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Mathematicians Are Chronically Lost and Confused 114

Posted by Soulskill
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.'"
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Mathematicians Are Chronically Lost and Confused

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  • by immaterial (1520413) on Wednesday March 05, 2014 @05:13PM (#46412715)
    I was the best mathematician in my university math classes. Who knew?
    • by ackthpt (218170)

      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 subtraction of beer from a case, addition to empty bottles, dividing time between drinking and the necessary room and multiplying the number of pink elephants surrounding me.

      Heady times.

      • 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.

  • Sounds like learning is not necessarily a linear process. Makes me feel better about my learning experience!
    • Re: (Score:3, Interesting)

      by CapeDoryBob (204240)

      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.

  • Now to learn what the question is...

  • "Trivial" (Score:2, Interesting)

    by oldhack (1037484)
    Everything, and only things, that math people do is "trivial".
    • by Jeremy Erwin (2054) on Wednesday March 05, 2014 @05:40PM (#46412979) Journal

      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.

      • 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.

        • 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.

  • by Anonymous Coward on Wednesday March 05, 2014 @05:28PM (#46412849)

    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.

    • by jones_supa (887896) on Wednesday March 05, 2014 @05:36PM (#46412941)
      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.
      • by esldude (1157749)
        I think you are right. Psychology learning investigations even with toddlers found in a natural environment (non-classroom) people learn by watching or interacting with those just very slightly more advanced than them. People who know just a little something they don't. This is actually obvious. Two people like that can communicate very easily as they are at a similar point of learning. The person knowing the extra thing or two learned it recently. Making it easy to help someone else replicate the aha
      • That is a good argument in favor of instructors giving time for students to work together in class (though this is often difficult to do in large lecture sections in a university setting, where contact hours are limited), and for students to form study groups outside of class (something that I strongly encourage my students to do whenever possible). I remember a time when the concepts that I am teaching were difficult to understand, but, frankly, that was 15 or 20 years ago, and I have forgotten what I had
        • When I was in high school, I was very good at math but couldn't be bothered to actually apply myself and take the highest level classes. I picked up everything pretty quickly, and I remember it being hellish when the teacher would say "break into small groups." Nobody cared to actually do the work. Most of the time it would be me and 1 or 2 other people in the class who actually got the concept, and everyone else would beg us to let them just copy off of me. I never consciously let anyone copy off of me.
      • by jandersen (462034)

        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

      • I think the same. Accordingly, math teachers may not teach as well because they are the select few who naturally understand and enjoy the subject. Most of their students have a very different perspective.
    • An alternative explanation is those math teachers didn't actually understand the concept, and therefore were unable to properly explain it.

      • by techno-vampire (666512) on Wednesday March 05, 2014 @06:14PM (#46413311) Homepage
        What's worst is a teacher who defines a new term in a way that only makes sense if you already understand the concepts behind it. As an example, Rudy Rucker once defined a cardinal number (in a book) as, "A number is a cardinal number if it doesn't share its cardinality with any other number." Now, if you know what a cardinal number is, and what "cardinality" means, that's true. If you don't, as most of the readers of that book wouldn't, it's useless.
        • by ColdWetDog (752185) on Wednesday March 05, 2014 @06:41PM (#46413603) Homepage

          Wow. He must write a lot of computer documentation.

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

          • It's been decades since I read that book, but unless my memory's worse that I think, he didn't define it at all.
        • 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

          • 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.
    • by royzeng (3157739)
      I remember when I was studying in Ireland, one of my maths lecturer created a question that he can not answer.....and finally he asked help from students... http://www.szwelder.com/ [szwelder.com]
    • by Galilee (90424)

      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

  • by Anonymous Coward on Wednesday March 05, 2014 @05:32PM (#46412889)

    There are two types of theorems: trivial and unproven.

  • Bizarre advice (Score:5, Insightful)

    by AthanasiusKircher (1333179) on Wednesday March 05, 2014 @05:42PM (#46412995)

    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)

      by vdorie (1106873) on Wednesday March 05, 2014 @07:32PM (#46414053)

      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 do math. 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.

      • by lgw (121541)

        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.

      • 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.

        .

        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

      • 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.

    • by Anonymous Coward

      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.

      • 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

    • by reg106 (256893)

      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

    • "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 pr
  • by Anonymous Coward

    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

    • by Anonymous Coward

      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!

  • I think the "lost and confused" applies to both...
  • the constipated mathematician ?

    He worked it out with a pencil.

  • 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

    • See, there's a difference between knowing what you don't know and living in a sea of ambiguity the way the OP seems to imply. In mathematics especially, there is a very tall and elaborate edifice of deductions and axioms from which all exploration takes place.

      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
  • 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

  • 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.

  • 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 thr
  • 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

    • by Anonymous Coward

      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

    • by Agent0013 (828350)

      (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 always true. 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

  • Cute headline, anon, but it's kinda misleading and reeks of BuzzFeed's desperate "read me!" attempts.
  • Andrew Wiles [wikipedia.org] made the following comment [pbs.org] that has always stuck with me:

    Perhaps I can best describe my experience of doing mathematics in terms of a journey through a dark unexplored mansion. You enter the first room of the mansion and it's completely dark. You stumble around bumping into the furniture, but gradually you learn where each piece of furniture is. Finally, after six months or so, you find the light switch, you turn it on, and suddenly it's all illuminated. You can see exactly where you were. Then

    • by jnana (519059)
      Oops, that's what I get for just reading the comments before commenting and not looking at the article until after posting! The exact quote is prominently mentioned in the article.
  • [satire] What, math researcher have ADHD, what a surprise, maybe we will some Asperger among them, who know. [/satire]

  • Felt much the same throughout my sojourn as a doctoral candidate in Electrical Engineering.
  • Have they considered Hari Krishna? - apologies to Kermit the Frog

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