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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)

#101

Post by Jtsummers »

nothis wrote: post #60

That old topic. Ironically, I find people who work with math ( chosen to work with math, that is) to probably be quite unqualified to truly understand the problem. He opens by saying that learning avoiding mistakes before any bigger concepts is a major root of this but then again, people do make mistakes in math (even math professors) and it's a field where any mistake can completely invalidate the work. Also most pe…

> opens by saying that learning avoiding mistakes before any bigger concepts is a major root of this but then again, people do make mistakes in math (even math professors) and it's a field where any mistake can completely invalidate the work.

The problem is that an emphasis on avoiding mistakes leads to paralysis. Though he phrases it differently. Students don't want to be wrong so they just don't instead.

As he points out later in the piece, learning from our mistakes is key. If I never learned to walk because I fell the first time and failed walking class, I'd be crawling even today. An overly negative assessment of the students' works, an overemphasis on right and wrong, rather than correction and expansion to different methods when some method has failed a student, ends up producing many students who can't do math.

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

#102

Post by senekerim »

lisper wrote: post #86
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…

Oh, this. 10,000 times this. Instead of memorizing multiplication tables, you instead have to memorize derivative formulas, or the formulas for the Fourier and Laplace transforms, or the difference between a bijection and an injection, or all the conditions on an elliptic curve that make it suitable for cryptography. I could go on and on and on. The stupid notations that mathematicians use doesn't help either. All ma…

Math notation is actually pretty great and is a result of multiple iterations. Earlier forms were much worse. S-Expressions by themselves are obviously not enough, you have to introduce at least first-order logic notation, set notation, etc. It's not obvious that the end result will be anywhere near as readable as the current notation.

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

#103

Post by lisper »

senekerim wrote: post #102
lisper wrote: post #86

Earlier quoted context omitted.

Oh, this. 10,000 times this. Instead of memorizing multiplication tables, you instead have to memorize derivative formulas, or the formulas for the Fourier and Laplace transforms, or the difference between a bijection and an injection, or all the conditions on an elliptic curve that make it suitable for cryptography. I could go on and on and on. The stupid notations that mathematicians use doesn't help either. All ma…

Math notation is actually pretty great and is a result of multiple iterations. Earlier forms were much worse. S-Expressions by themselves are obviously not enough, you have to introduce at least first-order logic notation, set notation, etc. It's not obvious that the end result will be anywhere near as readable as the current notation.

Current math notation may be the result of multiple iterations, but they all took place before the widespread availability of computers. Current math notation is optimized for hand-writing and manual rather than automated proof checking. It may be great for paper and pencil, but times have changed.

And BTW, the notation actually sucks for paper-and-pencil too because of its ambiguity. See:

http://mitpress.mit.edu/sites/default/files/titles/content/s...

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

#104

Post by maker1138 »

I found his note at the end most interesting and mirrors my observations since. Governmental standards are making our children math-illiterate. There should be a myriad of different schools with different specialties. There should be schools for the learning impaired as well as the gifted. If every school was private it would allow experimentation with every form of teaching and promote innovation and effective methods to be discovered and thrive. One size fits all education works as well for education as it does for clothes.

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

#105

Post by senekerim »

lisper wrote: post #103
senekerim wrote: post #102

Earlier quoted context omitted.

Math notation is actually pretty great and is a result of multiple iterations. Earlier forms were much worse. S-Expressions by themselves are obviously not enough, you have to introduce at least first-order logic notation, set notation, etc. It's not obvious that the end result will be anywhere near as readable as the current notation.

Current math notation may be the result of multiple iterations, but they all took place before the widespread availability of computers. Current math notation is optimized for hand-writing and manual rather than automated proof checking. It may be great for paper and pencil, but times have changed. And BTW, the notation actually sucks for paper-and-pencil too because of its ambiguity. See: http://mitpress.mit.edu/sit…

It's definitely not perfect, but it's much better than any alternatives I have seen. Would love to take a look if there are any good proposals based on S-Expressions or other computer-era notations.

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

#106

Post by sn6uv »

j2kun wrote: post #56
crispyambulance wrote: post #42

Earlier quoted context omitted.

Disagree. Kids disengage from math for many different reasons. There is no single pattern to it. If it were that simple, it would not have remained an issue since schools were commonplace. Minsky solidly makes the point that when math is taught as nothing more than an endless series of tedious drills with no purpose in sight, of course it demotivates kids. No one likes pointless Sisyphean tasks. Many students, howeve…

> Many students, however, "rediscover" math when they get to their first class that involves theorems and proofs (often in high-school geometry) High school geometry (at least in the US, and in my experience) is the worst of them all! Two-column proofs should be banned.

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 becomes tedious quickly.

[1] http://research.microsoft.com/en-us/um/people/lamport/pubs/p...

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

#107

Post by YeGoblynQueenne »

anon4 wrote: post #71
j2kun wrote: post #56

Earlier quoted context omitted.

> Many students, however, "rediscover" math when they get to their first class that involves theorems and proofs (often in high-school geometry) High school geometry (at least in the US, and in my experience) is the worst of them all! Two-column proofs should be banned.

Two-column proofs I had never heard of that monstrosity before and having googled it, it looks worse than programming in COBOL.

Programming in COBOL is vastly underrated. If you're not speaking from experience, Google "working storage" (one of COBOL's nicest features) .

Also: it runs on machines that look like Cylons. You can't beat that. :p

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

#108

Post by simonebrunozzi »

sago wrote: post #32

I had dinner with Prof Minksy, about 15 years ago. He asked what I was working on (evolutionary algorithms) we got talking about domains of knowledge. He said all knowledge has a half-life, the time it takes for half of what you know to be redundant or wrong. Math, he said, is the longest, measured in centuries or millennia. One should feel sorry for neuroscientists: they can go to the bathroom and half their knowled…

This is super interesting. I tried to look for any reference of this, but found nothing. Do you happen to know if this thought of his is available somewhere, in a book, a paper, etc? Or is it a concept borrowed from someone else?

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

#109

Post by jrcii »

sago wrote: post #32

I had dinner with Prof Minksy, about 15 years ago. He asked what I was working on (evolutionary algorithms) we got talking about domains of knowledge. He said all knowledge has a half-life, the time it takes for half of what you know to be redundant or wrong. Math, he said, is the longest, measured in centuries or millennia. One should feel sorry for neuroscientists: they can go to the bathroom and half their knowled…

>neuroscientists: they can go to the bathroom and half their knowledge will be out of date

Web development seems to be even worse than neuroscience in this regard. I feel like I have to perpetually study to be abreast of best practices.

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

#110

Post by derefr »

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…

Note that kids don't reach the formal operational development stage until they're ~11 on average. Until then, expecting them to do abstract (formal) reasoning will be about as successful as expecting a chimp to do so.

And, on the converse, in previous centuries, you'd often see people only start learning math from the beginning in their late teens or early twenties—the time when they first attended "university" without previously attending a grammar or seminary school. These 20-year-olds would be able to pick math all up quite quickly, proceeding from addition and multiplication to calculus and beyond within a year or two.

This suggests to me that what we think of as "mathematical aptitude" is far more a measure of developmental age than anything else. I think a large part of why kids hate math is that we try to cram concepts into their brains when those brains aren't yet the right "shape" to take in those concepts. I expect that offsetting every part of the mathematical curriculum at least two years upward—if not far more—would do wonders for attrition rates.

My belief is that we could even fit all the same curriculum in by the end of public high school: although you'd be "squishing" more of the learning into the later years, the stronger minds kids would have at a higher age would allow them to acquire each successive concept both more quickly and more thoroughly, serving as a better base for the learning going forward, which would in turn be accelerated by that base.

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