
Is Algebra Necessary? - ColinWright
http://www.nytimes.com/2012/07/29/opinion/sunday/is-algebra-necessary.html?_r=1&smid=tw-share&pagewanted=all
======
Steuard
It's easy to recognize that the author's arguments could apply just as well to
any academic subject: literature, history, you name it. ("We should just teach
'citizen reading', where students learn to read recipes and furniture assembly
instructions.")

But the real surprise to me is his ignorance of actual college math curricula.
He says, "Why not mathematics in art and music — even poetry — along with its
role in assorted sciences? The aim would be to treat mathematics as a liberal
art". I don't know about his college, but the math requirement where I teach
can be met with courses such as "Math in Art and Nature" or "Liberal Arts
Mathematics". It's not as if his suggestions there are novel! But here's the
kicker: for both of those classes, proficiency in algebra is a prerequisite.
It turns out that you can't really describe those topics that he likes without
actually using some math.

On the other hand, I _have_ come around to agree that statistics is more
broadly useful than calculus. Here's a TED talk by one of my old math
professors making that case:
[http://www.ted.com/talks/arthur_benjamin_s_formula_for_chang...](http://www.ted.com/talks/arthur_benjamin_s_formula_for_changing_math_education.html)

~~~
daleroberts
I don't get that TED talk. For example, how are you meant to properly
understand the ubiquitous Gaussian distribution without learning Calculus
first?

~~~
Steuard
I may not entirely understand your question. My wife has taught statistics
college statistics many times (including Gaussian distributions), and her
classes have never had a calculus prerequisite.

Are some topics easier to understand if you already know calculus? Sure. (I
assume she has to do the same sort of brief "area under a curve" explanations
there that I have to do when I teach algebra-based physics.) Can you
understand the topic in greater depth using calculus? Of course. But for a
first exposure to basic statistics I think you can mostly dodge the issue.

(And really, apart from already knowing the concept of an integral, does
knowing calculus really buy you much when studying Gaussian distributions? You
can't even _do_ those integrals! That frustration might be even more annoying
to a calculus student than to others.)

~~~
kaiwetzel
I've taken an introduction to statistics course with social science students
and based on that experience, I can relate to what you are saying, 100%.

However, I think there is a broad group of students[1] for which a
significantly earlier exposure to calculus would be beneficial and make
learning statistics (and physics) a lot easier or at least faster.

When I took introduction to statistics as a math major, I found the subject
extremely confusing because the discrete and continuous case where taught
completely disconnected and useful anchors for understanding such as basic
measure theory and Lebesgue integration where left out. That's certainly a
good way to teach for many but for some it doesn't work.

A similar case was physics for me (classical mechanics in particular). From
grade 5 to 10 (after which I avoided the subject) there was little insight
gained (e.g. heavy things fall down, there may be some friction, memorize all
those seemingly random formulas and if you use a long lever, make sure you
pick a strong material). Then I was exposed to an introduction to physics
course at university (for non-majors) and the revelation that all those random
formulas have a strong grounding in just 3 general principles and can then be
developed with some help from calculus was liberating. Just too late in my
case. Maybe I would have loved physics and actually study it, had they told me
in 7th grade that there is something tying all of it together, and the
ultimate goal of the class was to reach that summit. Just trying to show the
other side of the coin which should be integrated into the way math and
science is taught in schools in my opinion :-)

[1] Say, the top 5-10% of middle school students.

~~~
Steuard
Oh, don't get me wrong: _I_ was served very well by today's standard math
sequence. I learned fascinating stuff in precalc, and calculus was a
revelation and a profound joy. Prof. Benjamin's argument favoring statistics
instead was a tough sell for me.

But I'm a theoretical physicist. As much as I hate to say it, structuring the
entire standard math curriculum so it works best for kids like me (or even for
the top 10% of students) just isn't reasonable. (Ideally, a solid gifted
program could fill that gap.) I think that we agree on that.

I'd like to think that there are ways of introducing concepts from physics or
statistics that do highlight the underlying structure of the field, even if
the students don't yet know all of the math they'd need to work through the
details themselves. If I find a perfect way to do it, I'll let you know!

~~~
bunderbunder
One approach I've seen and thought was interesting is the course schedule
offered at the Illinois Math and Science Academy.

Their pre-calculus courses have been somewhat radically reorganized into a
curriculum called "mathematical investigations" which orders the topics
according to more of a practical progression. So, for example, bits of linear
algebra are pulled all the way up into precalc because they're useful in
geometry, and will also work better with the science curriculum. Perhaps
physics teachers inheriting students who already understand vectors, for
example.

Then the calculus curriculum is split into two tracks, one more basic, and a
more intensive one for students who anticipate going into fields that require
more calculus.

------
bluekeybox
Recent example. A family member who is an artist asked me about submission
rules to a guild competition (now, of all people, the artists are commonly
believed to require algebra knowledge the least). The flyer indicated, "the
photos/scans you submit must be 300dpi" (dots/pixels per inch). Which is
perfectly confusing because even a 100x100 pixel thumbnail can be considered
300dpi if you set its width to 1/3 inches. When I tried to explain said family
member that the guideline is confusing, that whoever wrote it probably failed
their high-school algebra, he started to mistrust what I'm saying. Words can't
explain how dumb the situation was: one idiot writes a guideline that can't be
followed (or rather one that can be satisfied even by a still from a security
camera), another takes what is written on blind faith and refuses to use logic
to understand why the guideline is insufficient.

Now imagine dealing with people who never completed high-school algebra but
who are trying to calculate dosage of a drug. According to a pharmacist I
knew, a child died on his watch because of incorrect dosage.

~~~
benzofuran
Hold on there hot shot, they're likely saying the required scanning resolution
of submittals. While not incredibly clear, 300dpi is the general high-
resolution setting on most scanners and they're just indicating as such.

------
benzofuran
This is an interesting read, but the author is also a political science
professor. If this article had been by a mathematics educator, it might be a
bit more persuasive.

I'd argue that algebra is fundamental - the excerpts taken involve the
quadratic equation and other supposedly tedious tasks, while ignoring that
these are foundations for higher level problem solving. In almost any field,
somebody will ask at some point "how many X do I need for Y" - and it's
usually not that clear.

I would almost posit that the author is trolling the NY Times, but today
that's probably not the case. While math is hard for some students, it's an
essential part of education for just about anybody that hopes to be a mildly
functional member of society.

------
Thrymr
I was forced to read _The Scarlett Letter_ in high school, and I have never
once had to use that knowledge in my professional life.

I memorized dates for history exams in high school, and that knowledge has had
no practical impact on my life whatsoever.

When introducing myself to (non-math/science) academics and telling them I was
a (now former) physics professor, I can't tell you how often the first thing
out of their mouth was something along the lines of "I was never very good at
math" or "I never liked science." I frequently have had doctors tell me how
much they hated physics as a pre-med student. That kind of pride in ignorance
is quite rare in the opposite direction. The most broadly educated people I
know are scientists.

[edit: typo fix]

~~~
Thrymr
As a followup, a nice blog essay on the theme of intellectual contempt for
math and science:

[http://scienceblogs.com/principles/2008/07/26/the-
innumeracy...](http://scienceblogs.com/principles/2008/07/26/the-innumeracy-
of-intellectual/)

I often wanted to call bullshit on those people too, and reply with something
like "I always hated reading" or "What use is Shakespeare, anyway?" Of course,
I never did, out of both social politeness and an actual respect for the
humanities and what they bring to civilization.

------
Tooluka
Yes it is.

You can easily swap solving math problems for writing essays in that article
and get the same conclusion. Also the whole article completely lacks any
statistic basis. 42% of students didn't pass their bachelor exams and 57%
students of one university (not saying which faculties were included in
statistics) didn't pass algebra course (one math course only is mentioned). So
what? How is that connected? What are the long term trends? What about other
courses and other universities? Same applies for all the article - throw
together random numbers that SEEM to be related (they may be, of course).

Very shallow and probably incorrect article.

~~~
nextstep
I agree with what you are saying, and I think algebra is an incredibly
important skill. However, the article touches on an underlying issue that I
think is true: the US education system does not prepare people for certain
jobs that are in high demand. Many jobs that could be learned at a vocational
or "trade" school don't require much high school math.

~~~
gersh
You need Algebra even for "trade" school. According to
<http://www.njatc.org/training/apprenticeship/index.aspx>, you need "One Year
of High School Algebra" to become an electrician. I think the same is true for
other trades.

------
tsahyt
This is a perfect example of how Betteridge's Law can fail. Yes, algebra is
necessary. Very much so. It's basically the first time students are required
to operate on a formal system, which requires strictness and analytical
thinking. As another poster has pointed out, algebra is a gym for your brain.
Also, algebra has tons and tons of applications. Anybody claiming they "never
needed to solve for x" has completely missed the point of symbols. They have
certainly solved for x but haven't realized it yet.

The article lists arguments which can be applied to about everything else you
learn in secondary education. How comes nobody ever complains about learning
literature, arts, music or whatever, but people seem to insist that they'll
never need math in their jobs, when in fact, math has made their lives
possible as they know it.

</rant>

------
photon137
"But there’s no evidence that being able to prove (x² + y²)² = (x² - y²)² +
(2xy)² leads to more credible political opinions or social analysis"

Nothing else leads to that either. Political opinions and social analysis have
zilch value in understanding the Universe we live in. Math does.

------
sreyaNotfilc
Hmm.. Algebra is not hard. Its actually pretty easy if you know what you're
looking for. Heck, the professor even tells you the answers before every test.
Just follow the steps.

That's the problem though. There are many steps to be learned. Things like
"order of operations" and "FOIL" are necessary to get the correct answer. Its
not that learning Algebra is hard, its the discipline that comes from the.
Trouble is, most young people don't see the rewards in learning this
discipline.

For example here's a few other disciplines that takes time but seems more
rewarding to a young student (or anyone)

Swimming - can have more fun at the beach Martial Arts - self defense and
confidence, and proof that you're a bad ass Learning a FPS - bragging rights,
more fun online against friends and other gamers Guitar Lessons - sing your
favorite songs, perform live, get lots of friends/admirers

Algebra - learn about long term goals and ... processes?

Kids are not into the "long term". They are into the short term. Thus, they
are kids. Adults needs to show them why it is important to learn how to find
"X". Unfortunately, most adults would rather stay far away from that stuff
because they've had a hard time with it as a child as well. Its painful to
them. Its easier to tell the child "because it is important" than explain why.

I personally think algebra is very cool. I like how things can workout just
because you know a proven formula for solving that problem. But then again,
that's why I (and other CS) make the big bucks. We do the things that no one
else wants to deal with.

Using your brain is hard. Solving a tough equation is probably equivalent to
running a mile. It can be done. It can be fun. Yet, you don't see many people
with long slender bodies running around everywhere do we?

Everyone has their niche. Its really up to them to find it.

------
irreverentbits
I find it challenging to believe that individuals incapable of basic, entry-
level algebra are really qualified to attend a university in the first place.

Having many good friends in disciplines which will never require the use of
any sort of mathematics, I can safely say that none of them have had trouble
with high-school level algebra.

Portions of the article strike me as a depressing appeal to entitlement
amongst the lowest common denominator of students to a university education.

------
stcredzero
True story. I was having trouble sleeping and I went to a big box home
improvement store for some custom cut roller blinds. I measured the blinds to
the nearest 1/8 inch so less light would leak around the edges. I grab the
blinds and go to the blind cutting machine, which has an arm marked off like a
ruler to 16ths of an inch, with the cutting blade held by a screw clamp. There
are no detents or grooves to force discrete measurements, just a screw clamp,
so this machine can do any length between its min and max length.

Well, the young woman who comes looks at the measurements I wrote down and
looks at me and says, "The machine doesn't do fractions."

I was floored. She was lying to cover her innumeracy, because _she_ didn't do
fractions. If this is indicative of a trend, it's bad news for the future of
the United States.

~~~
Steuard
More charitably, she wasn't lying, she was just ignorant (and therefore
wrong). An innumerate person might simply not recognize all those extra marks
as having any meaning at all, or at least not as having anything to do with
those dreaded "fractions" from school.

------
jayferd
Making a subject mandatory is a great recipe for sucking all the life out of
it, and mathematics is one of the best examples. Is algebra awesome?
Absolutely. Does every member of our society need to be able to factor
polynomials? ...not really, especially not at the expense of a child's natural
love of learning.

------
shin_lao
See algebra as the gym for your brain. Although you might not need algebra,
you will benefit from the exercises.

~~~
kalid
I think that's a dangerous line of reasoning. Why not learn to memorize
ingredients on a can of soup? That is also exercise for your brain.

Algebra lets us describe relationships between unknown quantities. Nearly
every physical law is expressed as an algebraic equation (or Calculus, which
requires it). It's the lingua franca of describing the world. That's why we
need it :).

~~~
shin_lao
Memorizing trains memory, that's not the same exercise.

------
s_baby
Understanding variables is a hallmark of the "formal operational" stage of
development.[1] It's the same reason why we hear some people "just don't get"
programming. Same reason why languages like LOGO avoid this construct.

If grasping Algebra is actually about attaining this developmental stage, we
need to be approaching the problem on a more fundamental level. Kids will move
through these stages at a different pace and if you're on the tail end of
developmental pace you're going to fall through the cracks.

[1]
[http://en.wikipedia.org/wiki/Piaget%27s_theory_of_cognitive_...](http://en.wikipedia.org/wiki/Piaget%27s_theory_of_cognitive_development#Formal_operational_stage)

~~~
kaiwetzel
I had totally forgotten about this :-(

Searching a little I found this interesting document addressing some of the
problems which I have put on my reading list: "The Science of Thinking, and
Science for Thinking: A Description of Cognitive Acceleration through Science
Education (CASE)" (Philip Adey, 1999).

Assuming the idea of a "formal operational" stage of development applies, the
situation looks abysmal (at least in the US and Germany, can't say much about
other countries):

From the little I gathered so far (on the internet, so it has to be taken with
a grain of salt) it seems that (1) a vast majority (over 60%) of people never
reach formal operational maturity. (2) Ideas of how teaching can actually help
with it are in it's infancy. (3) Application of said ideas is not very far
along. (4) Educational systems keep leaving many (or most) students behind
early, especially in math and science, while other students get bored and
waste their time in class, being taught a mind-choking curriculum.

Ok, I guess I'm ranting now :-) Saddens me greatly, though.

[1]
[http://www.ibe.unesco.org/fileadmin/user_upload/archive/publ...](http://www.ibe.unesco.org/fileadmin/user_upload/archive/publications/innodata/inno02.pdf)

edit: spelling, grammar

------
stcredzero
There was a recent discussion on reddit following some Khan Academy
controversy, where some education people mentioned some research about the
concept of slope.

It turns out that a significant number of people don't understand speed as a
rate. They think of it as an intensity, as in volume of sound or brightness.
(Those are flux, which are related to rate, though that's not how we perceive
them.)

People like that are alien to me. I think it also explains why freeway drivers
in Houston often have about 0.3 seconds of separation between cars.

------
yummyfajitas
Math literacy is valuable even for regular people.

Case in point - an innumerate relative of mine has been duped into selling
"Nu-skin" products for virtually no money. Nu-skin gave her documents
explicitly stating her odds - 99.5% of active nu-skin sellers make <
$15k/year. But they also showed her videos of people who won won carribean
cruises and made thousands/millions.

Guess which one she bought into?

A math literate person would recognize that her odds are better working at
chipotle and reinvesting some of the proceeds in vegas.

~~~
telemachos
I agree with your main claim about the value of mathematical literacy, but
your example is terrible. It's not really about math at all. It seems to be
about decision making and many people's tendency to choose very faint hopes,
no matter how well they understand (on some level) that the odds are massively
against them. I very much doubt that's about math _per se_. That is, your
relative didn't misunderstand how little 99.5 leaves from 100. She simply
isn't a logical machine.

tl;dr What a person recognizes and what a person acts on are _not_ always the
same.

~~~
yummyfajitas
Having actually had conversations with her about the matter, I can tell you
that the fundamental problem is she just doesn't get what 99.5% means.

A numerate person looks at things like this as a math problem. An innumerate
person doesn't.

~~~
UK-AL
It's hard not to understand 0.5% is not a lot. It's seems to me she knew
risks, but ignored them.

~~~
Someone
Not a lot? As odds in a lottery that pays millions to the winner (original
remark said thousands/millions, but that 'thousands' would seem less than the
<15K that the 99.5 would get) and still pays out to everybody, it looks
incredibly enticing.

The lie with these kinds of things is not as much in those percentages as it
is that it is not a lottery. There are people who can sell anything; they are
the winners. Also, typically, there is some kind of pyramid scheme involved.
You get stats on the early birds, but you cannot become an early bird
yourself.

~~~
UK-AL
When you talking income, you don't want high risk/high payout, because your
going to have serious cash flow problems, if you don't win, which the most
likely case.

A small chance of winning, but with otherwise serious cash problems? Or high
probability of a nice stable income?

I know what I would choose.

~~~
Someone
I know what I would choose, too, but I have an income. The target for such
schemes is people who don't have that, or have a really low one. For them, the
options look like "about what I get now, and I can choose my own working
hours, and no longer have to listen to a boss" and "bingo".

And that likely is true. If you choose any 20 hours to work each day, 7 days a
week and have some talent for sales, you likely will make that 15K.

------
evoxed
> Mathematics, both pure and applied, is integral to our civilization, whether
> the realm is aesthetic or electronic. But for most adults, it is more feared
> or revered than understood.

This is very true, but lowering expectations is hardly a solution. Ignorance
(in the form of oh-well-that's-the-class-smart-people-take) is no better than
fear. Before we set the bar too low, let's keep pushing to make mathematics
_less_ intimidating, _less_ foreign and easier to learn for everyone.

------
simonster
I agree that algebra in and of itself is not particularly useful. However, an
in-depth understanding of statistics is vital to basic scientific literacy and
most certainly "leads to more credible political opinions or social analysis.
I've often wondered why high schools don't place a bigger emphasis on
statistics, but I'm not sure if you can teach basic statistics without algebra
(or calculus).

~~~
Steuard
Most basic statistics classes that I've seen don't require calculus. My wife
(who's taught stats in college many times) claims that they don't _really_ use
algebra all that much, either (you aren't generally solving quadratics or
anything), but they certainly require the comfort with complicated symbols and
formulas that algebra classes teach.

Have a look at the TED talk linked in my top-level comment for an argument
along the lines of what you're making here.

------
ggwicz
I think algebra is necessary, and calculus, and geometry, and trigonometry,
and everything else I don't understand in the realm of mathematics.

But they're important in the context of real life.

So, yes, we should keep these classes, keep having kids go through algebra and
calculus and geometry...

...But you shouldn't be bound by arbitrary rules. Algebra is important, but if
you find "x" by doing something other than some arbitrary thing where you
subtract both sides, etc., you shouldn't get an F in the class.

Same with calculus, geometry, everything. The importance is the thinking and
the logic, and what real-life application you can take from your knowledge.
Making hard rules for these math courses, for example, definitely hurts this
and does some of the things this article claims.

But true, honest exploration in math and thinking about it is very important,
and if it's free and done in an honest way there is no question about whether
it's necessary or not.

Richard Feynman covered this better than I ever could:
<http://www.youtube.com/watch?v=5ZED4gITL28>

~~~
simcop2387
> ...But you shouldn't be bound by arbitrary rules. Algebra is important, but
> if you find "x" by doing something other than some arbitrary thing where you
> subtract both sides, etc., you shouldn't get an F in the class.

In middle school at least this kind of thing was allowed as long as we could
show why what we did worked. The idea being that we would have to understand
what we did in order to know either when it would work or when it would fail
so that we could apply it correctly. If we could do that we weren't given full
credit for anything because the homework/tests were meant to check that we
understood how to get the correct answer (other than copying from the nerds
like me).

------
jeffdavis
It's hard for me to comprehend that someone could understand any subtle (or
even not-so-subtle) distinctions or complex arguments without at least a basic
understanding of algebra and statistics.

I could be wrong though. What about famous thinkers like Jefferson or Lincoln
-- did they understand algebra and statistics at all?

~~~
Thrymr
Jefferson was quite well educated in science and mathematics for the day:

<http://www.math.virginia.edu/Jefferson/jefferson.htm>

Geometry was prominent in those days, but as a student he also learned physics
with Newton's _Principia_ and _Optiks_ as textbooks.

Lincoln was mostly self-educated, but I have no idea what his background in
math was.

------
jeffdavis
It seems likely to me that algebra during school is a cause for dropout not
because it's harder, but because it's more objective.

If someone doesn't know anything about history, you can pass them along,
claiming that they wrote a few essays or something. It's fraud, and any expert
can see it, but it's easy enough to ignore it if you try.

But with algebra you can't ignore failure. It's obvious. Even the most basic
tests will reveal ignorance quickly.

I conjecture that those people who drop out "because of algebra" are not
proficient in any subject.

I observe that the alternatives suggested are less objective and more "hands-
on". What does it mean to learn about the consumer price index without
learning basic algebra? The only explanation I can think of is that it offers
more opportunity to ignore educational fraud.

------
abtinf
How is this not an article from The Onion?

------
heycosmo
To the extent that algebra courses consist of solving # + x = # over and over
again, I agree that they are mostly useless. In life, algebraic questions
don't come at you in symbols and numbers, they come in words. And I would
argue that these questions surround us! It's just a matter of people
recognizing that they are there. Therefore, in my opinion, a good algebra
course (which is essential!) focuses on problem solving. Necessarily, the
course would involve some mindless equation solving to learn the framework of
algebra.

------
bickfordb
I'm biased since I'm a software engineer, but I believe algebra is extremely
valuable. I can't imagine not knowing it! If I were to revise math education I
would balance the amount of time spent on geometry (1 year HS), trig (1 year
HS) and calculus (1 year HS, 2.5 yrs undergrad) better with discrete math,
linear algebra, probability and statistics. The current system seems to be
disposed toward creating 50's NASA fodder and economists.

------
artlogic
I rarely comment, but I feel compelled to add to this discussion.

Algebra, as it is taught today in the U.S., is not necessary, and is probably
detrimental in many of the ways stated in the article. I can't speak for non-
traditional schools, but public schools, with their focus on standardized
testing, have effectively destroyed the original spirit behind teaching
mathematics.

Mathematics, much like other academic topics (e.g. literary analysis), was
taught not as a practical skill, but as a way to improve your abstract
thinking and problem solving skills. As another commenter stated, Mathematics
(along with most later academics) should be a gymnasium for your brain. I like
this metaphor, because if you think about solo athletics, all students are not
expected to achieve at a pre-defined level. Rather students are evaluated
based on improvement in performance over time.

Sadly, mathematics in recent years has become less about abstract thinking and
problem solving and more about rote memorization, computation, and
application. I taught basic algebra to college students for a semester. One of
my most interesting experiences was with the dreaded story problem. Most of
the students were simply unable to apply math to solve a problem. They were
all very good all computation, but when faced with a problem in a form they
didn't recognize they instantly began flailing.

Standardized testing has turned math into the process of recognizing a form,
plugging in the numbers, and computing. We're now starting to see the first
generation of math teachers that are a product of standardized testing, and
the results are frankly, frightening. I've spoken with younger math teachers
who couldn't explain the practical importance of their subject. While they
loved math, they couldn't tell me why they were teaching it, other than: "It's
on the [standardized] test."

This is the kind of Algebra we don't need.

* * *

Incidentally, the only way I avoided the shocking deficiencies of standard
high school "math" was by fighting my way into an advanced program at the
local University. It was there I was introduced to Euclidean geometry and
really learned what math was all about. I think everyone should learn Geometry
using the Euclidean method. I learned from a simplified book (Geometry, by
Moise/Downs) which starts out with a few more postulates than Euclid did. I'd
go as far to say that geometry (properly taught) will make you a better
programmer/problem solver/thinker.

------
kghose
Yes it is and we need to train teachers to teach it better.

------
serichsen
Yes.

------
alpine
I stopped reading the moment I encountered 'I say this as a writer and social
scientist...'

~~~
esrtbebtse
Do you believe that writers and social scientists have nothing to add to this
discussion? Not everyone is a scientist, not everyone is a mathematician, not
everyone is an expert in education.

Equally, not everyone's opinion is equally valid, but they too may have
something to add. In particular, find people with full and meaningful lives
who have _not_ done algebra, and that will show that it's not essential.

Showing them that their lives would be better _with_ a working knowledge of
algebra would be a challenge worth considering.

~~~
alpine
_Do you believe that writers and social scientists have nothing to add to this
discussion?_

Mostly, Yes.

------
moron
Telling that students struggling with a subject is considered evidence of a
problem with the subject. Couldn't be the teachers or the curriculum or the
students.

~~~
bluekeybox
In math, a good teacher goes further than in any other subject. For example,
in Eastern Europe, the teaching facilities has been very poor in the past
century, but it is ridiculous how many influential 20th century mathematicians
were Hungarian. I couldn't understand this phenomenon for a long time until I
realized that it nearly entirely has to do with the fact that most of those
mathematicians studied under eminent Austro-Hungarian mathematicians who
studied under eminent German mathematicians who studied under eminent French
or Italian mathematicians all the way back to the Renaissance. Look up Paul
Erdos (<http://genealogy.math.ndsu.nodak.edu/id.php?id=19470>) for example,
and keep clicking on "advisor" until you reach 15th century Italy.

This is why I love online learning and have good hopes for the Khan academy.
Projects like that can set us/our children free from the plague of bad
teaching.

