Sunday, 7 October 2012

Why does it take so long to learn mathematics?

I'm teaching graph theory this year.  It was one of my favourite areas of mathematics when I was a student.  It contains many gems, ranging from with Euler's solution to the problem of the seven bridges of Konigsberg to the power of Ramsey's Theorem.  The arguments seem to me to be unusually varied, and often sufficiently elementary that great depth of study is not required.

I have had very little contact with graph theory in the time since I graduated.  As an undergraduate I used Robin Wilson's Introduction to Graph Theory, and I am now using it as the basis of my course.  I remember enjoying the book in my youth, and finding it approachable, but I don't remember finding the material as straightforward as it now seems.  (My students aren't finding it entirely straightforward, either, but that may be my fault.)

Why is this?  I don't think I'm a better mathematician than I was 35 years ago.  In terms of solving exam questions, I would not perform as I did when I was twenty.  Even with practice, I am sure I could not get back to that level, and not only because I no longer value that kind of cleverness enough to put the effort in.  I now have a much better general understanding of mathematics and how it all fits together, but I no longer have the ability to master detail that I once did.  

Perhaps Wilson's book (which has gone through four more editions since my undergraduate days) has improved, but, with all due respect to its distinguished author, I doubt if it has really changed sufficiently to make a difference.  (Pure mathematics doesn't change much: theorems that are proved generally remain proved, the Four-Colour Theorem notwithstanding.) 

Learning mathematics takes time, and it has always astonished me how much better I understand material when I go back to it, months or years later, than when I first studied it.  As John von Neumann is said to have told a student who complained that they didn't understand a piece of mathematics, "You don't understand mathematics, laddie, you get used to it."  Even if I haven't looked at Philip Hall's Marriage Theorem, for example, for 35 years, the proof seems much simpler to me now than it did when I was first immersed in the subject area. 

Perhaps I am misremembering my difficulties as a student: perhaps I didn't find it as difficult as I now remember it.  Certainly I had little understanding of how an area of mathematics fitted together: my learning at University consisted of reading strings of definitions and theorems, with little idea where it was all going, making sure I understood each result before going on to the next one, until, perhaps, in the last lecture of the course the lecturer would say something like "and so we have now classified all Lie algebras" and I would suddenly find out what the point of it all had been.  I now feel that I would have been a much more effective mathematician if I had read more superficially, skipping proofs until I understood the context, but since got good marks as an undergraduate I had no incentive to adopt what I now feel would have been a much better strategy.

But I think it is the case with mathematics, much more than with many other disciplines, that time is essential to understanding.   Things we struggle with become much simpler when we return to them months later.  This is why modularisation of mathematics studies is so pernicious.  Examining students in the same semester as they have learned an advanced mathematics topic is, I feel, grossly unfair.  It forces our exams to be superficial and makes it impossible to test deep understanding.  At least, although my graph theory course finishes in December, the exam is not till May.  I suspect my students don't like that, but they are likely to do much better than if they faced the same exam immediately after the final lecture.

21 comments:

  1. mathematics grosses me out. i cant really get the differentiators and integrators. i skipped all my math courses and took business. now i make $500k/yr and never been happier. just sahring my insight.

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    1. I'm sure that's what you tell people when you get home from your shift at McDonalds.

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    2. For someone who makes $500k/yr, your ignorance as reflected by your "insight" is bewildering.

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    3. I don't think a person who trolls a blog on mathematics with a comment filled with typos, anonymously, earns $500k/yr. Aren't you busy doing what you are paid $500k/yr for?

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    4. People like you gross me out. Maybe you or other people earn a lot of money without math or science with only "business classes"; but I don't understand why you have to troll - without mathematicians and scientists be sure the life for all of us would be much much much poorer and without any kind of technology - just like dark ages used to be. I doubt you would be as happy as you pretend to be in the dark ages.
      I don't know much math either now because a lot of years passes since I learned it but I have a deep respect for mathematicians and scientists and so should you.

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    5. You can also earn $500k/yr by backstabbing people or being an asshole.

      Even ignorant, insensitive buffoons, that everybody hates can earn $500k/yr under the right circumstances and in the right line of work.

      That does not mean it's bad to be sensitive or studied.

      Or that knowing Maths or not has any bearing to what you make...

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    6. Read this, please: http://xkcd.com/1050/.

      You're the worst kind of ignorant person: someone proud about his ignorance.

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  2. WEll done - I'm glad things worked out for you.

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  3. Maybe because we have to learn to (ab)use part of our brain that were not evolutionarily meant for math ?

    For example our innate number system is logarithmic: http://web.mit.edu/newsoffice/2012/thinking-logarithmically-1005.html (see the link in the comments at the bottom for a nice audio review)

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  4. I'm not sure that deep knowledge can be tested since it depends on connections formed with ideas from *other* topics which necessarily are outside the scope of whatever is purportedly being tested.

    This is just one way in which our entire education system is faulty!


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  5. For me the main problem with higher-level mathematics is that I can't envision how to use it in 'real-life' which makes it difficult for me to apply myself to it.
    The only math course I ever truly liked was linear algebra because it finally taught me how to derive a quadratic equation from raw data, before that I never had a concept of how one got from the raw data to the equation, and mathematics didn't mean anything to me other than something required as a prerequisite for some other course.
    I have been programming and designing computer systems for more than 30 years and have never had a use for anything more complex than the standard buttons available on a dumb calculator.

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    1. Wait, Linear Algebra? Huh? We derived the quadratic equation in Algebra I, if I recall.
      As for the programming, it depends on your project. e.g. I have a minor in Mathematical Finance, and I'm currently working on stochastic systems requiring PDEs and non-linear systems and optimizations.

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    2. I felt the same about some maths I've done while studying. But there are a few areas in computing where advanced maths can make live a lot easier: 3d graphics, artificial intelligence, simulation and cryptography to think of a few.

      It was really an eye opener for me the day I realized that I can do an transformation on a 3D object by changing to the basis for the vector space the object lives in!

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  6. Mathematics, like most skills, requires a lot of training. When I was younger I though mathematics was just a matter of understanding. The truth is that just understanding a concept means nothing if you don't know how to apply it consistently. That is why solving math problems is the only way to really master it.

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  7. Long compared to what? What field have you studied that you learned any more quickly?

    Do you know English (I'm assuming this is your first language) better than when you were in school? If you picked up a violin again after 30 years, would you think that you would be much better, and appreciate the depth of the music more? Would an advanced high school chemistry or history class seem trivially simple to you today?

    I would bet that the answer to all of these is "yes, of course". Everything is harder the first time, and especially when you're younger. Maths involves thinking, and you're better at thinking, so it seems easier. But in this respect it's not unique: everything that involves thinking is similarly easier for you.

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  8. No doubt it takes time to absorb complicated subjects, but also consider how our brain changes as we age. Studies show that our brain is most effective at high-level reasoning in our fifties. Graph theory may seem easier to you now because you're in the prime of your life for understanding that subject. I found this book very informative in this area: http://pragprog.com/book/ahptl/pragmatic-thinking-and-learning

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  10. Some of the comments above hint at what I am about to say. I believe that topic 'A' cannot usually be well understood without a background in 'B'. However, 'A' is a prerequisite for study of 'B'. Therefore, we first study 'A', then we study 'B', then we have a better understanding of 'A'.

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  11. I am reminded of my own experience studying Maths in school and later, when majoring in B.Sc. ....

    In classed 11 and 12th, I learnt Calculus as a set of formulae, theorems and lemma. The fundamental concept of the limit was so difficult understand and so less stressed upon that it just went over my head. I still remember my Maths teacher telling me how to use the L'Hospital rule - if basic limits dont work out in a numerator & denominator case, simple, just replace with their respective derivatives, till things work out .....calculus just seemed to be a magical, eerie place, with no explanation.

    In B.Sc. enter analysis and the fundamentals that I was looking for, became clearer. And I did go back to the calculus class in engg maths later. Things fell in place and not so eerie after all.

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  12. I think Mathematics, in the same way as most aptitudes, obliges a great deal of preparing. When I was more youthful I however math was simply a matter of comprehension. The reality of the situation is that simply understanding an idea means nothing in the event that you don't know how to apply it reliably. That is the reason tackling math issues is the best way to truly ace it.
    5th Grade Math Practice

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  13. If we choose any subject but we have no interest in that subject it'll definitely take more time to learn. I have also done my BSc with mathematics and I enjoyed math because of my interest. I never thought that mathematics is difficult subject. I agree it needs more concentration and practice but if you are doing this with full concentration and interest you can score more in mathematics rather than other subjects. My younger brother has studied math in board exam and we are waiting for Kerala SSLC Results and he is sure to get more grade because of mathematics.

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