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America's top "mathletes" have won the first place once again this year in the international Math Olympiad.

The team's group picture, however, is as racist as it gets...

The winning team picture has been archived @ https://web.archive.org/web/20...
 
There were altogether 7 members on the stage - consist of 5 yellow (4 male and 1 female) plus 2 brown males
 
Taking a cue from Slashdot's anti-Chinese sentiment - that the Chinese are all thieves and cheaters - the 5 Chinese thieves and cheaters must have stolen everything from their 2 brown skinned friends.
 
Next time we should just send those 2 brown skinned guys. It would save us some money.

America's top "mathletes" have won the first place once again this year in the international Math Olympiad.

The team's group picture, however, is as racist as it gets...

> USA is the bastion of freedom and democracy in the world

Bullshit

(Clip from HBO's "The Newsroom")

It used it be. And can be again.

Wake me up when:

* music, home economics, shop, and finance are mandatory classes in school again,
*we stop teaching math by rote which kills all curiosity in the subject,
* we stop idolizing sport stars and actors who make millions -- who will be forgotten in a few decades and instead have more and better Teachers who struggle to make a decent wage
* we stop spending Billions fighting another man's rich war
* we stop the insanity of Imaginary Property and the obnoxious duration
* we stop corporations hijacking culture for the sake of profit
* we stop the visual pollution of advertising
* we stop tracking everything fucking thing a person does and selling the data to the highest bidder
* we take security breaches serious and enforce fines for when personal information is hacked / stolen
* we stop censoring people who think different
* etc.

--
Main St. built America,
Walls St. destroyed America.

> There is no "fun" way to learn calculus.

Bullshit. Go read A Mathematician's Lament

First off, having one teacher for ~30 students is an absolutely shitty way to teach. The fast learners are bored while they wait for the rest of the class. The slow learners are always struggling as they try to understand concepts. The best kind of teaching is one on one, self-directed learning.

When I was in high school one of my classmates was struggling to get 50% in Trigonometry. I spent one hour with him and he got 80% on the next test. The teacher thought he cheated until I said I tutored him. He was NOT stupid -- he just learnt a DIFFERENT way from HOW the teacher was teaching. A good teacher MUST use different ways of learning: Algebraic, Visual, Tactile, Auditory. The focus is HEAVY on symbol manipulation with some visual learning, and almost zero tactile or auditory modalities of learning.

The fact that Music is not a required course shows how brain-dead the education system is. Music and Mathematics go together like a ball and glove. So what happens? We neuter Mathematics and then wonder why the kids are bored out of their fucking minds. Gee, lets ignore 80% of the OTHER fun ways to learn.

Kids used to learn cyclic addition aka Number Theory's modular arithmetic when they were taught how to read an analog clock. No one ever stopped to tell them that they were doing advanced math. Gee, who knew!

Calculus is differentiation (sub-dividing things into infinitesimals) and integration (summing infinitesimals up) -- it isn't rocket science. The secret to teaching all mathematics is make it _engaging._ You can make a game out of ANYTHING. But that involves work -- and teachers don't have time to prepare for that. Their schedule is already over-loaded -- they don't have time to personalize and individualize learning because we don't value it and give the excuse that we can't afford it. But yet we can make missiles that cost a million each. Our financial priories are completely fucked up so we are left with a crap teaching modality.

> but the raw mechanics of calc and differential equations aren't things you can master by doing anything other than rote work

Again bullshit. That's because that is used as a Litmus test to tell if the students know how to _apply_ the concepts. Having students repeat mind-numbering, boring, exercises is the symptom of a shitty teaching methodology. That's not to say "practice" is bad, but practice for practice sake is stupid. The difference is musicians and great teachers know: perfect practice makes perfect. You need to be practicing the _right_ things. Doing the same set of dumb exercises over and over doesn't teach kids critical thinking -- only formulaic, no-original-thought, regurgitated memorization.

When you engage students one on one, and go at _their_ pace, they are able to learn, and apply, FAR faster more then the traditional, indoctrination, mode of "teaching".

If learning isn't fun -- then you are doing it wrong.

The accuracy commitments do not apply to GPS devices, but rather to the signals transmitted in space. For example, the government commits to broadcasting the GPS signal in space with a global average user range error (URE) of 7.8 m (25.6 ft.), with 95% probability. Actual performance exceeds the specification. On May 11, 2016, the global average URE was 0.715 m (2.3 ft.), 95% of the time. GPS Accuracy

User accuracy depends on many factors in addition to range accuracy, so the result is GPS Accuracy Levels it is possible now to get very accurate positions now, with differential GPS.

The really big change is that less expensive hardware is now able to handle the more complex math, and it is getting to market. Global Positioning System: The Mathematics of GPS Receivers

It does not seem to be well-known that Alan Mathison Turing (1912-1954) spent two academic years at Princeton University, from the summer of 1936 to the summer of 1938.

Alan Turing returned to the U.S. during WWII as a liaison between the two communities of cryptanalysts for about four months, from November 1942 to March 1943. He arrived in New York City on November 12, 1942, before heading to the headquarters of the U.S. Secret Service (now the CIA) in Washington, D.C.

Source

The tablet doesn't really contain trigonometry as we understand it today. There is no concept of angle, for instance. Some have convincingly suggested alternate interpretations. That paper, by the way, dates from 2002, so this isn't really news.

Can you elaborate a little?

Sure, read, e.g. this https://www.maa.org/external_a...âZ

It gelled heavily with what I saw.

and I confess to being ignorant to how mathematics are taught abroad

I don't really remember much from my schooldays in that regard. The US maths tutoring is a recent scar.

I'm not a mathematician, but - from many years ago - I do have a mathematics degree. I'm also not a musician, but I have a very strong interest in many types of music. (But not Disco! Nor Herb Alpert & The Tijuana Brass. But I digress.) And I have a very strong interest in dance: but I'm not a dancer.

Anyway, what you write resonates with me. At my university (Warwick) mathematics students could choose whether to be awarded a BSc (UK - BS in USA?) or a BA. Most chose BSc, but a few of us chose BA. My reasons were partly that I felt that most of the mathematics I chose to study was - although rigorous - in some ways more of an art than a science, and partly because I rather preferred having an "arts" degree to a science degree.

I read Francis Su's address in full, and I recommend it, particularly the sections on the importance of play (not just in mathematics) and beauty. (And if you read Andrew Wiles's account of how he finally saw how to solve the serious difficulty that was preventing his approach to proving "Fermat's Last Theorem", you'll appreciate the joy of creation.)

As an example of beauty in mathematics, I want to cite Muntz's Theorem, also known as the Muntz-Szasz Theorem. I came across this while taking a course in Topology: the set book was "Introduction to Topology and Modern Analysis" by G F Simmons. The appendices weren't included in the course but I read them, with not much understanding. But I was delighted when I read a description, without proof, of Muntz's Theorem. It didn't give me the aesthetic pleasure of the greatest music or dance, but my aesthetic pleasure in seeing this theorem was - and still is - maybe similar to that given by a good relatively minor piece by Beethoven or Chopin.

I think part of its appeal to me is that the theorem is a combination of the expected and the unexpected: if you were asked to guess at the correct form of the theorem, then you might well choose what is actually the theorem, but it's still in some ways a surprise, something which is perhaps also true of some of the greatest music - it can be both familiar and strange.

Think of a "continuous" function, say sin(x), and consider it defined on a restricted interval a <= x <= b.
* The Weierstrass Approximation Theorem says that any "continuous" function defined on a restricted interval a <= x <= b can be approximated as closely as we wish by (carefully chosen) polynomials of a sufficiently high degree.
* Muntz's Theorem says that suppose we don't allow all powers of x in the polynomials, and instead use only a restricted set of powers of x: for example
**maybe (1) only x**0 and x **i where i is a multiple of 3,
** or maybe (2) only x**0 and x **i where i is a prime number,
** or maybe (3) only x**0 and x **i where i is a power of 2:
then Weierstrass's Theorem is still true if and only if the infinite sum of 1/i diverges, where i are the powers of x allowed in the polynomials.
So polynomials of type (1) or (2) are all we need to approximate any continuous function, but for polynomials of type (3) there are some continuous functions which they can't approximate well.

In the literal sense, they are retarded compared to children of similar age 40 years ago. Their grammar and word usage is worse, their punctuation is worse. Their grasp of mathematics is worse. Their knowledge of history is worse. Their cognizance of current events is worse.

Citation needed. I think you're wrong. Here are charts of A-level performance (national exams taken in the UK at the end of 12th grade) which have shown steady and significant improvements since the 1960s. (Source = http://www.buckingham.ac.uk/wp..., and a further report of data since 1990 = http://www.bstubbs.co.uk/a-lev...)

http://i.imgur.com/RWdWAjx.png
http://i.imgur.com/gJZ5rbb.png

I picked A-levels because they've been the same kind of exam for a long time (as opposed to say the 10th grade O-levels which were changed out for GCSEs).

On the subject of maths, my understanding is that calculus used to be a college course, but now it's taught to loads of high school students. Here's another graph showing increased earlier uptake of calculus:
http://www.maa.org/the-changin...

In fact, hearing the shapes of objects was proven impossible some years ago. But I'm sure this biotech company has worked out a method to market this tech regardless.

> But learning how to think, and solve problems is important.

Concur 100% as does Paul Lockhart's A Mathematician's Lament agree with you: (I've included an exert)

The first thing to understand is that mathematics is an art. The difference between math and
the other arts, such as music and painting, is that our culture does not recognize it as such.
Everyone understands that poets, painters, and musicians create works of art, and are expressing
themselves in word, image, and sound. In fact, our society is rather generous when it comes to
creative expression; architects, chefs, and even television directors are considered to be working
artists. So why not mathematicians?

Part of the problem is that nobody has the faintest idea what it is that mathematicians do.
The common perception seems to be that mathematicians are somehow connected with
science -- perhaps they help the scientists with their formulas, or feed big numbers into
computers for some reason or other. There is no question that if the world had to be divided into
the "poetic dreamers" and the "rational thinkers" most people would place mathematicians in the
latter category

By concentrating on what, and leaving out why, mathematics is reduced to an empty shell.
The art is not in the "truth" but in the explanation, the argument. It is the argument itself which
gives the truth its context, and determines what is really being said and meant. Mathematics is
the art of explanation. If you deny students the opportunity to engage in this activity -- to pose
their own problems, make their own conjectures and discoveries, to be wrong, to be creatively
frustrated, to have an inspiration, and to cobble together their own explanations and proofs -- you
deny them mathematics itself. So no, I'm not complaining about the presence of facts and
formulas in our mathematics classes, I'm complaining about the lack of mathematics in our
mathematics classes.

If your art teacher were to tell you that painting is all about filling in numbered regions, you
would know that something was wrong. The culture informs you -- there are museums and
galleries, as well as the art in your own home. Painting is well understood by society as a
medium of human expression. 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?

The cultural problem is a self-perpetuating monster: students learn about math from their
teachers, and teachers learn about it from their teachers, so this lack of understanding and
appreciation for mathematics in our culture replicates itself indefinitely. Worse, the perpetuation
of this "pseudo-mathematics," this emphasis on the accurate yet mindless manipulation of
symbols, creates its own culture and its own set of values. Those who have become adept at it
derive a great deal of self-esteem from their success. The last thing they want to hear is that
math is really about raw creativity and aesthetic sensitivity. Many a graduate student has come
to grief when they discover, after a decade of being told they were "good at math," that in fact
they have no real mathematical talent and are just very good at following directions. Math is not
about following directions, it's about making new directions.

Exactly. See also A Mathematician's Lament a.k.a. Lockhart's Lament. A bit long, the first 5 pages or so get the point across, though I also like the "summary" of math education in the US at the end.

Math IS a chore. Learning IS a chore. People need to realize that not everything in life is "fun". You need to do the chores in order to get work done. Too many people don't want to put in the work.

While that might be partly true, it is also true that Math education is a chore because it was treated as a process of memorizing, not discovering - memorize process x,y,z so you can answer contrived questions a, b, and c. There is an excellent essay on this topic: A mathematician's lament.

Actually knot theory, quantum mechanics, and statistical mechanics (not to mention other fields) have deep connections. Knot theory is unreasonably effective.

Just to whet the appetite for our readers .....

Unreasonable Effectiveness of Knot Theory
Mathematicians Link Knot Theory to Physics

This lament is probably relevant.

There's not much "special" (don't know why you're using scare quotes here, but I'll follow) about the requirements for maths, but it is an incredible failing of high school maths curricula that you simply don't get to do what most mathematicians would consider maths until you get to university. See Lockhart's lament - https://www.maa.org/external_a... for someone who puts it far better than I can. It is a problem in science if students simply learn "facts" about physics, chemistry or biology instead of doing experiments to find out why we believe those facts. Or conduct an experiment following a cook book description instead of thinking about how we could find out if breathing rates depend on CO2 rather than nitrogen or oxygen contents of the lungs.

Following a recipe isn't doing chemistry any more than following the instruction manual with your Ikea furniture is engineering. The mathematics we teach at high schools is basically "Do this because it leads to the answer" - integrate this equation, rearrange and make X the subject, apply the quadratic formula etc - not "work out how to find out X about Y". There's nothing special about maths in this regard, other than that it seems to be where this is most blatantly done.

With apologies for the delay - Slashdot appears upset that I would like to respond to you in less than an hour...

Making something "mandatory in all grades" breeds dislike. Young kids often like programming, (or math, or art, or language, or music) and understand right away that it can be fun. Then the schools mess it up. If you haven't read it, I recommend the essay known as Lockhart's Lament:

A musician wakes from a terrible nightmare. In his dream he finds himself in a society where
music education has been made mandatory. “We are helping our students become more
competitive in an increasingly sound-filled world.” Educators, school systems, and the state are
put in charge of this vital project. Studies are commissioned, committees are formed, and
decisions are made— all without the advice or participation of a single working musician or
composer.

My wife, an educator, just heard me ranting and popped into the room: "Preschoolers need to play. That is the developmentally appropriate thing for them to be doing." She also reminded me that Steve Jobs didn't want his children looking at screens - he wanted them talking and reading.

After reading a interview with Randall Munroe (XKCD) i find myself wondering if what is needed is a computer engineering course alongside existing computer science courses.

http://www.maa.org/publication...

"And there's another distinction: There's coding, and then there is computer science. The best explanation I've ever heard of that is that coding is writing programs, and computer science is the study of computers only in the sense that astronomy is the study of telescopes. I think that's a really concise summation, because computer science isn't the study of computers, it's the study of what you can do with a computer and what stuff you can explore with a computer."

For those who are interested in math and/or card tricks, Colm Mulcahy is a professor of mathematics who often writes about math, cards and card tricks. He writes a blog called Card Colm for the Mathematical Association of America (MAA). He has written a book, Mathematical Card Magic: Fifty-Two New Effects, published by CRC Press.

A web site contains other interesting information about Mulcahy and his work, including links to past Card Colms.

Enjoy!

Hopefully you've read Paul Lockart's A Mathematician's Lament. If not, you'll find it quite resonating, I think.

Mathematicians and Physicists have been pointing this out for years ...

* A Mathematician's Lament

* Cargo Cult Science

> A jail or prison consists of a school, dorm, library, ...

Some would say school is a jail of individual, creative thought. (A Mathematician's Lament by Paul Lockhart)

It was was probably inspired by Cargo Cult Science by Feynman.

This is exactly the greatest danger of modern schooling what Feynman called "Cargo Cult" Education:

  "There is only _one_ way to get the right answer, all other paths are wrong." which is the very definition of a "cult"

The lack of critical thinking is not new. :-(

"A Mathematician'ss Lament"
* http://www.maa.org/sites/defau...

Today's society over-engineers everything.

"The Caleb Bonham Show: Common Core Math"
* https://www.youtube.com/watch?...

100% Agree. I've shipped numerous games and am completely disgusted at "eSports". Sitting on your ass all day gaming is not a "sport." Sports involving, you know, going gasp outside getting some physical activity.

Students should be doing a balance of mental and physical activities: Mathematics, Philosophy, Science, Reading, Writing, Thinking, Music/Singing, Checkers/Chess/Go, Sports that involve physical activities -- even Martial Arts/Yoga; all with a focus on:

* Inspiring people to pursue their passion
* Critical Thinking

not chasing after the latest dumb fad(s).

Who are overpaid the most in society? Entertainers that no one will give a shit about in 50 years.
Who are the most important people in society? Teachers that inspire thousands of students.
What is the pay difference between entertainers and teachers? Why does pay tend to be proportionally to how useless you are to society??

This is nothing new of course:

A Mathematician's Lament
* http://www.maa.org/sites/defau...

And

Underground History of American Education
* http://www.johntaylorgatto.com...

Relevant: http://www.maa.org/sites/default/files/pdf/devlin/LockhartsLament.pdf

You are the problem. You think that getting the right answer is all that matters. You think that merely memorizing procedures and patterns means having an understanding of the material. You are wrong. Understanding math requires imagination, and oftentimes, you don't need to make a specific effort to memorize any facts because that will happen naturally as you try to understand the logic behind what it is you're working with. Merely repeating something over and over is not real learning, and does not an educated person make.

I don't understand where people suddenly got the idea that learning was supposed to be always fun, enjoyable, and easy.

Easy? No. But if you think it's not supposed to be fun and enjoyable, then I once again must say that you're part of the problem and most likely do not understand what actual education looks like.

Plenty of countries (Japan comes to mind) have education systems that are as bad or worse than the United State's, and that is sad. Rote memorization is not the solution.

No. Mathematics is absolutely NOT taught in schools AT ALL:
A Mathematician's Lament -- by Paul Lockhart

Math is reduced to pre-chewed problems where students are looking for identifiers so they know what formula to use and what values to put where.

Tell me when this wasn't true? I think you are nostalgic for an era that never was. Mathematics in Engineering has always been about number crunching. Students in engineering have always been taught "mathematics" as if it is a tool to solve a problem, no more and no less. I suggest you read Lockhart's Lament to get a better understanding of what mathematics is like, and how it is taught..

I went to undergraduate (and graduate) school in Electrical Engineering. You would be shocked at the number of students (even Ph.D. level) who think of complex numbers like vectors (complex numbers are scalars, and form a field). You don't need measure theory to be a good engineer. Real analysis, functional analysis, etc. are never offered in the Engineering department - only in the Mathematics dept.

Most engineers don't take Mathematics - they take a dumbed-down "here is how you use the math as a tool" version without rigor, proofs, and analysis.This has always been true - and for a good reason. Do you need to understand Hilbert spaces and Banach spaces to design a filter? Not really (it helps to get a deeper understanding, but you don't really need it)..

> Schools are at least as much about social engineering as they are about indoctrination

FTFY. Specifically, numerous people have pointed out the problems of a public indoctrination system:

* The Underground History of American Education
http://www.johntaylorgatto.com/chapters/index.htm

You can sue a doctor for malpractice, not a schoolteacher. Every homebuilder is accountable to customers years after the home is built; not schoolteachers, though. You can't sue a priest, minister, or rabbi either; that should be a clue."

" by 1840 the incidence of complex literacy in the United States was between 93 and 100 percent, wherever such a thing mattered. Yet compulsory schooling existed nowhere."

* A Mathematician's Lament
http://www.maa.org/devlin/LockhartsLament.pdf

* Here's what schools don't teach kids:
1. Anything about money.
2. How businesses work, so that they enter the game with no knowledge of how it's played.
3. Basic psychology, so that even if they understand the game, they can be effectively gamed. Obviously, psychology would be very useful in raising kids.
4. Parenting, other than what they learned by living (courtesy of parents, teachers, ministers, coaches, police ..) so they repeat all prior mistakes.
5. Collaboration and team effort.

Here's what they learn.
1. There is only one right answer to each question.
2. Your success is entirely based on your grades and obedience/attendance.
3. There are no new ideas. Everything you know is in books, according to a curriculum approved by committee.
4. Creativity, taking your time and questioning authority and status quo are punishable offenses.
5. Sharing information with others is punishable by expulsion.
6. Ethics are OK to talk about, but in real life, everything's fair; just don't get caught.

You can see the result. Roughly 10% of people are "successful" and innovation comes from roughly 1%. 90% of work is meant to make the boss happy, and 10% towards customers, teamwork is unheard of and requires expensive consultants to achieve at a minimal level, and you're paid almost entirely for your paper certificates and longevity.

Reference:
From the book "Children Learn What They Live" by Dorothy Law Nolte.

I believe you were looking for this: http://www.maa.org/devlin/lockhartslament.pdf

I think you are confusing calculus and the other FAIL one does in school with math.

Which is like confusing color-by-the-number and learning how to hold a paintbrush with actual painting.

The saddest part is, that I was naturally good at math, and did it for fun, but got told is school to "stop playing around" and “learn your math". Which fucked-up my natural sense for math forever.

I just hope Lockhart finally gets his stupid book done, which he promised us a long time ago, and which supposedly is a learning book that teaches actual math.

It's the unintuitive ways in which it's taught ... that is the problem

Lockhart put this quite elegantly in his A Mathematician's Lament. Treating math as a rote subject (as it is now) is the moral equivalent teaching art as paint by numbers.

I haven't seen anything mentioned here in this discussion that isn't mentioned in the career pages of SIAM or the MAA.

Honestly, I view the whole system as rather poor in general. Perhaps this will allow you a better insight into why I say that automated essay grading is a good thing for the American system... it certainly can't make it any worse...

Oh, but it can make it much worse:

1. you identified the problem as "rather our education system overloads teachers with students". Now, "automated essay grading" will not address the problem, but only keep it out of sight, out of mind better.

2. FTFA

Computers also have a hard time dealing with experimental prose. They favor conformity over creativity.
"They hate poetry," said David Williamson, senior research director at the nonprofit Educational Testing Service, which received a patent in late 2010 for an Automatic Essay Scoring System.

Yes, I hear you saying: "Well, it's already that bad, how can it get worse?" My answer: a "bubble sheet" system is neutral to the creative minds (most of them will even use their creativity to game the system). An automatic essay scoring will punish the creative minds, without even addressing the real problem, which is not enough educators.

The result of it: a system that is as much education as coloring between the lines is painting - how many painters/musicians/mathematicians (or even excellent trade persons) will survive such a system?

almost half the working population of England have only primary school math skills

For the U.S. at least, http://www.maa.org/devlin/LockhartsLament.pdf places the blame with the education authorities themselves.

http://www.maa.org/joma/Volume8/Siegrist/RedBlack.pdf

If you want to see the types of questions on the FCAT, you can look at the item sampler here. [fldoe.org]

And that wasn't that much harder either. Unfortunately, this guy has an inflated ego - he seems to think students should want to be like him, and he isn't very good at math, so they don't need to be - and is on the school board as well.

While I believe the school system (and especially mathematics curriculum) needs reform, I'd rather people who actually use real math for a living be consulted. You don't ask engineers how to reform art class, or artists on how to reform gym. I wish Lockhart's Lament be required reading for anyone who is trying to "reform" school mathematics.

The summary takes a cheap shot at the anonymous school board member, and a lot of comments mistakenly assume that the board member took, and failed, the 4th grade math test for which there were sample questions, rather than the 10th grade test. I think the school board member's criticisms are well founded, and many here are missing the point.

There's the often-referenced essay (in PDF format), A Mathematician's Lament, which argues that the method of teaching mathematics in the US is arbitrary, rigid, and fails to teach mathematics -- and that furthermore, not all students actually need or want to learn advanced mathematics, and the rigid math curriculum is a hindrance to those students who do need or want to learn it.

In practice, much of the way our education system works is not about teaching practical skills, providing the background knowledge for full participation in a democracy, or enabling a rich and rewarding life. It's about sorting out who goes in which social class. Tests are designed so that kids will fail -- and increasingly, so that teachers will be fired. If enough teachers and students rise to the challenge, and more students pass the tests, they'll just make the tests harder.

Honestly, how many people have studied calculus? How many people have sweated over integration with hyperbolic functions, and yet never have to cope with mathematics more complex than simple algebra in their daily lives? Certainly, mathematics is important, and certainly, it would be better if people knew more about such an important field of human endeavour -- but there are other things that are important to know as well.

Yep :) http://www.maa.org/mathland/mathtrek_5_7_01.html

You're still working from the assumption that standardized testing is actually a meaningful test of whether someone has learned a subject. That's widely disputed. There's abundant evidence that tests and grades fail to predict anything other than future success on tests and future grades. They don't predict future professional success or future happiness.

Educators have lots of suggestions for alternative models of education; I've often heard something like my suggestion for teaching writing, for instance, and that's not even a radical suggestion. Educator's suggestions for improvements to education are generally ignored by politicians, at least for public and most private schools in the US -- it's an open secret that the only institutions where modern education theory is taken seriously and is actually implemented are elite private schools. Grading and testing are most rigorous in public schools, and the poorer the community, the more rigorous the testing. Teachers are stuck "teaching to the test", even though they know it's grossly counterproductive and makes it all but impossible to actually teach students; if they don't keep up test scores, they lose their jobs. Some rebel, and lose their jobs. I've never known a public school teacher who didn't loathe "teaching to the test" for this reason, and they all love to describe their covert efforts to actually help their students learn.

There's a popular essay, A Mathematician's Lament, in which Paul Lockhart argues that the math curriculum standardized throughout the United States is completely ineffective at teaching mathematics. One upshot is, given that most people graduate from school and promptly forget the mathematics they learned in school, presumably we can get by without that many people learning advanced mathematics; anything we do that actually involves at least some people actually learning what mathematics is really about, before graduate school, would be a significant improvement.

Given that tests, grades, certificates, and even college degrees are notoriously almost useless as predictors of actual competence and productivity on the job, the only value of such things is that they are quantifiable, and thus seem like rational means to decide who gets to advance to higher levels of education or which job applications escape the circular file. But they're rational only in the sense of being quantifiable; they aren't actually fair.

Overall, I increasingly suspect that many of us are very busy doing things that are essentially useless; that much apparently productive work amounts to supplying resources to others rushing to work to supply resources to others rushing to work to sit there and do nothing useful; that our entire global economy is based, fundamentally, on the fallacy of the broken window, and overall we'd all be leading richer and better lives if we stayed home more and only worked about ten or so hours per week. But that's a suspicion, and even beginning to test it would require radical changes.

Your comment has broad application around here. At the same time, one does expect a certain level of competence across the board in basic life issues - which these days includes at least an acquaintance with the nature and operation of computers.

Cue car analogy in 3...2...1...
Not everyone needs to spend their weekends modifying stock engine blocks, but we do expect normal adults not to cringe timidly when faced with a steering wheel and PRNDL lever.

It's not that difficult to understand how an OS matters to a computer system. People do equivalently difficult things regularly, yet somehow the sight of a kbd and monitor conjures Jacob's ladders and mad cackling.

Many decades (or even over a century) ago, it wasn't like this. A kid finishing 8th grade (about 12-13 years old) had roughly the education of a typical high school graduate these days.

This is just wrong. In the area of math, you can look at studies like http://www.maa.org/features/faceofcalculus.html that show that the level of calculus education in high schools has tripled over the last 30 years, and has actually reached the point where a majority of incoming freshmen math students have already taken calc; in 1950, that was almost nonexistent at the high school level (let alone 8th grade). The state of science education in US middle schools and high schools was even more pathetic prior to the 1960s; a combination of Sputnik-inspired funding efforts and the legal demise of prohibitions on teaching of evolution and the like were among the key movers in stimulating science education. More generally, the AP program didn't even exist until the late 1950s.

One enlightening thing to do is to flip through math assessment tests like the American High School Math Exam from 1950 through present; the difference is pretty stark. In the 50s and 60s, the limit of difficulty is the kind of "a train leaves Chicago going X miles an hour while another leaves Los Angeles going Y miles an hour" questions that are more common for 7th graders (or even bright 5th graders) today.

And that's ignoring the fact that in 1960 over 60% of the population didn't even make it to high school graduation, compared with about 20% today; see for instance http://www.livinghistoryfarm.org/farminginthe50s/life_12.html

I'm sorry if you still prefer your new and improved oil lamp over electric light, because you can't comprehend the value of creativity.

There's a reason Einstein said: "Imagination is more important than knowledge.â (He also said that oil lamp thing. :)

Not saying we don't need engineers. Hell yeah we do. But they wouldnâ(TM)t progress at all without us. (And we would get nothing made without them.)

I have to say Lockhart's argument is true for programming too.
It is ultimately a creative task. The actual fleshing out is something you automate away more and more, as it is mostly algorithmic work.. (Case in point: Haskell, QuickCheck, etc.)

Ok, I realize you might be might offended by this. But to be honest, we all are a bit, when we see that not only can the fleshing-out be automated, but it is actually really possible to automate the entire scientific process with algorithms [Automated abduction. See chapter 2.]

Because it sucks to realize that we too are only machines after all. But only because we have been so arrogant before. As if we had something special inside us. Some "soul" / god sauce.

Conclusion: We wouldn't have had the idea for a plane in the first place without creativity. (Not denying at all that we wouldn't have build that idea without engineering. But... you know what I want to say.)

If only...

If you (and others here) haven't already seen it, I'd highly recommend reading the essay, A Mathematician's Lament by Paul Lockhart. He does an excellent job explaining what math is, the misconceptions surrounding it and how these things are reflected in the state of math education. You can find the PDF on the MAA website: http://www.maa.org/devlin/devlin_03_08.html

We were offered four classes, to be taken in strict order, Geometry, Algebra, Pre-calculus (I guess the word "trigonometry" doesn't fit in a CHAR(16) column or something stupid like that), Calculus.

That reminded me of "A Mathematician's Lament" (PDF) by Paul Lockhart.

Here is a calculation to illustrate the main idea

Define the function f(x) = (x+2)/(x+1)
a function of this type is called a fractional linear transformation

Set
x0=1
and iterate using f(x)
x1=f(x0) = 3/2=1.5
x2=f(x1)= 7/5=1.4
x3=f(x2) = 17/12 ~ 1.4167
x4=f(x3)=41/29 ~ 1.4138
x5=f(x4)=99/70 ~ 1.4143
x6=f(x5) =239/169 ~ 1.4142

These fractions approximate, indeed converge to the square root of 2

It turns out that in this particular case these fractions are the best possible approximations for sqrt(2)

We know that 1.4142 are the 1st 6 digits of sqrt(2)

sqrt(2) ~ 1+f1 where
f1=0.4142....
1/f1 = 2+f2 where
f2=0.41421...
1/f2 = 2+f3 where
f3=0.41421...

actually f1=f2=f3=.... ad infinitum

This means that
sqrt(2) = 1+ 1/(2+1/(2+1/(2+...))))
The last expression is called a continued fraction

the numbers in it are obtained by subtracting away the whole part taking the reciprocal, subtracting away the whole part, etc

The amazing thing is that for square roots this process isn't random but repeats cyclically. For sqrt(2) the cycle is particularly simple
we start with 1 and after that all the #s we get are 2

The numbers
1
1+1/2=3/2
1+1/(2+1/2) = 7/5
1+1/(2+1/(2+1/2)) = 17/12
are the fractions that result from truncating the continued fractions
These fractions are the best possible approximations for sqrt(2) ("best possible" has a precise meaning that we don't need to get into here)

Now here is the punchline.

Notice how we get the same fractions by iterating f(x) and by doing the continued fraction expansion

This doesn't happen for all square roots.
More interestingly, there can be an infinite overlap in the two sequences of approximations
O'Dorney figured out when this happens

http://www.maa.org/abstracts/mf2010-abstracts.pdf

See page 13

I'm sure his raw mathematical talent exceeds mine, but he's still mathematically naive. From the Mathematical Association of America:

“Math is neat: A statement is either true or false,” O’Dorney says. “In science, any theory can be overturned by experiment because science is founded on experiment. But in math, there are theorems that can never be overturned because they have been proved with logic.”

I don't really see much value in celebrating geniuses of this sort - clearly mentally (or physically) gifted winners of prize X, Y, Z. It's like celebrating a particular race - you were born that way, and you can either rejoice that you're like that and have the opportunity for fame and fortune, or you can get depressed that you're not.

I prefer celebrating the love of learning and hard work for the sake of advancement of humanity. There are lots of people who fit this goal, and maybe O'Dorney will reach that status. But he's not there yet.

It says that the assignment is testing knowledge, not understanding.

Of course testing understanding is again more difficult; The instructors in our training facilities (aka school) can't measure understanding, so we teach a series of useless facts without context, preparing the next set of cogs to man the wheels of the corporate machine.
This is why Are You Smarter Than a Fifth Grader works: the kids haven't yet forgotten the useless information we expect them to (temporarily) memorize.
I think we need to massively revisit how we educate. This and (pdf warning) this are about math, but I believe it actually applies quite generally.