9 thoughts on “Weekend discussion

  1. I learnt geometry in lessons at school.

    Unless you’re proposing some kind of Platonic anamnesis doesn’t that bring the discussion to a close?

  2. I also learnt about geometry at school, but I learnt it much more quickly and easily than any of my fellow students. I certainly had better intuition than my peers. (On the other hand, I never did understand the rules of cricket!) My question is Why? Why do some people learn geometric ideas much faster than other people? Is this because of something innate, something there about geometry in their heads before they learn anything formally, or is it taught?

    Scientific American a few years ago had an article about the sophisticated geometric problems which were widely popular in Japan in the 18th century. The people who solved these problems were not mathematicians and usually had no formal training in geometry. So it is possible to think deeply about geometry without formal training in the subject.

  3. So you’re actually asking why it is that people who are presented with the same course materials learn at different rates. Wouldn’t it be more surprising if everyone learnt at the same rate?

    Talking of Japanese puzzles, I recommend Hanayama puzzles. In my opinion the chain puzzle, for example, is a work of genius. I’m astounded that anyone could invent such a puzzle. On the other hand, it might be a mistake to assume it’s the invention of a single person. It might be the result of a tradition of puzzles with each designer making incremental changes to an earlier puzzle. Amazing puzzles nonetheless.

  4. Apparently some researchers think that understanding geometry is innate in humans.

    My 10 month old baby clearly wants to understand how objects are oriented spacially. Her discovery of a new object follows the same course: she picks up said object, rotates it and looks at it from each angle and then puts it in her mouth. Not exactly geometry but I thought it was interesting from a psychological standpoint (and I think everything my baby does is interesting).

  5. All of the people in that research had spent at least 6 years of their lives actually embedded inside a nearly Euclidean 3D space. I’m having trouble understanding how it was a test of innateness.

  6. Sigfpe (two posts back): My question was about geometry specifically, although the answer may apply to education more generally. But here’s a conjecture, specific to learning geometry:

    Neuroscience says that our brains have two hemispheres, one side (the left side in most right-handed people, and the right-side for most left-handers) processes speech while the other side processes images and non-speech creative activities (eg, art, music). However, there are some oddities: People who have studied music for several years typically process sound in their left hemisphere (the language side), rather than the right hemi, as non-musicians do — the sound-processing function seems to switch hemispheres after some years of training. Perhaps this is because music is also a language (a symbolic representation of something with formalized rules of transformation).

    Now, in order to master geometry one needs both a good spatial awareness (the sort of ability which Megan’s baby is acquiring through experiment) AND good logico-deductive abilities, the ability to reason using a formal language and defined rules of inference. That would mean using both hemispheres of a normal brain simultaneously, or at least swapping between the two hemispheres repeatedly. If so, this could explain why some people find geometry easier than other people, since their brains are wired to do this more effectively than average. It may also be the case that training in geometry (or in math more generally) leads to one or other of these functions switching hemispheres, as with music training.

  7. An interesting thing about mathematics is the way that “logico-deductive” abilities can be substituted for abilities like “spatial awareness”. For example, in physics, gauge theory is about physics on vector bundles. But some physicists do good work in this area while blissfully unaware of the geometric meaning of what they are doing. As long as you have a good intuition for manipulating the symbols you can make good deductions. So if there really are ‘centres’ in the brain for manipulating symbols that are distinct from the centres for doing geometry (detectable uisng SQUID experiments say) then it’d definitely be interesting to investigate these in mathematicians and physicists.

  8. Bits and pieces of information on the development of the brain have appeared in the news. I have a paper to finish up today so do not have the time to go back and find references.

    An article in Sciam suggested that the visio-spatial areas of the brain develop earlier in males than females. Depending on the timing, this may explain some differences in learning geometry – if it occurs earlier, do more boys than girls appear to understand? There was also an article (Sciam again) reporting on differences in the male and femal brain. I have this flagged as info to follow up on when I get some time. (Note: I have poor visio-spatial skills and can only deal with geometry if I can find a method that is more analytic or possibly algebraic and doesn’t rely on understanding spatial relationships.)

    The CNN piece referenced by Megan is interesting enough to warrant looking at the Science report in order to see the “real” results (regular news seldom gets study results entirely correct).

  9. I teach Algebra to eighth graders. This past summer, however, I taught Geometry at a high school. Forty of my forty-four students were “repeaters,” some for the second or third time. Can you imagine what it was like, for them, to walk into a room knowing bad things were going to happen?

    I teach Algebra as if I were teaching French; it’s a language. So I taught Geometry the same way. I emphasized vocabulary and all the patterns I could find. We did very few proofs. My lessons were based on “this is how this works. It’s the same as this but different from this. Let’s see if you can repeat it.”

    When they would talk to me privately (these are kids, ok?), these fourteen-through-seventeen-year-olds told me that no one had ever taught them math they way I did. Most definitely my Geometry class was “different.”

    I don’t think learning Geometry is innate. I think some people (teachers) act as if it’s innate and that shuts out students who don’t learn they same way as others. I also think some people (teachers) are so advanced in their knowledge of mathematics that they’ve lost perspective on how difficult learning “the basics” can be. It’s analagous to a star ballplayer coaching Little League. Personal abilities don’t always translate into teaching abilities.

    BTW, student success rate was about 75%. I think a few of those kids showed up for school in September because of it instead of quitting.

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