
Does mathematics have a place in higher education? - cottonseed
http://mathbabe.org/2012/07/29/does-mathematics-have-a-place-in-higher-education/
======
DanielBMarkham
The future is computers, and computers work through tagged languages and
symbolic systems. Everybody needs to understand formal symbolic systems. Math
is the best way that I know of to do this.

It used to be that to be a composer, first you learned the piano. The piano
was thought to give one an appreciation of the nature and nuance of music.
From there you could much more easily learn other instruments and then learn
how to put them together into something beautiful.

Math is the new piano. Learn symbolic manipulation -- and I'd say that means
first-year calculus -- and you can work with the kinds of symbolic systems
you'll find everywhere else in the modern world.

I used to do some tutoring when I was a kid, and I think this level of
understanding is achievable by 90% of the population _given the right
environment_. The real question is "why are the structures and environments
for education that we create so unable to accomplish this?"

Dumbing down the system isn't the answer. It's not like you can dumb down the
world to make up for your inability to prepare students for it.

~~~
gizmo686
Why is first year calculus the standard for symbol manipulation? Algebra is
also pure symbol manipulation, and the applications of it are more universal,
whereas calculus is symbol manipulation over 2 functions, plus algebra; but
those two functions have a lot of conceptual baggage that is not nearly as
useful.

~~~
sanxiyn
I think abstract algebra (group, ring, field) is a better way to teach symbol
manipulation. Or even logic. A proof of well-ordering theorem beats
integration by parts if your goal is to teach deduction and symbol
manipulation.

Let's be honest: calculus is taught for its application in engineering, not to
teach deduction and symbol manipulation. If that is your goal, there are more
suitable maths to teach.

~~~
gizmo686
You need to change the way math classes think of proofs. Every times I have
seen a proof in math class, they managed to take a beautiful and elegant
argument, and distill it down to the least number of characters that formally
proves the statement. Presently, I am doing research work in (applied)
chryptography, which as you might imagine involves reading a fair amount of
modern math papers. All but one of the papers I read were easier to understand
than the textbook proof for Pythagorean's thuerom.

~~~
sanxiyn
I agree. More motivations, intuitions, insights behind fewer proofs would be
more useful than lots of proofs, if the goal is to teach mathematical
thinking, not to teach particular math results.

I think proofs are compressed because of lack of margins. Don't laugh! Authors
seem to think they need to cover lots of essential results, and if you do
motivations, intuitions, insights for all of them, already long math books
will be impossible to hold in your hands. In my opinion the obvious solution
is to omit most proofs and do far more non-formal discussion around important
proofs. Which won't happen because their goal is to teach particular math
results, not mathematical thinking, contrary to what they say.

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ef4
"School mathematics" is something almost completely different from
mathematics. (For an elaboration on this idea, see
<http://duckduckgo.com/?q=lockharts+lament>).

Meanwhile, being good at (actual) math is rapidly become a prerequisite for
having a bright economic future. School math is worth increasingly little
(computers do it better than you). Whereas true math is deeply creative, and
computers suck at it.

This separation is also why so many programmers say "programming doesn't take
math". It doesn't take _school_ math. It rests deeply on true mathematical
reasoning.

~~~
mseebach
My personal anecdote-backed theory is that the lack of proper math teaching
(and yes, a little bit of old school drilling) lays the foundation of more
advanced "school maths" subjects being perceived as hard: If you struggle with
in-your-head multiplication, then the FOIL mnemonic for binomial
multiplication is going to be hard, which makes algebra hard. If you can read
the language of maths, you can see patterns and understand things in the same
way you can when you can read an article.

This continues to spill over until it becomes a negative selector, driving a
preference for liberal arts type subjects over science and engineering, not by
the positive virtues of the former, but by the perceived (and, probably at
this point, real) difficulty of the latter - and further augmented by the
fallacy that _any_ college education is better than none.

We're happy to drill reading in schools, but loath to drill even the simplest
maths. Only the most abject failures are allowed to go through school without
being able to read a newspaper article at a reasonable speed and give a
summary, but if someone breaks down having to do 12 * 9 in his head, it's
fine, he's just not very "sciency".

Rant over.

~~~
Someone
I had to google "FOIL mnemonic" to learn what it is. My reaction would be "why
in ???? would you teach your pupils such a trick, given that it breaks down
when generalizing to e.g. (x+3y+z)(2x+5y+z)? (anybody who could say "that's
just ((x+3y)+z)((2x+5y)+z), so I'll just apply it multiple times would be able
the much more general "just add all pairs (something from the left, something
from the right; there ar #items on the left times #items on the right such
pairs"

Do they really teach that somewhere?

~~~
lukeschlather
Yes. It's a common teaching method (and not just in math) to teach simplified
methods so that people can see them in action, and once students have some
experience experimenting with them, delve into the underlying theory. I think
it's a perfectly valid teaching method. Students should probably be past the
point where they really think "Now I'll use FOIL" by the time they finish a
class where it's used as a teaching method. But in any class, there are going
to be students who don't really end up grokking the theory.

~~~
saraid216
What _is_ the underlying theory? I've taken a lot of math courses
significantly past the point where I learned FOIL and I don't remember it
being explained.

~~~
Daniel_Newby
Multiplication distributes over addition.

------
colanderman
She makes a good point about algebra students not understanding prerequisites.
I worked for two years at a high school where students were having trouble
with algebra. Almost invariably this was due to (a) an abysmal grasp of
arithmetic; many students couldn't divide or even subtract reliably, leading
to insurmountable frustation, and (b) the hand-wavy means by which algebra,
and especially its archaic conventions (such as order of operations and
elision of the times symbol) were taught.

I once solved these problems for a small group of these children to whom I
taught algebra from first principles: all grammar was explicit (no elision of
parentheses or operators) and we used no numerals. The reduction rules then
were able to fit down one side of a whiteboard, and after five one-hour
sessions, self-professed math phobes were solving nontrivial equations.

~~~
gcv
Interesting! This same problem occurs in calculus as well. Leibniz's notation
is pretty offensive, for example: in basic calculus, d/dx is an operator. In
differential equations, you suddenly begin treating dx as a separable
algebraic term — and way too many teachers suck at explaining why this is
suddenly permissible.

~~~
sanxiyn
I recommend Elementary Calculus: An Infinitesimal Approach to those interested
in calculus. It treats dx as a term from the start, and matches my (and I
suspect most people's) intuition of the subject much better than epsilon-
delta. And it is actually completely rigorous, not hand-waving infinitesimals.

~~~
vlad
The book is available as a free online pdf:

<http://www.math.wisc.edu/~keisler/calc.html>

Checking it out now.

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doc4t
What a marvelous stupid idea.

Math and Applied Math is more than numbers and math for the sake of math. It
is a way to bring your level of abstract thinking higher - much the same way
learning languages does.

Math is so much more than algebra and calculus. It will teach you pattern
recognition and deduction - two highly sought after skills in engineering -
but can be applied to any profession.

If anything I believe we should bump up the level of math that high school
children learn.

~~~
sanxiyn
You seem to imply that actual mathematical content is rather less important.
Then why stick with algebra and calculus?

Frankly, I agree with Hacker: quadratic equations are boring. Teaching number
theory will do better job of teaching pattern recognition and deduction. More
Fermat's little theorem, more Chinese remainder theorem, less quadratic
equations. I found undergraduate number theory to be far easier and more fun
than calculus, and you need no calculus to start on number theory.

~~~
doc4t
> You seem to imply that actual mathematical content is rather less important

No. The point was that it is much more than what you call the content.

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impendia
"Nobody ever brags about not knowing how to read, but people brag all the time
about not knowing how to do math."

This is not just math, it's virtually every subject you learned in school.
Imagine the following at a cocktail party:

"I had to read Joyce, Whitman, and Hawthorne, but I never got the point of any
of them."

"I took a lot of history, it was a bunch of dates which I could never
remember."

"I took three years of Spanish, but I don't remember a word."

Et cetera.

~~~
toomuchcoffee
All perfectly understandable. It's called "the path less taken."

A lot of people just don't click with formalized education at all in their
early years. They just couldn't throw themselves behind all those authors and
facts they're "supposed to" be up on. Some of them are too busy learning
_informally_ by dealing with the world around them -- i.e. with things like
nature, work, and you know, _other people_.

Then slowly grow to form their own view of the world, and seek their own path
to obtaining a more structured comprehension of it -- which might well mean
starting (on their own volition) formalized education sometime in their 20s or
30s -- or perhaps not.

A great many who choose this route, or have it forced upon them may fail or
flounder -- but a great many also don't, and end up with accomplishments for
others to read about or hear about in those books, classes and received
intuition that some people think are the only valid starting points to
figuring out how to make a dent in the universe.

------
pirateking
My dream educational system destroys the concept of "subjects" - no more
siloing knowledge within artificial boundaries. Students think things like
"Algebra is Math and I hate Math so I hate Algebra" or "Math is my worst
subject." It is an easy label to associate their failures with.

An example of learning without boundaries:

One day, I decide I want to play the guitar. I get my hands on one and fiddle
around. Perhaps a brief history on the instrument will give me some better
context as to why the instrument is shaped as it is, and what all the features
are. An exposure to famous guitarists and their music improves my appreciation
of music and gives me context to work with. As I continue to struggle with
notes and basic music theory, it may be a good time to take a brief detour
into the realm of physics - a lesson in simple harmonics. If interested, I can
dive further into the required algebra relevant to harmonic equations. Hmm...
maybe learning to use a computer program can help me figure out how messing
with these waves changes the sound...

And so it goes. A familiar story to most hackers, but sadly an approach that
most educational systems do not encourage.

Is this exploratory form of learning superior? My theory is that the
associations formed between nodes of knowledge are necessary for any sort of
actual deep learning to occur. By sandboxing subjects, the standard school
curriculum limits the possible associations that students think are allowed,
thus limiting the probability of successful links between nodes. Even an above
average student can graduate from college, and be left with nothing but a
bunch of barely reachable random islands of knowledge, floating further out to
sea every day.

In other words: school sucks, learning rules.

~~~
kaiwetzel
I really love that idea. I think a lot of students which are bored by school
(and/or feel left behind) would regain their natural curiosity in such an
environment. I think the technical possibilities we have now make much more
individual approaches to learning possible and computers can help with this
cross-subject approach.

Based on your example they could then go on to construct some instrument (say,
a flute) by writing a little program and print the instrument with a 3d
printer or have a rough version cut by a laser cutter. The question is: how to
integrate ideas like that in the current school system? ...

~~~
pirateking
I can see an immersive virtual world being amazing for encouraging curiosity.
This world, the information stored within it, and the humans who frequent it,
would attempt to encapsulate all human knowledge. Sound familiar?

We already have such a world - it is the Web. And we know it is a mess, a time
sink, and a minefield. However, with focused effort, it does function as an
amazing learning environment.

Unfortunately, it is not a suitable alternative to school yet. Telling kids to
drop out of school and hit the 'Net instead is most likely a bad idea.

Perhaps the future role of Teachers, will be to function as Guides in this
virtual world of information. Most awesome field trip ever!

------
jayferd
Nobody seems to understand what college-level math _is_. This isn't balancing
checkbooks, folks. I loved being a math major, and _I_ have trouble doing
simple math. College-level mathematics is about as far removed from the "real
world" as you can get. We convince ourselves that this learning is valuable
because those who have certificates saying they've learned it get better-
paying jobs -- because we've convinced ourselves that it's valuable.

Don't get me wrong - math is awesome, and everyone, especially young people,
should have the opportunity to explore it. Perhaps it's helpful to think of
math as a cultural activity, which is mostly removed from practical concerns.

~~~
eshvk
I don't think she is talking about Math majors. She is talking about some
requirements that other majors have to complete as part of finishing up their
degree. Having said that, I know people in say computer science who struggled
through Calculus not really caring for the subject. I can't imagine how things
were for people in less technical disciplines. While Calculus is certainly
important for CS majors (well probably about as much as say randomized
algorithms), I can't help but ask why these classes are not replaced by a
class in first order logic, especially if the end goal is to train people to
think mathematically.

~~~
sanxiyn
Calculus is certainly not important for CS majors. Compared to, say, linear
algebra.

I always recommend [http://steve-yegge.blogspot.com/2006/03/math-for-
programmers...](http://steve-yegge.blogspot.com/2006/03/math-for-
programmers.html) on math for programmers.

~~~
shrughes
How the heck is linear algebra important for CS majors? Where in CS is linear
algebra useful where calculus isn't?

~~~
UK-AL
Machine Learning, Graphics etc

On the other hand calculus is also used in machine learning...

Plus optimisation via calculus is sometimes a better choice then other cs
specific methods.

~~~
shrughes
And calculus is used in graphics.

------
podperson
Well, it's not an example of Betteridge's law ;-)

The political science professor in question might consider the fact that while
many college students struggle with basic math, they struggle just as much
with basic literacy (most undergraduates can't write grammatical sentences,
use punctuation, don't know how to write an essay, and have a hazy grasp of
the idea of citation). Extending his argument we might consider removing
History and English from the curriculum.

------
sanxiyn
Go read "How should mathematics be taught to non-mathematicians?" for a
mathematician's take on the topic.

[http://gowers.wordpress.com/2012/06/08/how-should-
mathematic...](http://gowers.wordpress.com/2012/06/08/how-should-mathematics-
be-taught-to-non-mathematicians/)

------
jopt
Is this problem unique to mathematics? Consider natural sciences, history,
geography.

It's easy to make up real-world examples of when each could be an advantage,
but it's hard to say anything is necessary. It is what you make it.

~~~
sanxiyn
The article is not really talking about mathematics. It is talking about
numeracy, which is on par with literacy in importance in my opinion.

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ekm2
So...why shouldnt we phase out Poetry too?Is there a definitive study showing
that training in poetry makes for more competent assembly line workers and
lower unemployment levels?

~~~
peschkaj
I think we should commission such a study. Those of us with English degrees
will be perfectly suited to conducting this survey and delivering the results
in epic poem format.

------
sharms
I would say that the basic college level math courses have helped me
tremendously in non-obvious ways. Companies run on spreadsheets and formulas,
and to better predict and analyze situations, it is easier if all parties
involved have a common understanding.

Our economies are huge, number driven in almost every aspect. To ignore that
or turn out graduates are less well rounded would not be more productive as a
whole.

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Tycho
The thing about mathematical literacy and financial contracts (credit
agreements etc) is that if those contracts are constructed in such a way as to
deliberately confuse the average customer/borrower, then increasing the
average level of education wont actually help. The contracts will just get
even more confusing to compensate.

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comicjk
> Yes, young people should learn to read and write and do long division,
> whether they want to or not.

Ironic, since in our modern computerized world algebra is important and long
division is useless. The writer himself probably carries a powerful computer
in his pocket.

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PaulHoule
people say they want science and math skills but do they really?

when i was a 10 yr old kid in 1982 i wanted to create email and newsgroup type
systems and my computer science teacher was like that's a big waste of time
like video games, get thinking about data structures and algorithms instead

10 years later I'm in college and the guy who runs the computer center got a
source code license for SunOS and fubar-ed our whole system because he wanted
to block our access to the internet, which would stop our academic pursuits.

hell we were just trying to be matt zuckerberg a decade before his time.

or there's this guy who was the one bright kid who remembered what he learned
in organic chemistry class. all the other muggles sold their textbooks and
forgot about it, but he applied what he learned to invent a way to make a new
drug with stuff you can buy at the grocery store.

this grad school dropout got his work cited in the "journal of emergency
medicine" because some kid made a batch of this vile brew and chucked all over
his bed and the doc in the ER is like "what the f'uh?", "i mean you read this
on the f'en internet and really did it?"

And all this time this guy and his crew are worried they're going to get
called to testify about this in Congress and they were just so happy when the
statue of limitations came up.

If students studied science and engineering and got both the ability and the
will to confront technological society, we'd have a revolution.

~~~
apl
A little coherence goes a long way.

~~~
PaulHoule
...and James Joyce showed you can get lit profs to take what you write
seriously without any coherence

~~~
gcv
...only in Finnegans Wake. And let's be fair to English professors, many of
them don't take that book too seriously. :)

------
photon137
Yes.

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pliny
No.

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alpine
'...people brag all the time about not knowing how to do math'. This is true,
I think, mostly of Anglo-Saxon countries. It is quite a disgusting attitude.
In what way is someone's life-chances or overall happiness degraded by a
better understanding or skill in the use of numbers and higher mathematical
concepts? I consider this anti-maths meme one of _the_ most pernicious aspects
of our culture.

~~~
sanxiyn
I think this is misunderstanding of cause and effect, and correlation does not
imply causation. High math skill does correlate with nerdiness, and ceteris
paribus, nerdiness does degrade your life chances or overall happiness. It
does not mean high math skill alone, if you are not nerdy, will degrade your
life chances, but well, they do not have high math skill to understand that.
:)

~~~
alpine
I wasn't so much thinking about innate mathematical ability, more the
mathematical education most kids should receive. Disparaging attitudes towards
maths skills, which is a general rule, is pretty harmful to society, I think.

