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Mathematicians Are Chronically Lost and Confused
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.'"
I was the best mathematician in my university math classes. Who knew?
Sounds like learning is not necessarily a linear process. Makes me feel better about my learning experience!
Now to learn what the question is...
Everything, and only things, that math people do is "trivial".
Fuck systemd. Fuck Redhat. Fuck Soylent, too. Wait, scratch the last one.
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.
There are two types of theorems: trivial and unproven.
"This is something that has been bred into me after years of studying mathematics."
He must be a mathematician if that's how he thinks breeding works.
EOM
love is just extroverted narcissism
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."
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.
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 have left 10 or 15 minute early.
when the results were in i learned i'd got an A. on an exam that was supposed to be 3 hours and i'd completed every question in a little over 2. that was 1987 and i've never forgotten what happened. the point is: i know that once you get started, and get into the mindset, anything is possible: questions you couldn't answer suddenly become easy.
I think the "lost and confused" applies to both...
At least it would have been if I hadn't been lost and confused.
the constipated mathematician ?
He worked it out with a pencil.
...omphaloskepsis often...
Those are called teaching assistants. I mastered a lot of stuff when I spent my senior year being a teaching assistant. The review was great and I understood all the stuff I had just hurried over in previous years.
I tell my students this regularly. Mathematics is confusing. The stuff you know already is "simple" but at every level, the stuff you are learning is "hard". Here's a wonderful quote I read in a book by Norwegian mathematics Berndt Oksendal, who apparently found it posted around the Tromso University math department:
We have not succeeded in answering all our problems. The answers we have found only serve to raise a whole set of new questions. In some ways we feel we are as confused as ever, but we believe we are confused on a higher level and about more important things.
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 implications.. Forgive me for bringing up religion, but most (albeit not an overwhelming majority of) physicists do not believe in any higher power. If you're comfortable with not knowing things, then the answers provided by belief in a higher power doesn't provide additional comfort. You have no need for that hypothesis. (I'm not saying that people like religion simply because it provides easily digestible answers -- although religious groups *cough* young earth creationists *cough* certainly go for that. Many religious Jews spend their lives studying and debating the meaning of the bible; it's anything but easily digestible.)
Math is great.
Proof is left as an exercise to the reader.
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 going to be easy, and it should not be easy. It ought to be hard. It ought to be radical, it ought to be restless, it ought to lead you to places you'd rather not go." - Henri Nouwen
...omphaloskepsis often...
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.
Myu:
I don't see what is so special about mathematicians skimming over stuff and not sweating the small stuff. Many project planning meetings we treat lots of stuff as black boxes and proceed with some simple assurance that it would do what the team says it will do. The post processing guy would have a very nebulous idea of the geometry core team's claims. Nobody understands what the mesh maker says anyway. Then there are the mathematicians from the solver group. Upside down triangles, dots crosses some time three integral signs lined up like sails of some old ship .. Eventually we understand enough of it to make it work most of the time. Even after the project is done and the feature has been shipped and the user story has been marked complete and independently verified by the user proxy, nobody understands how the mesher meshed it nor how the solver solved it.
sed -e 's/Chuck Norris/Rajnikant/g' joke > fact
http://en.wikipedia.org/wiki/Proof_by_intimidation
And yet, according to your other postings you're quite a nutcase...
"The likes of Facebook and WhatsApp are free to those whose privacy is of zero value."
It's a good thing to be confused, reach understanding, then rinse and repeat... If ignorance isn't part of your day, then your not pushing yourself hard enough...
I find the advise to "sleep on it" a great way to solve problem. It was actually start from a completely different angle to attack the same problem. I also developed the habbit of going to bed early before exams, so that my brain wasn't tired from cramming the night before, and had my wits when I need them most. So, when in doubt, and getting in to a bind, sleep.
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 knowledge, and the skills needed to learn the rest during the project. This will be mostly stuff related to the project, but may include more programming tools as well.
Designing/architecting a program or system is like handling a black bag with the solution inside, in the form of blocks connected by strings. You squeeze it around until you get it into two lumps with very little string running through the thin neck. Then you it into two bags and document all the strings that went through the cut. Repeat unti the bags are small enough to understand easilyj and keep the entire explanation in your head. (In the case of a program that means the code itself fits on a page, with over half of the page being comments.) Then you can open the little bags and grok each one - which by now will be either trivial or maybe embody a single deep concept or "neat hack". (But avoid "neat hacks" if they're not obvious or if something straightforward does the job just fine.)
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 question: "how?". Math is answering the question "what is?".
http://abstrusegoose.com/395
If it weren't for deadlines, nothing would be late.
Oh Maths, your just the tool physicists use. Like a hammer, a bath tub or a Large Hadron Collider.
Becoming a mathematician is like becoming someone who is fascinated in shoes, or briefcases or watches or hammers. Not the application of shoes, briefcase or watches, but the objects themselves. People need to never use them, never get them dirty, never do anything with them, leave them and their nice neat proofs alone, vacuummed packed and hidden away. Black and white, so pure, its like a wacky religion thats out of touch.
These stupid silos. I work at a university where they abolished the physics department to keep the mathematics department. Physics was part of Science, where Maths was part of Law and Accounting (which is where it should obviously be, because you know, its not a science). Law and accounting had so many graduates, and they all had to do these maths units, you know, a whole subject on percentages, a whole subject on addition.
So while the physics guys were going out to high schools, teaching physics and maths, and inspiring minds, Maths were teaching basic addition and subtraction to uni people. It was in their interest to keep high schools stupid, because if students could do the maths, they wouldn't need these stupid feeder units and they wouldn't exist.
So now everyone wants to be a lawyer, business graduate (you know, to do the business) and none of them can do maths. No one can of course apply maths, not even the mathematicians. I ask them all the time, where might you use that? And they say with an abstract question like this. Give me an applied example, in any field, they scratch their head and say don't be silly I'm not a trades person, I'm a mathematician. Why are you teaching that then? Because they need to know it.
Of course I'm using the same maths to solve lots of problems, and because I don't rote teach it, my student can apply it to stacks and stacks of problems even outside the field I'm teaching them in. Some mathematicians think the same way I do, until they become the head of the mathematics department, which they then think this is an awesome job, don't blow the case.
Mathematicians aren't confused, well they shouldn't be, there entire world is fabricated and unrealistic.
Cute headline, anon, but it's kinda misleading and reeks of BuzzFeed's desperate "read me!" attempts.
[satire] What, math researcher have ADHD, what a surprise, maybe we will some Asperger among them, who know. [/satire]
Ceci n'est pas une Signature !
It's difficult sometimes trying to throw yourself into a direction with some Math to guide you I suppose. I wouldn't know but I suspect software developers have similar problems too, especially for cutting edge software. I wouldn't know about that yet either. Some day.. lol
Felt much the same throughout my sojourn as a doctoral candidate in Electrical Engineering.
It's the standard state of being for a Ph.D student.
I once worked at a large computer manufacturer and one day a tech told me that a new guy was a "wall toucher". I asked him what he meant and he said the guy was a mathematician and all mathematicians run their fingers against the wall as they walk down the halls. This place had many mathematicians on staff and I noticed it was true. You could tell if a guy was a mathematician because he would touch the wall as he walked down the hall. The physicists, mech engineers, comp sci engineers, electronics techs didn't do it but every mathematician did.
"Guess and go" -- the modern teaching technique. Not in a good way. This is seriously what they are teaching kids today.
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 problems. If I don't understand a group of them, THEN I dig back into the theory.
I come here for the love
Group work is fine, but in my opinion when you're trying to get to grips with mathematics (and I'm talking about undergraduate/graduate level mathematics here) there is no substitute for going away and studying the material ON YOUR OWN - reading it, re-reading it, trying some examples, turning it over in your mind. That's how definitions and theorems turn from words on paper into concepts and relationships that you can understand intuitively. And when you understand them, then you can work with them.
Yes there are good teachers and bad teachers, and a good teacher can HELP you to learn mathematics, but at some point you are going to have to put in the hard work on your own. And as the original post says, a big part of that is being comfortable with the fact that you don't understand and prepared to keep going until you do.
That's why mathematics is hard, and also why the best mathematicians are the kind of people who are happy in their own company.
Therefore, you have to be a nutcase to get an A in nonlinear differential equations.
Have they considered Hari Krishna? - apologies to Kermit the Frog
The headline: "Mathematicians Are Chronically Lost and Confused"
I'm surprised the OP was modded down.