Does mathematics have a place in higher education?

http://mathbabe.org/2012/07/29/does-mathematics-have-a-place-in-higher-education/

by cottonseed • 14 years ago

70 106 14 years ago

mathbabe.org

photon1372• 14 years ago

Yes.

pliny• 14 years ago

No.

jneen2• 14 years ago

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.

photon1372• 14 years ago

Your are confusing Math versus Applied Math. We take what we want to use in the study of engineering, physics, economics, finance and political science - pure Math develops those tools and they're highly useful.

Is Math of a college level necessary for everyone? Certainly not. Should algebra be necessary for everyone - I think so (I think the same way about calculus).

Nothing is necessary to perform your day-to-day duties - not even arithmetic - people have been getting by and been leading fruitful lives for thousands of years without that sort of stuff. So when people talk about learning Math as a life-skill they're dead wrong.

But if you want to anything more that just get by - if you want to ask questions and support your arguments, analyze problems and come up with answers you can trust you gotta, oughta, shoulda learn Math.

eshvk4• 14 years ago

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.

_delirium2• 14 years ago

I agree that formal logic is probably a more relevant math-y course than calculus for the average CS student. But how to design these requirements is a longstanding problem, partly due to disagreements over what should be in a "technical core", and partly due to staffing and issues of uniformity.

Calculus is probably more relevant than symbolic logic for many kinds of engineering, for example, so the course is already being offered. It's also seen by (most) mathematicians as a more central subject, so it's what goes in the core if the mathematics department is given responsibility for designing the core. Additional complications arise over what to do with things like statistics (in some places it's in the math department, while in others it's a separate department).

sanxiyn16• 14 years ago

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... on math for programmers.

jneen2• 14 years ago

Spot on. All I know is the situation at Berkeley, but I imagine it's similar at other universities. The lower-division math curriculum (particularly intro calculus and linear algebra) is under pressure from a bazillion different departments to cover such a wide range of topics that 90% of the material is useless to 90% of the students.

For example, a friend of mine was a forestry major, and the only reason she needed calculus was for population modeling. She knows exponentials and logistics better than I do! But she had a hell of a time passing calculus because she had trouble with the other 90% of the course. Keep in mind that this is a discipline that until maybe a decade ago required zero math.

Maybe at least at the lower-division level, applied math needs to be a bit more targeted...? But that would require us letting go of the notion that math is intrinsically edifying, and that people "ought to", "must", or "should" know it.

colanderman• 14 years ago

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.

pacala• 14 years ago

+1 on explicit grammar. In my recollection, that was the biggest source of frustration in studying math subjects: decoding the actual meaning of a math text fragment and separating random elisions from errors.

podperson3• 14 years ago

It's funny how in grade school the multiplication symbol is omitted, only to be (more confusingly replaced) by a period later on when the expressions get complicated.

I suspect it's the lack of arithmetic skill more than the conventions that are the heart of the problem. I think kids should be required to learn to do mental arithmetic before they're allowed to use calculators, but this may be impractical.

sanxiyn16• 14 years ago

Why is it impractical? (It's the way we do it in South Korea.)

toomuchcoffee4• 14 years ago

I think kids should be required to learn X before they're allowed to do Y,

Clearly you were never a kid.

gcv4• 14 years ago

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.

sanxiyn16• 14 years ago

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.

gizmo6864• 14 years ago

My high school has a relatively progressive math program, and we never learned d/dx as a function. We began calculus with limits, then started solving Lim[dx->0](dy/dx), then we started using d/dx as a shorthand so we did not need to pull a nested expression out to take its derivative. I never thought to think of d/dx as a function until I started doing functional programming.

ekm2• 14 years ago

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?

sanxiyn16• 14 years ago

We probably should.

streptomycin• 14 years ago

If there is anywhere where some poetry class is a requirement for graduation for all majors, that requirement probably should be phased out.

peschkaj• 14 years ago

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.

ef4• 14 years ago

"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.

guscost5• 14 years ago

Math became ludicrously easy for me as soon as I realized how to think about it. It wasn't the same method that they were suggesting in school, and I don't know that my method would work for everyone.

ThomPete• 14 years ago

As someone who still haven't found a way to think about it I would be curious what you ended up with.

drieddust2• 14 years ago

I will be very much interested in learning what method did you use.After dropping from regular education 12 years ago, I have tried to learn mathematics multiple times with moderate success only.

Please try to share your methods via a blog post/elaborate comment if you can spare some time.

gcv4• 14 years ago

Nicely put. I have increasingly come to the view that computer science really is a branch of applied mathematics. My most intellectually rewarding recent work has been on (programming) projects where I get a chance to use some computational geometry and linear algebra.

It goes without saying that almost every non-trivial project requires serious thinking about time and memory complexity costs, and I've been surprised to find theory of computation useful when working even on simple parsers (nothing fancy from compilers classes, just straightforward reading of proprietary formats).

mseebach2• 14 years ago

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.

sanxiyn16• 14 years ago

This is completely shocking to me. Is it really allowed not to be able to do 12 * 9 in your head after completing school? I am serious. (Since this is not the case in South Korea.)

stan_rogers• 14 years ago

What's odd here is the line of demarcation between mathematics (or technical subjects in general) and liberal arts -- two of the quadrivium are arithmetic and geometry. And while we can probably agree that the influential aspects of astrology are no longer worth serious study (except, perhaps, from a historical, psychological or sociological perspective), the periodicity -- the astronomy -- of astrology was also considered part of a well-rounded education. (Music was the fourth element of the quadrivium.)

Perhaps it would be best to refer to a completely non-technical education as trivial since it ignores more of the liberal arts than it incorporates.

Someone2• 14 years ago

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?

spockofborg2• 14 years ago

I started programming when I was 9, mostly taught myself to convert between hex, binary and decimal at about 13. I say mostly out of fairness: high school did spend a day or two on it one year. Needless to say I dropped out in 9th grade; I learned things that would be applicable in the real world better out of school than in. I was making 45K/yr, starting salary, by my mid twenties.

Despite that, I'm going to college currently for my degree. I just got an "A" in intermediate algebra, though it was an up hill battle the whole way. The struggle, aside from it being a 6 week summer class, was trying to convince myself that I wasn't wasting my time; that I would learn something applicable to life outside that or the next three requisite classes. Something, that is, besides helping my kids with their math homework.

I've come to the firm conclusion that college math exists purely to perpetuate itself, like a virus in our education system; The only job it prepares someone for is to teach the damn class!

spockofborg2• 14 years ago

side note: As a programmer I specialized in both data compression and cryptology.

sethammons• 14 years ago

I've found that it is nearly impossible for students to find any math useful until they really understand it. As a simple example, you don't need to know nor understand how to multiply. After all , you could just repeated addition and figure it out. However, there is no need for addition either; you can just count up for a while. point being, the more math you know, the more tools you have. you may never use the quadratic formula outside the classroom, but I use algebra all the time to make my life at both work and home easier. please excuse typing errors as I'm replying from my phone.

wcarey• 14 years ago

It would be interesting to understand how "school mathematics" displaced "actual mathematics" in our educational system.

I suspect that part of explanation has to do with computers. The word "computer" was originally a job title (see, for example the Harvard Computers: http://en.wikipedia.org/wiki/Harvard_Computers). It changed meaning in the middle of the twentieth century when mechanical (and later electronic) computers displaced human computers.

It seems like much of "school mathematics" is vocational training to work as a computer. Unfortunately, we don't employ people to do computation anymore because the cheapest and worst computer you can but will be thousands of times more productive at computation than a human being.

If our mathematics curriculum was designed to train people to be computers, and that profession no longer exists, it's no wonder that school mathematics seems pointless and disjointed.

sliverstorm• 14 years ago

I think "actual mathematics" was used to refer to abstract and/or high level mathematics, the sort a math major might take in college. While "school mathematics" was used to refer to arithmetic and algebra.

grrabear• 14 years ago

If by "school mathematics" you mean algebra, then I completely disagree with you. Algebra is essential for almost every other mathematical concept, not to mention everyday tasks. Just because a computer can do it easily, doesn't mean that people should rely on them to do simple calculations when they could have easily learned how to do it on their own. That is terribly inconvenient and a waste of resources.

jopt• 14 years ago

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.

sanxiyn16• 14 years ago

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

doc4t2• 14 years ago

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.

sanxiyn16• 14 years ago

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.

doc4t2• 14 years ago

> 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.

podperson3• 14 years ago

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.

sharms• 14 years ago

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.

PaulHoule2• 14 years ago

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• 14 years ago

A little coherence goes a long way.

PaulHoule2• 14 years ago

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

impendia• 14 years ago

"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.

quacker• 14 years ago

You're right, but he's claiming this at a lower level. I imagine it's more embarrassing to have to ask, "What does this say?" than to have to ask, "How much change should I receive?"

toomuchcoffee4• 14 years ago

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.

Tycho• 14 years ago

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.

DanielBMarkham• 14 years ago

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.

stdbrouw• 14 years ago

> Math is the best way that I know of to do this.

But is it the best way educators know of to do this? Because at the end of the day, whether math is a good basic skill that pays dividends in logical thinking and symbolic manipulation (dividends beyond what other methods could deliver) is an empirical question. What you say sounds plausible, but theories can be plausible and dead wrong.

swordswinger12• 14 years ago

That is a really pretty analogy. My compliments.

toomuchcoffee4• 14 years ago

But is also flawed, because its premise is flawed.

gizmo6864• 14 years ago

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.

sanxiyn16• 14 years ago

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.

billsix• 14 years ago

Because calculus is the first class where functions (e.g. derive, indefinite intergration) take a function as its input, and outputs a function.

nekojima• 14 years ago

> Everybody needs to understand formal symbolic systems.

Everybody does not need to understand this and I dare (with fingers crossed) say a good percentage of those on HN don't either. There needs to be rationalization within the educational system for what is taught, with relevance to education for the sake of knowledge and for how material can be used for future employment prospects & general life skills. That may play to the lowest common denominator or the student mean, but that's largely and unfortunately what public education has become over the past fifty years.

kaiwetzel2• 14 years ago

I'm dreaming of a curriculum where students start to program their own computer algebra system in 9th grade (or earlier), say, using Python or Clojure. Starting by teaching it to solve simple equations for x, then proceed over the next 4 years by extending it to know about binomial formulas, solving quadratic euations, perform gaussian elimination, symbolic differentiation and integration, etc. I guess the world isn't ready for that idea, yet ;)

sanxiyn16• 14 years ago

Automatic differentiation by dual number! Really, after I learned it, I wondered why symbolic differentiation isn't taught this way by default. Good use case of operator overloading, by the way.

toomuchcoffee4• 14 years ago

The future is computers,

Let's rather hope they're a passing phase, please.

pirateking3• 14 years ago

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.

alpine3• 14 years ago

So you aspire to be a Renaissance Man?

TheAmazingIdiot• 14 years ago

I guess you could put it that way.

I see in his description more of a tech tree. It'd be more akin to what you'd see in Diablo 2, but expanded by a 1000. The tech tree method would also explain prerequisites. Ideally, if you can prove you have the knowledge of a node, you could pass through it and continue learning.

No clue how we would go from factory farmed schools to that though. It's another educational pie-in-the-sky thought.

kaiwetzel2• 14 years ago

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? ...

pirateking3• 14 years ago

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!

alpine3• 14 years ago

'...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.

sanxiyn16• 14 years ago

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. :)

alpine3• 14 years ago

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.

sanxiyn16• 14 years ago

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...

comicjk• 14 years ago

> 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.

Does mathematics have a place in higher education?

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