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A Mathematician's Lament — an Indictment of US Math Education
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."
second!
From what I can tell, they all look to be the same length and size and hopefully are not older revisions of this paper.
My work here is dung.
Pwned.
Evidently, someone didn't do the server math.
Have gnu, will travel.
The problems with K-12 education go WAY BEYOND mathematics.
Mooniacs for iOS and Android
Having not read the actual PDF, I wonder if having a bunch of mathematically disinclined women teaching math to young students would have something to do with it? Note: I'm not trying to be misogynistic, just my anecdotal observations. If the shit really hit the fan, I think I'd rather enjoy being a high school advanced math/computer science teacher. Aren't school districts hurting for qualified people in those positions?
A black hole is where God divided by 0
From the blog:
I defy you to read and find a single sentence that isn't permeated, suffused, soaked, and encrusted with truth.
Very well, here is an excerpt from the PDF:
Mathematics is an art, and art should be taught by working artists, or if not, at least by people who appreciate the art form and can recognize it when they see it. It is not necessary that you learn music from a professional composer, but would you want yourself or your child to be taught by someone who doesn't even play an instrument, and has never listened to a piece of music in their lives? Would you accept as an art teacher someone who has never picked up a pencil or stepped foot in a museum? Why is it that we accept math teachers who have never produced an original piece of mathematics, know nothing of the history and philosophy of the subject, nothing about recent developments, nothing in fact beyond what they are expected to present to their unfortunate students? What kind of a teacher is that? How can someone teach something that they themselves don't do? I can't dance, and consequently I would never presume to think that I could teach a dance class (I could try, but it wouldn't be pretty). The difference is I know I can't dance. I don't have anyone telling me I'm good at dancing just because I know a bunch of dance words.
Now I'm not saying that math teachers need to be professional mathematicians--far from it. But shouldn't they at least understand what mathematics is, be good at it, and enjoy doing it?
Well if you're not asking for teachers needing to be professional published mathematicians, what was that paragraph about?
... everywhere. That art teacher that actually made you think about what 'art' is? Not going to find many of them in the political science department, are you? Of course, for any subject, someone who puts their heart and soul into the subject is the best teacher! In this respect, math is not special.
I'm sorry man, you're asking for the perfect math teacher. You know Robin William's character from the movie The Dead Poet's Society? You want a guy like that for math
The paragraph I quote is not the truth, it's wishing for the impossible. I wish I had a math teacher like this all my life but come on. The public school system is more worried about getting someone that actualy cares about the students at all. They can't even find those people let alone people who care about the students and live/eat/sleep/bleed math.
I'm right their with you in wishing for this but the expectation is unrealistic. 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.
My work here is dung.
High school students are forced to write proofs as part of geometry class. However, they are never taught the rules of logic before being asked to write these proofs. That is just one example of how horribly, horribly stupid the HS math curriculum is in the US.
A slashdotter who didn't build his own computer is like a Jedi who didn't build his own lightsaber.
... 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/
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
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.
I don't know Scott.... I'm getting mixed messages from you about the article. Why don't you open up and tell us what you really think ;-)
In university, I was taking an intro philosophy course on critical reasoning.
We had covered the concept of statistical significance. The example we'd used was a case of "0.05" meaning we had 95% confidence in the statistical results. On the exam, the professor made a typo, and the question read "how much certainty with a statistical confidence of 0.5", to which the correct answer is 50%.
I was marked as wrong, and when I complained, the professor indicated that since we'd never covered that example, and only covered 0.05 in class, it was assumed that was what she meant.
I informed her for someone teaching critical reasoning, she wasn't demonstrating any. I also insisted I get the credit for giving the actual correct answer (which I and everyone who answered it correctly did).
If that's indicative of how math is taught nowadays, we're all hosed. :-P
Cheers
Lost at C:>. Found at C.
It could be the sad state of science education. Science is largely taught as memorization of facts, rather than a process for discovery. We turn out high school graduates who are easily suckered by such frauds as homeopathy and creationism...the latter of which in some places is actually taught as being science rather than its antithesis.
Found it here: http://plato.asu.edu/LockhartsLament.pdf
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.
Solve for education.
I implore you to read his essay with every atom of my being.
Well, OK, seeing as I can use *your* atoms.
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 : ! & %
Specialists in every field complain that educators get their field wrong or don't stir the passions of kids for their field as much as they ought to. What they fail to understand is that they're coming at the whole problem from the perspective of someone who is obviously gifted at and highly passionate about the field. They don't seem to get that most people don't pick up their field as easily as they do, and don't care enough to put in the effort it would take to get even half as good at it as the specialist.
Instructors of just about every field at any level of compulsory education (K-12) have to battle against entrenched biases against their fields, and against education in general, that have been fostered for years before the student ever gets in their classroom. Further, their task is to teach the curriculum provided. If they inspire their kids to love the field, that's great, but if they spend so much time inspiring the kids that they don't have enough time to teach the kids what they need to pass the state-required tests, they're still going to lose their jobs.
Teaching math, science, or anything else is HARD. Teaching it to people who don't care and don't want to be there is even harder. Teaching kids to love the field when the only metric used to judge your performance is pass rates on a standardized test is harder still. It's all well and good for professional mathematicians to bitch and moan about the state of education, but until they're ready to step in with some realistic and implementable ideas that don't presuppose that all kids have some inherent interest in these things that just needs to be tapped into, it's not helpful in the least.
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."
Atheism is a religion to the same extent that not collecting stamps is a hobby.
I myself have gone through the US school system starting at grade 7 (lived in Switzerland and The Netherlands before then), I am currently in uni for a software engineering degree. While I have read only part of the article (the blog post) I wanted to post my experience compared to that of my cousin who went through school in The Netherlands.
Math at the schools I went to was catered to the lowest common denominator, the slowest person in the class, the person who would just not get it got the most attention and the rest of the class was stuck at that level until that person tagged along and finally got moving. Whereas in Europe and other places they place those students in various levels of math dependent on their skill level so that those that don't need the extra time are able to get to the higher level maths faster. This creates a gap between the math that is considered required at age 18 in the US and The Netherlands. My cousin was going for a degree in hotel management and food preparation (chef). He at the age of 18 had a better understanding of math, and had more knowledge of high level math (Linear Algebra, Calculus and others) than I did when I graduated High School, and the classes he were in were considered the slower less demanding classes since it was not as much of a requirement for the degree he was going to be pursuing.
This is the same with a lot of the classes though, history, english, and science classes. Especially for English, you don't get to think for yourself anymore, you have to follow exactly what the teacher told you. If the teacher says this is important for this reason, and you attempt to argue it differently in a paper you fail, everyone coming out of high school has been passed through a cookie cutter, there is no innovation left, there is no real thinking for oneself anymore.
It is sad, and the state the US educational system is currently in will not allow it to compete in the global market, it will not allow it to be innovate and provide new ideas, but what it will provide is people who are like sheep and are more than willing to follow the crowd and just do it because everyone does. These people will be easy to govern and control since they won't ask questions and least of all will they rebel and fight for their beliefs. In other words, the US education system as it currently stands is making zombies.
cat
For instance the ancient mayans used shapes for numbers, instead of 1, 2, 3
Psst! The numerals "1", "2", and "3" are shapes too!
F***in' indocentrists...
Information theory is life. The rest is just the KL divergence.
The United States is being outclassed in math and science education by a host of other nations. Those nations, for the most part, teach the subject in an exceedingly traditional format. Asia, for example, is still really keen on rote learning. The failure of American pupils is probably not due to the way the subject is taught, but rather because they don't feel the pressure to excel like students in other cultures.
...that Roget ever compiled his damned, accursed, infernal, confounded thesaurus!
If "beauty is in the eye of the beholder" and "it was beauty that killed the beast" then "please stop staring at me".
that math is better taught as an art than as a pragmatic problem-solving toolset when you can convince me that Pablo Picasso should have been forced to paint the Golden Gate bridge.
Society needs math as a tool in far greater quantity than math as an art. Socially-funded education serves the greater need of society. QED.
I survived public school mathematics. I still appreciate the beauty of patterns, especially the relatedness of art, music, and math. (Godel, Escher, and Bach really resonated for me. But that didn't make me a mathematical artist, any more than a musical composer or a woodblock printer.)
Lockhart's essay is an interesting read, really, but on some level it boils down to "Those unworthy schlubs treating Mathematics as a tool don't deserve it. It belongs to the artists, the dreamers, the purists!"
It's a pretty common arrogation in the math culture, it seems. I dont' recall sculptors ever being pissed at concrete workers or ironworkers. And I've never heard of any artist painter getting mad at the other kind of painter for not employing good artistic composition principle while painting the side of the barn.
Seriously. Math is both an art and a tool. The best artists find their art by themselves; they're not turned out by artist factories. School mathematics is to turn out the mathematical equivalent of bridge painters and ironworkers, because society needs those more (in greater quantity).
Welcome to the Panopticon. Used to be a prison, now it's your home.
...I know I'm supposed to say "Things ain't like they used to be!" but the fact is, they never were. K12 in the fifties and sixties tried hard to convince me that I was to hate math and science and treated me as wierdo when I didn't. Instead I learned to despise classroom education, which did me incalculable harm at university.
Basically, public education sucks.
Warning: this article may contain humor, sarcasm, parody, and perhaps even irony. Read at your own risk.
From his critique of Algebra II:
Students will learn to rewrite quadratic forms in a variety of standard formats for no reason whatsoever.
I guess he's exaggerating, since he must be aware of the deep connection between algebra and geometry which is realized via manipulating equations. And this provides lots of approaches for a good teacher. I dunno, he just comes off as a garden-variety teacher with strong opinions.
While I like our system, I think there are two things that need to happen in the US.
I swear to God...I swear to God! That is NOT how you treat your human!
You must have attended a very very small school. Most US schools have different courses based on skill level. Your conclusions about the US school system are therefore wrong. They are merely conclusions about very small schools.
A slashdotter who didn't build his own computer is like a Jedi who didn't build his own lightsaber.
The depiction of math education in Star Trek was great - you know, the scene where the youthful Spock is answering math questions prompted by a screen in front of him, instructors observing overhead. Something akin to this would be pretty sweet. Where you could whiteboard out stuff all day in a high fidelity environment that uses OCR and AI to keep testing you on your weak points until you become stronger in each particular subject area. Something like this would ensure that you truly do have an understanding of everything before moving on. It could also use this information collected about you to introduce you to new topics in other subjects like physics based on your current understanding. Concepts could be masterfully articulated and narrated by famous voice actors like Morgan Freeman ect. A taxononomy/hierachy of subjects and concepts could be traversed to create unique learning programs when achievement is unlocked through true understanding rather than letter grades. Kind of like leveling up in an RPG.. would make things fun.
Two mathematics professors are having lunch at a restaurant. The first mathematician keeps complaining about how ignorant the typical American is and how he's suprised that the average person in this country has enough mathematical prowess to balance a checkbook.
The second mathematician says, "Don't you think you're being a little harsh? The average person surely has more mathematical ability than you give them credit for."
The first mathematician responds, "Absolutely not! I'm sure if you asked the first person you met on the street to solve a basic algebra problem, they would have no idea where to start."
The second mathematician says, "Okay, I'll make a bet with you. At the end of the meal, I'll ask our waitress to solve a calculus problem. If she can solve it, you pay for lunch. If she can't, I'll pay."
"Thanks in advance for lunch!" the first mathematician says confidently.
Later, while the first mathematician is in the bathroom, the second mathematician flags the waitress down and says, "Listen, when you bring us our check I'm going to ask you a math question. I want you to answer, âone-half x-squared.' Can you remember that? If you do, I'll leave an extra big tip." He encourages her to write it down phonetically and practice it so that it seems natural.
At the end of the meal, after the waitress puts the bill on the table, the second mathematician says, "Oh, could you answer a little question for me? What's the integral of x with respect to x?"
The waitress looks unsure at first, but says, "One-half x-squared."
With a grin, the second mathematician slides the bill over to the first mathematician.
As the waitress is walking away, she turns back and says over her shoulder "plus a constant!"
Technoli
This man is a beautiful dreamer. I don't think his rather Platonic vision of the perfect math class will ever be acheivable. But there are a bunch of half steps that I think would really help math and address his fundamental point that math, as it's currently taught, is boring as all heck and does nothing for the vast majority of us who don't use calculus or even algebra in our day-to-day lives. I mean really, the last time I did anything more than basic algebra was tutoring others! And while learning math so that you can help someone elses' kids study for a test is a fine goal, I'm not sure it's really worth the thousands of hours I spent taking math!
First, *use* math to solve real problems and explain real scientific principles. Radio Lab (THE official National Public Radio show for geeks everywhere) had a great little episode where some student "discovers" that the periodicity of a pendulum forms a parabola when charted on a graph. Wow! That's heady stuff. (It's the first story of this episode.) Understanding the interaction of science and math -- the universe, really -- is something that we can teach. Integration of math and science gets us part of the way there.
Second, incorporate the history of math into math class. Math advances all occur because of some historical context. Combining the two is a half-step that will get students to understand "why" we created this math, even if they never quite get the quadratic formula down. Combine these two principles, and it would go a long way.
"Asia, for example, is still really keen on rote learning. The failure of American pupils is probably not due to the way the subject is taught, but rather because they don't feel the pressure to excel like students in other cultures."
Performance is unrelated to the overly jargonistic complexity, just because asia and india are harsh and drill their kids to perform does not mean they have any clue how to derive creatively go beyond what they are learning. They may make good workers but that doesn't mean anything.
One gains a fuller understanding of math when you realize how to start from the beginning and learn how to observe, if you look at the progress of mathematics over the centuries - systems of symbols and other systems were created to systematize a problem and break it down, most people didn't have to sit down and come up with calculus or algebra themselves, but you can teach kids how to observe and derive things themselves and not feel ashamed to get creative and "go outside" traditional symbolic jargon for leaps forward in creatively seeing underlying relationships between things beyond symbolic computation.
One can be a good performing mathematician and still be clueless about the deeper relationships and observational skills required to become very versed in what math is, outside of what one is taught.
Math has an enormous amount of dogmatism attached to it. Performance in mathematics is only one aspect, we should ask - besides performance, what about understanding? I mean everything I've learned about mathematics I had to teach myself, so I could see through the bullshit of the establishment. I learned I was a visual mathematician, that I understand math through more natural mode of thought: pictures and geometry.
Math is really the language of form and structure, and all structure is necessarily geometric in some way, even relationships (structures of information).
Stop hiring Education Majors to teach The Hard Sciences. Unless you include Historical Curriculum of famous and infamous Scientists into your early days of learning the Hard Sciences will forever be a mystery.
If you want a kid to know Euclidean Plane Geometry you better make ``The Elements Books I-XIII'', by Euclid part of the curriculum early on and gradually bring into view the history of its making followed by the actually application.
The same goes for Physics with Newton, Robert Boyle, Euler, etc.
Hell, I'm just getting all the backlog history of these giants and I'm a M.E. It would have made my days far more enriching to know how they came up with this crap outside of the Calculus derived explanations. I love Mathematics and it's endless Engineering Applications [mainly because I could always visualize their application--something innate and not taught] but reading the greats memoirs and more makes it come together.
Instead of just History over political events we need History over Mathematics, Physics, Chemistry, EE, ME, CE, etc.
You don't suddenly become educated in Paleontology without first knowing it's foundation, heavily grounded in History. Hell even Fine Arts requires a massive background in the history of the fields pioneers.
You are not truly a mathematician until you learn to abstract. Symbols are ultimate abstractions. Yes, you can invent your own symbols, but you will still have symbols.
Think about how children learn to count. They first count concrete objects, one finger, two fingers, ... one apple, two apples, etc. and then we abstract. Remove the object being counted and just have one, two. This is where the big abstraction happens. We arrive to the concept of oneness, without thinking about 1 something. Now add symbols for those abstractions 1, 2 and now you can do some neat things with symbols that translate into concrete objects when applied. This is the essence of math.
Yes, symbolism introduced is standardized so that we can talk to each other and exchange ideas. Otherwise, it would take a lot of time for you to explain all your symbols to me before we could have any meaningful conversation.
As the island of our knowledge grows, so does the shore of our ignorance.
You start by having someone like the gentleman who wrote that paper create a new textbook and teachers' manual to go along with it (or, really, a 'series' of textbooks that go through the different grades) that implements the different way of teaching mathematics which he is espousing. It then dies in state and local education department when there is resistance from comittees on doing things differently than they've been done before, and anyhow there is no funding for new textbooks anyhow.
. . . But if you can manage to get some school districts to attempt the material, you basically have the teachers go through a math 're-education' using the textbook and teachers' manual, then they just teach from the book (of course, there needs to be *some* creativity on the part of the teacher, in adding new examples or explanations that the textbook author might not have thought of, but is necessary to help students who aren't 'getting' the examples and explanations in the book). But, if the teachers themselves have read through the book and the manual(and hopefully there is good supplemental discussion in the teachers' manual about *how* to teach the material), they should be prepared to teach the new method themselves.
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.
Dude, now you're approaching xenophobia. Have you looked at the state of mathematics in American universities? A conspicuous amount of highly original researchers are the product of foreign educational systems. They aren't doomed to being tech support monkeys like you insinuate.
Yes they are but notice how
3 hides the fact that its actualyl THREE 1's
(1) (1) (1)
of couse all systems use shorthand to compress teh relationships but when we say 3, we mean *three distinct shapes/objects/things*
Our characters we use for numbers *hide* those relationships.
Not sure where you went to school, but in my high school, you took the class most appropriate to your abilities at the time. Some were taking basic math and algebra in 10th grade while I was in a precalculus class. In 12th grade I was in an early morning advanced calculus class taught by a professor from the local university.
There is no "math class" at that level anymore, at least in my experience. Even middle school was like that because I got my advanced algebra there.
As for the essay linked by the original article. it smacks of the "whole language" approach that swept through schools here in California in the 1980s and 1990s, much to its detriment. I actually went into the article expecting to agree with Mr. Mathematician, but wound up thinking the guy is a loon (after slogging through *TWO* overwrought analogies). I get what he wants, but you don't toss out the basic foundations of functional mathematics to accomplish it.
I wish I could find an electronic copy of an editorial I found not too long ago in a local paper. It made an excellent and succinct point about how teachers unions are bringing about mass idiocy in our educational systems and, as a result, our populations.
I'm sure there's many factors contributing to our declining educational systems, but I don't think anyone can deny that attempting to standardize teaching and "level the playing field" as it were may not be as great of an idea as it sounds in theory. For many, math can be a hard subject. All arguments about relative difficulties and complexities aside, maths and sciences are at least not as accessible as, say, literature, history, or art. So it would stand to reason, that with less people gaining a firm and passionate grasp on a subject, there would be less available who are qualified to not only teach it, but teach it well.
Now, with teacher's unions gaining the same benefits for all teachers, where a history teacher and a calculus teacher of the same level of skill and experience receive the same compensation for their troubles, how much motivation does the truly talented mathmetician have to teach in highschool when various industries will pay him ridiculously more to work for them instead? How many history teachers would get offered $100K plus by the private sector?
So with supply and demand being what they are and all teachers NOT being created equal, why are they being paid the same? This only results in lesser qualified and lesser motivated math and science teachers in highschool resulting in less motivated and less educated students who have less chance of going into post-secondary math and science. Less post-secondary math and science students means a smaller number of talented mathmeticians and scientists graduating. This means a smaller number of potential future highschool teachers who are talented and educated enough to guide and motivate students in those fields. The cycle repeats.
I can't see the current educational crisis improving at all if these unions continue to insist that a history major has the same value as a good mathematician. I'm pretty sure we would have never made it to the moon if all we did was research the details of the war of 1812.
The big buzz word/trendy strategy in elementary-level teaching right now is "differentiated instruction". What it means is teaching a single concept at several levels of difficulty simultaneously. It's sort of like ability grouping into different classes like you talk about, but all in one classroom and with one teacher (and maybe a TA or Para or some such).
It's actually pretty good when it's done right, but as far as I can tell most elementary school teachers are awful at coming up with effective differentiated lesson plans. Many just think it's impossible and refuse to even try. I expect it to go away in a few years, just like most of the other trendy teaching ideas since, well, forever ago. Maybe we'll move toward having separate ability-grouped classes after it fails.
US schools have Advanced Placement classes. Oh yeah, and there is no reason motivated students can't move at their own pace outside of the classroom (or inside).
All you are calling for is a system that offers more than one sized spoon with which to feed students. That's just a better version of a fundamentally flawed educational philosophy. Or rather, it stretches a fine philosophy beyond its logical limit and makes it flawed. A chef can get by just fine with a US HS level understanding of math, for example. More math won't make better chefs (the opposite, potentially).
If a person's excuse for having less education than they wanted (or than Europenas might have) is that the US school system only caters to the lowest common denominator, they are lazy, not deprived.
Most US schools have different courses based on skill level.
Um, no they don't, apart from a very limited portion of Honors/AP courses. The rest are one-size-fits-all, based on some sort of misguided egalitarianism.
The failure of American pupils is probably not due to the way the subject is taught, but rather because they don't feel the pressure to excel like students in other cultures.
Huh? Getting into college isn't an extremely pressing/taxing/competitive ordeal?
I'll tell you why US students are degrading in test-worthy performance: Grade-inflation forced down the throats of schools by bitchy parents who can't believe their kid got a C when in prior years they'd gotten an A (most likely due to grade-inflation having to slowly work it's way up and through college). This does a tremendous disservice to the children, as they are less and less prepared for each successive year, until the overwhelming feeling completely puts them off of any subject that has prerequisites (like math/science).
-Michael
This will sound rich coming from an Anonymous Coward, but an expert on exactly what is wrong is a fortiori an expert on what is right. So I hope the author publishes a teaching manual with his ideas. No doubt, it'll bring a revolution in math teaching.
so you want to teach math using base-1 ... that's... insane.
If you cannot keep politics out of your moderation remove yourself from the Mod Lottery.. NOW!
I think what he is saying is that numerals are purely arbitrary symbols, if you have no experience with the numbers a first glance at the sequence "1, 2, 3" would net you nothing. the Mayans used symbols that actually represented the number in a more universal way: ".", "..", "..." (I would continue the numbering but it's challenge using word processing) with the Mayan method the first 4 'numbers' directly correlate to quantity, and the bar can be inferred by a simple number progression. I'd say Genius.
DEMETRIUS: Villain, what hast thou done?
AARON: Villain, I have done thy mother.
Shakespeare invents 'your mom'
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).
Simple Unexpected Concrete Credible Emotional Stories
Ah, well, you see, when I was in school 40 years ago, classes were broken out based on ability - those students that were slower to understand, shall we say (boneheads, my mother called them) were in one class, and those quick to catch on in another. Then in the very early '70s, such tracking came to be seen as evil, discriminatory, and just not fair. So, everyone was lumped into one group, so no one would feel stigmatized. The results were predictable - schools got into a race to the bottom.
Fortunately, that nonsense seems to be fading away. My son's high school has an honors track (I think most do, now), and he's in it. At first, he hated it, because it meant extra work, then he realized that the clowns who caused him so much aggravation weren't in the honors classes, but his friends were, and now he's all for that kind of separation.
The central point of the paper is distilled nicely on page 16, and this appears to even escape the author, to some degree:
Sartre discusses this in a much broader way, employing a concept he calls "authenticity." To Sartre, the life lived by fulfilling social expectations and roles is an awful existence. A major piece of authenticity is finding the freedom that exists in our choices, when those choices are unconditioned by our social institutions' crushing expectations.
Lockhart is pointing out that mathematics education is nothing but training - and training is an inherently inauthentic, anti-human approach. We train people when we don't want to have dialog. We train animals when the conformance of their behavior is the most critical thing about their existence. And Lockhart is pointing out that we are training our children to know a twisted language that most will never be permitted to speak, to read, or to write.
Cheers to that point - as a human that's interested in mathematics, I've been crushed by this at all levels. But the author's point is much more salient than the article lets on. Seek out and eradicate all of your own authenticities. Help others to do the same. Humanity will be a better place!
The author suggests that math be taught more as an art than a dry requisite skill. This presents somewhat of a problem though: A great deal of a person's mathematical ability depends on how strongly they understand the foundation of a particular area of study. If we were to treat math courses as "elective", I could forsee all sorts of problems arising.
I've always done well in math and enjoyed it thoroughly, but I'm not to naive as to say that I would enjoy learning about differential equations without first understanding functions, slopes, derivatives, etc. Taylor Series, Fourier Transforms--these are all beautiful and intricate mathematical concepts, but without understanding the fundamental building blocks which make infinite series work, they are all but useless.
As someone who was in school during the transition in a California district from a traditional to CPM-based math curriculum, I'd have to agree that math education has gone way downhill. I had the last traditional Algebra II class in my school. The book was almost 300 pages. The chapters set out the theorems/rules, built upon them and Algebra I concepts as they progressed and had A LOT of rigorous nightly problems. (YUCK! at the time, miss it now) My sister had the first CPM-based Algebra II class. It was two sixty page booklets, one for each semester. It was ALL group work (meaning, in the end some poor schlub would be doing the group's work). It didn't mention the rules/theorems except very obliquely -you had to "synthesize" (if I recall the term) them. Assignments were week long elementary school-level experiments that the group would write a five or so page report about. It was the blind leading the blind. When we got to trig., our class was fine. When her class got to trig., it was a disaster. It turned into an Algebra II class, and it really set my sister and her classmates behind on math.
We have a state college nearby that many of us went to. Those who had done CPM ALL had to take remedial math. Those who had done traditional went straight to calculus or statistics and did mostly well. I would even go so far as to say that, in our small area at least, CPM is responsible to a slight degree for the financial lack of sophistication that led to the stupid loans that are causing our city/county grief.
K-12 math is bad. I'm a product of that system. Until 5th grade my parents were told that I was "slow" when it came to math. In 5th grade I had a wonderful teacher that saw me nodding my head during math class one day while working on a problem. She asked me what I was doing. I said I was "matching" the pattern in my head. It turns out I don't work math like most western people.. I don't memorize. I visualize patterns of 5 and 10, the 5 looks like the 5 on a pair of dice, the 10 looks like two rows of 5's laid out. She, luckily, was learned enough to realize that I naturally mimicked the eastern way of solving lower level math problems. She gave my parents literature and pointed them in the correct direction, even though she will still bound to teach memorization type math in class which never made sense to me.
Well I ended up becoming an Engineer. I can solve most math problems in my head that make people cry just thinking about them. I can usually add the grocery bill up faster at the checkout than the machine can... and add in the tax. I add in my head left to right, I use the "close enough" principal, all the stuff that drives teachers here nuts because its "wrong". Never mind I was always better at the math and faster than the teachers themselves.
I have three daughters. The first one started struggling with math in first grade. She did not understand why teachers kept making her use a numberline to add objects. It was hell working with her night after night with the idiotic numberline. We had fights about it every night. One night I kicked myself when I realized she might have the same issue I had... and was pissed since I should be one of the first to recognize it. I grabbed a bowl of dry cheerios and asked her to use them to work the problem. I removed the numberline. Sure enough, she geometrically laid them out. It was a different layout than I use, but none the less It was the same pattern like solving that I use. I almost cried.
I looked around and attended a few learning centers to see if any of them would teach differently. All of them were the same idiotic system until I found Kumon. If there is one in your area I highly suggest you go check it out. Its not for everyone because the learning is based on eastern principals... and some people jsut don't think that way. My wife for instance just shakes her head at it.
I guess the moral here is that you can't shoehorn everyone into the same system. Mathematics in its truest form is more an extension of logic than it is about numbers. K-12 teachers keep pounding the number thing into kids heads over an over. Memorize Memorize! My daughter is now going into third grade next year and can add and subtract multiple digit numbers in her head. She is able to divide in her head and get "close enough" by spitting out fractions or remainders, even though she has no idea how to do long division by hand. Quite honestly I don't think *I* remember how to do long division by hand since it never made sense to me. Sure, I understand the principal of it, it just seemed like a backwards way of doing it. Thats the thing... why do it if it seems wrong when there is a perfectly good way of doing it another way?
The reason is, as was explained to me, that the way I do it doesn't work for standardized testing. Well it works for the right answer but since you are doing the bulk of it in your head there is no "work" to write down. And when there is written on say adding two large numbers the work doesn't make any sense to the grader. There is ALOT of money riding on those standardized tests. So in the end it all boils down to money :(
I have to comply with 300 pages of regulations for the school I started in Denver. The cost of compliance is at least half the total budget.
Although this article did not touch once upon the issue of wages, it is a very good article -- perhaps the best I've read all year on the subject of education. The need to introduce mathematical intuition at a young age is something the Montessori Method has done for a century. In a Montessori school, the child progresses from concrete to abstract, working first -- from very young at two years old -- with physical objects that embody length, area, or volume, and only later attaching the abstract symbols we call numbers. The physical manipulation leads to visualization of how addition, subtraction, multiplication, division, and fractions work. A child who goes through all three years of "Primary", which is age 3 to age 6, by the end of it, the child will be multiplying and dividing, and have worked with manipulative materials that demonstrate fractions and even binomials and trinomials from algebra.
In the face of competition from government schools, it is a challenge. I have learned that the competition isn't so much for students as it is for teachers. By using tax dollars, they can pay so much more, offer more benefits, and provide stability stemming from a legally-guaranteed funding sources. Meanwhile, the government schools are there for the purpose of creating cannon fodder, with its flag worship every morning and the forced admission of military recruiters under No Child Left Behind for as early as third grade. And when they do grab a hold of an effective pedagogy like Montessori, they pervert it by adding standardized testing and segregating by ages (e.g. two-year age groups rather than the three-year age groups prescribed by Montessori).
By eliminating public education, and by reducing the morass of regulations for running a private school, the free market could decide how important math education really is, rather than hearing hot air about it from public officials and CEOs, or by listening to earnest mathematicians such as Paul Lockhart, the author of this white paper, attempt to influence curriculum, presumably in government schools. The century-long battle between phonetics and "whole word" in the area of language (and the resulting reading levels no matter what is done) should be evidence enough of the futility of this approach (to use an anlogy, which Lockhart seems to love).
You must have attended a very very small school. Most US schools have different courses based on skill level. Your conclusions about the US school system are therefore wrong. They are merely conclusions about very small schools.
Really? "Most US schools" have this? Maybe your school did, but my high school, which had over 1300 students in grades 10-12, most assuredly did not. Well, if your definition of "different" means "two", then mine did. My high school (that's "secondary school" for all you non-North Americans) offered one advanced level class in chemistry, math, English and Social Studies. Entry into those classes was restricted to the brighter students (I got in - lucky me). Then they had normal level classes in all those subjects that everyone else took. Granted, I graduated in the 1980s, but I don't know what the heck school you went to, but I tend to think that your experience is the atypical one here and not that of the guy who posted.
No. There's always some college that will take you, even if you got average grades (and below average, people probably aren't interested in college anyway). Sure, you might not get a scholarship and have to take out burdensome student loans, but when American culture now emphasizes that a college degree is for everyone, and universities are businesses after your money, it's a buyer's market.
All NCLB is, is an excuse to close down public schools in favor of private schools. What we should be pushing is not a greater centralization of our schools, but is to decentralize our schools further through the use of vouchers or tax credits to enable parents to CHOOSE where they send there kids. There is no reason why a child should be locked into failing schools based on geography.
Of course, the NEA would resist that, they are the other part of the problem...
I can vouch for that. my home town experimented with a "School of the Arts" system in one elementary school and one middle school - switching those schools 3 years after I was through them. So my senior year in HS we had these kids as freshmen. I was the student TA for the Earth Sciences teacher.
My earth HS's earth sciences teacher was an AWESOME guy (in fact.. all the science teachers were) and were great with demonstrations, etc. All kinds of things that got most kids involved in sciences they previously considered boring. I mean our earth sciences teacher yearly took his class out to hunt for marine fossils in a spot where the task wasn't finding one, it was extracting them from the small cliff face without damaging them.
anyway I'm off topic.
He had a few students my senior year who were "School of the Arts" graduates. His tests were simple - 1/3 was "circle the answer", 1/3 was "short answer - one word?" (IE 'What is the name given to molten rock after it is erupted onto the surface'), 1/3 was "Short answer - one sentance".
Several of these kids refused to do the second 2/3rds of the test.. but their parents yelled at the teacher when he failed them. I was in grading exams one day when they were yelling at them, I knew who's parents they were. I extracted their meat-head sons test from the pile, graded it (the 1/3 that was actually done), walked up to them and handed it to them. Then instructed them to kindly stop treating the best earth sciences teacher I've ever seen like he was an insect under their boots.
If you cannot keep politics out of your moderation remove yourself from the Mod Lottery.. NOW!
My large school got rid of all the courses based upon skill level shortly after I left because helicopter parents were too stressed out and causing too many administrative and political problems for the school when their children didn't qualify for the high skill level class. That has happened across a large number of schools in the US.
Also, the US tracking starts actually tracking kids in like 10th and 11th grade. Great, they're tracked and get the benefits for two years...yippee!
Doug
VisualPhysics.org
Working on new views of old physics at http://VisualPhysics.org
This is the biggest misunderstanding most people have of teaching. The general view seems to be that if you know your subject, you're qualified to be a teacher. Nothing could be further from the truth. The most important qualification a teacher has is the ability to relate to their students; the relationship between student and teacher is the single most important factor in how much a student gets our of a class. Second is the teacher's ability to encourage students to do the work, since people only learn by doing. Third is the ability to design a series of activities that will, if give students the practice they need to learn what they're supposed to learn.
None of this presupposes a knowledge of the subject beyond what's being taught. Obviously, the more you know about the subject, the better. But there are half a dozen more important things.
They teach you how to count in kindergarten, they even show you with blocks what 1 means, what 2 is and what not.
What your holding as a genius way to do things is no different then Roman numerals. It gets extremely unwieldily for anything other then simple addition and subtraction or basic counting, and they aren't any less arbitrary in their symbolism then Arabic numerals are.
"I use a Mac because I'm just better than you are."
I thought the summary was over-the-top, but after the reading the article I agree. The guy has a "artful" way with words....about math. I imagine that pretty rare. I enjoyed reading it. Suppose I never knew how much of a math guy I really was.
Trackball users will be first against the wall.
Are excellent for newbies. Well written with good examples and graphics.
When I was studying computer science I had a horrible time understanding recursion. Now understand that this was a long time ago, and structured programming was just beginning to be developed, so recursion was something new, both conceptually and technically.
But a big problem was the way it was presented. Basically we were thrown into a world of stacks, heaps, pointers, and so on, without a single word of why recursion was useful, and without any sort of introduction to recursion.
Years later, when I taught CS, I would take my students outside, have them pull a leaf off the tree, and trace the veins. We'd talk about the self-replicating nature of the structure of the veins in the leaf, at smaller and smaller scales, and finally stopping at some point.
Once my students understood this self-replicating nature of nature, we'd start implementing it in the classroom on a computer. And things like recursion, binary trees, and traversal became trivial.
It's all about tying real world observations to the science you're doing.
no one has cited this yet. This particular comic strip is the "smug" side of the mathematician mindset, whereas the essay cited in TFA is the "angry" side.
Welcome to the Panopticon. Used to be a prison, now it's your home.
You must have attended a very very small school. Most US schools have different courses based on skill level. Your conclusions about the US school system are therefore wrong. They are merely conclusions about very small schools.
Are there honestly any US high schools that teach beyond calculus? No curricula that I know of goes beyond the contents of AP Calculus BC in high school. That's single variable differential and integral calculus, with coverage of series/sequences, and some vectors.
I would very much like to hear of any US high school which teaches linear algebra.
oh wait...
Your conclusions are wrong on the grounds that "Anecdote, therefore all-encompassing statement" is a horrible argument. My school in North Carolina had "English" and "Honor's English", and then once you got to junior and senior classes "AP English" (and don't get be started on the AP board...).
Conscience is the inner voice which warns us that someone may be looking.
"Dude, now you're approaching xenophobia. Have you looked at the state of mathematics in American universities? A conspicuous amount of highly original researchers are the product of foreign educational systems. They aren't doomed to being tech support monkeys like you insinuate."
I don't know how the heck you got that out of my statement. I'm not saying these people are not smart, I'm not saying they are not mathematical prodigies, but they have all learned math in a particular way, many mathematicians don't even realize it because their mind is *naturally suited* to the symbolic form in which the were taught.
Drilling kids with a structure of math when they have no idea how to relate it to their own natural knowledge limits their ability to understand what 'math' is. Most people have never really looked into what mathematics is, where it comes from, how it is derived. I've got books I and articles I've slogged through doing my own research in my spare time and I've realized how disconnected and arbitrary how math is structured in our society really is, and I'm not discounting these peoples contributions to society.
I'm telling you math is much more rich then what most people have even begun to think about*, yes even the PHD's.
I'm talking about how mathematics is *structured* how it is represented.
I remember taking "gifted" tests in school that structured mathematical principles using colored shapes/empty shapes for patterns and principles.
Kids need a way to *connect* what they see as meaningless symbols and see they are *derived* from observations in the world, mathematics *began* as a way for someone to take their observations and format them in a systematic way, but there are many ways to do this and the way something is presented matters A LOT.
I wish I could find the article at about how someone built a physical model as a metaphor of mathematical principles that explained the principles better then the equations and graphics they had made.
Either way there are better ways to communicate mathematical principles and ideas then has been traditionally been taught in societies institutions because I have spent a heck of a lot of time researching this on my own time. As expected on slashdot I would meet a lot of resistance for people who are without my lifetime of experiences that I have yet to congeal into a work of origina lresearch.
I am a teacher, albeit not a a math teacher but teaching in general has a lot of problems in the U.S. The largest problem that I see in America is that we have a system of education that is largely based on talent. We recognize it, reward it, and care for it like a price flower. Effort on the other hand is culturally unappreciated and that cultural attitude is reflected in education. The talented students have the opportunity to shine, and they always have.
Would our culture demand effort from our students instead of recognizing talent we'd be much further along.
I'm not suggesting that talent should go un-nurtured but, at least from an educators point of view, the effort of the students should be the focus of rewards.
load "$",8,1
The problem with math is that it is taught with no reward. Imagine if we studied the formulas of physics w/o ever hearing about the cool stuff like General Relativity, the Uncertanty Principle, etc.. Understanding those interesting paradoxes is what makes physics interesting. Now look at math. How many people know that math can limit the very scope of a scientific theorem (as expressed in a piece of paper), or the odd patterns of primes, or that there is knowledge in the universe that cannot be summarized, etc.. Nobody ever gets to see the cool stuff.. they are just burried under the mechanics.
I believe school should teach people to communicate effectively. Initially this implies imprecise communication like English and Spanish, reading and writing.
But later as we try to describe things more fully we may employ the language of math. For example, lets take a table and see how we can draw it, we can measure it and precisely define its attributes. In fact we can do so so precisely that we can end up telling how heavy it will be, how much room it will take, and how much it will cost.
As school advances and the need to describe things increases we can use math to describe chemical and biological proceses, physical processes, and sociological processes.
In fact, as people advance in their education they tend to need math to precisely describe what happened, their theories of what will happen, etc..
In this manner, students can enjoy the benefits of math in all fields as they advance in its study.
I know so many people that have studied Differential Equations or even basic Algebra and have no idea how that could ever be useful to them.
As a CS student I had to take a lot of math. One thing that always struck me is that a lot of math is a lot like programming (this is not a coincidence) except that you're only allowed to use single letter (greek!) variable and function names.
A lot of math reads like extremely bad Perl programs too, with tons of functionality on every line and no documentation except for a giant paragraph at the top written by someone who is apparently from Mars.
On the other hand, a lot of math is just pattern recognition. Realizing when you need to use one transform over another is a fundamental part of mathematics. Maybe the language simplifies this task somehow? I'm not sure. It always seemed to obscure it more than anything else to me.
I read the internet for the articles.
Comment removed based on user account deletion
Assuming you could fix the teachers, you could always combine math and science more closely. After all, mathematics started as an attempt to quantify and understand the world through a more precise language and physics, chemistry, biology, economics etc. all make more sense when explained mathematically.
Since we're dreaming here, math at home could also help the process too. After all, no one needs to learn to speak by the time they hit kindergarten. Parents can really help there kids by explaining how they use math in their jobs.
On a side note, I took my Ph.D. in Ops. Research (which lived in Engineering at my university) and my masters in math, so my opinion is colored by too much university training. Also, I hated math until I started to teach myself out of old text books around 6th grade. Then it got much better. It didn't hurt that my dad was an engineer; hence the "math at home" comment.
so you want to teach math using base-1 ... that's... insane.
Maybe he works in the school supplies industry?
... with the ability, knowledge and inclination. The real problem is that they can all make twice or more money by doing some other line of work. This is a matter of paying what is necessary to compete with the other possibilities open to mathematically able, knowledgeable and inclined people.
The interactive way to Go -- http://www.playgo.to/iwtg/en/
I remember this story, only it had dragons, miniature and large, in it as well.
I only look human.
My mother is a halfling and my dad is an ogre, so that makes me an Ogreling
Excellent argument on the frustrating habits of culture... and well written too.
You could substitute nearly any area of study into this analysis, and find a great deal of truth in the result, and in this, The Lamenting Mathematician has uncovered a very subtle and elegant habit of culture. The fact is that there are a great many musical technicians, incapable of creating the art of music, just as there are a great many mathematical technicians, who will never contribute to The Masterpiece. Software, Politics, History, Leadership... All have their share of artists and technicians alike. The key element is that the cultural perception of mathematics is that there is no art; that it is but a technical discipline.
The truth is that all disciplines are both artistic and technical in nature, and that society would do well to discover this and promote this duality through education.
The first advanced math course I took in college consisted entirely of proofs and abstract discoveries such as described in the article, and it was eye-opening. The clever approaches and solutions discussed gave that intuitive appreciation... no less artful than capturing a feeling with a photograph or instilling instant familiarity with a speech.
This is exactly what should be done. (Wish I had Mod Points today).
Peter predicted that you would "deliberately forget" creation 2000 years ago...
Honor classes are pretty widespread nowadays, and offered to whoever wants to get in.
You must not be in the Los Angeles Unified School District where there seems to be less willingness on the part of Admins. to seperate kids according to their abilities. Pisses off the parents of the dumb kids. Can't have that happen.
I prefer: There are 01 types of people in the world - those who know what 'little-endian' means and those who don't.
It is sad, and the state the US educational system is currently in will not allow it to compete in the global market, it will not allow it to be innovate and provide new ideas, but what it will provide is people who are like sheep and are more than willing to follow the crowd and just do it because everyone does. These people will be easy to govern and control since they won't ask questions and least of all will they rebel and fight for their beliefs. In other words, the US education system as it currently stands is making zombies.
I am sorry to say, but this is very much by design. The system was designed to by the powerful to perpetuate their own interests, not those of children. It is designed not to teach children how to think, but to prevent them, insofar as possible, from ever doing so, or even realizing that they can. After all, the easiest way to enslave people is to keep them so ignorant that they don't even realize that they are slaves. And, sad to say, that is exactly what they have done. It is probably among the greatest crimes of all of human history.
Nonaggression works!
and most of them can be traced to certain groups (*cough*fundamentalists*cough*) waging a 30 year war on public education
Depends on what you mean by fundamentalists. Honestly, I have my doubts you can trace all our problems back to creationists and prudes. You'd have to get the market fundamentalists, the "one curriculum to bind them all" fundamentalists, the Fabians, the Rothschilds, the Rockafellers, and probably more in there to get a really good idea of why we've ended up so mixed up.
That said: I got a fantastic high school education. I learned quite a bit and could have gotten a lot more out of it if I'd had the inclination.
Tweet, tweet.
TFA doesn't make any sense unless you start with a spherical educational system...
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.
While there may be great modern artists, there is no rigor in modern art and great art is no longer appreciated. Wishing mathematics could enrich society is nice but misguided. There are reasons our education is the way it is, and although lamentable, it cannot be fundamentally changed without culture experiencing a true rebirth. Not likely.
No you don't get it, it was just to demonstrate there are many ways of looking at things.
I lol because you're making the same type of generalization that you're looking down your nose upon.
Sony ha
Disclaimer: I am not familiar with the American High School system, since I first came to the US to attend an Ivy League school. I may sound critical, but please remember, I love being in America, and I am merely pointing out what I came to realize over the years.
I found that, even in an extremely prestigious American college, the mathematics taught in freshman and sophomore courses was at a similar level to what was broadly taught in the public high school system back in Europe where I came from, as early as 9th grade. I found most my American's colleagues math knowledge to be largely absent, but more importantly, many of them who had not entirely abandoned mathematics were nonetheless not even be aware of what they did not know .
My impression of why they got to be in this state had to do with the teaching method. The mathematics textbooks used in 'mainstream' courses even in my $45k/year college (i.e. over 20 students in attendance) are useless because they adopt a teaching style entirely devoid of insight and entirely too focused on mindless calculations. I suspect similar methods / textbooks are employed in high school. I found, over time, that I can recognize such a book right away, because they tend to be filled with examples with specific numbers. It doesn't matter if the topic is linear algebra, calculus, multivariate calculus, differential equations, Fourier analyis, they manage to insert 'exercises' with 'insight-building' arbitrary values, e.g. 'integrate 12.51 x^3 / (2.98 x + 1) from 0.1 to 2.31.'
There's of course nothing wrong with practicing integration (or other) techniques, and in fact back home we've all had to spend a massive amount of time doing just that. There's nothing wrong to being able to quickly and correctly do algebra, with large natural numbers or even arbitrary rational numbers - but that is something that is ingrained early on to the point where it doesn't need revisiting. In fact I would argue I am faster, more accurate, and can perform more complicated algebra, although the last time I was asked to work on it was in 5th grade. By the time you make it to calculus and beyond there's no need to test whether you can evaluate a function at specific values of its parameters (if you cannot do that, you would have failed a long time ago), so looking at the integral of x^3 / (a x + b) is allowing you to focus on the essence of the problem at hand, and not on mindless algebra. (Again, the algebra is mindless because you are supposed to know how to do it by the time you're 10, not because you can outsource it to a calculator or India.) Having examples with actual values is hardly the worst flaw, but it is strongly indicative of the mindset of the author and the teaching method, which I would characterize succinctly as 'lacking insight'. I would not have been able to understand and learn math had I only been exposed to such methods.
Math is a combination of art and hard labor, and both components are important. Good professors are absolutely essential, more important than even good textbooks. In fact all the good textbooks have already been written, many of them decades and sometimes 100+ years ago, it's just a matter of knowing about them and using them.
Rant time. From the original post: "I defy you to read and find a single sentence that isn't permeated, suffused, soaked, and encrusted with truth."
Challenge accepted, for example, from the article:
"The area of a triangle is equal to one-half its base times its height." Students are asked tomemorize this formula and then "apply" it over and over in the "exercises." Gone is the thrill, the joy, even the pain and frustration of the creative act. There is not even a problem anymore. The question has been asked and answered at the same time--there is nothing left for the student to do."
Item (1) I have an MA in math and teach at a community college in NYC (previously Boston; algebra, trigonometry, statistics, etc.) (2) As an academic, when you start teaching, you are in for a rude shock. All throughout school, I was engaged, getting "A"'s almost all the time, and considered a "B" to be a signal of failure. The shock is to discover that the majority of people in most classes (including, unknown to you, all of your prior classmates) are unengaged, and are more-or-less comfortable with doing C/D/F work. (3) The problem discussed here (exercising area of a triangle) is, yes, trivial to someone who "gets it". However, it is very difficult to the majority of community college students that I see. For students who fundamentally can't grasp the concept of a variable, repeating algebra for years and years, and who can't "get" the idea of substitution, it's possibly overwhelmingly difficult.
Yes, to you and me, "there is nothing left for the student to do", I agree fully. But what I've learned since starting as a teacher is that the exercises are an ongoing attempt to prove mastery of the "substitution" concept, and it's actually an enormous struggle for most people who aren't posting on Slashdot.
I've learned that I can hand out a complete "practice test" in advance of an exam (passingly similar to this proposed exercise), and give an exact duplicate of that test in the next class, with only the numbers changed, and still have the majority of a class fail the test.
Now, that's not all I do, but I do include examples of this just to check my own sanity all the time. What I also do now is to always include one or two "concept questions" requiring actual analysis of ideas, and the level of frustration and aggravation from the students for those is far, far more enormous. Frequently people just stop trying those by the end of a semester, leaving them blank, and are happy to walk away with a "B" or "C" from the rest of their tests.
In summary: I now consider my #1 job in all my classes to be an effort to make students comfortable with abstraction. Give me or you a formula and then, indeed, "there is nothing left... to do". But for most students, whose brains fundamentally cannot abstract enough to grasp substitution, there is an enormous skyscraper-sized obstacle still standing in front of them. That is in fact the fundamental goal of most math classes for most students, and they certainly can't do creative exploration or problem-solving until they at least "get" that, and are able to express patterns coherently when they see them.
Unlike mathematicians like these, my claim is that mathematics is not art; it is a desperate battle. For your consideration, the AngryMath Manifesto: http://angrymath.blogspot.com/2009/01/angrymath-manifesto.html
We know where leadership by an anti-intellectual "strongman" who scapegoats minorities and likes boisterous rallies goes
Yes you understand, many posters can't seem to grasp what I'm getting at.
Yes, and the Mayan use of a horizontal bar for "5" hides the fact that it's actually FIVE 1's. The only numeral system that doesn't use some sort of compression is one which has only one symbol (perhaps a second just for zero), and every number is shown with that number of that symbol. Thus, one dot for "1", three dots for "3", 10 dots for "10", 1000000 dots for "1000000", etc. Obviously, this isn't a very good system unless you're dealing with quantities 10, so humans developed shorthand systems. There's simple ones like the Mayan and Roman systems ("I", "III", "V", etc.), and then there's more efficient ones like the Arabic-numeral system we have now. The numerals we use now might be harder to grasp for a 5-year-old just learning math, but are far more efficient for someone familiar with math needing to do more complex operations than just 3+8=11.
Part of being intelligent is being able to understand abstract concepts, so unless you're an idiot, using "3" to represent three instead of needing to see three separate dots or bars or whatever shouldn't be difficult. If you ever hope to progress to using exponentiation, sines, cosines, integration, differential equations, vectors, curls, dot products, etc., then the symbol for 3 better not be a problem for you.
If you read the Wikipedia article on Maya numerals, linked to above, 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).
when there are so many people out there claiming that failure isn't their fault. Let alone a government which essentially pats them on the back and tells them that the government will make it all right, the government will take care of them, the government will take money from other, more successful people, and give it to them?
When you have schools which decry any form of testing or proof of ability? When schools and the unions fight tooth and nail to ignore or subvert proof of the schools upholding their education ability?
When you can pass kids because they tried... because "trying" is so easy to prove.
* Winners compare their achievements to their goals, losers compare theirs to that of others.
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.
HIGH SCHOOLS (and some junion high ones) tend to have various levels of math courses. However, in grades 1-6 (and somewhat 7-8), there's really only one math course for everyone. SOME very few kids are allowed to take the next-grade-up's math course (my best friend was one, 'twas how I met him), but for the most part, nearly all third graders (at a school) are studying the same thing, a nearly all sixth graders are studying the same thing.
The trouble comes when, in your fifth or sixth grade class, the teacher is going over (yet again) how to do long division or multiplication with one more digit than the previous year. The students who Don't Get It still don't get it, and are frustrated. The smart kids are bored stiff because it's months of crap that they learned two years ago, and thus they either screw off, are disruptive, or (some few) are lucky enough to have teachers who let them go out in the hall and doodle or read or work on homework while the rest of the class covers Yet More Long Division.
I learned how to multiply and divide in third grade. In fourth, we did it a little bit more, with two and three digit numbers... that might be when we were introduced to long division. We then repeated that for two more years, with the digits increasing. More rote-work, rather than finding interesting ways to USE the math. I realize that practice is important, but there are better ways than "OK, 50 more problems, this time with 4 digit divisors". I was fortunate to have compassionate teachers that let me play Oregon Trail in the hall. ;)
Yes but you're missing the point by quite a lot: Mayan's used geometric distinct shapes directly for a reason.
For instance if I take a cirlce and half a circle, that can be expressed as 1.5.
For instance in the stock market, companies often do stock splits to keep their stocks "cheap"
Google is $410 and something dollars at the moment.
If you buy a stock at say 410, and it goes to 440, and you buy a stock that's 4.10 and it goes to 4.40 its the same difference but many people don't intuitively grasp this (hence why some companies do stock splits to give the perception of "cheap")
Now say you express 400 as a single circle (base 400) and the change was expressed as a fraction of a circle, and did the same for the other stock cheaper stock (base 4)
Base 400 and base 4 have a relationship that can be communicated more clearly and concisely using visual figures and representation.
Now I know this is a simple model but I'm saying as you go up the mathematical pole their's way to take complexity and simplify it like this that hasn't been realized so many students get lost in symbolic jaron that seemingly has no meaning.
Also you and interpret the entire number system as merely distinctions in surfaces,
For instance there's a direct relationship between our ability to detect differences in objects in reality and mathematics itself, math is merely a codification of our natural way of thinking
http://www.danicamckellar.com/
I can't believe Summer Glau is the chick geeks are hot after. Danica is Hot, has her name on a physic theorem, mathematician, and has written math books for girls.
Her acting career is full of geek as well.
Not to say either one of them is a geek, just that I scratch my head over why geeks prefer Summer.
The Kruger Dunning explains most post on
So you've attended most US schools then?
Just because an idea is popular doesn't make it right.
I know towards the end of my school career in high school, the school system I was going through was getting rid of the lower end of most of the classes. My best guess is they were doing it for funding reasons. So while it may not be overly indicative of the entire country, the lack of different levels of classes for different skill levels is definitely not something to be taken for granted in the US.
Creativity can neither be taught nor guided. The analogy with painting and music is flawed; there are an infinite number of ways to create a painting or musical composition, but relatively few ways to create a logically consistent mathematical system. While discovering mathematical truths on your own may be fun for the author (it was for me as a child), allowing everybody to write their own Principia Mathematica is simply unpractical and would result in mathematicians being unable to communicate their precious ideas to each other. Learning math is more like learning english; while the author is correct that we shouldn't confuse the language with the beautiful ideas the language is intended to express, it is also true that we can't discuss Shakespeare without a common language for communicating the abstract ideas contained within. I feel the same way about software that this guy feels about math (some programs are much more aesthetically pleasing than others), but his worst mistake is assuming that everybody else should feel the same way about math he does. Unlike art where you can just fake it until you make it, math actually does consist of many layers that build upon each other and must be learned in progression. (There are some notable exceptions to this, e.g. Set Theory has been successfully taught to 5 year olds. Binary Arithmetic is really just a trivial case of Set Theory where only null set and unity set exist; it could be taught more easily to children BEFORE they learn decimal arithmetic, but our culture has a decimal-centric bias (in The Simpsons cartoon universe, do they count in base 8?)) Where was I? Most of us can't even make it all the way through Godel, Escher, Bach. Just because you enjoyed it is no reason to assume everyone else in the world thinks the same way you do.
I've abandoned my search for truth; now I'm just looking for some useful delusions.
Oh, I understood the logic and the rules which could be used in a proof. But no hints were ever given of how to select which rules to use to perform a proof. It was as if one had to just try all the rules and combinations to find a path to the proof.
I have a PhD in Math from a top school.
I was unemployed for a few months after graduating.
Employers were far more interested in my BS in CS.
actually, if you look closely at the numbers, you can see how they were formed.
1 - is a single downward stroke.
2 - horizontal stroke, connected to another horizontal stroke below it.
3 - (I'll ASCII art this one.)
---- stroke
| connected to
--- stroke
| connected to
---- stroke
4-9 are a little more esoteric, though.
(sorry, numerals, not numbers.)
Lockhart is dead on.
I have had one "mathematics" teacher in my life. The rest made me memorize crap. They never taught the essence of the subject, only the solutions.
In this teacher's class he would write a brand new problem on the board which I had never seen. By the time he was making the last stroke of the last variable, I would have the solution, and promptly raise my hand. My answers always correct. He would make me come up to the board and put what I did in my head on the board. At first he was astonished to see that I had done the problem in an entirely new way which he hadn't seen before and would validate that I had solved it correctly. Often the solutions I presented were extremely simple as well as being new to him.
He inspired me to love math, which subsequent teachers promptly crushed.
Every other "teacher" I had made me do it their way, memorizing everything they did and would fail me if presented the solution any other way but the way they expected. I'm convinced that very few mathematics teachers should be teaching this subject. Most are snotty pseudo intellectuals with 0 imagination. I would do my best to not be noticed by them.
I've only met one that understood what math was about.
That's why I'm a software engineer instead of a mathematician (or historian). I'm really really good at solving problems but am terrible at memorizing. I did what I had to to get by in the math classes I did have to take. Usually it involved "cheating". God forbid I should actually solve the problems because usually I was "wrong" when I did it my own way and produced the expected result. You can only take hearing "you can't do it that way" so many times when you CAN do it that way.
I'm no genius but I also don't need some teacher forcing my hand to "help" me solve a problem which I've already solved in my brain by the time I finish reading it. Coding seems to be a lot easier for most people to understand so the instructors give you more latitude. That was just fine with me.
-AC
The US system is something of an anomaly in that we have a secondary schools that are supposed to cater to EVERYONE. A person earning a high school diploma could be either on their way to MIT or an exciting career at McDonald's, whereas in many other places (like Europe), people are pretty well separated out by the time they hit high school or even middle school. In Germany, for example, the students earn different types of "degrees" from different types of secondary schools and only the hardest one (abitur) will get you into a regular university. It's not expected that everyone "goes to college" like they do here and in many cases people pick special areas of study early on. If you have a classroom full of kids who are planning on becoming engineers, it's a lot easier to push them harder when it comes to math since they know that they need it and thus will be more responsive. By the same token, if you're a teacher used to dealing with kids who don't have much interested in or aptitude for math, you will eventually become good at figuring out ways to get them to be more responsive and try harder. Here everyone is mixed together and teachers have the challenge of having to teach kids with widely varying abilities and plans for the future and many simply give up and point to the university bound students and say "Look, the curriculum and my teacher methods are sound!" while the kids who are not as gifted slip through the cracks.
Though I don't need the rhetoric, this hits it on the head, in every aspect.
I'd like to try teaching math like English -- Math 1, Math 2, Math 3, Math 4, with curriculum determined in part by such apparently meaningless factors as what might be useful in other classes or what's happening, you know, outside of my room.
The textbook comments are particularly right on -- step 1, burn them. If teachers complain that they won't know what to teach, fire them on the spot.
Geometry is also a lousy place for proof. Teach deduction all the time, in every topic -- and in classrooms other than math. "Here's a bunch of fake stuff you don't know anything about that's hard to draw. Now let's think really abstractly about how we're thinking about it!" And induction doesn't get taught at all.
The practical deal-killer, the one that drove me out of the profession, is that the barrel full of math teachers is so close to empty that you're pretty much scraping bottom from day 1. This kind of instruction -- and this kind of critique -- can only originate with someone who likes math, and is sort of good at it. You'd be amazed (or maybe you wouldn't) at how few public high school math teachers this describes.
America has gotten the math teaching instruction it asked for when it decided to prop up bad teachers with lousy but easy-to-use texts, and to boot it got the benefit of not having to pay very well for people willing to go through these motions. (It's not about money, but really, it's a little bit about money. I doubled my salary when I left last year.) It's a big, huge problem, and since you're going to have to convince parents that it needs the kind of dramatic overhaul this (great) article describes, and since parents were largely victimized by the existing system, I'm pretty sure it's a losing battle.
god is just pretend.
Let's not argue about anecdotes!
The Simpsons had the best summary of education ever: "Let me get this straight: I'm behind the rest of the class, and I'm going to catch up by going slower?"
If students aren't grokking concepts as quickly as other students, the students who learn slower need to spend more time on the concepts in order to compensate. Putting them in a separate class of equal length doesn't accomplish anything. They need a unique -tract- of classes that spends more time on concepts, and they need to remain in that tract until they can learn at the same speed as the other students.
More time is really the only way to help struggling students catch up. Logistically, this is a nightmare for even large schools. It's difficult enough to schedule students, teachers, and classrooms into basic timeblocks. Trying to add more classes of different lengths that progress at different speeds is borderline impossible.
I believe that the best, most feasible solutions are online courses and self-guided curriculum:
- Individual schools don't have enough teachers to have nine different Pre-Calculus courses, going at different speeds, each offered several times throughout the day, nor would they have enough students to fill all those sections...but if you connect schools with online learning, then you can share teachers and students among many schools.
- Self-guided curriculum allows individual students to schedule their time as necessary. If two students are both taking English I and Algebra I, one could spend 2 hours a day on English and 1 on Algebra, and the other spend 2 hours a day on Algebra, and 1 hour on English, and they'd both complete the year at roughly the same level in each.
Both of these ideas as 100% possible right now, but each has an obstacle caused by cultural inertia:
- Schools and districts very rarely work together. Yes, we can all find a thousand examples of when they work together, but these are less than a drop in the bucket compared with the shear number of classes and projects going on every school day.
- The education industry simply doesn't change at such a core level, mostly because training for educators is pathetic. Educational schools are largely guided by conservative former educators who perpetuate the status quo. School- and district-run professional development suffers from the same problems facing classes for students: all the teachers are at different levels of aptitude, so the classes sink to the lowest common denominator and most people don't learn anything new.
So, add another layer to the top of the two ideas above: before they can be implemented, schools need to develop individualized professional development for teachers. Then teachers will be equipped to grow and change, and then students will finally get an education in a style that is different from and improved over education from the first half of the last century.
You must have attended a private or EXTREMELY large school. Most US schools are nowhere near the described Netherlands system. At best, you've got three tracks - "honors" which targets the cookie-cutter wrote memory college tracked kids, standard for those who aren't fighting or don't care about math scores WRT university applications, and "essentials" for poor suffering masses who are not picking up or don't care to do the work. This is the situation in Washington State, Kent School district which is the 4th largest district in a High School with over 2600 students. Even this delineation of "skill" is still cranked through the un-inspired compulsory process Lockhart complains about. If you want to know why, check out John Taylor Gatto's "The Underground History of American Education" (http://www.johntaylorgatto.com/underground/).
Saying knowledge comes from a schooling about as correct as saying milk comes from a store. When you understand in both cases it's just simple packaging and processing, you can start asking questions about what it is, why it is, and how you can get it on your own, and how to evaluate the quality of the sources you get it from.
*** Sigs are a stupid waste of bandwidth.
In the "High School Geometry" section, Lockhart talks about how a fairly simple idea - that when two lines cross, the angles on opposing sides of the crossing point will be equal - is turned into a complex and ugly chunk of notation... Lines must be identified as "AB" (with a bar over the top), there's notation for identifying the angle formed by three points, and the whole, simple idea is then backed up with a hefty "proof" in place of a simple, natural-language explanation.
Now, I don't quite agree with all of this. Maybe that's because it worked for me, and because I enjoy the idea of mathematics and logic having their own "language", and their own notation. I mean, sure, back then I always used to wonder why they threw all those Greek letters at us - in some cases it seemed totally arbitrary. Some of it is just long-standing tradition: like the capital Sigma or the long-s glyph used for summation and integration...
But what I enjoy about the use of these symbols is that they provide a compact way of identifying precisely what one is talking about. That these characters aren't part of everyday English writing means that they can be set aside to encapsulate powerful, specialized ideas.
The whole "proofs" thing worked out just fine for me, too. I think it makes sense: as part of teaching people how to build up simple ideas to form an argument in support of a more complicated idea, provide examples of how to do this: even with the simplest of ideas, the things that lend themselves most readily to intuitive understanding... If a proof is provided for a problem people naturally understand to begin with, then it will help them to understand how the proof works.
With regard to line and angle notation I think Lockhart is dead wrong. In the context of the "crossed lines" example he argues that the lines could be called "line a" and "line b" or something - and that the whole idea should be presented in a more conversational style. This could work for certain problems - especially really simple problems like that one - but there are other problems with expressing things in a conversational style. For starters, natural language is imprecise - at least the way most people use it. Precise natural language is the domain, for instance, of lawyers and logicians. It tends to be very heavy, often with its own specialized vocabulary that people don't readily understand on first exposure. Codifying the problem makes it less accessible, but also much more precise and concise.
Now, if I'd learned "High school geometry" in high school instead of, you know, 5th grade or whatever, it's possible things would have played out differently... As it stands, however, I'm a big fan of taking advantage of domain-specific vocabulary and symbols where they are available and useful.
Bow-ties are cool.
Asian students also go to school 250 ten-hour days a year. They not only learn math, they learn the value and satisfaction of hard work.
I'm a Programmer. That's one level above Software Engineer and one level below Engineer.
Interesting and creative summary of the article. You might want to read it...
I went to a rather small high school, graduating class of less than 300, in a smallish town and we still had multiple math tracks. Your senior year you could be taking any of the following: Calculus, Precalculus, Trigonometry, Advanced Algebra, Geometry, or Algebra 2. Which track you were on depended on how you'd done in previous years. It started in 6th grade where kids were tested in to either normal or advanced math. It then just continued diverging based on how you did. Algebra 2 was the second half of the Algebra that the most advanced kids took in 7th grade. So if you were particularly bad at math, you'd get some basic algebra before you graduated, and have it presented at a slower pace. If you were the best, you took a real calculus class, equivalent to about Calc 1 at a university, your senior year.
This sort of thing seems fairly common. Indeed our school did it not other for math but for many subjects. They weren't all a direct linear progression as math was, but there were choices based on your skill and interests. You could take normal English, College Prep English, or AP English your senior year, for example.
It wasn't a perfect system, but then nothing will ever be. However it did do a reasonable job of allowing those that were good at a subject and interested to progress, without denying those that needed a slower pace the opportunity to learn.
After all, I get a little tired of the idea that education should be targeted only at the top 10% and all the "dumb" kids should just be left behind. No, I think the opposite is true. See if you are smart, you have the ability to learn on your own to a great deal. You can take the initiative to teach yourself. How many times have we heard geeks talk about their valuable self education in programming and such? However the lower performers don't have that option, they need more help. In particular, if they don't get help, they may not be able to be productive members of society. They can't just "learn it themselves."
So really I think education needs to be setup to help those with troubles first and foremost, and worry about the top achievers second. That doesn't mean ignore the top achievers, it just means their needs aren't the most paramount.
Now this is primary education, of course, university is different.
Dude! Three is one of the few symbols that doesn't hide that it is made of three ones. Look how many end points it has, 1, 2, 3. Not like that bastard 7, now that one is sneaky.
You should probably think that through a little bit more.
The performance levels you're talking about are measured by standardized tests, so increasingly intensive standard teaching methods will exhibit symptoms of increased performance.
Should probably read the article.
You write:
This reminds me of the major theme of the well-regarded book Understanding by Design wherein the authors ridicule schools' mandate to "cover material" rather than designing means to have children understand the material.
No, but it lessens the probability that the idea is "absurd" as so accused by the original response to my post. Further support is that the U.S. went without public education for most of its first century.
That matches very closely with my math education, but it wasn't typical. A lot of students stopped after Algebra II, since only three years of high school math were required. In college, I actually enjoyed discrete mathematics quite a bit, since it's easy to think in those terms, and it has concrete uses that I was interested in.
It is pitch black. You are likely to be eaten by a grue.
Do you hold any degree in mathematics, i.e. are you qualified to make the criticisms in your post? Or are you a crank?
Yo dawg, I heard you like the Ackermann function, so OH GOD OH GOD OH GOD
but rather because they don't feel the pressure to excel like students in other cultures.
And why do you suppose that they don't feel that pressure? American pupils may not excel at math, but even among the math underachievers there are smart ones who see the low pay and level of respect that engineering and science receive in our culture. In the United States we glorify and shower wealth upon the athletes, lawyers, businessmen, and politicians who are often willfully ignorant of math and science. These smart students see these wealthy and powerful individuals who couldn't write a proof to save their lives and yet they somehow end up with the best houses, the nicest cars, the most money and generally look down upon scientists and engineers while gleefully outsourcing their jobs to those low-wage Asian countries where rote-memorization scientists and engineers become the next wage slaves of American international corporations. If we want good students to become scientists, engineers, or yes even mathematicians then we have to start rewarding those positions in our society instead of outsourcing their jobs and treating them like dirt.
No honest educational professional would be behind many of the debates (evolution baiting anyone?) we see all over the US.
I think you're missing the nature and magnitude of the problem if you believe that's true. Indeed, I find it quite likely that education professionals could be behind a lot of the initiatives, and evolution baiting, annoying as it may be, is the least of our problems.
The biggest problem is that people think that education is something you can treat as a program, and everybody's got some set of beefs with the system, and they think that if you can *just* solve their particular issue, you'll get a quality (and maybe even ideologically correct!) education system for everyone.
There might be things we could do to improve the educational philosophy for our culture, but I have my doubts that we can make those improvements programatically.
The only thing that should be done nationally is a standardized curriculum
I'm not even sure this is true, although I can see it's useful to have a minimum bar that high school education represents.
Tweet, tweet.
Why aren't all textbooks open sourced already? Allowing teachers to take educational texts, modify them for their own needs, and distribute the changes makes even more sense than open source software does. And yet it rarely happens. Case in point: Beaverton School District wants to start a new math curiculum; with 32,000 students they will be spending $70,000/year on new text books for the next 12 years... I want to know the name of the teacher they are firing so that they can afford this!
I've abandoned my search for truth; now I'm just looking for some useful delusions.
Perhaps you went to a crappy public school with no advanced or "gifted" track, or you weren't slotted for it. (This is not to say that all public school in the US are crappy, but you can certainly find them.) I attended public school and went to college with a better education than a lot of students who went to the best private academies.
Actually, the Mayan numbering system is very much akin to Arabic and not so much to Roman numerals. They use a positional system (base 20) where the number from 1-20 in the n-th position represents a value multiplied by 20^n. The only major difference (besides base), is that instead of using ten different symbols to represent digits, a tally system using three symbols is used. This makes small arithmetic easier to think about, but makes the system less compact than the modern Arabic system.
On the other hand, the Roman numeral system is simply a glorified tally system: position is (mostly) irrelevant, and magnitude is indicated with new symbols (i.e. X=10, C=100, M=1000). Not only is such a system less compact, but also requires new symbols whenever numbers start getting too large, and multiplication is a painful experience. The widespread use of the Roman numeral system was one of the main factors that held back the development of European Mathematics until Fibonacci introduced the Arabic system in 13th century.
I don't dispute the premise that US math education sucks, or that it would be useful to add Discrete Math to the list. However, your assertion that all those courses are "just plug and chug" is just absurd. Geometry is generally centered around proofs - this is the first class most students ever have in which they are expected to learn a set of axioms and theorems, and construct new theorems from them. Trigonometry places some emphasis, at least, on trig identities, e.g. prove that tanx sinx = secx - cosx. Not extremely hard, but it's not just plugging numbers into formulas.
Calculus is similar: you work on limits, learning symbol manipulation rules like the chain rule and integration by parts, etc. Heck, even in algebra I you learn factoring polynomials, which is not at all plug and chug.
Again, don't get me wrong: I'm all in favor of more proofs and reasoning about math. But you're still mischaracterizing the rest of high-school math.
...following the principles of Heisenburger's Uncertain Cat...
I am in agreement with the above. However, I think the use of discrete mathematics needs to be spread throughout the entire curriculum. Expecting kids to progress intellectually just because you force feed them formulas is a bit on the ridiculous side.I was one of those who never managed to "get" mathematics in school. Mostly because of the teaching methods utilizing rote memorization. My conceptual learning style just didn't allow me to absorb the information without proper applications. It took a very dedicated college professor to show me how easy mathematics can actually be when you know exactly what you are trying to do, and why. I think that the proper implimentation of conceptual teaching methods would solve at least some if not most of our math problems in primary and secondary education. The author's idea that math be treated as an optional subject kinda bothers me. Yes you can treat it as a form of art, and it helps if you don't stifle the creative tendencies of those few who are already interested in math, but a general understanding of math is a necessary component for understanding many other things in this world.
Math For America does exactly this. If you want better math education, the solution starts with retaining better math teachers. How do you do this? Simple.. Pay them a competitive salary.
Those who excel at math, rarely stay to teach it because it's more lucrative for them to take higher paying positions in the private sector.
Read more at http://www.mathforamerica.org/home
From the MFA Website "We are a nonprofit organization with a mission to improve math education in secondary public schools in the United States by recruiting, training, and retaining outstanding mathematics teachers."
This is interesting particularly coupled with a posting earlier about Wolfram Alpha and all of the trouble rising over its place in mathematical education. Under the math system that this article seeks to indict the introduction of an accessible mathematical tool such as WolframAlpha would be poison to everything they're trying to teach. However, in the system that he tries to forward, such technology would be a great boon. In the general attitude of /. being "Technology giveth and technology taketh away" wouldn't this be simply another variation on the same theme? Mathematics education needs some revitalization and what better way than to put power to teach oneself into the hands of eager students?
I like losing arguments, it just means that I can take your point and make it my own.
I actually went to two different high schools. I switched the summer before my Junior year in school. This was because the high school I had been attending was having major issues with minorities fighting (the african americans and hispanics were bringing weapons to school), we had several knife stabbings at the school, and it caused many issues in general because of the way the school was run and where it was going. My dad figured it was easier to move, both closer to work, and into a new town than to have to worry about his kids going to a high school where fights were the daily norm and where no attention was being given to the rampant issues with drug use, alcohol abuse and fighting.
The second high school I went to was a higher class, more expensive town, and while the education there was generally better in terms of people wanting to be there it was not that much better. There were different levels of classes offered, but in those classes they were still catered to the lowest common denominator. This was a mostly chinese and white school compared to my previous school. The issue at hand was that the school and the school board were more willing to put money into sports and the sports programs and buying new football fields and tracks rather than improving the class rooms, providing new books, new computers and new equipment.
The first school was 1500 students in size, the second one was 1200 students in size, small? Not really. Not as large as some of the bigger US public high schools.
cat
I think the use of discrete mathematics needs to be spread throughout the entire curriculum
Sorry, said discrete when I meant applied.
I was just pointing out the differences between the two educational systems, and while I agree that a chef does not need more than the US math, the fact that it was required was something I wanted point out as interesting.
I was just comparing and contrasting the two and providing examples as to how the two differ, not using it as an excuse. Most if not all of the math I know and use I have taught myself, mainly because I have difficulty understanding math like most other students and I had to learn it my own way for it to stick.
cat
For folks interested, another interesting view on mathematics teaching is in the book "Knowing and Teaching Elementary Mathematics: Teachers' Understanding of Fundamental Mathematics in China and the United States" by Liping Ma. It's very thought provoking. It doesn't persuade me to any particular solution, but definitely gives me more data to think about how we teach people things like mathematics.
... two HS degrees, seriously? Is that necessary? Where I'm from (Calgary, AB, Canada) there is enough leeway in HS for most kids to get a HS degree even if they aren't very strong at one or two subject(s) (they can earn extra credits through options, community service/work experience, take 'easy math' courses), and they still get the satisfaction of getting a HS diploma like everyone else. Considering that the majority of kids get a HS diploma (including equivalent programs for those with learning disabilities) and have the satisfaction of graduating with their peers, is it necessary for the requirements to be even easier? Does unleashing a bunch of relatively uneducated 10th graders (in my area; they are kids at 16, are just out of junior high and are still very immature with only a year of high school to temper that, and just got a driver's license) seem like a good idea?
Interesting.
Even small schools almost always have different math courses based on skill level (I went to a tiny high school, and we certainly did). You misunderstand the issue.
Unless you have classes with only 2-3 students of equal ability you're going to have this problem. Even in advanced classes there are some people who learn things faster than others, and the people who learn faster are almost always forced to sit through lectures and do work that is for them pointless.
The prevailing attitude in US education is that people who learn slowly are most helped by being in the same classes as those who learn quickly. This isn't wrong, but it does mean that those who learn quickly are slowed down to help others keep up.
This isn't a problem unique to math education though--it's an issue for almost everything. Unsurprisingly, things like art classes and music classes are least susceptible to this problem. The people who excel can do so, and the people who don't are still able to learn from those who do.
Sounds like *YOU* were the "lowest common denominator" or you might've been in a higher level class. There's no limit to what math you can take in school here. AP and honors classes are common and many students take college courses for high school credit. Hell, I studied group theory (just a rough overview really, but still) in the 5th grade. More recently, I took a graduate university course on non-linear dynamical systems (aka "chaos theory") with a high school student.
Success in American academics is achieved the same as everything else here: work hard at it and the sky is the limit but you can always just "get by".
Wealth, yes. But who's glorifying lawyers and politicians? Sure we get the occasional exceptions (e.g. presidents Kennedy, Reagan, Obama), but two of your four careers generally don't get much respect at all. Agreed on the rest of your post, but that bit was way off the mark.
- T
The notion that public education should be provided to all in the USA came about during the industrial revolution. This system of education was designed not to stimulate intellect as much as it was to create a workforce with the basic skills required to work at repetitive, menial tasks day after day. While much improved over the years, public schools are still chartered with the same task of generating a pool of semi-skilled labor that simply cannot compete in the current global economy.
For a community that's so infatuated with Occam's Razor there sure are a lot of conspiracy theorists among it.
Do you really think that over 200+ years of US history that none of thousands of people that would need to have knowledge of the plan wouldn't have turned on the conspiracy?
The "Dumbing Down" of the country theory is too complicated. There isn't enough short term gratification for the government to be that interested in education Those running the government's tenure is usually shorter than would be necessary to show results and so they don't need to try, it will just make their successor look good.
"You are not truly a mathematician until you learn to abstract. Symbols "
Nonsense, you're just proving my point that your particular mind is suited to the way the current mathematical systmes have been systamatized.
Everyone on earth in some way mathematician (even if not a very good one) the very act of perceiving *this is not equal to that* is mathematical in and of itself, one could not much less think or navigate the world, let alone function if one could not determine *this* is NOT EQUAL to *that*. As you can see there are no numbers there and yet there you intuitively sense the 'matehmatical' nature of these relationships, but not ice how they are expressed without any kind of formal system of abstractions, and people do this *every day*.
Symbols only come in *after* you learn that math is a subset of observing relationships.
Say you want to create a system for caclucating the bounce of a ball or the motion of someones arm when they move it, etc.
Math started first with basic observations and people then used little ticks, symbols and whatnot to create systems of instructions to record, order and systemtize those observations so they could reproduce them.
You can do math without symbols at all using shapes and objects in reality and just thinking about it conceptually.
For instance one can interpret 1.5 as being the same as 15, and the same as 150 within a particular perspective context of looking at it.
Sure you notice they are all 10 times the distance apart. But you'll also notice they are also *reflections* of the similar ratio. By converting these numbers into visual figures or objects and alignin them on a scale or horizon you can see the relationship of scale and size along something much more natural and intuitive then simply just juggling sybols, sure it's simple example but there the point being there are ways to take down the complexity of mathematics for kids by teaching them that multiple ways to think about these things conceptually outside of the strict bounds of what the are normally taught.
I have some of the same attitudes to some mathematics. But there are two strong forces against me.
First, the curriculum is crammed with gumpf. There is a small set of mathematical knowledge that I think is important for citizens of a developed democracy to know, stuff around finance and statistical reasoning mostly. But I could probably cover this in one semester in Year 10. And there is a small amount of foundational number knowledge that makes it possible to teach much of the rest of mathematics - times tables, an understanding of place value. Again if this were done carefully it could be done in about a semester - I'd prefer if it were done in primary school. But I have to spend an awful lot of my time teaching other stuff that is not in any sense necessary or useful, coordinate geometry, trigonometry, calculus, volumes of complex shapes, multi-variable algebra etc etc etc. Any one of these would be fun to go into in some depth but the necessity of covering them all means that none of them are covered properly and the connections between different areas of mathematics are totally obscured.
Secondly, my students all come to me with a history of mathematics classes. Mostly, this history teaches them that there is a right answer and they are too stupid to find it. They wait to be told, they attempt to memorise formulae and they lack curiosity about how things work. I make attempts to reverse this but when the rubber hits the road and I need to cover content quickly, I reinforce it despite my best intentions.
If someone wants to found a charter school where I can use Godel Escher Bach as my only maths textbook just tell me where - I'll catch the next plane.
- Just trying to survive until the nanobots make me immortal -
You've missed the point: it isn't mathematicians who've made it overly complicated. It is people responsible for /teaching/ math.
Mathematicians have made it exactly as complicated as it needs to be, no more and no less. But many textbook authors have taken that complicatedness and introduced it into areas where it isn't needed, out of a lack of understanding as to why it was needed in the first place.
An example from TFA: mathematicians prefer "|x-5|2" over "x is between 3 and 7" because the former generalizes naturally to arbitrary metric spaces (like R^n). But until somebody is ready to talk about distance in R^2, and circles and such, the latter should be preferred. It uses less notation, and requires less thought to really grok.
And our numbers come from the Arabic world whereby the number of lines represent the count, i.e. 1 - one stick 2 - two sticks (one curved) 3 - three sticks (one straight across the top) 4 - four sticks 5 - five sticks 6 - six sticks etc.
I didn't see anybody give their experience with the "new math" experiment that was done a while back. If you think about this from a little distance you can see that there are two camps in mathematics education, the "Creatives" and the "Pedantics", the "new math" was an attempt by the "Creatives" (who Lockhart is clearly a member of) to inject "thinking" and "creative thought" into the mathematics curriculum. It was a total bust, primarily because the teachers teaching it really didn't understand the intentions behind this new curriculum and they reduced it to rote. Those in favor of "back to basics" would be in the "Pedantics" camp and have been making a comeback recently.
So here in a nutshell is the two opposing camps arguments.
Creatives argument against the Pedantics -- The Pedantic curriculum is a soul destroying exercise in rote and memorization leaving no room for a child to feel any inspiration or creativity.
Pedantics argument against the Creatives -- The Creatives assume the world to be filled with inspired teachers that won't reduce any curriculum to a pedantic exercise. If the quality of teachers is such that they can only teach pedantic material, you might as well have the children learn something useful and constructive even if it is boring and soul destroying.
I am an ex-mathematician and I am firmly in the "Pedantics" camp. I hate to see children that cannot add two digit number to two digit numbers without a calculator. That is the world that well meaning "Creatives" create.
Also, is there really that strong a correlation between the percentage of students that pass standardized tests on calculus and the overall success of the community? Russia has a very strong educational system, see what that got them. The general population of the U.S. would be considered be woefully uneducated by the standards of many other countries. But if you were to take any country with as large an immigrant population, I suspect you would see similar numbers. Over time the immigrants are absorbed into the main stream and their children do better. But could it be possible that these immigrants are also the source of the vitality of the U.S. economy and their education (or lack of it) is not the primary reason for why they make this nation so successful?
"But who's glorifying lawyers and politicians?"
I think he means they have a semi respectable image that people want to aspire to, usually people in such positions have social skills and some kind of charm and social network they benefit from.
Scientists, engineers, mathematicians and philosophers tend to be disproportionately nerdy, they may be alright socially but they aren't that agressively social in their personalities and truth is culturally they are shown in a negative light. (scientists = glasses + white lab coats, hunching over an experiment, John nash in a beautiful mind = genius = mental illness, etc).
Lawyers have always had both respect and disrespect. From a parents perspective dealing with their kids going into law - they make good money (and there are good lawyers, think EFF, civil rights lawyers, etc, etc) and then their are the bad apples in the profession that give the law profession a bad name and become the butt of endless lawyer jokes.
Mr. Tiber, St. Joseph's High School, Kenosha WI â"ÂI'm sure he's retired by now.
I'm an artist and very visual so math really wasn't my bag. In most of my math-related classes all through high school I would usually average a "C." I was taught, reluctantly but it was required, Advanced Geometry by this man and received an "A" for the entire semester. This man taught math the way my better art teachers taught painting and drawing. He was very passionate about mathematics, I think the true problem is that most teachers have no passion for whatever subject it is they teach. They're substitute teachers that happen to teach a specific subject all the time.
I don't think I ever thanked him for teaching me so well. I don't even think I was smart enough back then to even realize what he had done.
:3
This unconventional work initially involves the creation of a revised multiplication in which the revised product of two negative factors equals a negative real number, contrary to conventional multiplication. This precludes the existence of the unit imaginary number and thus, the complex number system.
Takes mathematics back at least 300 years, if not a couple thousand, and prevents the development of most modern technology: useful alternating current circuits (complex impedance), GPS (relativity), solid-state memory (quantum mechanics), etc. Complex numbers made these advances possible, and you'd throw them out because "mathematicians have made them hard to understand." Fantastic.
In revised algebra, a binomial, linear equation to the nth (any) degree is solvable since after revised cross-multiplication, it is reducible to the original, first degree equation. In conventional algebra, a binomial, linear equation to the fifth degree or higher is generally impossible to derive solutions for.
For every n > 4 there are polynomials of degree n which are not solvable by radicals. If this person has discovered otherwise, I'd like to see a concrete example. Give me a root of f(x)=x^5-x-1 using only radicals. If you can, I know a few awards committees that would like to speak with you.
To the parent poster: you have a valid point that math should be taught with a stronger connection to the real world. I agree 100% with you.
However, real science is damned hard, and demands results, not complaints. If you've got a better way of thinking about math that will make it easier to do real science, by all means, let the scientific community know, they'll love you for it. But you aren't going to convince people by linking to quacks. When you come at the problem with "throw out complex numbers" and "the Mayans were way smart", you need to back that up with "here's why science is better without complex numbers -- because I can predict X" or "thinking about numbers using shapes has these advantages..." Otherwise, you have no credibility, you're just whinging.
This strikes as me as bricklayer who looks at the rock stars getting laid any saying "Hey! Bricklaying is an art too! Why don't I get the girls? Why don't people love what I do? I just want to be loved! IS THAT SO RAWWWNG?"
I read the entire paper, and I agree with most of the statements. However the central thesis -- people don't like pure math because it's not taught properly in school -- is a load of bull.
Given no prompting at all, people will draw pictures. It's fun. They will sing and create music. It's fun. A three year will do it, and good luck stopping them, as it seems to be built-in to humans.
They won't look at a box and wonder whether a triangle takes up half the area and then carefully ruminate upon the chain of pure deductive reasoning towards the clever orgasmic bliss of enlightenment. Otherwise, people would spontaneously get together and have math parties where they talk about hypercubes and whatever stuff mathematicians talk about. No, instead they get together, have a couple of drinks and listen to music.
I love pure math, despite being terrible at it. It's obviously a great thing, but not everyone's built that way. The kind of thinking that he likes is just not as common as he would like it to be.
I can explanate how to administrate your network. You must configurate and segmentate it, so it can computate.
It's worth noting that this type of creative math education would be extremely helpful in coding and development of all kinds. The process of writing a program to perform a task is very similar to devising a proper mental solution to a mathematical problem -- at least that's how I've always felt.
Lockhart's paper was referenced March 2008 on the monthly Mathematical Association of America (MAA) online column of Keith Devlin, here
it's a thought-provoking essay, by someone with the credentials to be taken seriously. Nice to see it finally being slashdotted :)
"You've missed the point: it isn't mathematicians who've made it overly complicated."
Actually it is, go read the authors of modern textbooks "So and so, PHD of mathematics university of so and so", I know because I've got a heap of textbooks written by these complainers that I've gone through and marked with notes on my own time doing my own research.
I wouldn't have posted my OP if I didn't have genuine beef with many in the establishment in the first place.
This is certainly a generalization. The Singapore mathematics curriculum is very problem driven and not at all rote memorization. Singapore regularly appears in the highest places in the TIMSS. Indeed a study of student performance on statistics questions showed that the Singaporean students did better than the US counterparts despite the fact that the US curriculum contains statistics as a topic whilst the Singaporean curriculum does not.
I don't buy the "pressure to succeed" arguments either. I think that it is the system that is failing the students and not the other way around.
Math has been gutted of meaning, but this is changing. There are solid curricula out there that are being used, such as IMP ( the Interactive Math Program) or PBL (Project/Problem Based Learning) style lessons. An example of PBL that I used last year with my 8th graders was in modeling a bride. They were given a plausible scenario (school buildings are getting a 2nd story added on to reduce the number of portable classrooms, they had to design and model a bridge between these 2nd stories.) So, we went out and measured distances, built newpaper bridges and tested how much weight they could hold to find relationships for thickness v. load and length v. load, calculated needed load support based on population, class flow, 8th grader mass, etc., graphed some data in Excel, and used their formula and data to built a cost-optimized bridge. They had fun exploring some rich problems (and some frustration, as it did require some thought) and gained a better grasp of linear relationships, a key concept in 8th grade.
This type of teaching isn't widespread, but it was being advocated by my college advisers. One of the problems with doing this kind of math is the lack of public support. In the school district I'm in, about half the high schools were giving an option to use IMP to students, but parents complained and such, and now only a few charter schools use it. Still, support is starting to spread some, so the more interesting approaches are being slowly revived.
For those interested in this topic, check out What's math got to do with it?" by Jo Boaler (new edition out later this month.)
As a high school math teacher (who has a BA in Math and MSEd in Educational Psych - by the way, Masters are NOT required for teaching HS, at least not in most states - and I did find the MSEd valuable in helping me learn how to teach math, probably more valuable than an additional degree in math, as at that level, I would not be using it in HS), I believe there is a lot of validity to the article and the comments made here. But I think it is important to recognize additional problems (which may be listed in some of the 400 comments I didn't have a chance to read)...
1) Though I would ideally love to teach the history and art of math, we are limited by time and curriculum. We are held slave to standardized tests that require we focus on required material in a limited amount of time. I do my best to incorporate history and passion, but sometimes it is lost due to other constraints, but not by lack of desire.
2) One of the absolute largest problems is that our grade level standards are absurd. While the top performing countries will focus on 10 or less standards per year (thus allowing them to go into more detail, understanding, passion/art/history and mastery and thereby requiring less review each subsequent year), there are several grade/math levels in the US that have 20 or more standards to be covered. What that usually results in is the need for LOTS of review each year because it is not resonable for us to expect students to really "get" something if they spend 9 days on a topic, where as their international counterparts are spending 2 times that on the same material. And the more we review, the less time we have for new learning. If we, as simple teachers, cogs in the machine, could narrow those standards down so that we could really internalize the material, the students would better learn, understand and maybe even love the subject. Unfortunately, we cannot make that decision on our own - we answer to many other people. Ideally, we could revamp the entire system starting with Kindergarten.
There are many other things to consider as well, but what we can do right now is NOT demoralize the teachers that are working their tails off to help their students learn and succeed and instead, find ways to bring this important educational deficit to the front of government minds. We do need change, and it has to start at the very beginning.
Well, sorry but I have to disagree with you here.
Personally, I am 16 and currently at High School in the UK. What you have described is exactly the same situation we have here.
Also, I have spent this week at a trial event for what is a section of the proposed new physics course for the whole of scotland. Easy, does not even begin to explain it. If you cannot, at my age, or even a few years prior to it understand that course then you deserve to spend your life cleaning bins I personally feel. The worst part is that it is for the new "Higher" course which is regarded as one of the highest levels that can be achieved in High School here. Maths is the same.
Maths at my School is taught as a series of completely useless garbage that has absolutely no relation to anything in the real world. I think the following conversation with another boy in my class last year about sums this up. This was during a lesson on Quadratic Formula:
Pupil: Miss, what exactly are we EVER going to use this for?
Teacher: To pass your exams!
Pupil: No, I mean like, actually use it when we get a job or something?
Teacher: You will need to pass your exams to get a job.
Very, very sad in my opinion. The only teacher Ive ever found to go against this was a previous English teacher of mines, fantastic guy. He realised that being the top english class, it would take a fraction of the time to actually learn the course than the time we had. So he would bassically spend a very small amount of time doing "lessons" and then the rest of period debating things, class discussions. REAL interesting topics where we weren't learning anything by the stats definition, but I certainly learned far more in that time than any other subjects in that school. Physics was similar but I think Physics teachers tend to be the most laid back, anti-disestablishmentarian type of teachers around.
I'd like to just spin off some sort of society where the "best practices" can be pursued, current government be damned.
Problem 1: You can't get a society of 1 person. Who's going with me?
Problem 2: Easiest way out is to pilot a ship into international waters and then ground it, Sealand-style, but nobody makes that sort of platform that *I* know of.
Problem 3: Internet connectivity is a must. Attracting the attention of one of those big fiber-optic-cable-laying companies will be nigh impossible for a micronation.
Problem 4: How do you make a sea platform expandable?
tl;dr: I want to fork society. Who's with me?
Note: I was 13 when I wrote most of this. Take with several grains of salt.
Cool. Let's see your elegant non-symbolic proof of the Fundamental Theorem of Galois Theory. That way I don't have to slog through several chapters of Hungerford's Algebra. Bring it on, man! Your audience awaits.
Now, that is a very inflammatory subject title, so let me explain what I mean.
I was glad to see a previous comment referencing John Taylor Gatto. I do not see Gatto's name in the PDF document. Neither do I see John Holt's name. The fact is, the purpose of "schooling" (which is not the same as "education", and you would expect a mathematician to be more precise in a use of terms) is precisely to do what the mathematician decries at the end: "And there you have it. A complete prescription for permanently disabling young minds-- a proven cure for curiosity. What have they done to mathematics! There is such breathtaking depth and heartbreaking beauty in this ancient art form. How ironic that people dismiss mathematics as the antithesis of creativity. They are missing out on an art form older than any book, more profound than any poem, and more abstract than any abstract. And it is school that has done this! What a sad endless cycle of innocent teachers inflicting damage upon innocent students. We could all be having so much more fun."
Education in the USA will not improve until people like this mathematician accept that what he said is the intentional purpose of schooling in all subjects for almost all children. See things like:
"The 7-Lesson Schoolteacher" by John Taylor Gatto, NYS Teacher of the Year
http://www.newciv.org/whole/schoolteacher.txt
or:
"The Big Crunch" by Dr. David Goodstein, Vice Provost Caltech
http://www.its.caltech.edu/~dg/crunch_art.html
or:
"Growing Without Schooling" about John Holt's work, including failed attempts to reform schools
http://www.holtgws.com/
At this point, it is people like Paul Lockhart who are the problem. People who think school is about education, when it is about socialization in a certain way intended for the most part to produce compliant workers, obedient soldiers, and mindless consumers. School is for fish. Curriculums are race tracks. And "class rooms" are literally to build social classes through selective breeding by genetics. Those are the origins of all those terms, at least according to Gatto, and, again, you would expect a mathematician to be precise about the origins and use of terminology.
With all that said, of course Paul Lockhart is right about how to improve mathematics education. But, it will never work within a Prussian-derived school system with no interest in truly educating children, despite every person who works at a school calling themselves an educator, and despite the truth that most of the people in schools might be fine educators if given the chance and a few years of untraining of their bad habits. ... Resistance"
"The Emergence of Compulsory Schooling and
http://web.archive.org/web/20071014123355/http://www.social-ecology.org/article.php?story=20031028151034651
Anyway, sorry to be so harsh on you, Paul. Read "Disciplined Minds" and start building a social network to help you and them and others break out of the prison around you:
"Disciplined Minds: A Critical Look at Salaried Professionals and the Soul-Battering System That Shapes Their Lives"
http://www.disciplined-minds.com/
The good news is, you have already taken the first step of getting out of the prison others have forced you to build for yourself.
A 21st century issue: the irony of technologies of abundance in the hands of those still thinking in terms of scarcity.
But he got to base 2...
A patriot must always be ready to defend his country against his government. -edward abbey
"Bring it on, man! Your audience awaits."
Note what you suggest would take a large chunk of ones lifetime to put together, by restructuring the theory and mapping them onto observations from the real world.
Consider:
The birth of Galois theory was originally motivated by the following question, whose answer is known as the Abel-Ruffini theorem.
"Why is there no formula for the roots of a fifth (or higher) degree polynomial equation in terms of the coefficients of the polynomial, using only the usual algebraic operations (addition, subtraction, multiplication, division) and application of radicals (square roots, cube roots, etc)?"
Now I would force the original author to translate his equations into real world objects and manipulations, no more abstract jargon, this *exactly* what I'm talking about, where people take jargon so far away from observations in the world and *de map them* to such an extent they've gutted the original sources persons from whom they are derived.
Most people don't keep a perfect record of their insights and intuitions and the ideas that lead them to systems of equations, you'd see a lot of what looked like 'gobbleygook' to you because unless you have direct access to that persons mind *you simply can't grasp the relationships* and intuitions they have been building up for an enormous amount of time.
I am only one man and I would *happily* take your challenge. There is only so much time and mathematics is an enormous discipline.
I think the pdf was quite good, albeit repetitive. In any case, in the end, I think he makes an interesting point about abstraction:
On the surface this seems fairly innocuous; why not make some abbreviations so that things can be said more economically? The problem is that definitions matter. They come from aesthetic decisions about what distinctions you as an artist consider important. And they are problem-generated. To make a definition is to highlight and call attention to a feature or structural property.
Learning to make good abstraction when programming is a difficult challenge, the right choice can have huge downstream consequences. An educational system which allowed students to get comfortable with the exploration of different abstractions, as well as, the forestalling of notation and rigour until the right rigour was appropriate would be a huge win to reasoning across the board.
Damn you! Damn you to hell! You made me like math again. Now I have to get Mathematica out again to get funny looks just like in high school.
I always knew that I did poorly in classes that required formal proofs, ranging from Geometry to Analysis of Algorithms. I always had problems because I could not tell what was "obvious" and what was not. As an intuitive thinker, I'd either not be "rigorous enough", or, after being told I wasn't rigorous enough, I'd try to compensate and end up trying to prove "obvious" things. (They were somehow defined as "obvious", but I could certainly see lots of nooks and crannies in them that could contain problems.)
And I still cringe when someone says they "hate theory". I love theory, because it's theory that actually illuminates things. But most people have all kinds of unpleasant experiences with "theory classes" taught by people who do not understand the subject matter. The result is that the teaching is brittle: if you stray away from the teacher's guide in your question, you are herded back onto the straight-n-narrow path with confusing hand-waving and hurled jargon. It makes no sense to you, but the teacher says it with authority and you assume that you're too dumb to understand it, and eventually come to hate "theory classes".
This starts at early ages with math education. And it might be called "the hard place". Opposite this hard place is the rock of boring, rote repetition.
Some of us manage to get through this relatively intact. I guess we have a strong attraction to underlying explanations ("theory") and enough school-smarts that we get good grades, encouraging us that perhaps we are smart and what we don't understand is in fact understandable if we apply ourselves.
The trick is how to balance the ideal math education with the abilities/training of the huge number of teachers required to teach it. (Who have themselves been warped by their own math education.) And to balance the need for rote things (multiplication tables come to mind) with curiosity and enticement to learn.
Your cousin took linear algebra in high school?
The U.S. is being outclassed in mathematics because the average IQ of Americans
is lower. Period. End of story.
That's not even remotely true. Most major metropolitan school districts in my state (California) cater mostly to the mediocre, and offer advanced classes to some kids who are better at following instructions, and minimal training to problem kids. The control ought to be more fine-grained than this. The problem kids, for example, are all just lumped together, even if no rational person would think that prudent. For example, at one school I know of, the mentally disabled kids are placed with kids who are just too far behind other kids of the same age, and both groups are blended with kids who are just disciplinary issues, with no ability on the part of the teacher to deal with each situation in an appropriate way. Thanks, standardized testing!
News Flash: Godzilla hates infrastructure.
Funny you should say that here, you are aware that many significant results and definitions in computer science are based on unary arithmetic? If you want to know more I strongly suggest looking at definitions of computability, turing completeness, recursion/computation, the halting problem, etc.
Now I would force the original author to translate his equations into real world objects and manipulations, no more abstract jargon, this *exactly* what I'm talking about, where people take jargon so far away from observations in the world and *de map them* to such an extent they've gutted the original sources persons from whom they are derived.
And, I put to you that MOST mathematics cannot be translated into real world objects and manipulations. What concrete objects would be used in a discussion of infinite dimensional vector spaces, the Banach-Tarski paradox, the Radon-Nikodym Theorem, or transfinite cardinal numbers?
i.e. second base...oh forget it.
I went to a school about your size, just more recently. My graduating class was somewhere in the mid 700's. There was more than 2 levels of math and science. Granted there was only two with the same name, AP calc 2 and calc 2, or AP calc 1 and calc 1 etc. but not everyone even made it to calc1. By 12th grade students were spread out into any of the following math classes: Algebra 2, Algebra 2 AP, Pre-Calculus, Calc 1, AP Calc 1, Calc 2, or AP Calc2. Thats 7 different "senior" levels of math in a single highschool. If you're goal is just to graduate then Algebra 2 is where you finished. Depending on your academic ability and how far you really wanted to go you could be as much as 7 "levels" ahead of that by the time graduation rolled around. Thats not to say that you'd have to take the other 6 to make it to AP Calc2. The highest "path" you could take would look something like Alg 2AP -> Calc1 AP -> Calc2AP. But in order to do that you'd need to start Alg 1 in 7th grade (Alg1 in middle school was split into two years taking 7th and 8th grade to complete allowing you to go straight into geometry/trig your freshmen year in highschool, again this was optional and a "level" above what most 7th and 8th graders take, as well as 8th graders who took the first part of alg1 in order to breeze through it freshmen year). There was also summer school if you wanted to get ahead although that option wasn't very popular. Most kids I knew also jumped around different levels taking Alg 2 and Calc1 before taking AP Calc2 or say they took AP Calc 1 but not AP Calc2. It was surprisingly close to college where you're just told to take classes in these categories and this is how many years of these classes you had to take and certain classes require other classes to already have been taken or to be taken at the same time. Science and Math crossed paths like that a lot.
Science was pretty much the same but you could further specialize ending in a physics or chemistry path or a more basic level of both if you wanted and basic levels of each were required before graduation.
English had less options but still ended up with 4 different levels you could end up taking your senior year, one of those being a "college" level course taught by a professor that drove in from a community college 3 days a week. We didn't have anything called "social studies" past elementary. We had geography, poly sci-ish classes (not called that but the name escapes me and the subject was basically the same), history etc. They each had 3-4 levels you could end up with your senior year as well and again you could sort of 'specialize' in your favorite.
This is ~5 years ago in a public school in Texas. The district currently has 6 or 7 highschools all about the same size with the same curriculum. I think its 7 now, they keep building more and I dont live in the area anymore and my parents rarely talk to me about it. Also just like to point out that you calling into question someone elses experience, given he did say "most", as atypical is kind of funny seeing as how you only really have on view point to look from as well and it is admittedly ~20 years old if not closer to 30.
Moral of the story is things change, and public education across the US varies WILDLY in terms of quality and choices available.
Also just remembered that for each of those AP classes there was a "Pre-AP" version. So make that 3 sublevels for each level.
Because:
A. The US has only about 5% of the world's population; thus we have only about 5% of all math whizzes.
B. Compared to other fields in a given nation, math pays poorly in the US.
Those who keep blaming it on our education system ignore these facts.
Table-ized A.I.
... at which point the authour appears to give science the same deconstructionist treatment he has been arguing against for mathematics???
Qoute TFA (my emph.)- "Likewise, if your science teacher tried to convince you that astronomy is about predicting a person's future based on their date of birth, you would know she was crazy-- science has seeped into the culture to such an extent that almost everyone knows about atoms and galaxies and laws of nature. But if your math teacher gives you the impression, either expressly or by default, that mathematics is about formulas and definitions and memorizing algorithms, who will set you straight?"
Modern science is the communal pursuit of truth and beauty via observation, imagination and critical thinking.
And did you exchange a walk on part in the war for a lead role in a cage? - Pink Floyd.
The fact that you had to post such a reply to someone whose opening sentence was, "I'm excellent with logic" is ironic. I point this out because so many Slashdotters are irony challenged. This example is priceless humor. It's hard to make this stuff up.
Superintendent. A perfect buffoon. If you dont believe me, listen to him whine and cry like a girl, on the radio and on TV, as he "earns" 300K per year and has 5 assistants making 100K EACH per year - DOING NOTHING. Not one damn thing, to further our city's education. Meanwhile, he wonders where the good teachers are? Yea, they want to earn 25K and put up with brats who curse them and hit them, and when the teachers complain, they get fired. Sounds like a better job than 20K a year at Borders to me! Not.
Math, my friends, is NOT the only problem with the US education. Look deep within the system. Is so far gone it needs to be replaced - top to bottom.
Ugh...
I defy you to rigorously prove -any- statement without the use of the "symbolic format" of mathematics. The great power of mathematics is to turn unclear intuitions into clean, precise symbolic statements, and to -understand- why these statements correspond to intuition. If you believe you can greatly simplify the notations currently used to do mathematics, then i believe you would be a runner up for the next field medal award.
Er.. the "symbol manipulation rules", _are_ plug and chug. Aka, recognize the specific cases shown in your text book, and whenever one of those cases comes up, apply the rule.
Geometry is worse than plug and chug, for the reasons mentioned in the article. And it's not primarily intuitive thought that is used, instead students get to memorize the rule list, and get trained to prove certain things a certain way, and these things they are "proving" are mind-numbingly obvious.
Also, factoring polynomials is plug and chug. In algebra classes they actually provide formulas for doing it: There's a formula for factoring ax^2 + bx + c. And a rote procedure students are trained to follow to reduce the polynomial to multiples of trinomials of that form, which involves [a] finding the GCF of the polynomial, [b] applying grouping, etc.. . there is a list of tasks students are trained to go down and deal with in a certain order.
And when they implement all the tasks in the list to completion, the result is factored.
I've done some rather math-heavy programming. I'm one of the people who made ragdoll physics work, a painful exercise in geometry, differential equations, and error control. (If you're not real serious about the error control, your ragdolls will fly apart or launch themselves into space for no reason visible to the end user. This gets you nasty writeups in game magazines.)
I've also done proof of correctness work, using and working on automatic theorem provers. And I've done some work on sensor fusion for inertial navigation systems.
Despite this, I've never had to do a classic high-school type geometric proof since high school. High school geometry is taught that way because Euclid taught it that way two millennia ago. (A century ago, schools were still using Euclid's Elements as a textbook.) It's only taught because it's locked into college entrance exams like the SAT.
If you want to teach mathematical reasoning, that's fine. But there's no reason to teach it in the geometric domain. It's a skill that's used very, very seldom.
The other big problem with the teaching of mathematics is the emphasis on "puzzles". That's all wrong. Mathematics is a tool for design and analysis of things you might want to build or understand. It should be taught that way. In particular, high school calculus and high school physics should be integrated. Teach calculus as a way to understand mechanical systems and electrical circuits, and it makes much more sense.
I have (somewhere) a U.S. Navy textbook from WWII which teaches calculus from exactly that standpoint. During WWII, the Navy needed engineering technicians in a hurry, and they set up a crash training program without much input from the "educational establishment".
Worse yet, the AP/Honors courses are not only at a higher level, but for some baffling reason most would assign an order of magnitude more work. While I could've handled the material, I simply didn't want the increased workload and so avoided AP classes like the plague. The threat alone of so much more homework was enough to make me avoid AP classes like the plague. Then, using the handy loophole, my non-AP-enrolled self took and passed 3 AP tests for the college credit.
-- I prefer the term "karma escort."
Several posters touched on other good points, but I'd like to point out that those "better performing" countries, especially in Asia, do not share our "total graduation" philosophy. If a kid doesn't do well in school, he is encouraged to leave. This significantly bumps up their test scores, making comparison between those countries and the US a false one. It's why their scores are always better than us -- but that makes those scores useful to the academic sector, since they can always try to argue for more money. (Which can then be sucked into school administration's pockets. If they do it fast enough, teachers and students will never see a dime's worth of improvement!)
On the first day of class, my college history teacher asked for a show of hands on how many had been in a high school history class taught by "Coach What's-His-Name". Almost all the hands went up. The situation is quite similar for high school math. I had one HS math teacher who wasn't a coach, and she was even worse than the ones that were.
-- I prefer the term "karma escort."
On the other hand, he wrote an essay/article based upon his reasoned judgments about subject he was taught poorly by people
that didn't know anything about it while claiming that this education provided no means of making the argument he just made.
If you only knew what really happens in the classrooms today.....
Yeah but the rote learning style only produces robots, not critical thinkers, decision makers and game changers. I agree though that North America is producing neither the former or the latter.
"In our tactical decisions, we are operating contrary to our strategic interest."
fourth!
It's more because we in Asia (I can talk about India, at least) still have teachers who can take notice of the kid who's questioning that rote learning and guide him/her further. Unfortunately, that's being compromised too
American pupils feel the pressure to make money, which is done best by the American model through business or law schools. To get a business degree or MBA, one needs very basic algebra, even in 'good' business programs. We have a problem of culture obsessed with making money first, ahead of many other values including being creative. The MBAs and lawyers then go an run the country and promote their lifestyle: "I made it well without math or sciences, others can, too." Not sure when/how it will change, but this model puts us at great risk in a democracy: if the majority don't create any value, making money in itself is not creating value, then they'll run us into the ground. This was the very reason public schools were created; forgot what president in the 19th century figured this out. Pretty sad story we are living.
The author of the paper is a math nerd who can actually write and communicate ideas; that's an oddity in itself.
As a working mathematician I had a great deal of sympathy for many things Lockhardt had to say. In particular he couldn't be more right about the total uselessness of most of the math curriculum to most students. Go ask a working professional (doctor, lawyer, etc..) to solve a system of linear equations in 2 unknowns and it's immediately apparent they got no direct practical benefit from their math classes.
I quibble with his ragging on epsilon-delta and other precise definitions. I finally realized math was elegant and exciting precisely because I was so disgusted with (ugly) intuitive arguments about smoothness I went and found a book that taught me the elegant formal definitions that made calculus all fit together. Not that I would recommend this for everyone but I personally find it one of the most aesthetically aspects of analysis.
-----
However, where he really totally blows it is when he assumes that math can be a fun exploratory intellectual adventure for everyone. Yes, virtually everyone has the innate intelligence to do this but no matter what you do math is going to make some people feel dumb and frustrated. There are right and wrong answers in math and not everyone can be above average.
Sure, everyone might be lackadaisical in HS art class but that's because few (no?) people's future depends on their ability to do well in the class. On the other hand the best and the brightest signal their ability by performing well in math. Sure, these students succeed because they are curious and interested but all the other students will struggle to look like the mathematically advanced kids and those who fail will feel bad about themselves for it. No matter how you teach you can't eliminate the economic pressure on the students to appear as if they are good at math.
People don't like doing things that make them feel stupid or frustrated and learning real math requires genuine curiosity and thought. You just can't force people who resent the subject to think.
Perhaps we should simply accept that math is going to be like literature or art. A small percent will have the desire and interest to pursue it in highschool and we should just try to avoid turning off the rest enough they might return in their own time.
If you liked this thought maybe you would find my blog nice too:
Isn't this fairly old? I know I have seen it before.
The public schools I attended in the US were set up very similarly to the way you describe European schools - sections based on performance levels.
Incidentally, they stopped this about two years after I graduated high school.
I come from a small rural town in western Maryland. Redneck city. Many of the kids I graduated with left town, went to college are all over the world doing all kinds of things. I currently design airborne hyperspectral sensors for instance.
The kids afterward? Much lower college attendance. Many of them still living with their parents. A lot of unemployment in that group.
Oh well.
I am very small, utmostly microscopic.
He said:
"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."
"Math is a very rich subject which unfortunately has a lot of cultish like people"
You said: ... that's... insane.
>> so you want to teach math using base-1
Boy, what a way to miss and prove his point.
-dZ.
Carol vs. Ghost
If you think about it a little bit more, you'll realize that it's the other way around. Throughout centuries we've learned to deal with the various patterns and concepts in mathematics by creating shorthand symbols and methods that expressed them. It is therefore no wonder when people educated in such mechanims create computer languages similar in expression and syntax.
A "function" in Perl (or C) is essentially the same as one in Algebra, compare:
sub foo(bar) {
return (2 * bar) / 10;
}
2x
f(x) = ----
10
A set of parameters are transformed into an answer directly related to them. The concepts, terms and syntax were borrowed liberally because they were already there and it was convenient. This is not an accident.
-dZ.
Carol vs. Ghost
If I were going to change this sequence, I would drop geometry before I added anything. I loved geometry, for all the reasons that the essay identifies: an opportunity to create something beautiful out of ideas alone. Frustrated when the teacher asked, "How would all this change if we were doing it on the surface of a sphere, instead of on a plane?" and then didn't pursue it. But it's simply boring as hell for most people, and at least from my anecdotal experience, chases more kids out of applied math than anything else in the curriculum.
As a quick test, would your mother have been bragging to her friends if you had gone to law school? Would she have been thrilled if you married a lawyer? If yes, then we value / respect lawyers. Thankfully, some value engineering and science, too (thanks, Mom!)
>> The author wants to a priori assume that everyone will love math if only the beauty of it is shown to them. This is mistaken.
Mistaken? Perhaps. But his point is that the student should be exposed to its beauty to at least have a chance of it sparking his interest. He contends, and quite eloquently in my opinion, that the current curriculum precludes this beauty and prevents the student from even considering the notion that there could ever be beauty and elegance, or any interesting things at all, in Mathematics.
-dZ.
Carol vs. Ghost
Because all your *other* base are belong to US!
Any sufficiently advanced intelligence is indistinguishable from stupidity.
Math may be an art, in the same sense that programming is art, but it's not an art form. I like Wikipedia's definition of art: "Art is the process or product of deliberately arranging elements in a way that appeals to the senses or emotions." In this sense, programming is not an art, but computer games are. Yes, the work 'art' has other meanings, some of which apply to math and programming, but it's not the same meaning applied to music or painting.
That's why to me this article doesn't make sense or propose a real solution. Math is no different than history or science, or, for that matter, literature. They are all taught mainly as a collection of facts, with just glimpses of the way these are arrived at. Putting math on a different pedestal IMO makes it more difficult to reach a better solution to teaching it, rather than seeing where the problems of teaching lie.
Frankly, everything that's taught in school is boring. You need a good teacher who really likes the subject to make it feel interesting. Even if he or she teaches the exact same material, it'd feel more interesting. That's in my experience, at least.
Read "The Seven Lesson Schoolteacher" by John Taylor Gatto. Gatto is an award-winning schoolteacher in New York State, and he takes much the same anaylsis as Lockhart does for math to the entire industry of education. Only Gatto wrote his piece in 1992. http://www.newciv.org/whole/schoolteacher.txt
... is whot bwings os tugevza tsuzay.
The author doesn't claim that the point is to get everyone to love it. The author claims that the point is to get everyone to realize what mathematics really is.
From TFA:
SALVIATI: If everyone were exposed to mathematics in its natural state, with all the challenging fun and surprises that that entails, I think we would see a dramatic change both in the attitude of students toward mathematics, and in our conception of what it means to be "good at math." We are losing so many potentially gifted mathematicians -- creative, intelligent people who rightly reject what appears to be a meaningless and sterile subject. They are simply too smart to waste their time on such piffle.
SIMPLICIO: But don't you think that if math class were made more like art class that a lot of kids just wouldn't learn anything?
SALVIATI: They're not learning anything now! Better to not have math classes at all than to do what is currently being done. At least some people might have a chance to discover something beautiful on their own.
SIMPLICIO: So you would remove mathematics from the school curriculum?
SALVIATI: The mathematics has already been removed! The only question is what to do with the vapid, hollow shell that remains. Of course I would prefer to replace it with an active and joyful engagement with mathematical ideas.
... is whot bwings os tugevza tsuzay.
Nope, he's directly addressing both of your goals. He claims that the current system fails at both. Rote memorization and robotic application of cryptic rules doesn't stick with people. So we'll give someone 12 years of math education, but after a few years in the real world they'll only remember maybe the first 5. So what was the point of all that time? Secondly, because math is presented so poorly, many people who might love math are turned off. Worse, some people who enjoy and are good at rote memorization and robotic application might erroneously think they want to be math majors, only to discover that upper level math is a very different beast. As he notes, it's like teaching students musical notation without actually listening to music as a way to discover who might have a love of music. He believes that his proposal may cover less material, but students will actually understand it better and will retain it and will be better able to learn whatever else they need to know. And because they're learning "real" math, the people with a predisposition to loving math have a better chance to discover it.
Search 2010 Gen Con events
Read about it, it states (with clinical research going back to early 60's), that math is best learned before 3 years old. So all you old morons can't benefit but for the few of you that will reproduce, you can make your children into geniuses.
That's not true if you exclude Hispanics and Blacks. Exclude Hispanics and Blacks, and US White/Asian kids do the same or better as world leaders. As someone who's taught in barrio schools, most Hispanic kids (and I assume Blacks) have severe problems with the US educational system.
1. Single Motherhood -- this is the big one. Single motherhood is rampant in both these groups, increasing sadly in the White population, and it leads to low resources particularly time for parents, and kids acting out. Boys being hypermasculine or withdrawn, and girls being generally too interested in boys and not enough in school around puberty.
2. Negative cultural attitudes towards education, which is perceived as "White" and a "sell-out" to racial/ethnic values.
3. Lower median IQ. It's not very popular, but it is true, that every study ever done, shows consistently a lower median IQ for Blacks (around 85) than Whites (around 100) and Asians coming in around 105. Hispanics come in around 89. This does NOT mean all Blacks and Hispanics have low IQ, but enough of them do that they drag down instruction to their level, making most classes slow, boring and remedial. A huge frustration for the few bright kids, and really a huge drag on the US educational system.
Let's get real. Mostly White/Asian schools do well because the kids are mostly orderly, behaved, come from two-parent families where the parents are involved and care about the kids, and are committed to Education as a gateway to upward mobility. Mostly Black and Hispanic schools are pits, because the students don't care, gangs are rampant, their parent is a single mother (often chasing the neighborhood thugs), nobody cares, and no one views Education as a gateway to upward mobility. In Ghetto/Barrio schools, the only upward path is through athletics. Which is the only effort kids put forward (particularly for boys, being seen as a "schoolboy" equates to being less masculine, and unworthy of girls). White/Asian boys face this too, but it's not as strong. [Girls face little penalty for being "smart" -- they are either hot or not, it's purely physical.]
Shrug.
There have been a FEW schools that have done well with Black/Hispanic kids, they are outside the Public Schools and screen out troublemakers, and emphasize fairly strict discipline, rote memorization, and group identity. They are not replicable because they require highly dedicated and charismatic leaders, and are not "community friendly" i.e. featherbedding for local political machines, and swayed by parents demands. They don't allow gang members for example, require uniforms, and Marine Corps like group indoctrination/spirit. Few communities would stand still for that -- they want niceness rather than results. [Black/Hispanic kids probably can do far better, but it is not politically possible to construct strict discipline schools they need to perform at higher levels, which requires walling off in effect the dysfunctional outside community.]
But exclude Black/Hispanic schools and US White/Asian kids do in the top of nearly all categories, it's just impolitic and UN-PC to say it. Even though it is in fact true. This should shock no one. By and large, the society in Black/Hispanic communities is dysfunctional and a complete failure. In White/Asian communities, mostly successful. It would be shocking in fact if their schools did not reflect the failure and success of their wider communities.
I am unsure about the size of my school (freshman class of 500, senior class of 300, 1800 students total) compared to others, but even though we had three different levels of math at my school, some students still did not do well (there own failings) but others couldn't do well enough (there was nothing offered after AP Calc AB - which I finished junior year technically two years ahead of people I was told were "just as smart as I was", and AP Calc BC was not available at my school).
I tend to think that the problem is above individual schools - at the school boards, NEA, and state and federal testing requirements. The school administrators are being told to run them as cheap diploma mills (meaning no classes less than a specific size no matter what - say goodbye to advanced classes) where no students fail a grade, good teachers aren't rewarded any more or any better than bad teachers (meaning not even the good teachers will bother to help or challenge advanced students that are obviously interested), and yet every student that graduates has to just barely pass a standardized test developed by people who are completely divorced from the concept of teaching or learning (meaning that the focus of the school isn't on promoting the best students to see as far as they can go, the focus is on babying the lazy students - they were very smart about things they were interested in, so they weren't dumb by any stretch of the imagination - so they simply don't fail the test outright).
But above all, I think the blame lies squarely on the shoulders of the parents of students. They were not interested enough in the future of their kids to care about their kids' intellect or education. The saddest part is that their grandkids will turn out the exact same way, and the cycle continues.
And there isn't anything you, nor I, nor teachers, nor the government can do to stop the endless cycle. Another brick in the wall, indeed.
I'm a math teacher and what he's saying is completely true. Feedback from a student, like this, is critical to teachers. I completely agree with his ideas about internalizing math and bringing out the artistic side in people. More importantly, he gives us a clue to what and how the learner experiences modern math in a real-world context. I feel his frustration, as do many other teachers, with the standard curriculum of math and the teaching methodologies which have become common in schools. Why can't we paint in painting class? This has the connotation that students are asking: Why can't we calculate in math class? But it's true! My students are always asking why we do things this way or that, why we don't just do math like the world suggests... Instead, we are trapped inside of a regimental rigmarole of successive problem-solving skills which are never expressed in the real world until it's too late! Tomorrow dawns a new era for mathematics teaching(at least at my school).
I have a M. Sc. in math from a highly respected university. Unfortunately, it is damn hard getting a job which really uses my knowledge. Sometimes, it has been a determent. I have been disqualified from jobs because I would be bored after a while. Granted, this is true, but after you have been out of work for a while, you will take a job doing most things.
So, what do you do with a degree in math, especially in a down economy. Maybe go back and get some training as a auto mechanic? Yes, there will be times when you are not very busy, but on the other hand, you will never be completely unemployed.
The other problem is that there is no real place for someone without a Ph. D. There probably are lots of Ph. D. out there looking for work tutoring, or doing things using considerably less than all that they know.
So, what's the sense of really getting people all inspired and fired up with real math, if all you are doing is setting them up for a lifetime of disappointments? Tell them up front that most mathematicians aren't going to amount to a hill of beans, and that is is time to rethink their career choices. I am convinced that mathematicians talk young bright people into studying math so that they will have classrooms full of people in order to justify their career choices, without which them might be in a position to look for alternative choices like a short order chef or rest room attendant.
Now, I wish that life were different. I wish that every educated mathematician would have plenty of career choices in front of him. Actually, I wish that on all students, but the obvious truth of the matter is that society would rather spend money on rescuing failed banks and auto manufactures developing obsoleted products than using the same money to stimulate the "arts and sciences". I have read that the Soviet Union would do that, but I guess they have failed. Too bad that we haven't realized a capitalistic society when in comes to rescuing billionaires who have made bad bets.
I'm all for "math is cool," and "let's explore," but this guy seriously torques me off. My kids are currently suffering from the influence of people like him. They have taken the basics out of math and substituted this useless "math explorations" curriculum. Other folks have written better criticisms, but suffice it to say that the vast majority of kids don't benefit from all this "exploration" and "visualization" and the ones who do would have had those epiphanies anyway without any help whatsoever.
The BETTER way is to stick to the basics and train teachers to recognize kids who have a mathematically artistic talent and then remove them to an environment where it can flourish. That's tough for a couple of reasons. First, those kids might not actually get good grades. The author of TFA is entirely correct that the basics bore them which results in inattention and lack of motivation. Second, when they ARE good, removing them will lower the overall test scores of the class. Since teachers' pay and bonus structures are based on their students' test scores, there would be a strong monetary motivation to intentionally fail to recognize them.
Assuming that those two problems can be overcome (big assumption there), you continue to train the "artistic" kids in the basics, but only just enough to get by. The rest of the time, you motivate them in a way that would make the author of TFA happy.
The problem is, people have this wonderful but sadly mistaken belief that ALL kids can benefit from artistic mathematics when in fact most can't. Compounding the problem is the bizarre theory that teaching the artistic mathematics will somehow magically result in the basics becoming trivially easy. It doesn't. And unfortunately, our kids have to fail spectacularly in order to teach the education system this simple fact. "Luckily," that's what they're doing in droves.
Try doing long divison in Roman numerals.
Consistent cardinality and scale of symbolic digits is a major step forward in maths.
I didn't see in the comments, and the story submitter doesn't mention, that this essay, which is from 2002, has blossomed recently (April, 2009) into a book.
No wonder they are among the world's biggest opium producers...
I know tobacco is bad for you, so I smoke weed with crack.
I just got around to reading the whole 25-page PDF today, so I see the author of TFA is using a different meaning of "plug and chug" than I was. You see the importance of agreeing on definitions ahead of time.
If a student insisted on calling right angle "pigpens", I think I'd tolerate it for about 5 minutes unless he was the second coming of Ramanujan. :)
...following the principles of Heisenburger's Uncertain Cat...
I gotta agree, calling right angles "pigpens" is nuts. I would find "corner" acceptable, unless they're doing something like repeatedly calling right triangles, "a triangle with a corner".
If a word has a generally agreed upon definition, then the student shouldn't define it to be something else.
Also, if a generally agreed upon word to refer to something exists, the student must pick one of them, or provide a very good reason for using a different word. Vocabulary is just as important in the study of mathematics as it is in English class.
The field of Mathematics isn't just about solving problems, it's also about effectively and accurately communicating the ingenious solutions to problems that you find.
An understanding of common definitions used by mathematicians is fundamental, and students should learn to speak the same language, even if their dialect varies a bit from problem to problem.
Oh come on mods! Clearly there are 10 responses to this joke: either you understand it or you don't.
Submitter's claim: "I defy you to read and find a single sentence that isn't permeated, suffused, soaked, and encrusted with truth."
Well, Scott Aaronson, your defiance is unsuccessful. I found one. Bottom of page 8:
"Here is a simple and elegant question, and it requires no effort to be made appealing."
I didn't find the question appealing, so it WOULD require some effort to make it so. The sentence is a claim based on the subjective experience of the reader and can't be truth.
So yeah, I'm being a pedantic ass, but Scott, DON'T BE A LYING SENSATIONALIST. Next time just say you agree with the entire essay and you think everyone else will too, which is what you tried to say, but FAILED due to wanting to sound cool.
http://www.maa.org/devlin/devlin_03_08.html
"Paul is a mathematics teacher at Saint Ann's School in Brooklyn, New York... "
and
'After several years teaching university mathematics, Paul eventually tired of it and decided he wanted to get back to teaching children. He secured a position at Saint Ann's School, where he says "I have happily been subversively teaching mathematics (the real thing) since 2000."
'He teaches all grade levels at Saint Ann's (K-12), and says he is especially interested in bringing a mathematician's point of view to very young children. "I want them to understand that there is a playground in their minds and that that is where mathematics happens. So far I have met with tremendous enthusiasm among the parents and kids, less so among the mid-level administrators," he wrote in an email to me.'
That in order to avoid plug-and-chug we develop the corresponding mathematical theory to solve an equation or make an integral?
Once you are done and dusted with the theory behind a concept then you proceed to apply the concept as needed.
IANAL but write like a drunk one.