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Mathematics As the Most Misunderstood Subject

timothy posted more than 3 years ago | from the philosophical-engagement dept.

Math 680

Lilith's Heart-shape writes "Dr. Robert H. Lewis, professor of mathematics at Fordham University of New York, offers in this essay a defense of mathematics as a liberal arts discipline, and not merely part of a STEM (science, technology, engineering, mathematics) curriculum. In the process, he discusses what's wrong with the manner in which mathematics is currently taught in K-12 schooling."

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Inital comment (0, Funny)

Anonymous Coward | more than 3 years ago | (#34639278)

Commenting on something near the top of the list.

he's right (1, Insightful)

FuckingNickName (1362625) | more than 3 years ago | (#34639294)

Mathematics is the foundation for philosophy, not technocracy. What a better world we'd be in if we were motivated by the former rather than pursuing the latter.

Re:he's right (3, Interesting)

Anonymous Coward | more than 3 years ago | (#34639314)

Yes, the problem teaching Math(s) and programming (applied Math(s)) is that it's just about intelligence - which you can't teach. The smarter you are, the better you'll be able to figure it out. The problem with teaching is all the generalists who think because they have a "Degree in Education" they are able to teach any topic. Traditionally dry disciplines need to be taught by specialists with passion and enthusiasm for their topic, not by generalists who happen to have a gap in their timetable.

Re:he's right (5, Interesting)

FuckingNickName (1362625) | more than 3 years ago | (#34639358)

The brain can be trained and the processes of problem-solving can be generalised - see Polya's How to Solve It. But it doesn't help much to just read the book: you've got to practice, and practice, and practice some more. You must make mistakes and learn from them. You must be prepared to accept multiple inputs rather than merely those which reinforce your strengths and/or prejudices. You must sometimes, as the old 9/11 troll used to say, get some perspective - don't count the angels on a pinhead while Rome burns, even while the most secure of academic positions involves the former and there's such an alluring spirit of mental masturbation in many disciplines and departments.

Meanwhile a good teacher has spent enough decades on some area that he knows both where to provide you hints on specific complex problems and which direction to guide you in when you're contemplating your whole professional life. But, again, don't just choose the teacher who happens to share your academic and ethical prejudices.

Re:he's right (2)

gilleain (1310105) | more than 3 years ago | (#34639574)

...the processes of problem-solving can be generalised - see Polya's How to Solve It. ..

"How to Solve It" also talks about more general problem-solving than just mathematical problems - crossword puzzles, for example. Prof. Lewis's article talks about the universal question "Why did they teach me the quadratic formula when I will never use it?" and this is really the answer; doing mathematics (should) teach people how to solve any problem logically. Well, any problem that can be solved logically, of course.

Meanwhile a good teacher ...knows where to provide you hints

Heh. Although a bit dry, one fun part of the book is where Pólya talks about giving hints to students : "Yet the teacher should be prepared for the case that even this fairly explicit hint is insufficient to shake the torpor of the students; and so he should be prepared to use a whole gamut of more and more explicit hints".

Re:he's right (1)

kiddygrinder (605598) | more than 3 years ago | (#34639668)

heh, that is one of the most unrealistic comments i've seen in a while... specialist teachers with a passion and enthusiasm for maths? i've only ever met one, and he sucked balls as a teacher. maybe we should start recruiting fairies or goblins, maybe they'll get the job done.

Re:he's right (1)

digitig (1056110) | more than 3 years ago | (#34639692)

I had teachers with a passion and enthusiasm for maths all the way through school. Condolences for your experience.

Re:he's right (2)

tehcyder (746570) | more than 3 years ago | (#34639750)

I had teachers with a passion and enthusiasm for maths all the way through school. Condolences for your experience.

I had various teachers with passion and enthusiasm for chemistry, geography, German, English literature, physics, Latin, history, PE and maths, and none of them were PhDs or anything, just good teachers.

Re:he's right (2)

Stooshie (993666) | more than 3 years ago | (#34639854)

A Ph.D. tells you nothing except that the holder did some original research at an early point in their career.

Re:he's right (5, Interesting)

CProgrammer98 (240351) | more than 3 years ago | (#34639738)

You should have had Mr Burton, my maths O level teacher. He was brilliant. He was totally passionate about his subject and he was also a fantastic teacher. he encouraged us to think about maths rather than to just blindly follow formulae. I still vividly remember the lesson where he taught us differential calculus from first principles.

He encouraged us to study outside of lesson time and his door was always open during lunch, or after school. almost every one in his class passed their maths O level with at least a B, over half had A's

It's no exageration to say I owe my career as a developer to him and his enthusiastic teaching.

Re:he's right (0)

Anonymous Coward | more than 3 years ago | (#34639336)

The other way around Brodo. Philosophy is the basis for Logic, which is exactly 33.3% of Math. (And because you are just dying to know: the other 66.6% is evenly divided between Set Theory and Numbro Theory, the remaining permil is magic).

Re:he's right (1)

FuckingNickName (1362625) | more than 3 years ago | (#34639384)

OK, define philosophy without logic.

Re:he's right (5, Funny)

Anonymous Coward | more than 3 years ago | (#34639398)

Okay.

Philosophy is the process of speaking greek and stroking beards. Therefore by stroking a grecians beard, I shall become a philosopher.

I have defined philosophy (badly) and applied a complete absense of logic. Is that not what you meant?

Re:he's right (2)

FuckingNickName (1362625) | more than 3 years ago | (#34639422)

It's time to stop posting.

Re:he's right (0)

Anonymous Coward | more than 3 years ago | (#34639424)

Philosphy: The Love of Wisdom. You don't need syllogisms to ask "What does it mean to be?" and think really hard about it.

Re:he's right (1)

FuckingNickName (1362625) | more than 3 years ago | (#34639548)

You can't think really hard about anything without syllogism. Try it.

Re:he's right (1)

CProgrammer98 (240351) | more than 3 years ago | (#34639756)

(V)--We demand that you cannot keep us out.
"(Priest)--Who are you?
(M)--I am Majikthise [pronounced Magic Thighs].
(V)--And I demand that I am Vroomfondel.
(M)--You don't need to demand that.
(V)--All right. I am Vroomfondel and that is not a demand, that is a solid fact. What we demand is solid facts.
(M)--No, we don't. That is precisely what we don't demand.
(V)--We don't demand solid facts. What we demand is a total absence of solid facts. I demand that I may, or may not, be Vroomfondel.
(Priest)--Who are you?
(M)--We are philosophers.
(V)--Though we may not be.
(M)--Yes, we are!"

Douglas Adams, what a guy!

Re:he's right (1)

Stooshie (993666) | more than 3 years ago | (#34639862)

Try defining logic without philosophy.

Re:he's right (1)

digitig (1056110) | more than 3 years ago | (#34639706)

Try doing Set Theory and Numbro (Numbro?) theory without logic.

Re:he's right (5, Interesting)

ShakaUVM (157947) | more than 3 years ago | (#34639388)

>>Mathematics is the foundation for philosophy

Eh, kinda. Advanced logic is the foundation for a lot of modern philosophy, but Wittgenstein and the rest of the 20th century analytics were just responding to the tremendous success of physics at figuring shit out, and wanted to smear some of that patina on themselves. Well, logic has always been a part of philosophy (think Socrates and his syllogisms) but reading the Tractatus is like reading a modern computer science proof.

Which isn't surprising, either, given that computer science is essentially applied philosophy in a lot of ways. (cf Bertrand Russell, etc.) If you've ever sat through a class where philosophers have sat there talking themselves in circles about how an object can't both be is-a and has-a at the same time, you (if you're like me) feel like leaping up and just telling them to fucking encode whatever paradox they're trying to create in a object hierarchy, and be done with it. I've long longed to write a book called "Computer Science has figured a lot of your shit out in practice, Philosophers".

It does kind of bug me though, that a person who graduates with a degree in mathematics (which is a fairly difficult, hard-nosed subject) gets a wishy-washy BA degree, whereas a hippie with a degree in "environmental engineering" gets a BS, but ultimately I think there's a lot of problems with our current conception with categorizing things into "science" and "not-science". Economics and Climatology are very analogous in terms of what they do - gathering tons of data, running analyses on it, and projecting things out into the future, and both are essentially "empirical studies of the world about us" (i.e. a sort of base level of science, though with the testing, replication and confirmation bits left out), but we consider one to be a social science and another to be hard science. There's also a huge debate now over Anthropology, after the American Anthropology Association dropped "science" from its official bits.

Re:he's right (3, Insightful)

ifiwereasculptor (1870574) | more than 3 years ago | (#34639644)

Economics and Climatology are very analogous in terms of what they do - gathering tons of data, running analyses on it, and projecting things out into the future, and both are essentially "empirical studies of the world about us" (i.e. a sort of base level of science, though with the testing, replication and confirmation bits left out), but we consider one to be a social science and another to be hard science.

Well, economics is, especially in its present state, largely influenced by individuals, who can be a lot harder to predict than wind currents. You may identify trends, constants and correlations, but mostly in hindsight. Accurate predictions are as scarce as in cartomancy and useful controlled experiments are hard to imagine. While Climatology shares some of those characteristics, I think we have a much higher chance of predicting a storm than the stock market. Unless tons of people start walking around with nuclear powered, oversized fans. If you catch my drift.

Re:he's right (0)

Beale (676138) | more than 3 years ago | (#34639714)

This will be the next big plan to discredit climate change.

Re:he's right (5, Interesting)

grouchomarxist (127479) | more than 3 years ago | (#34639686)

If you've ever sat through a class where philosophers have sat there talking themselves in circles about how an object can't both be is-a and has-a at the same time, you (if you're like me) feel like leaping up and just telling them to fucking encode whatever paradox they're trying to create in a object hierarchy, and be done with it. I've long longed to write a book called "Computer Science has figured a lot of your shit out in practice, Philosophers".

I understand where you're coming from, but for many philosophers, what they're doing is not just trying create a practical solution to a problem, but describe reality. Your object model might solve the problems from your point of view, but it includes many built in assumptions about the thing modeled.

In a related way Wittgenstein later came to criticize the Tractatus. Part of the criticism is that if you assume the universe can be fully described with formal logic (logical atomism), then you are already subscribed to a certain type of metaphysics.

Re:he's right (1)

digitig (1056110) | more than 3 years ago | (#34639748)

If you've ever sat through a class where philosophers have sat there talking themselves in circles about how an object can't both be is-a and has-a at the same time, you (if you're like me) feel like leaping up and just telling them to fucking encode whatever paradox they're trying to create in a object hierarchy, and be done with it. I've long longed to write a book called "Computer Science has figured a lot of your shit out in practice, Philosophers".

The philosophers are way ahead of you. The trouble with your method is that it only shows whether something is possible or not under one particular model of reality, the object-hierarchical model. If you show that it's possible under that model then fine, job done: it's possible. But if you show that it's impossible under that model you've not shown that it's impossible in general. And frankly the most likely outcome is that you can't work out how to do it but can't prove that it's impossible without sitting down with the philosophers and listening to their arguments.

Re:he's right (4, Insightful)

tehcyder (746570) | more than 3 years ago | (#34639828)

I've long longed to write a book called "Computer Science has figured a lot of your shit out in practice, Philosophers"

Well, go on then, if it's that fucking simple and obvious. Put those silly old philosophers in their place, what do they know?

I'm thinking of writing a book called "Why do so many students of Computer Science think they have solved all the riddles of the universe because they know how to write a sorting algorithm?"

Re:he's right (5, Insightful)

ultranova (717540) | more than 3 years ago | (#34639434)

Mathematics is the foundation for philosophy, not technocracy. What a better world we'd be in if we were motivated by the former rather than pursuing the latter.

Well, we would likely all be malnourished, due to lack of fertilizers, at least those of us who hadn't died at childbirth or soon after. There wouldn't be an Internet to talk on, but that would be okay, since we wouldn't have time to use one due to the lack of engines and the resulting need to do backbreaking labour 16 hours a day. In short, our lives would be miserable, but due to lack of medicine, they would at least be short.

Missing these kinds of little details is why I have very little respect of philosophers. As far as I can tell, most of them chose their field because it doesn't punish sloppy work. And then there's idiocy like the Chinese Room, which assumes that a system cannot have properties its components don't have, yet hasn't been laughed out like it should had been.

Philosophy means you accept the human condition. Technorcacy means you try to do something about it. Hope for a better world in the future lies on the latter, not the former.

Re:he's right (2, Insightful)

FuckingNickName (1362625) | more than 3 years ago | (#34639538)

So over the past two millennia we have cut the working day by 1/3rd and doubled the average lifespan at birth (if you ignore infant mortality, our lifespan hasn't increased that impressively).

Meanwhile we have turned the majority of Western humans from independent men into chair-warming consumers singing in lockstep for trinkets. We've made up for the opportunity to live a life of leisure surrounded by virtually infinite resources by blasting our population beyond 6 billion.

Technocracy is for the lazy man who wishes to be controlled and for the fascist who wishes to control others. The technocrat only has to think about one thing. But philosophy regards technology as one of many tools, not as a master. The philosopher-ruler (for philosophy is a basis for living, not an alternative) must not let prejudice cause him to dismiss the possibility that he can do better and for more.

Re:he's right (-1, Redundant)

durrr (1316311) | more than 3 years ago | (#34639642)

So your advice is what exactly, that we all adopt an ill-defined philosopher-lifestyle/society and suddenly have the world turn utopian from our current day dreary and grey existance? To me that sounds like every other ideology out there that promises freedom, comfort and an enjoyable life for all and then fails to deliver while turning into some facist derivate.

Philosophy is only a pretty word for wild speculation/daydreaming/brainstorming, it's a narrow tool, not a fundament of human life. The people who fancy themself as philosophers are only lazy people who wish to get paid for just thinking random thoughts and producing fancy pants arguments and ideas with little to no real world value. They also fancy their fact-devoid opinion to somehow be worth more than anyone else, because hey, they are philosophers.

The only true philosophy is the one that a philosopher holds, which manifests in literally endless arguments about philosophy between philosopher, it is argument for arguments sake and there's no solution to it which manifests in people debating subjects ranging from hundred to thousands of years old, trying to find support for their view.
Guy 1:"Hay guise, Plato support my view!"
Guy 2:"Lol no, my better analysis of his philosphy says you're wrong"
Guy 3: "You're both wrong because Plato was wrong, this have been proven by neo-philosophistic logicism!"
That's more or less the core of philosophy.

Re:he's right (2, Insightful)

FuckingNickName (1362625) | more than 3 years ago | (#34639710)

Philosophy is the path by which every man continually asks questions of his condition and can thereby strive to improve it. It is something practiced while living, not instead of living (as "pursuit of happiness" is the ongoing enjoyment of happiness, not the singular and final goal of happiness). You may as well argue that man should not breathe because people who breathe are wasting their time only breathing when they should be doing other things.

Philosophy does not give a single solution to the world's ills and it does not force you to do anything or to make others subordinate to your will. I'm not sure what you're afraid of, but it's not philosophy.

Re:he's right (5, Insightful)

ifiwereasculptor (1870574) | more than 3 years ago | (#34639672)

Why are people even debating philosophy vs technocracy? Why should someone have to choose one over the other? How do people get dragged into such nonsense? Here a new subject for you: tomatoes vs rainbows. Go.

Re:he's right (1)

CProgrammer98 (240351) | more than 3 years ago | (#34639774)

Is that Tom-aaaaah-toes or tom-eh?-toes...??

What a load of crap (5, Insightful)

Viol8 (599362) | more than 3 years ago | (#34639782)

"Meanwhile we have turned the majority of Western humans from independent men into chair-warming consumers singing in lockstep for trinkets."

I suggest you take off your rose coloured glasses and go read some history, in particular just how "free" your average serf was in feudal times and even later. Don't like what your overload or king does? Tough. Complain and you'll probably at best end up homeless or at worst end up swinging from a tree.

People in the west have NEVER been as free as they are now.

So get yourself a fucking clue!

Re:he's right (5, Insightful)

CRCulver (715279) | more than 3 years ago | (#34639592)

due to the lack of engines and the resulting need to do backbreaking labour 16 hours a day.

While agriculture requires backbreaking labour, hunter-gatherer societies only worked a couple of days a week. Not that I advocate a return to it, but backbreaking labour all the livelong day was not universal in ancient society.

As far as I can tell, most of them chose their field because it doesn't punish sloppy work.

Philosophical journals have the same rigorous standards for papers as journals for the various sciences. Your view of philosophy is about as valid as a grizzled mountain man who mutters about hard science being all book-learnin' and mumbo-jumbo.

Philosophy means you accept the human condition. Technorcacy means you try to do something about it.

Even that is a statement of philosophy. Furthermore, you seem unaware that many calls for improving human lives came from works of philosophy: More's Utopia, Kirkegaard's questions of metaethics, even what is often called the beginning of the Western tradition, when Socrates hung out in the agora and asked passersby "What if what you comfortably believe is wrong?"

Re:he's right (1)

Nrrqshrr (1879148) | more than 3 years ago | (#34639600)

The very source of disagreament about this subject comes from this question "What is a better world?". Philosophy has it's answer, technocracy have got a different one.

Re:he's right (1)

digitig (1056110) | more than 3 years ago | (#34639842)

Not quite. Philosophy proposes lots of possible answers, technocracy selects one of them.

Re:he's right (3, Insightful)

Kashgarinn (1036758) | more than 3 years ago | (#34639670)

Philosophy means you accept the human condition.

No.. Philosophy means questioning the human condition. it's confronting the status quo and asking "why?"

So exactly the opposite in every way of what you think it is.

You're also wrong in your assumption that philosophy and technocracy are mutually exclusive, in fact if they aren't mutually inclusive, then as a technocrat you're trying to find solutions when you don't even know what the problem is.

Philosophy is a very powerful way of thinking, and in no way whatsoever does it represent conformity or acceptance, it represents freedom of thought to think critically.

In fact philosophy really should be tought in schools, it's the basis of how we view the world today, and if the future is bright, it will be philosophy to thank and the people who dared to question the status quo.

Re:he's right (3, Interesting)

horigath (649078) | more than 3 years ago | (#34639694)

Missing these kinds of little details is why I have very little respect of philosophers. As far as I can tell, most of them chose their field because it doesn't punish sloppy work. And then there's idiocy like the Chinese Room, which assumes that a system cannot have properties its components don't have, yet hasn't been laughed out like it should had been.

There's plenty of philosophy-types who think that Searle is an idiot, too, for the Chinese Room and other things. Guy loves to position himself as a defender of rationality and realism because it lets him belittle poststructuralists with oversimplifications and straw men while acting like a hero of a scientific worldview that he clearly doesn't know that much about.

In some ways his antagonistic materialsm is quite similar to your dismissal of philosophy in general, actually.

Re:he's right (1)

Lundse (1036754) | more than 3 years ago | (#34639776)

Mathematics is the foundation for philosophy, not technocracy. What a better world we'd be in if we were motivated by the former rather than pursuing the latter.

Well, we would likely all be malnourished, due to lack of fertilizers, at least those of us who hadn't died at childbirth or soon after. There wouldn't be an Internet to talk on, but that would be okay, since we wouldn't have time to use one due to the lack of engines and the resulting need to do backbreaking labour 16 hours a day. In short, our lives would be miserable, but due to lack of medicine, they would at least be short.

Missing these kinds of little details is why I have very little respect of philosophers. As far as I can tell, most of them chose their field because it doesn't punish sloppy work.

I chose philosophy as a field because it seemed the only one that did punish sloppy work. My original field of study turned out to hinge on theories that were unassailable (for us freshmen), and wrong (in my opinion). Philosophy was were I was allowed and encouraged to criticise the theories we learned - _provided I could argue for said criticism! The sloppy ones dropped out as soon as we started on formal logic (which is very close to mathematics, not incidentally).
There might be philosophy departments were you can get by with sloppy work and vague formulations of personal opinions. Mine wasn't one of those.

And then there's idiocy like the Chinese Room, which assumes that a system cannot have properties its components don't have, yet hasn't been laughed out like it should had been.

Exactly the criticism I, and loads of philosophers before us, offered (yup, you are doing philosophy when you criticise philosophy, weird, hu?! Which raises new questions about whether such a system works the same way as our internal one (which, this time incidentally, was the stupid assumption that had me leavning English Lit.). If a person internalised all the trappings of the Chinese Room, does he understand Chinese (as the Chinese do?) The thought experiment is useful for studying our concepts of syntax and semantics, and what constitutes understanding and consciousnes.

Philosophy means you accept the human condition.

It really, really doesn't. Not at all. It means you study it, whether with a mind to change it or not has absolutely nothing to do with it.

Technorcacy means you try to do something about it.

Yes. With a certain toolset and a certain idea about the initial conditions and a certain idea about the kinds of improvements that can be made. I'm all for those, but I like the fact that we have fields that come up with alternative ways than technology, to change the human condition. That's how we discover new fields (of improvement) - no constitutional democracy without Locke and Montesquieu, no analytical engines without Frege and Leibniz, no reformation of the jail system with Bentham, etc. etc.

Re:he's right (0)

Anonymous Coward | more than 3 years ago | (#34639814)

Technocracy and technology are too different and independent things. Additionally, we are definitely motivated by the philosophy (religion, politics, enlightenment, science) rather than "motivated by technocracy" (slashes at dawn for not performing according to the system requirements).

Re:he's right (4, Insightful)

digitig (1056110) | more than 3 years ago | (#34639816)

Missing these kinds of little details is why I have very little respect of philosophers.

They don't "miss" those details, they're not in scope.

As far as I can tell, most of them chose their field because it doesn't punish sloppy work.

Philosophy does punish sloppy work. relentlessly. Philosophical work is subject to more scrutiny and criticism than any discipline I know of, and that includes pure maths.

And then there's idiocy like the Chinese Room, which assumes that a system cannot have properties its components don't have, yet hasn't been laughed out like it should had been.

Laughing something out doesn't work in philosophy. Unlike whatever discipline you work in, it seems, in philosophy you have to show the reasons why something is wrong. And if you think the issue of emergent properties hasn't been considered in excruciating detail in connection with Searle's Chinese Room thought experiment then you clearly have no idea what philosophy is doing.

Philosophy means you accept the human condition.

Say what? Some philosophy is abstract, but so is some maths. Lots of philosophy (philosophy of science, political philosophy, ethics) is concerned with changing the human condition. Maybe you criticise philosophy because it didn't discover antibiotics (although it did lay a lot of the foundations), but do you criticise biology because it didn't invent democracy? Both changed the human condition, in ways appropriate to their respective disciplines.

Re:he's right (1)

SirGarlon (845873) | more than 3 years ago | (#34639818)

From TFA:

Isn't it interesting how the mention of these two most important goals of learning--truth and beauty--now evokes snickers and ridicule, almost as if by instinct, from those who shrink from all that is not superficial.

Re:he's right (0)

Anonymous Coward | more than 3 years ago | (#34639794)

Mathematics is the foundation for philosophy, not technocracy. What a better world we'd be in if we were motivated by the former rather than pursuing the latter.

Don't you mean the other way around?
Mathematics is one way of applied philosophy, but philosophy is not limited to numbers, you can also use it to reach conclusions in other abstract (And practical.) fields.

The problem with philosophy is pretty much the same as with math. People who pursue the pure form tend to make a lot of errors and misinformed assumptions because they are unable to see if the axioms they have based their reasoning on is valid.

Re:he's right (0)

Anonymous Coward | more than 3 years ago | (#34639822)

The history of mathematical discovery is exciting along with corresponding social movements such as The Age of Reason.

HERE IS WHAT YOU NEED, KIDS !! (0)

Anonymous Coward | more than 3 years ago | (#34639334)

2 + a = 3
a = 3 - 2
a = 1

Know that, and the world is your oyster !!

Re:HERE IS WHAT YOU NEED, KIDS !! (4, Funny)

Pseudonym Authority (1591027) | more than 3 years ago | (#34639348)

let A=1, B=1
A^2=B^2 because A=B, so
A^2=AB and
A^2-B^2=A^2-AB , next we factor
(A+B)(A-B)=A(A-B) , divide like terms
(A+B)=A
substituting our variables for their values we learn that
2=1.

Re:HERE IS WHAT YOU NEED, KIDS !! (1)

Anonymous Coward | more than 3 years ago | (#34639374)

Dude, this is too trivial. You cannot divide by (A-B), because a-b = 0.
You cannot even shock a highschool kid with that lame attempt. At least try derivations or something.

Re:HERE IS WHAT YOU NEED, KIDS !! (0)

Anonymous Coward | more than 3 years ago | (#34639376)

Exception: Zero Division Error

Re:HERE IS WHAT YOU NEED, KIDS !! (1)

Joce640k (829181) | more than 3 years ago | (#34639378)

Error: Division by zero at line 50

Re:HERE IS WHAT YOU NEED, KIDS !! (4, Insightful)

AliasMarlowe (1042386) | more than 3 years ago | (#34639394)

(A+B)(A-B)=A(A-B) , divide like terms

Divide by zero error! After this point, every conclusion is invalid since the results are undefined.
Depressingly, some people (adults as well as kids) would not spot that.

Re:HERE IS WHAT YOU NEED, KIDS !! (1)

Anonymous Coward | more than 3 years ago | (#34639494)

Actuallly:
      - if A=B, then result is undefined
      - if A!=B, then B is 0

Cheers,
G.

Re:HERE IS WHAT YOU NEED, KIDS !! (1)

dcollins (135727) | more than 3 years ago | (#34639578)

1st line assumption: "let A=1, B=1".
Thus, second case is a contradiction.

Re:HERE IS WHAT YOU NEED, KIDS !! (0)

Jesus_666 (702802) | more than 3 years ago | (#34639674)

That's how comment folding/hiding can bite you. Without the GGGP you don't know that A = B = 1. For instance, I came here from the RSS feed where the "division by zero" comment is featured but the "1 = 2" comment isn't, making my picture incomplete.

The same can be a result of C2's filtering as "1 = 2" is score 1 while "division by zero" is score 4.

Re:HERE IS WHAT YOU NEED, KIDS !! (0)

Anonymous Coward | more than 3 years ago | (#34639474)

Wow - you're really smart. Will you be my friend?

FYI:

If, in fact, as you say, A=1 and B=1, then what you *really* have is:

1^2 =1^2 because 1=1, so
1^2=1*1 and
1^2-1^2=1^2-1*1 (which, by the way means 0=0, but let's keep going and see where we end up...), next we factor
(1+1)(1-1)=1(1-1), divide like terms (this is the good part)
(1+1)(1-1)/(1-1)=1(1-1)/(1-1) which is, of course
1 * 0 / 0 = 1 * 0 / 0 which is
0 / 0 = 0 / 0

aaaaannnnnnddddddd... you fail.

But -- not completely! You do demonstrate the point that it's not *math* that's hard, it's *thinking* that's hard. Most people (as is observed elsewhere in the comments) don't like
to (think, that is), and will do pretty much anything to avoid doing so. But you can get better at it - practice, practice, practice (again, observed elsewhere in the comments).

We *believe* math (and other "difficult" activities) are the way they are because we are trained and socialized that way. Whether or not that is *actually* the way things are is another
matter.

(Other problems with the above chain of "logic" are left as exercises for other readers.)

Re:HERE IS WHAT YOU NEED, KIDS !! (1)

Pseudonym Authority (1591027) | more than 3 years ago | (#34639514)

(1+1)(1-1)/(1-1)=1(1-1)/(1-1) which is, of course
1 * 0 / 0 = 1 * 0 / 0 which is

Excuse me, but last I checked 1+1=2, meaning you should have written:

2 * 0 / 0 = 1 * 0 / 0

But don't let me get in your way of being a fucking prick.
Other problems with your post (including your FAIL at line breaks) are left for others to marvel at.

Re:HERE IS WHAT YOU NEED, KIDS !! (0)

Anonymous Coward | more than 3 years ago | (#34639552)

I'm sorry you're sad. I did ask you to be my friend...

Re:HERE IS WHAT YOU NEED, KIDS !! (1)

Pseudonym Authority (1591027) | more than 3 years ago | (#34639582)

You're right, I'm sorry. Let's be friends!

Math misunderstood because it's hard (0)

S3D (745318) | more than 3 years ago | (#34639344)

Math misunderstood because it's hard, and that's why people have misconceptions about it. Understanding of math require considerable effort and concentration which most people tend to avoid if possible.

Re:Math misunderstood because it's hard (5, Insightful)

Joce640k (829181) | more than 3 years ago | (#34639390)

Basic math is easy enough for nobody to have an excuse for not knowing it.

Re:Math misunderstood because it's hard (1)

kiddygrinder (605598) | more than 3 years ago | (#34639848)

the excuse is that it's pretty much pointless to know, much like basic grammar or how to make a bomb from fertiliser.

Re:Math misunderstood because it's hard (5, Insightful)

Palmsie (1550787) | more than 3 years ago | (#34639418)

This is exactly the kind of thinking that has got us into the mess we're into now.

Learning math is just as difficult as learning any other subject or content material. Deciphering poetry, learning programming, studying psychological theory, and learning calculus all involve concentration, study, and struggle from the learner. No one is born knowing any of those things, therefore they all must be learned. The entire point of the OP is to say that the way we go about teaching math is wrong and that people need to reconceptualize how they teach the information because it doesn't make sense to the learner. In the end, its all difficult to some degree. It's when you have that "A-Ha!" moment, it clicks, and you get it. But if you have some terrible algebra teacher who doesn't understand advanced math or someone who doesn't care that you learn, only that you can complete problems 1-50 in a mechanic fashion, then of course it's going to seem difficult (or more difficult than it should be).

Re:Math misunderstood because it's hard (5, Insightful)

txoof (553270) | more than 3 years ago | (#34639460)

The way math is taught in schools is atrocious. Most math texts that I've used with 5th and 6th graders emphasize learning processes and methods for solving a set of problems. The texts do not hold all of the blame, however. The texts are written to follow state and national standards. The standards are written in such a way to emphasize process and not necessarily apprehension of greater concepts. For example:

5th Grade Level Expectation 1. Differentiate between the term factor and multiple, and prime and composite (N-1-M)

While these vocabulary items are important and these skills are definitely useful, learning this skill in isolation (which most texts teach) is pretty useless as students do not connect these skills to a greater picture.

A revision of mathematics standards and teaching methods will go a long way to improving the quality of mathematics education. A holistic approach that includes some wrote learning of basic skills and lots of real application problems. Real application problems are not word problems. How many "real" word problems have you had to solve in the last ten years?

Some texts such as Every Day Math from the University of Chicago does a much better job at integrating all sorts of skills and teaching in a much more holistic method. It includes some excellent modeling exercises, games that rely on a real understanding of mathematical principals for mastery and interesting lessons. But even the best text can't help a kid if they don't have a good teacher that really understands mathematics. Watching an uniformed teacher try to explain what a prime number is, or a different method for division (such as repeated subtraction) is painful. They simply can't do it. Unfortunately, in my experience most of the teaching candidates that were in my classes thought that math was "hard" and "didn't really matter." They scraped by with the lowest possible scores in the required math classes and one even told me she "wasn't going to bother teaching math." While this is pure anecdotal evidence, the declining math scores in the US show that we really do suck and producing math teachers.

The problem stems from bad math teachers badly teaching math which of course leads to more poorly instructed math teachers. Placing a real emphasis on reading and mathematics, with highly qualified and well-supported specialists is the only way we're going to solve this problem. Unless we have some real political will akin to that found during the space race, we're not going to solve this problem any time soon. We'll just keep cranking out kids that think that math is done by computers and a few nerds that wave their magic math wand over problems to find solutions.

Re:Math misunderstood because it's hard (1)

jimicus (737525) | more than 3 years ago | (#34639724)

Not necessarily so. I'm given to understand - though I'm not a mathematician myself by any means - that the problem is not so much maths is difficult as teaching is difficult.

While it's relatively easy to teach a subject to someone who's been blessed with a pretty innate grasp of it, it's damn difficult to teach that exact same subject to someone who doesn't have such a grasp.

Re:Math misunderstood because it's hard (1)

dtmos (447842) | more than 3 years ago | (#34639764)

Math [is] misunderstood because it's hard, and that's why people have misconceptions about it. [The u]nderstanding of math require[s] considerable effort and concentration which most people tend to avoid if possible.

Look, that's just flat wrong. When I was in grade school, the same people who wouldn't do their math homework would then go to the gym and shoot baskets for three hours every day. When I asked why, they would say "math is hard, it's either right or its wrong, and to be any good at all takes considerable effort." When I told them I felt the same way about shooting a basketball, and if they spent the same amount of time in a math book as they do in the gym they'd be stars at math, all I would get was funny looks. I never could understand how such people would work so hard at learning one thing -- basketball -- that they'd sweat, be out of breath, and have to take a shower afterwards, and then turn around and say that learning math was hard!

Like learning anything else (including basketball), if learning math is hard, you're learning it incorrectly, and need better instruction.

Being a mathematics undergraduate... (4, Interesting)

pieisgood (841871) | more than 3 years ago | (#34639380)

I can attest that "true" math is very removed from computation. The computational classes are all regarded as the "easy" classes. This is in contrast to the "hard" classes, real analysis and abstract algebra. Being thrown into real analysis after just one quarter of study in proofs is extremely rough going. If proofs were introduced as puzzles or just introduced earlier in education the whole of America would be better off for it.

My own motivations for being in math are for the challenge and because of the lack of concrete answers in calculus. Trigonometric functions especially are always treated as little boxes that magically calculate what you need.

In any case, at least math attracts the curious.

Re:Being a mathematics undergraduate... (1)

martin-boundary (547041) | more than 3 years ago | (#34639684)

My own motivations for being in math are for the challenge and because of the lack of concrete answers in calculus. Trigonometric functions especially are always treated as little boxes that magically calculate what you need.

Trigonometry predates calculus by a long time (see Ptolemy's table of chords [rutgers.edu] which were calculated purely geometrically, since algebra wasn't invented then either). Trigonometric functions are incredibly rich and important, there are so many different ways of looking at them, and so many mathematical fields which are related to their various properties.

Re:Being a mathematics undergraduate... (1)

pieisgood (841871) | more than 3 years ago | (#34639796)

I understand that, but there is a regular function for both.

sin(x) = (1/2i)*(e^ix - e^-ix)

cos(x) = (1/2)*(e^ix + e^-ix)

while this is covered in undergrad calculus courses, I was really happy when, in my real analysis course, we covered how e is defined. It was more to study the properties of series but still interesting, more interesting that a series converges to an irrational number.

e = Sum from 0 to infinity of 1/n!

so a sum of rational numbers converges to an irrational number... not too mind blowing if you take into account the Cauchy criterion and the fact that every irrational number is a limit point of the rationals... but it was interesting to me at least.

blah blah blah blah...

Re:Being a mathematics undergraduate... (1)

Hylk0r (1295086) | more than 3 years ago | (#34639732)

Being a chemical engineering undergraduate I started to enjoy mathematics more and more when I went to university. In highschool you were given a set of theorems and assignments. With these assignments you could practice those theorems. Now at university, lectures are used to prove theorems and I have to say, I absolutely love it. From the derivation of the integral to the hydrodynamic profile of a laminar fluidum: They're not a bunch of characters scrapped together, they're elegant relations fit together in one single formula. But you can't see this beauty if you have not seen the prove. Maybe theorems just need to be proven and derived at highschool in order to show the real beauty of mathematics.

Re:Being a mathematics undergraduate... (2, Insightful)

Anonymous Coward | more than 3 years ago | (#34639740)

There is a whole world of fascinating computational mathematics out there, young learner. Try reading Trefethen's Numerical Linear Algebra.

Re:Being a mathematics undergraduate... (1)

pieisgood (841871) | more than 3 years ago | (#34639868)

I'm not saying there isn't. Just that proof, usually, proceeds calculation and that proof is a bit more difficult.

I looked up the book, seems to be for graduate classes. I'm currently stocked up on books to read, I'll likely encounter it one day though.

Why math is worth doing in the first place (5, Informative)

LambdaWolf (1561517) | more than 3 years ago | (#34639402)

I've seen the following link in many a Slashdot thread before, but it certainly bears repeating here: "A Mathematician's Lament" by Paul Lockhart [maa.org] It's mostly known as an insightful critique of what's wrong with K-12 math education, but I've always liked it as an explanation of why people who enjoy math do it in the first place: it's satisfying in an artistic way. I think it would be great if more students saw math as something worth doing for its own sake, like art or athletics, and hey, it lets you do science and engineering too.

In fact, this summary sounds similar enough to "Lament" that I wouldn't be surprised if this Dr. Lewis was inspired by and/or cited it. But this is Slashdot, so I'll let someone else check that out.

Re:Why math is worth doing in the first place (1)

avatar139 (918375) | more than 3 years ago | (#34639558)

It's mostly known as an insightful critique of what's wrong with K-12 math education, but I've always liked it as an explanation of why people who enjoy math do it in the first place: it's satisfying in an artistic way.

Good for you, but for the rest of us, (aka people who don't enjoy or care about math that much) I'm afraid it's merely so much futility and frustration!

The point that seems to be lost here for so many people who talk this way about Math is that in the end anything is an "art" for higher end professionals and enthusiasts of a particular field of study. Personally, I often say that the way you can tell a master from an apprentice is that experience gives them the means to solve problems intuitively, hence for the masters of said field, the subject really can be viewed as much of an art as it is a science!

Let us remember the (slightly paraphrased) immortal writings of Dave Barry on this subject, "One man's vision of art is another man's view of an insanely overpriced "modern art piece" that looks suspiciously similar to the rusted remains of a helicopter crash!"

Re:Why math is worth doing in the first place (1)

LambdaWolf (1561517) | more than 3 years ago | (#34639742)

It's mostly known as an insightful critique of what's wrong with K-12 math education, but I've always liked it as an explanation of why people who enjoy math do it in the first place: it's satisfying in an artistic way.

Good for you, but for the rest of us, (aka people who don't enjoy or care about math that much) I'm afraid it's merely so much futility and frustration!

True, but the remainder of the Lockhart article addresses that. To paraphrase, students take to math class with a lot less friction if they understand that math at least can be satisfying. Plenty of students dislike their high school art classes too, but they can at least sit through them understanding why certain other people think it's pleasant and important—and therefore not futile, even if it is frustrating. By contrast, too many high school students are ready to dismiss math as something that other people use for techie things but will never to themselves be of any value, intrinsic or otherwise, other than as a prerequisite for college. (And they're mostly right because of the way those classes are taught, but that's a separate complaint.)

The point that seems to be lost here for so many people who talk this way about Math is that in the end anything is an "art" for higher end professionals and enthusiasts of a particular field of study.

The Lockhart article actually does address the issue of making that side of math apparent to novices and laypeople, and makes a pretty persuasive argument that it is possible (if beyond the capabilities of most public school classrooms).

Let us remember the (slightly paraphrased) immortal writings of Dave Barry on this subject, "One man's vision of art is another man's view of an insanely overpriced "modern art piece" that looks suspiciously similar to the rusted remains of a helicopter crash!"

Dave Barry is awesome. :)

Re:Why math is worth doing in the first place (1)

gilleain (1310105) | more than 3 years ago | (#34639744)

The point that seems to be lost here for so many people who talk this way about Math is that in the end anything is an "art" for higher end professionals and enthusiasts of a particular field of study.

Different meanings of the word 'art'. As you say, if an activity is 'an art' it means that it can be carried out at a masterful level - and perhaps intuitively. What I think the article is talking about is the similarity of Mathematics to Art. Again I should mention the 'Beauty of Equations' program (see below) where the presenter - who is an art critic - talks about Dirac's ideas on mathematical beauty. He said that it takes an experienced mathematician to recognise beauty in mathematics. So, just as with Art, it is essential to take the time to learn about and invest in Mathematics before you can recognise the beauty.

Re:Why math is worth doing in the first place (3, Informative)

dcollins (135727) | more than 3 years ago | (#34639734)

As a part-time college math teacher, I almost totally disagree with Lockhart's Lament. (Ironically, the K-12 school where he teaches is close to the neighborhood where I live.)

It's not that it's bad to see that math can be an art and a pattern-finding exploration (some part of the time), but someone has got to teach and be held accountable for the nuts-and-bolts of how to read and write mathematical vocabulary, notation, and justification (algebra and geometry, for starters). Knowing about the scientific method is necessary, but exclusively spending your K-12 time re-inventing the wheel is inefficient at best. It's the same problem as in English nowadays -- I was told last weekend that teachers in junior high schools are forbidden from teaching the rules of grammar. That is, it's exclusively about expressing "big ideas", no matter how poorly-formed or unreadable. The more this produces crippled students, the more we seem to run deeper in the same direction -- if you abandon teaching the basic structure of our shared communication systems, then we thereby just generate more and more unreadable nonsense as time goes on.

The remedial math I teach (basic algebra; about half my assignment load) is almost entirely about just reading & writing. Even the first unspoken step of simply transcribing symbols (i.e., an expression) from one page to another is almost impossible for about half my students, because no one has ever asked for any level of precision in their reading, writing, or observation skills (whether in English, math, or anything else). To me, basic math is an opportunity to focus on precision in thinking and writing -- applications belong in other classes! No, that's not what a professional mathematician works at on a daily basis, but frankly, not every K-12 class can be an independent research opportunity. At some point you've got to eat your vegetables, and if you run entirely away from that, then it truly is a monumental waste of time.

Re:Why math is worth doing in the first place (1)

Racemaniac (1099281) | more than 3 years ago | (#34639788)

I wish i had modpoints, that's an incredible text :). I sadly enough have the same feeling about computer science in school. I'm a master in computational CS, and if it had been from my experiences in high school, i wouldn't have touched either with a mile long pole >_...
Math was too often just learning stuff by heart, and i've had teachers give me bad grades because i didn't exactly copy their method, but just figured it out as i went along (luckily some other thought that was brilliant, but not all of them...). But just the fact that i "got" it, and even with all the crap around it it costed me the least effort to get what i hoped was a good education, and have passing grades.
computers was even worse at my high school... the computer teachers knew next to nothing about pc's, and except learning never used words for parts of the computer (i'm a fuckin CS master now and still haven't used some of the words they made up in our high school textbooks -_-), and our curriculum was of course word and excel, with the same emphasis on just learning certain operations (and excel functions) by heart for no good reason what so ever... We also had isolab as a method of learning programming and reasoning. While i support that it is a useful program, it gets the same treatment as everything else, which made it yet another excruciating experience (that and the fact that i figured out the entire thing after looking at it for 5 minutes, since i already had learned some programming on my own).

That text you just posted really reminded me of how i feel about the things i've chosen to do :). I know i like math and CS because i'm good at them, and indeed, they are a form of art :). They're a way of creating something, and in IT it's something far more real than in math, but still the exact same problems apply i think :).

Also something i think about when reading that text, is that i feel lucky that i'm rather smart, and that where i live high or low grades hardly make any difference at all. So while i was subject to the same poor education system, by just choosing what i was best at, and learning it enough to get passing grades, gave me a chance to keep my real interests in it alive and continue doing it now, without having been discouraged by the massive amounts of crap i would have had to go through otherwise...

Re:Why math is worth doing in the first place (1)

Speare (84249) | more than 3 years ago | (#34639890)

Last week, NPR had a shout out to this essay (which I had read before) and also to a blogger named Vi Hart. Check out her YouTube videos and blog at vihart.com [vihart.com] , especially the math class doodles. She talks very fast, cracks a lot of puns, and ridicules the established educational methods as she draws doodles that relate to math concepts. Explore your numeracy visually.

Mathematics as an art (4, Insightful)

Chrisq (894406) | more than 3 years ago | (#34639426)

I have a cousin who is great at mathematics, and really can see mathematics as an art. Whereas I am happy if I can solve a problem, he will look for an "elegant solution". I had a number of equations that I solved, trying to optimise the buffer size for various input queues. I shown him, and he quickly said that I had the right answer. A day later he came and shown me how he derived an equation that could simply solve all problems of this type. He also generalised it to allow buffer sizes that were complex numbers. The first part was very useful to me, the second absolutely useless - but to him it was all just interesting.

This is one way that mathematics as an art is unlike any other art. It gives useful results. I have heard time and time again about engineers going to the mathematics department of a University asking how they can solve a "new" problem - to be told that the solution had been discovered a century before. I am sure most of these solutions came from someone just wanting to find an elegant way of expressing something without thought of any use. So if its an art and is useful why do so few people follow it?

The answer is obvious, because its hard! In many forms of art you can slap anything down and convince someone that it has value and its art. This may not always have been true, before photography accurate representational art was highly valued - but today someone producing a lifelike portrait will not be values as much as someone slapping their name on an unmade bed! Mathematics has to be right, you can't just slap down a few numbers and call it an equation. This is the basic problem that anyone will have in persuading someone to follow maths for its art, there are a lot easier ways to become an artist.

Re:Mathematics as an art (1)

chichilalescu (1647065) | more than 3 years ago | (#34639530)

you had me at complex buffer sizes.

On a more serious note, you cannot discuss mathematics as an art without realizing that humanity's progress is a highly nonlinear process, and huge leaps are made from "hm, that's funny" moments. Society's problem is it wants a clear way to distribute money to mathematicians (and scientists generally), and we can't really decide which are the good problems to work on, in the context of "what'll be more useful 50 years into the future?".
This is also related to the issue of copyrights and patents: if something is truly useful, you will find all the relevant information in a public library or on the internet, but usually you still need to pay specialists to explain it to you. once you understand it, your teacher encourages you to propagate that knowledge, and expand it in anyway you can (the only problem is that it's hard to find people that have the patience to understanding it).

Re:Mathematics as an art (1)

alexhs (877055) | more than 3 years ago | (#34639836)

He also generalised it to allow buffer sizes that were complex numbers. The first part was very useful to me, the second absolutely useless - but to him it was all just interesting.

Not that useless, if you think about it. The choice of complex numbers is strange though, you better see it as a 2-dimensional array. You could then generalize to n-dimensional buffers.

Re:Mathematics as an art (1)

stuckinarut (891702) | more than 3 years ago | (#34639866)

When I am working on a problem, I don't think about beauty, but when I have finished, if the solution is not beautiful then I know it is wrong - R. Buckminster Fuller [wikipedia.com]

Not just maths (4, Interesting)

Schiphol (1168667) | more than 3 years ago | (#34639454)

I wish science in general was considered part of what a learned person has to know. I mean, if you want to pass for an intellectual you have to read your Dante, your Beckett and you at least need to know who Lautreamont was. But, apparently, you can very well get away with thinking that you can suck gravity out of a room the way you suck air, or with not having even heard about string theory. That divorce makes no sense, and it was impossible in the history of ideas till very recently. And Euler's formula is more beautiful than most poems.

Excellent. (2)

dtmos (447842) | more than 3 years ago | (#34639462)

This is by far the best defense of mathematics I've ever read. It's a shame that the poor quality of grade school math education has made it necessary, though. Can one imagine a similar essay on any other subject? Only math is so poorly taught.

Copy edit quibbles (1)

dtmos (447842) | more than 3 years ago | (#34639470)

-- The parenthetical comment "(if it was done right!)" in "Ready For The Big Play" should, of course, be, "(if it were done correctly!)"

-- References in "Cargo Cult Education" to the "south Pacific" should be to the "South Pacific"

-- Also in "Cargo Cult Education", "But of course nothing came. (except, eventually, some anthropologists!)" should be, "But of course nothing came (except, eventually, some anthropologists!)."

Math is a tool, not a art (1)

TheDarkMaster (1292526) | more than 3 years ago | (#34639506)

Simple that... I honestly can not understand where there can be "beauty" in a mathematical expression that covers the entire blackboard. And more so when the teacher fails miserably to show practical uses for the expression.

Re:Math is a tool, not a art (1)

gilleain (1310105) | more than 3 years ago | (#34639622)

There was a BBC4 program recently called "Beautiful Equations" where an art critic went round various mathematicians asking about E=MC^2, F=G(m1m2/r^2), S=A/4, and er the Dirac Equation.

The point about most of these examples they chose - apart from being conveniently in the UK - was that they were short. Also that they are directly related to important ideas about how the Universe works. So mass can be converted to energy, bodies attract each other, black holes can shrink, and antimatter exists. Dirac was particularly chosen because he believed that if you are given a choice between two possible formulations of (or equations for) a problem, you should chose the more elegant, shorter, more beautiful one.

I must quibble (1)

Peter (Professor) Fo (956906) | more than 3 years ago | (#34639792)

The original meaning of "art" was 'Man's work' as opposed to "nature" which was 'God's work'.

A tool is a man-made thing (even if it a rock to chip flints with it is selected and used in a way that is man-made). The cave-man who acquires a better hammer rock is naturally pleased and proud of it and will either imbue it with magical qualities (God-Nature you see) or appreciate its qualities as they matter to a flint-maker (weight, hardness, fit in the hand etc.). The latter is just a 'beautiful' as a clever team manoeuvre to score a goal, or the technology that goes into making an affordable, low maintenance, lightweight bicycle. Of course you have to 'know what beauty looks like' - Those ingredients that make you most proud of your tools and achievements.

I don't think anyone was claiming that 'expressions all over the blackboard' were beautiful... ...but the conclusion may be, and the lead-up to it may be a guide for our own explorations.

FWIW here is my analysis of levels:

  1. Reading number, counting and realising 'sums can do things' (Many are shamefully allowed to fail even this!)
  2. Basic facility with numbers. 'Arithmetic' (Failure here too. IMHO the key here is 'confidence'.)
  3. Maths for high-school science. Inkling of curious 'worlds' and strange coincidences (The best motivator here is 'being brainy is cool')
  4. Maths as a field of intellectual study in its own right. (A minority interest.)

Somewhere, possibly after school, especially in old age, people need a sense of 'be safe with numbers, statistics and graphs'

So, what should we do? (0)

Anonymous Coward | more than 3 years ago | (#34639510)

I don't know about the rest of the world, but where I live inspiring math teachers are rare. I've had one, but consider myself one of the few lucky. A new approach on teaching mathematics with base in understanding and application would be great. This would of course require much more from the teachers than before. They would actually have to understand the math themselves. Can't really blame the good mathematicians for not wanting to end up as low wage teachers. Maths really are hard too, and all about reasoning and understanding, which you really can't teach, only help by guiding. Even though I'm studying maths at university now, I wouldn't really trust my self to tell a kid what maths really are about (I have my own ideas though, like everyone else).

Also, maths aren't for everyone. While I believe many would enjoy a different approach on the subject, there are enough of those who just don't have any interest in it (or science at all). The learning curve is also much independent, and to include a whole class of students you would have to ignore both those who learn too fast, and those who just don't get it. You can't really hold all your students back to wait for someone to have their own little revelation. So, while I find the current approach on maths somewhat ridiculous, I can certainly see why it's ended up as it is. I feel he really nailed the problem, but what should we do?

Differenciation (2)

tonywestonuk (261622) | more than 3 years ago | (#34639568)

I remember been taught differentiation at school – One lesson, lecturer puts a parabolic curve, x=y*y, on the board, and asks the problem, determine the angle of the line

Then, he didn’t say anything else.. Just, for the rest of the lesson, responded with ‘Yes’, ‘No’, or ‘Maybe’. So, after a frustrating 20 minute discussion, trying to work out how the hell to do this problem, someone came up with the idea of adding a ‘little bit’ of x, to x..

We worked out, as a group, the concept differentiation, with only the smallest bit of guidance from the lecturer. This is how things should be taught – allowing people to discover concepts themselves, rather than preaching the correct ways to do things.

Re:Differenciation (1)

dtmos (447842) | more than 3 years ago | (#34639616)

Brilliant -- a live version of "A Pathway into Number Theory [slashdot.org] ". That's the kind of teaching for which awards should be given.

related article (1)

Trepidity (597) | more than 3 years ago | (#34639594)

"A Mathematician's Lament" [maa.org] , an article that's been making the rounds among mathematicians since 2002 (but was only published in 2008), expresses some similar views, and is also a good read.

Too abstract (1)

freeshoes (826204) | more than 3 years ago | (#34639676)

As a standalone subject Math is too abstract, the relevant areas should be taught along side the subjects where it is applied. I think a lot of degrees could be cut down to a year or two. Cut out all the crap that no one needs to know and make that part of a further research degree. Revolutionize the education system with my ideas, or fall behind when I move to China and they listen to me.

Mathematics consists of two parts ... (1)

golodh (893453) | more than 3 years ago | (#34639704)

The first part comprises the results of previous work by mathematicians; the finished product. That's what underpins most of physics and engineering nowadays.

The second part is the "live" Mathematics, i.e. the process of actually doing Mathematics in the sense of figuring something out. That's a slow, arduous, iterative and groping process. Starting with an observation that confuses or amazes us, incrementally and tentatively formulating concepts (definitions, constructs of previously known mathematics), their properties (sometimes axioms but mostly properties of known constructs), drawing inferences from those concepts, seeing if they throw light on the situation, and going back to changing the concepts if they don't).

Where the second part is like mental rock-climbing, the first part is like a list of views that were discovered by rock-climbers but which which can now be reached by cable-car (or bus).

For better or worse, the mental rock-climbing takes more talent and dedication on part of a student than about 75% of them have. And even talented and dedicated students will take thousands of years (about two millennia to be exact) to reinvent Mathematics on their own (so much for "Letting students discover Mathematics on their own").

We therefore tend to teach the finished results because they are (a) enormously valuable insights (b) useful in other subjects, and (c) accessible to someone with a modest amount of perseverance, an adequate memory, and ordinary talent.

The problems really start when people (education boards) fail to distinguish between the two forms of Mathematics and neglect to clearly set out the goals they want education to address. Which then results e.g. in them insisting on letting students memorise the square-root formula for quadratic equations instead of teaching them how to solve a quadratic equation through simple algebraic manipulation (which also gives people a bit of insight in what they're doing) and letting them look up the quadrature formula when they need it.

Mathematics in school and university (1)

mseeger (40923) | more than 3 years ago | (#34639726)

Hi,

in school mathimatics is mostly execution of algorithms provided by your teacher, learning when and how to apply them. This changes a lot with university. At first, mathematics is a language to be learned. You have to be able to express your problems in a normed language. This is the first art. If you read papers, you can distiguish easily between those peoples who truely have mastered that language and those who don't have. Later on, you learn how to prove things. The interesting things you cannot prove by just applying an algorithm. At that point you need a lot of creativity, which the second art form required by a true Mathemagician.

CU, Martin

1... (0)

Anonymous Coward | more than 3 years ago | (#34639728)

...is the loneliest number.

Re:1... (1)

chichilalescu (1647065) | more than 3 years ago | (#34639802)

yes, but seven ate nine

There is more to it. (1)

Kiliani (816330) | more than 3 years ago | (#34639784)

What is being said in the article is almost verbatim true for physics as well. Poorly taught, even more poorly understood by almost anyone (including teachers).

Mathematics is much more fundamental than physics, no doubt. Very good points are made here. But (ironically?), you could replace "math" with "physics" in the article and most of it would be true just as well.

I don't hold hope that especially Americans will ever get this, though, either for math, or for other STEM fields. Because it's "too hard". Being ignorant is just too damn easy, if you ask me.

Heck, I'd be happy if people at least would get math.

hmmm (2)

Charliemopps (1157495) | more than 3 years ago | (#34639810)

My highschool math teacher was a retired NASA programmer. According to her, teaching Mathamatics was about leaching logic and problem solving. If you forgot all the formulas taught in her class, she said, it wouldn't matter. The real skill learned was how to deal with an entirely new mathematical problem. WHY is area "height x width"? How to build your own sort of equations. Sure enough, decades later I have forgotten every single equation I had been taught there, but when faced with a logic problem I'm still able to work it out.

Yes, misunderstood and mis-taught (2)

dbune (1792602) | more than 3 years ago | (#34639876)

I can very well relate to this post.. the foremost reason for Mathematics being misunderstood is the problem with the way it is taught in schools. Right from childhood you are told to mug up the multiplication tables, formulas and everything is told be "it is like that.. just remember". The flaw is with the education system where stress is not to "understand" and see things logically but on how much can you mug up and pass those tests and get a so-called "good score". The teachers need to be trained to generate interest, talk concepts and not just ask to be ready with the multiplication table the next day for the test.

Not part of STEM? (0)

crow_t_robot (528562) | more than 3 years ago | (#34639886)

defense of mathematics as a liberal arts discipline, and not merely part of a STEM

Not part of "STEM?" It is clearly the "M."

Case closed.

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