Learning advanced math by drawing, playing with beach balls and knitting | Hacker News Reader

Learning advanced math by drawing, playing with beach balls and knitting

smithsonianmag.com

209 pointslebanon_tn41 comments

If this subject is interesting to you, and you haven't read Mindstorms yet, I cannot recommend it enough. The entire book is an exploration of the significance of mental models in learning.

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.'

Who is the author of the Mindstorms book? I see several different books with this title.

I think it's Seymour Papert. Mindstorms: Children, Computers, and Powerful Ideas, 1980, ISBN 0-465-04674-6

I can learn to understand basics of many advanced concepts outside engineering math to the level where I can read/skimp some research papers and understand the thinking behind it. But it's very hard to recognize the concept on my own and the utility 'on the wild' unless someone else is pointing it out and explains it.

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.

[1]: https://en.wikipedia.org/wiki/Sheaf_(mathematics)

[2]: http://www.drmichaelrobinson.net/20131024_overview.pdf

Oh, if only I could make a career of pointing math 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 before #strangeloop: I'm not a real programmer unless I know Haskell

Me after #strangeloop: I'm not a real programmer unless I knit”

https://twitter.com/adam_chal/status/914207020215042048

This depresses me, because there's a strange .. loop in society where people thinks advance means unaccessible and university lab only. Where in fact most concepts are uber 'simple' in form and can be explored quite easily. Similarly there was an article about J. Bose early radio research where you could see polarizing filters made out of a book with metal foil interspersed. I really feel bad about the distorted reality people live in.

> where you could see polarizing filters made out of a book with metal foil interspersed

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.

https://news.ycombinator.com/item?id=17271786

https://www.youtube.com/watch?v=JLfLSjbm4s8

never heard of it, thanks

We all live in distorted realities. Lots of people in laboratories think for part of the time, you need to be in those environments to be in sync with the research, because it's an active movement of pieces playing out and involves a dynamic that goes deeper than what research comes out through the surface, what gets published.

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.

Obviously a place called Strange Loop is going to biased towards knitting and biased away from Haskell.

https://en.wikipedia.org/wiki/Loop_knitting

https://stackoverflow.com/questions/21505192/haskell-program...

Every so often I'm reminded of why the first computers were based on weaving machines.

Same, but #joyofcoding. And apparently, Jon Skeet agrees[0].

[0] https://twitter.com/jonskeet/status/1005458230401748992

Yes, learning by playing!

A collaborative science-based games list: https://github.com/stared/science-based-games-list/ (and its discussion: https://news.ycombinator.com/item?id=14661813)

Is there a book full of this type of stuff that would be useful with young children?

Not exactly but YES!

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...

Matt Parker's "Things to Make and Do in the Fourth Dimension" is a good one too: https://www.amazon.com/Things-Make-Do-Fourth-Dimension-ebook...

https://naturalmath.com/

(The PDFs are pay-what-you-want and Creative Commons licensed)

https://csunplugged.org is not exactly but might be nice.

not a book, but Tadashi's Toys http://www.msri.org/web/msri/public/tadashis-toys

I knew a grad student who used a ball of yarn to learn knot theory.

It is quite interesting how our brains tend to deny the role of the rest of our body in the process of thinking. After seeing Alan Kay's "Doing With Images Makes Symbols", and his TED Talk "a poweful idea about ideas", I realised how silly that line of thinking was[0].

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.

[0] https://archive.org/details/AlanKeyD1987_2

[1] https://archive.org/details/AlanKay_2007

Brains think brains are the best problem-solvers.

Eyes see eyes as the most beautiful part of the body.

Skin feels that skin is the best part to touch.

For knitting, I expected links to https://www.americanscientist.org/article/adventures-in-math..., http://scientificamerican.com/article/the-stunning-symbiosis..., or http://www.toroidalsnark.net/mathknit.html

Reminds me of crocheted hyperbolic surfaces, which make it easier to visualize some less-intuitive properties of hyperbolic geodesics:

http://www.theiff.org/oexhibits/oe1e.html

I liked mathematics, but nobody got me to crave it like Vihart. That Youtube legend covers complex mathematics like no one else. The teaching method introduced in the article spontaneously reminded me of her chaotically creative ways...

Dirac independently rediscovered the purl stitch. https://books.google.com/books?id=Su-k9Kld5ooC&pg=PA232&lpg=...

I think the way you learn advanced math is going to class, reading a textbook, reading a paper, going to a talk, talking to a friend, etc.

Reminds me of this Richard Feynman story: https://thinkingwiththings.wordpress.com/2012/03/16/richard-...

Title is clickbait. Whatever you can learn from "drawing, playing with beach balls and knitting" is most likely not advanced math. (Otherwise professors in a math department would all be playing knitting and beach balls to get tenure.) Advanced math is hard. It is like toiling in the field under a scorching sun.

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.

It depends on your goals. Much of mathematics is training for specific goals (often narrow), while if your goal is simply to be exposed to many different things, you can get quite a lot of the intuition with knitting needles and drawings.

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.

I'll defer to the math professor on this one.

I'd be curious to see what a math prof would have on this.

But IMHO, math is learned by writing proofs. That is the only solid way for building an understanding.

The author of the linked article is a math professor.

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).

Fair enough. I didn't see the last line. I retract what I said earlier.

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.

Lots of people falsely equate the liberal arts with the humanities. This is a misconception that I try to fight in my in-person conversations with people. Math, chemistry, physics, and biology are all liberal arts. Business, engineering, nursing, and other types of professional training are not liberal arts.

1) Science and math (though not engineering, typically) fall under the liberal arts.

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).

>Still I am not very convinced that it is a good idea to teach STEM students that way.

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!).

------------------

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.

============================================

[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

[6] https://en.wikipedia.org/wiki/Rotation_group_SO(3)

[7] https://en.wikipedia.org/wiki/Grigorchuk_group

> Otherwise professors in a math department would all be playing knitting and beach balls to get tenure.

Your hyperbolic statement is not an argument.

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