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Everyone is capable of, and can benefit from, mathematical thinking

quantamagazine.org

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Re: Everyone is capable of, and can benefit from, mathematical thinking

#281

Post by auxbuss »

Shosty123 wrote: post #241
sourcepluck wrote: post #204

Earlier quoted context omitted.

> rotated shapes, slot machines, or origami Or gears (like Seymour Papert), or abacus beads, or nomograms, or slide rules, etc etc. Anyone have any more, throw them out! Is mathacademy good? I have been thinking of giving it a month of a try. You say "stressful", which I'm not sure is a mis-type or not. I ordered Mathematica at my local library by the way, and can now forget about it until I get an SMS one day inform…

I've had a MathAcademy subscription for some time and it's quite good. I'd say it's best at generating problems and using spaced repetition to reinforce learning, but I think it falls short in explaining why something is useful or applicable. I don't know, most math education seems to be "here's an equation and this is how you solve it" and MathAcademy is undoubtedly the best at that, but I wish there were resources…

If you're interested in how vector calculus developed, and who was instrumental, all the way from Newton/Leibnitz to Dirac or so, by way of Hamilton, Maxwell, Einstein and others, then Robyn Arianrhod's 'Vector' is brilliant.

But be warned, it gets progressively harder, along with the concepts, so unless you're conversant with tensors, at some point you will have to put on your thinking cap.

The reviews on Goodreads – including my own – are worth reading to get a flavour: https://www.goodreads.com/book/show/202104095-vector

Re: Everyone is capable of, and can benefit from, mathematical thinking

#282

Post by sethpurcell »

Given the sentiment in these comments, I figure this crowd might be interested in the book "Measurement" by Paul Lockhart (the guy who wrote "A Mathematician's Lament")

He's of the opinion that math should be taught not as jumping through hoops for "reasons", but as an art, enjoyable for its own sake, and that this would actually produce more confident and capable thinkers than the current approach. (I think the argument applies to almost all education but his focus is just on math.)

Re: Everyone is capable of, and can benefit from, mathematical thinking

#283

Post by Rendello »

gravypod wrote: post #253
mbbbackus wrote: post #180

I've been reading the author's book, Mathematica, and it's awesome. The title of this post doesn't do it justice. He shows that math skill is almost more like a sports talent than it is knowledge talent. He claims this based on the way people have to learn how to manipulate different math objects in their heads, whether treating them as rotated shapes, slot machines, or origami. It's like an imagination sport. Also,…

I really want to try MathAcademy.com. How quickly do you think someone doing light study could move from a Calc 1 -> advanced stuff using that site? In my case I could put in at least 30 minutes to an hour a day.

I can't speak to the advanced stuff but here's my stats on Fundamentals I:

Total time on site (gathered from a web extension): 40h 30m Total days since start: 32

Total XP earned: 1881

Since "1 XP is roughly equivalent to 1 minute of focused work", I "should have" only spent 31 hours. I did the placement test and started at ~30%, and now I'm at 76%. I'd say 75% is stuff I learned in HS but never had a great handle on, 25% I never knew before.

Overall, I'm quite happy with the course. I'm learning a lot every day and feel like I have stronger fundamentals than I did when I was in school. The spaced review is good but I do worry I'll lose it again, so I'm thinking of ways I can integrate this sort of math into my development projects. It's no Duolingo, you really do have to put in effort and aim for a certain number of Xp per day (I try for 60 XP rather than time).

Re: Everyone is capable of, and can benefit from, mathematical thinking

#284

Post by calf »

davidbessis wrote: post #189
tgv wrote: post #161

Earlier quoted context omitted.

> Not really. There's nothing inherently special about people who dedicated enough time to learn a subject. "You didn't work hard enough." People really blame you for that, not for lacking talent. > So far in human history there were less than 200 people who ran 100m in less than 10s. And many millions have tried. There may be 200 people who can run it under 10s, but there are thousands that can run it under 11s, and…

I'm the author of what you've just described as clickbait. Interestingly, the 100m metaphor is extensively discussed in my book, where I explain why it should rather lead to the exact opposite of your conclusion. The situation with math isn't that there's a bunch of people who run under 10s. It's more like the best people run in 1 nanosecond, while the majority of the population never gets to the finish line. Highly-…

This power law argument immediately reminds me of education studies literature that (contrary to the math teachers in this thread) emphasize that mathematical ability is learned cumulatively, that a student's success feeds on itself and advances their ability to grasp more difficult material.

As for my own half-baked opinion, I want to say that the Church-Turing Thesis and Chomsky's innate theory of cognition have something to add to the picture. Homo sapiens as a species essentially has the capacity to think analytically and mathematically; I want to argue this is a universal capacity loosely analogous to the theory of universal Turing machines. So what matters is people's early experiences where they learn how to both practice and, critically, to play, when they learn difficult ideas and skills.

Also, as an amateur pianist, most people don't know that modern piano teaching emphasizes not fixed limits of the student but that many students learn the wrong techniques even from well-meaning piano coaches. Just the other day I was watching a recent YouTube Julliard-level masterclass where the teacher was correcting a student on her finger playing technique, presumably this student had been taught the wrong technique since childhood. With music or sports a coach can visually see many such technique problems; with math teaching it of course harder.

Re: Everyone is capable of, and can benefit from, mathematical thinking

#285

Post by burnte »

jolt42 wrote: post #276
chankstein38 wrote: post #269

Earlier quoted context omitted.

Would you say the book ventures more into the practical side of learning this stuff or is it closer to the tone of this article? I found this article hard to gain anything from. A lot of just motivational cliche statements and nothing really groundbreaking or mind altering. If the book is better at that, I'd love to read it. If it's stories and a lot of fluff, I'd rather skip. So I'm curious what you are getting from…

Agree. The article turned me off as well. No specific example, felt like an ad.

Yeah, I quit reading it because it didn't talk about the book, it felt like a meta article.

Re: Everyone is capable of, and can benefit from, mathematical thinking

#286

Post by aleden »

In my high school we were basically only instructed to get good at applied math. Calculus. Which more often than not was simply "plugging it in". Most of that work is trivially automatible through Mathematica. When I reached a university, I took number theory and abstract algebra and it blew my mind that math was actually so beautiful in a way that defied explanation. When I took real analysis I finally saw the side of calculus that didn't seem like a waste of time.

One day, I went back to my high school and spoke to my computer science mentor back then [1]. I passionately asked him why we were never exposed to group theory. The answer, he said, was the SAT. None of that stuff is on the SAT, so it can't be justified teaching.

[1] The great Andrew Merrill

Re: Everyone is capable of, and can benefit from, mathematical thinking

#287

Post by goatlover »

wtetzner wrote: post #160
sdeframond wrote: post #137

Earlier quoted context omitted.

> I strongly believe that the average human being can be exceptional in any niche topic given enough time, dedication and focus. I respectfully, but strongly, disagree. There's a reason most NBA players are over 2 meters tall, and one does not become taller with time, dedication nor focus. It might be different for intellectual skills but I am not that sure. Almost anyone can become decent at almost anything though.…

> I respectfully, but strongly, disagree. There's a reason most NBA players are over 2 meters tall, and one does not become taller with time, dedication nor focus. Being tall isn't a skill. I suspect you could be skillful enough at basketball to overcome the hight disadvantage. However, I think most people who might become that skillful see the high disadvantage (plus the general difficulty of becoming a pro basketba…

Height is one physical attribute that helps, and professional players are mostly above average height for a reason. But also hand-eye coordination and fast-twitch muscles help even more. Many basketball players are very explosive athletes, because it's a sport with a relatively small play area and lots of quick movements are needed.

Track and swimming are where innate physical attributes have the most obvious benefits. Michael Phelphs had the perfect body for swimming. There is no amount of trainingg that 99.999% of the population could do to get close to what Usain Bolt ran. Most humans could not train to run under 4 minutes in a mile or under 2:30 in a marathon. They just don't have the right muscular and cardiovascular physiology.

Team sports are of course more complicated as other qualities come into play that aren't as directly physiological.

Re: Everyone is capable of, and can benefit from, mathematical thinking

#288

Post by davidbessis »

zyklu5 wrote: post #277
davidbessis wrote: post #220

Great to see so many reactions to my interview, thanks! I see that many people are confused by the interview's title, and also by my take that math talent isn't primarily a matter of genes. It may sound like naive egalitarianism, but it's not. It's a statement about the nature of math as a cognitive activity. For the sake of clarity, let me repost my reply to someone who had objected that my take was "clickbait". Thi…

I think you've simply redefined genius. Many years ago I read an article on youth football, if I remember correctly, and in it there was a bit about the writers visit to the Ajax Youth Academy. In it he writes of a moment during practice when a plane flies over and all the 7(?) year olds on the pitch look up to see it except for one kid who keeps his eyes on the ball. That kid (of course) grows up to be a very good m…

Indeed, it does involve redefining genius as a "state", or "flow", or "trajectory".

When I say it's not primarily genetic, many people wrongly assume there's an entirely explainable and replicable way of accessing this state. There isn't.

The 20,000 hours rule is a bit misleading, because who gets to invest 20,000 hours into something? How do you create this drive, this trajectory? You must have a good hope that it'll yield something worth the effort.

This is why the injunction to "work harder" so often misses the mark.

However, even if only a tiny fraction of the population will end up becoming a "genius", it's very important to debunk the myth, because the real story has valuable lessons for everyone: it gives concrete and pragmatic indications on what one should be on lookout for.

It's not fully teachable up to genius level, but the directionality is teachable and extremely valuable.

Re: Everyone is capable of, and can benefit from, mathematical thinking

#289

Post by j2kun »

brodo wrote: post #211
tracerbulletx wrote: post #9

A nice sentiment but clearly a large % of people never do learn even basic mathematical thinking and seem very confused by it. So is there some scientific study backing up the claim that all these people could easily learn it or are we just making it up because its a nice egalitarian thesis for a math popularization book?

The same goes for language skills, by the way. In the US, 21% of adults are illiterate, and 54% of adults have literacy below sixth-grade level.[1] This is higher than in other developed countries. For example, in Germany, 10% are illiterate, and 32% have literacy below fifth-grade level.[2] General intelligence also seems to have been trending downward since the 1970s (the reverse Flynn Effect)[3]. It has been measu…

What a weird comment. Are you trying to argue by analogy that a decent fraction of the population are not capable of literacy? It seems self-evident that low literacy rates have nothing to do with innate ability. I see no evidence to suggest that math is any different.

Re: Everyone is capable of, and can benefit from, mathematical thinking

#290

Post by FractalHQ »

hosh wrote: post #268
ericmcer wrote: post #254

Earlier quoted context omitted.

Sounds really cool. It reminds me of programming, that moment when new code starts to really sync up and code goes from being a bunch of text to more intuitive structures. When really in the zone it feels like each function has its own shape and vibe. Like a clean little box function or a big ugly angry urchin function or a useless little circle that doesn't do anything and you make a note to get rid of. I can kinda…

There's a lot of interesting discrete math that can supercharge programming at different levels of scale. What's pretty cool is that it reveals a layer of understanding when I watch my toddlers learn math from counting. One of the interesting things is being able to exactly describe how something is an anti-pattern, because you have a precise language for describing constraints.

I would love to learn about some of these anti-pattern proofs if you have any examples or references you can share!
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