific as discussing global weather. Is it cloudy or raining today? The education system is the US is quite federalized- most of the decisions about pretty much anything are made at the state and local levels.
I, personally, am quite happy with my 1st graders' (twins) math education. They've learned concepts like how to estimate, pattern detection, etc., as well as the rote mechanics of arithmetic. And they get more of it at home ("Here's a cookie. Tomorrow I'll give you twice as many as I did today. How many will you have in a week?"). But I live in a pretty rich suburb outside Boston, where the MIT professors live in the less-affluent neighborhoods.
We can bitch about the schools all we want, but it's a deeper cultural issues. School teachers get OK pay and benefits, good (though rigidly defined) vacations, and no respect. What kind of profile of person does that attract? In my experience, a real mix of people who are passionate about teaching (often with well-paid spouses) and those that mail it in 'til vacation starts. The balance of those (and other) groups varies widely by district. More than pay, this is really an issue of respect. I can't tell you how many teachers I know who report 'lack of respect for their profession' as the #1 gripe about their job. I wouldn't put up with that (not that I'd make a good teacher).
Re:Cue the other subjects (Score:1, Interesting)
If education is an investment in the future, that would imply that I'm going to get a payoff for my financial contributions.
When we couch the discussion in those terms, would you say that it's a "good investment" to spend 3x as much on kids with
brain damage brought on by Fetal Alcohol Syndrome than on kids that don't have brain damage?
Really, when you start talking about education as an investment in the future, that opens the door to discussions about how to
maximize the return on the investment. Maybe I spend shitloads on kids that show promise, a little bit on average kids, and jack shit
on 3rd graders that can't tie their shoes. How do you think the social justice set would take that?
Re:Several Proxies (Score:4, Interesting)
Bah, like we're going to RTFA on a Friday when there are much better ...
I know you're mostly joking but this was a pretty interesting albeit lengthy opinion piece. In fact, he even busts into dialogue between two fictional characters named Simplicio & Salviati to illustrate his point. It's a very Plato/Caroll/Hofstadter sort of way to illustrate his point. Hell, I love this format so much, half my posts are in it [slashdot.org]!
Anyway, after reading this, I am really eager for vdash.org [slashdot.org] to get its wiki up and running so that can be used to build engines and homework for students. Maybe even provide a hub for teachers to discuss interesting assignments? I'm sure the discussion pages will prove interesting if real academics get in arguments about proofs and math. I don't think the real payoff would be reinstitutionalizing the teachers but instead giving the students the free online resources to go the extra mile if they so desire. Save your Turings and Erdoses if you can't help everyone!
Lockhart is definitely a dreamer and this isn't going to change public schools. But it might change how you as a parent get involved with your children and math.
Re:I Sympathize With Him But Too Idyllic (Score:3, Interesting)
Now who's being the idealist? The public school system rarely has such concerns, or they wouldn't do everything possible to scare away the best teachers, and even moderately good ones. Standardized testing, nonsensical state mandates, psychotic district administrators, requirements to use ghastly textbooks, etc. So many headaches are thrust upon our public teachers that have nothing to do with teaching, that it's a wonder anyone sticks with it. I've known people who would have been excellent teachers (including one who was the "perfect" math teacher you speak of) who were scared off by the horrors of the system and ended up pursuing other fields.
The last thing our system is geared toward is finding good teachers, or ones who care about the students. It's geared toward finding teachers who are willing to put up with all the crap that our public school system shovels their way. Some do it because they love teaching or care about the students and will put up with the suffering. Some do it because the job offers an awesome 3 months off per year. Some do it because they were able to get tenure and love the job security. Some do it because that's the career path they started in and they don't want to make a change.
Re:It starts with the textbooks. . . (Score:5, Interesting)
And you get around the economic obstacles by subverting the system: Crowdsource the textbook to a group of interested mathematicians. Publish it online for free, with printed copies available for a price far below what a crooked textbook publisher would charge. Add value by posting demonstrations by mathematicians, math historians, and math professors on YouTube, linked to the relevant chapter of this comprehensive, global mathematics resource.
Re:Cue the other subjects (Score:3, Interesting)
Much earlier. Tennessee v Scopes.
While there was a moral victory for science educators at showing the issues in trying to restrict science education, most discussions I've read of the outcome of the trial point out that the end result was removing from textbooks significant material that was considered offensive.
This changed dramatically in 1957 with Sputnik. There was a brief rush to teach more advanced technical subjects. Beyond that date, I have anecdotes rather than more solid information about the state of education.
In math education, many people deride and criticize the New Math movement for focusing on correctness of technique over the answer. This despite the fact that in advanced math, all emphasis was on the technique. A sign error in a multiplication in a calculus class would likely lose a point or two, but would be unlikely to cost you all points in the problem if you showed understanding of the calculus involved. It also helped result in geometry being taught as a mathematics course instead of an engineering course (with theorems and proofs).
Yet despite that, New Math is often cited as the end of advanced math in schools.
I will agree that elementary math education has significant issues. I had extreme objections to the math that the public schools tried to teach in the past five years. I objected strongly to the fact that geometry was changed from a mathematics course to an engineering course (no work on theorem proof and studying math as a system of making proofs from axioms and previously proven theorems.)
Re:tl;dr (Score:3, Interesting)
Bingo, and that's one of the big problems with trying to do anything about the issues the paper raises: there are only so many people with the 1) ability, 2) knowledge, and 3) inclination, to do the kind of real mathematics he's talking about.
And not just the teacher training. This goes beyond what some students are capable of and can handle. What happens to them if they can't function inside a creative mathematical atmosphere?
Re:Cue the other subjects (Score:3, Interesting)
Re:The way math is structured is disconnected from (Score:1, Interesting)
People, people, people. If there's one thing outsourcing has taught us in the United States is that we don't have to invest in educating our children. We can always recruit the "best and the brightest" from around the world. They want to come to this country, right? Let the other countries do the heavy lifting and we'll all get jobs waiting tables using government paid health care. Listen to corporate America. Don't raise taxes on the idle rich, er, most productive segment of our country just to educate our children. We have foreign cars to buy. Professional sports stadiums to build (someone say something about bread and circuses?).
The sad part about being outclassed by other countries isn't that we can't afford to educate our children, it's that we simply refuse to do it. The culture in the U.S.A. simply doesn't value education or it's children anymore.
Re:Cue the other subjects (Score:1, Interesting)
If everyone was smart, who would work at mcdonalds?
You're right. See, we did an experiment once. A whole bunch of Alphas, and only Alphas, were put on an island. With no Epsilons to do Epsilon work, some of the Alphas had to do it. Well, they didn't like it at all, doing work beneath them. In the end, they revolted and full civil war broke out. Because of that, every society needs few Alphas, and more Betas, Gammas, Deltas, and Epsilons. Having too many Alphas would contribute to unrest and the downfall of civilization. Only by embracing the ideals of this Brave New World can we avoid such troubles.
Re:The way math is structured is disconnected from (Score:3, Interesting)
If you read the Wikipedia article on Maya numerals [wikipedia.org], linked to above [slashdot.org], you will see that it is not like Roman numerals. It is, in fact, a base-twenty positional system that happens to have logical symbols for its digits (zero notwithstanding).
Single Best Fix: Introducing Discrete Mathematics (Score:3, Interesting)
0. arithmetic
1. algebra I
2. geometry
3. algebra II
4. trigonometry
5. elementary analysis (includes some probability and statistics)
6. calculus
The above mathematics sequence is typically plug-and-chug: plug some numbers into some formulas and produce a result. No thinking is required.
What is sorely needed is a course in discrete mathematics between geometry and algebra II. Discrete mathematics teaches the most fundamental mathematical concept: methods of reasoning about mathematics. Not surprisingly, discrete mathematics includes plenty of proofs.
Discrete mathematics is not only a foundation of math but is a foundation of computer science. All the important ideas in data structures and finite automata require an understanding of discrete mathematics.
Re:Several Proxies (Score:4, Interesting)
Re:Cue the other subjects (Score:3, Interesting)
By teaching how, rather than why.
"Here is how to find some property of a right triangle" rather than "Here are the qualities of a right triangle. What can you find using that, and why?"
While that method is useful in learning how to apply some given formula, it is useless in learning how to derive a formula or understand which one to use and why. Modern US algebra students might be able to tell you the square footage of pen they can construct with a given length of fence. Very few would be able to reverse that rote equation and determine how much fence they need for a certain size of pen (or for a circular pen). If we were taught how to build that basic formula, we would recognize that it is the same problem with a different variable and be able to adjust the formula effortlessly and correctly.
Re:Cue the other subjects (Score:3, Interesting)
Of course, that group has a higher percentage than average of home-schoolers.
And those home-schoolers tend to get much more out of their education than average.
But go ahead with your beliefs.
Re:Several Proxies (Score:4, Interesting)
The Bible says Pi = 3, in the 1st Kings (7:23). That's OT, not NT.
Re:Cue the other subjects (Score:4, Interesting)
If everyone were generally qualified for interesting jobs, then boring jobs would pay much, much better. And interesting jobs would pay poorly.
That already happens with academia: the salaries at research institutions are often less than those in community colleges, simply because the former are more interesting jobs.
Re:Cue the other subjects (Score:3, Interesting)
Is this actually an argument for the promotion of ignorance? No, it's not. It's a way for us to confirm our belief in the American Meritocracy. I don't want to work at McDonald's, that's why I did my homework. Everyone had an equal chance in school. It's what we tell ourselves to help us sleep at night while others starve to death, a shame on our abundant society if ever there was one. But not everyone has an equal chance in school. The inequities are everywhere and in plain sight. If you go to visit schools in China, you will have an escort choose which schools to see. In the US, you can get a visitor pass from any school, any day of the week. You can visit the affluent, suburbian school and the rundown, ghetto school in the same day, with no special permission. At least the Chinese recognize the injustices as shameful and try to hide them. We, however, are shameless. To discuss a solution is to abandon our illusion. And, hey, somebody's gotta clean the toilets, am I right? Eh?
Finally, the comment betrays the truth of the education system. It's an economic sorting engine. It's a drawn-out college entrance exam. The truth is, we need factory workers. Why do you think we cram active children into seats in small, almost windowless rooms and drown them in rote, mindless exercises? We could ask why dropout rates are so high. We could ask why there are disparities in grades between economic and cultural groups. We could really question the goals of this machine we've built. We could ask you what you could possibly have against a smarter, more informed populace. But, hey, somebody's gotta wash the dishes, am I right? Eh?
Re:Several Proxies (Score:4, Interesting)