Learning advanced math by drawing, playing with beach balls and knitting(smithsonianmag.com) |
Learning advanced math by drawing, playing with beach balls and knitting(smithsonianmag.com) |
A quote from the opening:
'BEFORE I WAS two years old I had developed an intense involvement with automobiles. The names of car parts made up a very substantial portion of my vocabulary: I was particularly proud of knowing about the parts of the transmission system, the gearbox, and most especially the differential. It was, of course, many years later before I understood how gears work; but once I did, playing with gears became a favorite pastime. I loved rotating circular objects against one another in gearlike motions and, naturally, my first "erector set" project was a crude gear system. I became adept at turning wheels in my head and at making chains of cause and effect: "This one turns this way so that must turn that way so..." I found particular pleasure in such systems as the differential gear, which does not follow a simple linear chain of causality since the motion in the transmission shaft can be distributed in many different ways to the two wheels depending on what resistance they encounter. I remember quite vividly my excitement at discovering that a system could be lawful and completely comprehensible without being rigidly deterministic. I believe that working with differentials did more for my mathematical development than anything I was taught in elementary school. Gears, serving as models, carried many otherwise abstract ideas into my head. I clearly remember two examples from school math. I saw multiplication tables as gears, and my first brush with equations in two variables (e.g., 3x + 4y = 10) immediately evoked the differential. By the time I had made a mental gear model of the relation between x and y, figuring how many teeth each gear needed, the equation had become a comfortable friend.'
Take for example Sheaf[1]. The basics are not that hard if you spend some time. But once you have learned it in abstract. Can you see use for it [2] in data analytic, signal processing, or machine learning? How long you have to work for it to really click to the point where you can see and utilize the concept?
I think this is the reason why mathematicians are needed more in every area. They should walk around pointing things out.
Joking aside, I think a lot of researchers will oversell the applications of their technology, so it's just as important to be able to recognize when a particular math thing will help, and when it's more harm than good. The first rule of statistics is: if the decision maker already has their heart set on a particular action, you shouldn't waste your time and money designing a statistical study with fancy tests and inference techniques.
Me after #strangeloop: I'm not a real programmer unless I knit”
A collaborative science-based games list: https://github.com/stared/science-based-games-list/ (and its discussion: https://news.ycombinator.com/item?id=14661813)
One of my favorite books is Steven Caney's "Play"
https://www.amazon.com/Steven-Caneys-Playbook-Caney/dp/09111...
From an older edition, one of many projects was making a hammock out of old 6-pack rings. For many years, the plastic used in these rings is designed to degrade, but at the time the book was written it would have worked.
http://3.bp.blogspot.com/_Zkz9la3e96c/SmyCb2oFfvI/AAAAAAAABt...
(The PDFs are pay-what-you-want and Creative Commons licensed)
Using the right things to fiddle with physical manifestations of an idea lets "my hands" think along the more "abstract" parts of my brain. And doing that kind of parallel work is actually very useful.
Eyes see eyes as the most beautiful part of the body.
Skin feels that skin is the best part to touch.
Edit: Didn't see that the author of the article is a math professor. This method seems to work in a liberal arts college, but I doubt it would work in a STEM curriculum.
As a mathematician and engineer, I bet this class is more like a "film history" class than a "principles of optics" or "CGI techniques" course.
But IMHO, math is learned by writing proofs. That is the only solid way for building an understanding.
Your hyperbolic statement is not an argument.
A tangent. In case you missed the amazing documentary "Magic, Art, and Scanimation", it is all about the iterative process of "making" magic through hard work, creativity, and insight.
Many concepts are super simple once they've been absorbed by culture to the point that you can learn by osmosis. It's not the same clarity as coming to a conclusion without the awareness existing initially.
My dilemma often is, for some concepts such as math, computation, you can't really prove where such concepts originate from, or whether they exist independent of the human mind and are instead, discovered over time. The obvious implication being that some concepts may exist in some minds outside of the laboratory, and then what is the purpose of doing such research?
The reason often comes down to the fact that someone is funding you to be there 40-80 hours (or more, depends on how much sleep you need) a week doing nothing else. That's a lot of time to focus on just those things, and I honestly think that makes, not just a big difference, but THE difference, between who gets there first. Because otherwise, you have to be focused on other things, that might not be directly tied to whatever research concepts you may want to discover independently.
Concepts become uber simple because people work hard to spread them out and make them easily understood. There's a counter argument to this that you should also be able to decide whether you want to be influenced by those concepts, but that's just living life. I imagine a lot of people who are not interested in super simple concepts have some aversion to them that could be very valid. It doesn't make them less intelligent, it's just something you wouldn't understand unless you experienced the particular dynamic. Distorted realities indeed. We all have them.
https://en.wikipedia.org/wiki/Loop_knitting
https://stackoverflow.com/questions/21505192/haskell-program...
Additionally, not all math is writing proofs. It's also applying their results (a student doesn't need to prove 2+2=4 to use 2+2=4 in some computation) and developing intuitions about the nature of mathematical objects (which can then be used to guide future development by helping to discover new hypotheses to prove).
Still I am not very convinced that it is a good idea to teach STEM students that way. The author of the article works at a liberal arts college.
2) The course is offered in "J-Term" (January), it's a supplement and meant to connect dots for students, not explicitly teaching them advanced math theory (in the sense that they'd come out of a one-month course able to understand, intimately, graduate level maths).
Take it from a math PhD with quite some teaching experience that we do need more of what this professor is teaching, and less of number-mangling and rule-memorization, especially for STEM people. Understanding of what is going on is far more important than the formalism, which always comes later.
Also understand that the contents of a course aren't well condensed into a short article about it, and you won't get much out of the latter.
In the end, some complex mathematical notions have very hands-on representations that are faithful. The beauty comes from realizing that they are the same. Some examples:
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1a)The limit of the iterated dynamical system (f_1(z) = (1+i)z/2, f_2(z) = 1 - (1-i)z/2) in the complex plane
1b)The shape you get if you fold a paper over many times, and unfold keeping the angles at 90 degrees [1]
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2a)The algebraic field resulting from adjoining the roots of the polynomial x^2 + 1 = 0 to the real numbers
2b)All the ways you can move, rotate, and scale a flat shape on a desk [2]
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3a)The problem of classifications of embeddings of S^1 into R^3 up to ambient isotopy (a whole field of mathematics whose primary problem has remained open for over 100 years, and is connected to many others)
3b)Can you come up with a way to tell if you and I are tying our shoelaces the same way? [3]
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4a)The study of the following class: a set S with an associative operation ×, under which it is closed, such that every element is invertible, up to mappings that preserve × (that is, maps F such that F(g × h) = F(g) × F(h)).
4b)Study of reversible operations on an object that don't change the nature of it[4]. Like shuffling a deck of cards[5], spinning a globe on gimbals[6], or maybe swapping left and right children of some nodes in a binary tree here and there[7].
(The last example is more abstract, but hey, I made a thesis out of things like that!).
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The notions described in a) and b) are exactly the same. The way mathematics is taught is often you don't see b) while looking straight at it! And yet the formalisms in a) are much better understood when you know that they really are b).
If you have seen any definitions in part a), but part b) comes as a surprise - it's a problem. And yet that's the state of affairs.
That's the disaster that this professor is trying to fix.
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[1] https://en.wikipedia.org/wiki/Dragon_curve
[2] https://en.wikipedia.org/wiki/Complex_plane
[3] https://en.wikipedia.org/wiki/Knot_theory
[4] https://en.wikipedia.org/wiki/Group_(mathematics)
[5] https://en.wikipedia.org/wiki/Symmetric_group
