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Marvin Minsky: What makes mathematics hard to learn? (2008)

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Re: Marvin Minsky: What makes mathematics hard to learn? (2008)

#161

Post by toddan »

j2kun wrote: post #57
toddan wrote: post #52

Not everyone can learn math. I my self had brilliant and great math tutors that my parents hired for me. I event went to special private math classes with brilliant math teachers and i tried to the best of my abilities to get good at it and pass the advanced math courses in highschool. Still i suck greatly at math and i always will. I am very grateful for all the help my parents provided for me without it i would not…

Barring an extremely small number of cases, this is not true by any means. The cognitive skills required to learn math are the same as the cognitive skills required to learn language. If you're interested in learning more about this, check out Keith Devlin's "The Math Gene" [1] [1]: http://www.amazon.com/The-Math-Gene-Mathematical-Thinking/dp...

If it was as simple to read one book to be good at math I would be an engineer a long time ago with all the tuition and help i got thru my years in school.

Re: Marvin Minsky: What makes mathematics hard to learn? (2008)

#162

Post by unabridged »

Imagine if learning english required getting to college before you could even get past spelling. And that your teachers who majored in english, have only constructed a sentence a few times in a 400 level class and have never read more than a paragraph. How do you even begin to talk about literature? The payoff is so far away and unrelated to the subject taught in school that 99% of the people in charge have no idea what is going on.

Re: Marvin Minsky: What makes mathematics hard to learn? (2008)

#163

Post by TheOtherHobbes »

cmrdporcupine wrote: post #75
kazinator wrote: post #38

While kids may find arithmetic to be boring and demotivating, they're able to do it. It's when math gets symbolic that you get the permanent attrition. For instance, most people never get to calculus. When you progress from arithmetic to abstract, it hardly gets easier . Moreover, memorizing. You go from memorizing multiplication tables to other kinds of tables, like tables of equations giving various identities, row…

I failed math all through elementary school but somehow got into this industry despite that. The reality is I'm terrible at arithmetic, but fine at what most of CS is: logic in the form of boolean logic, symbolic stuff. As a kid I was much better with the symbolic than the arithmetic -- which I almost always got _wrong_. My brain stumbles over numbers. I'm terrible at it. I feel like my childhood education did a horr…

Some academic mathematicians are terrible at arithmetic.

There's no correlation at all between being able to add up a grocery bill in your head and being good at abstraction.

My accountant is very good at mental arithmetic. I don't expect him to win the Fields Medal any time soon.

This is more relevant than it should be. Too many people leave school thinking math is mental arithmetic, and have no idea what symbolic manipulation is, or why it's useful.

Re: Marvin Minsky: What makes mathematics hard to learn? (2008)

#164

Post by JadeNB »

throwupper247 wrote: post #35
qb45 wrote: post #6

Earlier quoted context omitted.

I think parent wondered how it's possible that she remembered 16+16=32. Children raised on positional decimal arithmetic are "supposed" to figure that 10+10=20 or 20+20=40 and then add/subtract 5+5. Of course it's easier to associate 15 with 16 than with 10 or 20, but the fact that she immediately knew 2·16 and was able to proceed further says something about either her experience with binary arithmetic or some tende…

In the decimal system it is just as easy to associate 5+5=10. 5xN is easier to remember than 16xN on account of 5 being the smaller number. 2^n is logarithmic whereas 5*n is linear. Arguably, logarithms are not too complicated, even if linear seems to be a degree easier, seeing that the decimal system is also logarithmic as that's a denser representation. edit: how to enter an aterisk as the multiplication operator s…

> 5xN is easier to remember than 16xN on account of 5 being the smaller number.

By that logic, shouldn't it be easier to remember 7 * n than 10 * n, because 7 is the smaller number?

> 2^n is logarithmic

Also, as Jtsummers (https://news.ycombinator.com/item?id=10973381) points out, the function `n \mapsto 2^n` is exponential: its growth is significantly faster, not significantly slower, than exponential.

Re: Marvin Minsky: What makes mathematics hard to learn? (2008)

#165

Post by JadeNB »

pas wrote: post #26
adrianN wrote: post #13

Earlier quoted context omitted.

The correct method might be the method that scales better. Counting on your fingers for example is a pretty decent method to add until you reach bigger numbers. If you don't use the more complicated methods, you'll never improved past some point.

Then ask harder questions that show the value of the harder (more complicated, less intuitive) methods. But at the same time, show the value of those answers, because on the face of it, knowing how much is 342x520+92 is not really valuable. Even knowing how much is any number plus any number is not that interesting for basically 90% of the population.

> Then ask harder questions that show the value of the harder (more complicated, less intuitive) methods.

It's not clear to me that there's an obvious right answer here, but I do believe that most obvious answers are, to some degree, wrong. Namely, saying "just ask harder questions" doesn't take into account that then the student is trying to solve hard questions and learn a new method at the same time.

Teaching a more general method on easy problems gives time to become acclimatised to the method, while also offering sanity checks (by easier, faster, and / or approximate methods) to make sure that it's not going wrong. On the other hand, as you say, this approach seems unmotivated, and bores or frustrates students—there's no obvious right answer! I think that all that can be said is that one must find the appropriate balance for each individual student, or class.

Re: Marvin Minsky: What makes mathematics hard to learn? (2008)

#166

Post by allemagne »

thomasahle wrote: post #2

> Anecdote: I asked a certain 6-year-old child “how much is 15 and 15”and she quickly answered, “I think it’s 30.” I asked how she figured that out so fast and she replied, “Well, everyone knows that 16 and 16 is 32, so then I subtracted the extra two 1’s.” Wait, is this girl some kind of base-2 native?

I remember knowing 16+16=32 because of doing doubles starting with 1. 1+1=2, 2+2=4, 4+4=8, 8+8=16... if I had to guess, I'd say this is how that little girl knew.

Re: Marvin Minsky: What makes mathematics hard to learn? (2008)

#167

Post by crispyambulance »

j2kun wrote: post #159
sn6uv wrote: post #106

Earlier quoted context omitted.

What's wrong with two column proofs? I think they're a great tool for getting into the absolutely rigorous mindset required for maths. Leslie Lamport even makes the case that professional mathematicians should use them to reduce publication errors [1]. My personal feeling (as a maths grad student) is that the utility of two column proofs depends on the field. For logic and algorithms sure but estimating integrals it…

I have never met a mathematician who feels that two-column proofs is the right way to teach high school geometry. Considering that little math taught in high school beyond basic algebra uses anything close to 'absolute rigor,' introducing rigorous proofs should be left for college when one actually has a reason to learn proofs for advanced mathematics (set theory, analysis, etc). Geometry is taught correctly when pro…

There is no single construct or pedagogical technique which is responsible for kids not learning math. In the hands a good teacher 2 column proofs are just fine. Calling them "awful" is a bit hysterical.

Re: Marvin Minsky: What makes mathematics hard to learn? (2008)

#169

Post by j2kun »

toddan wrote: post #161
j2kun wrote: post #57

Earlier quoted context omitted.

Barring an extremely small number of cases, this is not true by any means. The cognitive skills required to learn math are the same as the cognitive skills required to learn language. If you're interested in learning more about this, check out Keith Devlin's "The Math Gene" [1] [1]: http://www.amazon.com/The-Math-Gene-Mathematical-Thinking/dp...

If it was as simple to read one book to be good at math I would be an engineer a long time ago with all the tuition and help i got thru my years in school.

You misunderstand me. The book I referenced does not try to teach you math. The book I referenced surveys scientific studies on the question of "whether everyone can learn math." There is overwhelming evidence presented in this book that, in fact, everyone can learn math.

Re: Marvin Minsky: What makes mathematics hard to learn? (2008)

#170

Post by j2kun »

crispyambulance wrote: post #167
j2kun wrote: post #159

Earlier quoted context omitted.

I have never met a mathematician who feels that two-column proofs is the right way to teach high school geometry. Considering that little math taught in high school beyond basic algebra uses anything close to 'absolute rigor,' introducing rigorous proofs should be left for college when one actually has a reason to learn proofs for advanced mathematics (set theory, analysis, etc). Geometry is taught correctly when pro…

There is no single construct or pedagogical technique which is responsible for kids not learning math. In the hands a good teacher 2 column proofs are just fine. Calling them "awful" is a bit hysterical.

Yes, and in the hands of a brilliant artist, mud and coffee stains can make a lovely painting. That does not mean art teachers should use mud and coffee as the only tool for teaching one how to paint.

Unfortunately, most high school math teachers are undertrained, underpaid, overworked, and pressured to focus on exams. The two-column proof is a crutch. It's not the only awful part of high school math education, nor is it the sole bane of a student's math education (I never said it was). But it is the most egregious example of bad math education.

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