## Mathematicians Are Chronically Lost and Confused 114

Posted
by
Soulskill

from the dude-where's-my-cartesian-plot dept.

from the dude-where's-my-cartesian-plot dept.

An anonymous reader writes

*"Mathematics Ph.D. student Jeremy Kun has an interesting post about how mathematicians approach doing new work and pushing back the boundaries of human knowledge. He says it's immensely important for mathematicians to be comfortable with extended periods of ignorance when working on a new topic. 'The truth is that mathematicians are chronically lost and confused. It's our natural state of being, and I mean that in a good way. ... This is something that has been bred into me after years of studying mathematics. I know how to say, “Well, I understand nothing about anything,” and then constructively answer the question, “What’s next?” Sometimes the answer is to pinpoint one very basic question I don’t understand and try to tackle that first.' He then provides some advice for people learning college level math like calculus or linear algebra: 'I suggest you don't worry too much about verifying every claim and doing every exercise. If it takes you more than 5 or 10 minutes to verify a "trivial" claim in the text, then you can accept it and move on. ... But more often than not you'll find that by the time you revisit a problem you've literally grown so much (mathematically) that it's trivial. What's much more useful is recording what the deep insights are, and storing them for recollection later.'"*
## That settles it (Score:5, Funny)

## Re: (Score:3)

I was the best mathematician in my university math classes. Who knew?

Genius ahead of its time.

I know I was hard at work involving the

subtractionof beer from a case,additionto empty bottles,dividingtime between drinking and the necessary room andmultiplyingthe number of pink elephants surrounding me.Heady times.

## Re: That settles it (Score:2)

I got in a serious car accident and spent the second half of the year recovering in the residence I'd already paid for without going to class. Then, for a lark, I got totally hammered and wrote the Calculus exam with my mates.

Aced the exam, passed Calculus despite having not gone to class at all or done a single assignment.

I still find it hilarious, but my mom was not particularly proud of me.

## Learning != linear? (Score:1)

## Re: (Score:3, Interesting)

Math should not be taught as a linear process, but as a spiral. Visit the topics at first, so the student can understand why something is important when it is presented rigorously.

## So the answer is "42" (Score:1)

Now to learn what the question is...

## "Trivial" (Score:2, Interesting)

## Re:"Trivial" (Score:4, Funny)

I suppose you can reduce arithmetic and geometry (both quadrivia) to logic (trivia), but the liberal arts are seven in number for a very good numerological reason.

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the liberal arts are seven in number for a very good numerological reason

I thought the number of the counting shall be three? Well, at least since five is right out, you can't just get rid of arithmetic and geometry, you either have to get rid of something else as well, or keep one of those.

## Re: (Score:2)

The trivia: logic, grammar, rhetoric

The quadrivia: music, astronomy, geometry, arithmetic.

Those are the seven liberal arts. Note that under this scheme, astronomy and music both included a substantial amount of mathematics.

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Becoming a mathematician is like becoming someone who is fascinated in shoes, or briefcases or watches or hammers.

An important class of people inordinately fascinated by shoes, briefcases, watches, and hammers are manufacturers.

## Failing as a math teacher (Score:5, Insightful)

All too often I've encountered math teachers who failed to properly explain advanced mathematical concepts because to them it was obvious and trivial.

Gee, thanks.

## Re:Failing as a math teacher (Score:5, Insightful)

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Had a professor in college who taught only 1 class for the semester. After that, he assigned topics to each student in the class with a few suggested areas to starting researching their topic.

Then every class for the rest of the semester consisted of students going up and presenting their findings, while the professor questioned them to guide their presentation to key areas, and clarified/corrected/expanded on their presentation as needed to ensure the audience gets the information they need for the exams.

T

## Re: (Score:2)

I took some classes that way, and it worked well. Except for the chapter on business uses of artificial neural networks, which went to a woman fresh form Mainland China. That did teach me that there are some pretty sophisticated concepts that most of us in the US think just natural.

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I have sometimes thought that the best teacher might be another student who has just a moment ago barely (but still correctly) grasped the concept.

I'm not convinced. What you really need, in order to teach a difficult subject, is understanding of why it is difficult to understand for an initiate, and I suppose somebody who has just learned may be much closer to that understanding, but on the other hand, you also need a very thorough understanding of what the subject is all about, and you probably only get that with experience. I think what would really make a great teacher is somebody who has long, practical experience of what the subject is used for

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An alternative explanation is those math teachers didn't actually understand the concept, and therefore were unable to properly explain it.

## Re:Failing as a math teacher (Score:4, Insightful)

## Re:Failing as a math teacher (Score:4, Funny)

Wow. He must write a lot of computer documentation.

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One assumes that he then defined cardinality as "the quality that uniquely defines a cardinal number."

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This is how definitions work. Definitions would get absurdly long and difficult to read if we defined everything in terms of first principles. I could concisely describe a solvable group as a group having a subnormal serious whose factor groups are all abelian. If I have to go back and explain group and subnormal series and factor groups and abelian it ballloons to a page in length, and those are all concepts that are useful elsewhere is well.

Presumably that author wasn't just defining things cyclically and

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Presumably that author wasn't just defining things cyclically and had defined cardinality elsewhere.One would think so, but no. When I came across that book I was trying to learn about such things and I'd think that would have remembered it if he had.

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That comment brought back a bad memory from a calc 2 course I took in college. The professor was ancient. On the first day of class he refered to a silent film actress who never was able to make it in the talkies.

One day he was going over a problem on the chalkboard. A student asked how the professor got from one line to the next. The professor threw up both hands and exclaimed that he'd never be able to cover the course material if he had to go over every trivial detail. He then angrilly filled up the

## An old mathematicians' joke (Score:5, Funny)

There are two types of theorems: trivial and unproven.

## Re: (Score:2, Funny)

There are two types of theorems: trivial and unproven.

Actually, there are 3. Proof left for the reader.

## Obligatory Abstruse Goose (Score:2)

http://abstrusegoose.com/395 [abstrusegoose.com]

## Re: (Score:2)

There are two types of theorems: trivial and unproven.

Actually, there are 3. Proof left for the reader.

Does that include proofs that are too big to fit in the margin?

## Sound Like Software Development (Score:2)

EOM

## Re: (Score:2)

The two are basically the same [slashdot.org], or so the physiologists tell us.

## Bizarre advice (Score:5, Insightful)

He then provides some advice for people learning college level math like calculus or linear algebra: 'I suggest you don't worry too much about verifying every claim and doing every exercise. If it takes you more than 5 or 10 minutes to verify a "trivial" claim in the text, then you can accept it and move on. ...

While I agree that one shouldn't waste time questioning every statement you encounter, there's a very ancient and useful tradition in math pedagogy that emphasizes these sorts of things. See, for example, gradually building up geometrical theorems from a few axioms, a la Euclid.

Often, the process of working out complex proofs for yourself is crucial to understanding why things work, not to mention developing and practicing logic skills that are essential in math and elsewhere.

I'm not saying one should waste time trying every exercise or redoing every proof, but some of my greatest insights into the inner workings on math have come from exercises that took me a couple hours to work out or textbook passages I went over a number of times and really dug into how the details worked. If I skipped everything I couldn't do in 5-10 minutes, I doubt I'd ever have developed the more advanced skills and intuitions that would be necessary to see why some results are "trivial."

## Re:Bizarre advice (Score:5, Interesting)

I came here to post a similar sentiment. I think it is a terrible idea to just blow ahead every time an assertion is too confusing. Getting the big picture and developing mathematical intuition is great but it doesn't mean that you'll actually be able to

domath. For that, practice.I actually advise the opposite, to bang your head against a problem over and over until it breaks (the problem, that is). I don't think that we as a society or a species or whatever deal with confusion very well, and tend to take it as some sort of personal deficiency. We've also done a great deal of dumbing math down, so that when someone tries to make the jump from, say, AP calculus to real analysis, minds get blown and souls shattered. It's probably not that mathematicians enjoy crushing students, but rather that higher levels of math are just plain confusing for most people. They're based on abstractions that are pretty far removed from the human experience. None of this is to say that people who are good at math are better somehow, but it usually means that they put in a lot of time. I suspect that a lot of people who are math-phobic would get over it if you locked them in a room with nothing but math books to keep them busy.

One that is clear, however, is that most mathematicians have no fscking clue what the word "obvious" means. There are some brilliant, dead authors that I would love to punch in the face.

## Re: (Score:2)

I actually advise the opposite, to bang your head against a problem over and over until it breaks (the problem, that is).

But most people actually give up first, and quit studying. People "turning themselves off" to the study of math is a very common problem. Most people can only take so much frustration.

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even better!## Re: (Score:2)

.

I think that they know exactly what it means, but that you are confusing it with the non-technical meaning. In maths it generally means "I have managed to work this out, and I suspect that you will be able to (eventually) without my help. If you cannot, that I presume that you are an idiot and that you do not deserve my hel

## Re: (Score:2)

If you keep bashing your head against a problem, a time will come when you're not accomplishing anything. Try another one. Come back to the first when you're through with that one. Don't just give up on things, but be flexible in your approach.

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A lot of early math courses are trying to teach the mechanics of established principles so that you can solve specific, common problems and situations.

If the only point is to teach someone how to apply some precise algorithm to a specific type of problem, why bother teaching math at all? Just say --if problem type X, type numbers into computer and run program A, if type Y, run program B, etc. There's no point teaching someone how to act like a glorified calculator anymore... this thinking is a couple decades out of date.

## Re: (Score:3)

A lot of early math courses are trying to teach the mechanics of established principles so that you can solve specific, common problems and situations.

If the only point is to teach someone how to apply some precise algorithm to a specific type of problem, why bother teaching math at all? Just say --if problem type X, type numbers into computer and run program A, if type Y, run program B, etc. There's no point teaching someone how to act like a glorified calculator anymore... this thinking is a couple decades out of date.

There is a lot of basic stat you can do if you don't know why the standard deviation formula is the way it is,

For frack's sake, no! If you don't know how standard deviation actually works,

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It seems to me that we are approaching a brave new time when only the skills and knowledge which are economically valuable will be taught.

You can drop the qualifier, "economically". The only sort of value is economic value because if something has value, then that means you're willing to sacrifice something of significance for it. And that's all economics really is about, making choices with tradeoffs in order to get the stuff you value more.

This paragraph gives me the impression that you advocate educational institutions should resist giving what students and society wants out of education and instead deliver what some intellectual elite

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I read that as "we should only teach skills and knowledge that provides more monetary value for the society in the long run, compared to the resour

## Re: (Score:2)

There are other values. Moral values for example.

No, there aren't. If you aren't willing to do or sacrifice anything for those moral values, then they aren't actually valuable to you.

In fact, it often has negative economic value, as the economy is harmed by people trying to pursue morals.

Not to the person holdings those moral values. Value is always relative.

## Re: (Score:1)

The article says explicitly that there are times when digging your heels in is necessary (for the more important stuff). But a lot of math textbooks will leave oodles of exercises to the reader. If each one takes three hours you'll never get anywhere. I think his point is not to take the "do everything" attitude to the extreme, which he observes people doing.

## Re: (Score:2)

The article says explicitly that there are times when digging your heels in is necessary (for the more important stuff).

Yes, you're right. I was mostly responding to the quotation in the summary. But I still have a couple problems with this: (1) how exactly is a beginner to know what is "important"? And (2) the most insightful things that have happened to me were doing random exercises that interested me, rather than necessarily something "important."

I'd say his attitude is right that you needn't do every exercise or proof (and I actually already said this in my first post), but if you are interested and motivated to so

## Re: (Score:2)

I don't think the author is suggesting that details don't matter. Rather, he is suggesting that on a first pass through material, it is often better to focus on learning the material on a conceptual level (where is this material taking me? What does this theorem really tell me?) rather than focusing on the mechanical details of the derivations and proofs. To a certain extent, this is already built into the curriculum: freshman and sophomore mathematics coursework tends to focus on concepts and computatio

## The modern teaching technique (Score:2)

I'm not sure why the word "teaching" is still used. That movie title comes to mind: "I Can Do Bad All by Myself" -- interesting that both movies with that title on IMDB are rated in the 3's.

They don't teach phonetics. Kids in middle school don't even know how to do long division. WTF.

FWIW, I am not big at being able to derive things. My idea of studying is to work through the pr

## Any Good Scientist Knows This (Score:2, Insightful)

We must all learn to exist in that exquisitely uncomfortable place where everything we know is always up for reassessment. Otherwise, we miss change, and change is the only constant.

## Re: (Score:2)

Change is only constant if the degree of the leading term is 1 :-)

## "on a roll" (Score:2)

i had to be woken up at around 9:20am for a 3 hour A-Level Maths exam that had started at 9am and was to end at 12. starting at around 9:25 on the first question, after around 25 minutes i gave up and went onto the 2nd question. this one i did in around 15 minutes. from there i accelerated, completed *every* question, returned to the first and completed it in a few minutes. i then sat back for a while, then got some coloured pens and coloured in one of the graphs. i might even have been bold enough to

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Compare and contrast your amazing story with your granddad's from WWII and think about which one you'd rather hear.

And people say war is a bad thing!

## Mathematician or politician? (Score:1)

## you heard the one about ... (Score:2)

the constipated mathematician ?

He worked it out with a pencil.

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the constipated mathematician ?

He worked it out with a pencil.

Old School.

Nowadays, he'd work it out ... [*dons sunglasses*] ...

digitally.## Re: (Score:2)

Maybe but I'm not sure anyone other than the gotse dude could use an iphone to clear his bowels

I was not referring to electronics. [reference.com]

## Comfort with not knowing (Score:2)

I'm a physics graduate student, and while I'm not quite in the same boat as the mathematicians, I'm familiar with the problem. You spend a lot of time trying and failing to figure out what's going on. You have to be comfortable with not knowing things you want to know. I think that's a really useful ability because you don't demand easily digestible answers for everything. Such answers rarely exist, although many people seek them from short articles and soundbites.

It think it also has larger philosophical i

## Re: (Score:2)

For example, one of the more mind-bending exercises in undergrad abstract algebra is proving Peano's axioms for integers. On the one hand you could say "well, I thought I knew basic arithmetic, but now I have to question even that: I'm lost!" But on

## all thinkers are confused (Score:2)

Never trust the one who has the answers. The politician. The Preacher. The grammar school teacher. Seek those who have questions.

I'm a writer and inventor, I hope to come to understand things with my writing. I may draw a concept in an attempt to understand it better. I have written programs to unravel mysteries (you've seen the 'game of life'?) I try to reserve judgement when presented with an obvious 'truth' on Slashdot (as most of you do !).

Here's my email sig, feel free to share it:

"Your life is not goi

## RIP Philosophy of Mathematics (Score:2)

If this is how things stand, then the Philosophy of Mathematics to date is a catastrophic failure. When there is no better methodology than "fumble around in the dark a bit until suddenly you're convinced" then the project of attempting to guide students in understanding maths has done no work at all.

Is this the fault of the philosophers or the mathematicians? I'm inclined to think that the philosophers have at least failed in their advocacy, if not in their actual subject.

## Lots of us are confused (Score:2)

## Similarly with Engineering and Programming. (Score:2)

Some similar effects occur with engineering and programming. For instance:

An engineer is ALWAYS working on something that's broken. That's because, when he gets it fixed, he moves on to the next thing that's broken. (It's like the thing you're searching for always being in the last place you look. It's not Murphy's law, It's becaue, when you find it, you stop looking.)

A good programmer doesn't come to a problem with all he needs to solve it. Instead he comes to it with a big toolbox, SOME domain knowl

## Not following you... (Score:1)

A good programmer might know the exact solution immediately upon seeing the problem. It could have been something they've solved many times before. There are times when I'm quite bored implementing solutions, because they are the same (general) solutions I've implemented many times before.

Programming is really *nothing* like math, in my opinion. Programming is nearly always *pragmatic* in nature. Many mathematicians study things with no obvious practical application. Programming is answering the questi

## Re: (Score:2)

(It's like the thing you're searching for always being in the last place you look. It's not Murphy's law, It's because, when you find it, you stop looking.)

That is a nice trite little phrase, but when one thinks about if for a minute you can see situations where it is not

alwaystrue. Sometimes the thing was in the first place you looked for it, but you failed to notice it or overlooked it for some reason. I find this happens to me often when hunting a geocache. I have developed good instincts to where they may be hidden and look in the most likely places. When the cases come up where I still haven't found it and have to keep looking over the object or area at## Bait headline (Score:2)

## Andrew Wiles on exploring the dark (Score:2)

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## ADHD (Score:1)

[satire] What, math researcher have ADHD, what a surprise, maybe we will some Asperger among them, who know. [/satire]

## Sounds Like Any Advanced / Graduate Research (Score:1)

## "Lost and confused?" (Score:2)

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Therefore, you have to be a nutcase to get an A in nonlinear differential equations.

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The headline: "Mathematicians Are Chronically Lost and Confused"

I'm surprised the OP was modded down.