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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.
Evidently, someone didn't do the server math.
Have gnu, will travel.
The problems with K-12 education go WAY BEYOND mathematics.
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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.
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.
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.
I implore you to read his essay with every atom of my being.
Well, OK, seeing as I can use *your* atoms.
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
Bingo, and that's one of the big problems with trying to do anything about the issues the paper raises: there are only so many people with the 1) ability, 2) knowledge, and 3) inclination, to do the kind of real mathematics he's talking about.
We'd have to re-vamp our teacher training along the lines of what's talked about in the paper to try to increase the number of people who could do it, and hope Lockhart's right about this being an art with universal appeal so that enough of the teacher candidates "get" it. Even if elementary schools began using dedicated math teachers (some already do, but many don't) we'd still need a shitload of people trained in this "math as an art/math as play" style, and we currently have approximately zero in elementary education.
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 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.
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.
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.
I don't think that it has anything to do with the femaleness of the teachers, as I have also had excellent female math teachers. I think what grandparent is referring to is the fact that elementary education(and to a lesser extent middle school) draws heavily from the "Good with/likes kids" segment(which, among others, includes a lot of mathematically disinclined women) rather than the "strong knowledge of subject x" segment. This substantially abates at the high school level, and is largely absent in college.
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.
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!
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
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).
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.
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."
And you get around the economic obstacles by subverting the system: Crowdsource the textbook to a group of interested mathematicians. Publish it online for free, with printed copies available for a price far below what a crooked textbook publisher would charge. Add value by posting demonstrations by mathematicians, math historians, and math professors on YouTube, linked to the relevant chapter of this comprehensive, global mathematics resource.
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Bingo, and that's one of the big problems with trying to do anything about the issues the paper raises: there are only so many people with the 1) ability, 2) knowledge, and 3) inclination, to do the kind of real mathematics he's talking about.
And not just the teacher training. This goes beyond what some students are capable of and can handle. What happens to them if they can't function inside a creative mathematical atmosphere?
"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
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
... 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/
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.
1) Yes NCLB is an excuse to close down public schools - it was designed as such, and they intentionally fabricated a study [exposed as a fraud 2 years later] to get congressional support
2) Defunding them further with vouchers (most of which would be going to religious indoctrination centers that masquarade as schools) is not a solution
3) "Failing schools based on geopgrahy" is a problem with two things
a) How we fund schools [how about pool money state wide and dole out as needed instead of tying funding to their service areas land values.. that kidn of funding arraignment was obviously designed to serve only the rich neighbors]
b) home lives in disadvantaged areas are more often than not are harmful to getting an education.
The NEA would CORRECTLY resist #2. They would support repleaing NCLB and getting all schools funded better.
Vouchers are not a solution, they're just a furtherance of stripping funding from public schools so that they fail.
If you cannot keep politics out of your moderation remove yourself from the Mod Lottery.. NOW!
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).
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.
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
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.
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.
Wow, you didn't read the paper very closely if you think he could produce a textbook (or a series) to implement what he's advocating.
Yo dawg, I heard you like the Ackermann function, so OH GOD OH GOD OH GOD
But he got to base 2...
A patriot must always be ready to defend his country against his government. -edward abbey
I strongly disagree with the Adlerian precept. It is especially dangerous in science because knowledge is continually being added to the subject and long-hidden connections are continually being discovered. As a VERY relevant analogy, take complex analysis (square root of -1 and all that follows therefrom :P). If you read any books from the childhood days of this subject, they will seem incredibly complicated.
:). While this may seem a bit unromantic of me, I simply believe that the content and readability of scientific books is way more important than anything else.
:P).
While I appreciate where you're going with your statement: "the way the discoverer came to understand a principle is often more important to grasp than the principle itself", the hope that this will be clear by reading an original work is just too much to ask for. Science historians labor for years to try to grasp some of those original thought processes. I personally find it much more fruitful to read these histories or a good modern textbook with a historical bent (An imaginary tale and Dr. Euler's fabulous formula - while not textbooks - are excellent examples of this species) to obtain some understanding of how the scientist actually thought of doing this. It just doesn't seem like a good use of one's time to wade through obsolete jargon and obscure (and nearly always annoying) notation just for that one spark of inspiring genius, which can be found readily in modern treatments because modern authors usually worship these ancient masters and provide these little gems at no extra cost
Early notation is almost always ridiculous complicated (when you look at it in hindsight). Take the idea of vector notation that people use as a matter of course in nursery school math. It is remarkably elegant - especially the ideas of dot and cross products and the determinant form for the latter. Look at any old textbook on the subject and you'll get arcane and obfuscatory animals like dyads and triads. Tensor notation (relatively recent) revolutionized the way this subject (and it is used almost everywhere in physics, engineering, hell - even computer graphics, so it is VERY important as a practical matter).
Brilliant (often crazy) people give birth to a new subject - one feels only awe when one considers these people. Wiser people then consolidate the subject over the next N years until it hold together beautifully. Even wiser people then continue to find deeper connections between this new subject and others that have lain around for a while.
In fact, in physics, the only book from the horse's mouth (so to say) that I actually found halfway understandable was Dirac's treatment of Quantum Mechanics. Even so, more modern books (Sakurai for grads or Griffiths for undergrads) is entirely more clear because by then any redundancies and clumsy notations been polished away, things feel right because they are consistent notationally with the rest of physics. I cannot over-emphasize the importance of consistent, clean and meaningful notation in trying to convey scientific knowledge successfully. The Humanities can be wishy washy in this regard but science can never afford the loss the clarity that ensues.
Another example: for a graduate level introduction to General Relativity, one might try to read Einstein's original paper - historically significant no doubt. A better way would be to read the fearsome Landau for field theories (not bad at all but not easy) or Wald (1984) - even better and getting more modern in terms of things we know. Or one might do the wise thing and go straight for Sean Carroll (2003!) for what might the MOST lucid treatment of GR ever written. I have great respect for a man who spends time clarifying (and thereby making laughably simple) the ferocious tensor notation of GR. Indeed, it is so clear, that I wished it had come out before I graduated with my B.S. (coincidentally in 2003
Do you see a pattern here? I do no
Because all your *other* base are belong to US!
Any sufficiently advanced intelligence is indistinguishable from stupidity.
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.