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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)

#141

Post by kazinator »

Fordrus wrote: post #124
kazinator wrote: post #62

Earlier quoted context omitted.

Humanity as such absolutely loves pointless repetition, and abhors thinking about something hard, which involves focusing intensely on one subject in which where no repeatable progression of simple successes is taking place, leading to a mounting sense of discomfort. Just look at various cultural traditions, popular (and even serious) music, crafts, sports, ... sneeze ligion. Mindless repetition is everywhere. The sa…

I think your assumptions are fundamentally wrong. The examples of repetition you describe are specifically designed , whether by a video game producer, a musician, or a priestly class- to feel important and be generally pleasant . Video game creators spend thousands of hours researching the best ways to turn their systems into Skinner Boxes, priestly sorts have spent untold amounts of time preparing sermons to instil…

Yes, of course people don't like repetition per se abstracted from the particulars of what it is that is repeating. Someone likes repeated palm-muted metal guitar riffs; but thinks that someone else who likes knitting sweaters, loop by loop, should get a life.

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

#143

Post by tokenadult »

derefr wrote: post #110
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" witho…

Kids are ready for formal operations when their earlier experience has prepared them for formal operations. In some places around the world, students in fifth grade--including below-average fifth-graders--are already learning algebra. In quite a few whole countries around the world (for example, Taiwan, where my wife grew up when Taiwan was still a developing country) all seventh graders study algebra and geometry as part of their regular school math lessons, yes, including the below-average seventh graders. (In Taiwan, they also learned enough of the International Phonetic Alphabet to transcribe General American English at that age, but now that skill, which most United States reading teachers lack entirely even with a college degree, has moved into the elementary school curriculum.) Good education at the beginning produces better results in the middle grades.[1]

[1] http://www.amazon.com/Knowing-Teaching-Elementary-Mathematic...

http://condor.depaul.edu/sepp/mat660/Askey.pdf

http://www.ams.org/notices/199908/rev-howe.pdf

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

#144

Post by seba_dos1 »

marincounty wrote: post #119
kazinator wrote: post #62

Earlier quoted context omitted.

Humanity as such absolutely loves pointless repetition, and abhors thinking about something hard, which involves focusing intensely on one subject in which where no repeatable progression of simple successes is taking place, leading to a mounting sense of discomfort. Just look at various cultural traditions, popular (and even serious) music, crafts, sports, ... sneeze ligion. Mindless repetition is everywhere. The sa…

For myself, I found math difficult. Never understood, even now, the need to spend so much time memorizing multiplication tables. That said, I started over in a community college, and finished up to trig. with ease. I found trigonometry, surprisingly straight forward. I went on to finish a year of physics. I didn't need calculus for my major, and heard it was really hard. Plus, I didn't want to ruin my grade point ave…

For myself, I love math. I also never understood, even now, the need to spend so much time memorizing multiplication tables.

In fact, I've never learnt it. From time to time I still find myself multiplying on fingers; as a kid I used it a lot, now the table went into my head automatically simply from continued usage. The way to multiply two natural numbers from on fingers was one of the most important things my mom tought me :)

There's also a way as presented in the article (87 = 88-8), but I mostly use it when double checking, as I find it slightly more error-prone when doing it in head (and obviously very inefficient when done on paper). Works great for bigger numbers though.

Once in junior high school my math teacher caught me on multiplying on fingers and ordered me to learn the table for the next lession. Managed to pass it, but quickly forgot in following days. I never cared - I was always doing very fast and well on math classes anyway.

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

#145

Post by djhn »

mindcrime wrote: post #127
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…

It's just one anecdote, but this jibes with my personal experience. I hated algebra (high school algebra) because it was endless tedious, mechanical manipulations and there didn't seem to be any real elegance or beauty to it. It was just memorizing "stuff" like the quadratic formula, how to rationalize denominators, blah, blah, blah. But high-school geometry was actually interesting . Seeing how the various proofs we…

> the biggest challenge I had with math was being put off by cryptic notation. I tend to want to read things to myself in my head as I look at them on a page, and when I'm looking at (new/unfamiliar) math notation, and I don't know how to "pronounce" it to myself, my mind wants to just blank out and skip it.

Material that has an almost unnecessarily comprehensive vocabulary and notation practices guide at the beginning/end is the best kind.

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

#146

Post by derefr »

tokenadult wrote: post #143
derefr wrote: post #110

Earlier quoted context omitted.

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" witho…

Kids are ready for formal operations when their earlier experience has prepared them for formal operations. In some places around the world, students in fifth grade--including below-average fifth-graders--are already learning algebra. In quite a few whole countries around the world (for example, Taiwan, where my wife grew up when Taiwan was still a developing country) all seventh graders study algebra and geometry as…

"Learning algebra" doesn't necessarily imply understanding the abstract concepts of algebra. Without the developmental substrate, children instead lean mostly on rote memorization. This seems to "work" when the class is taught in terms of rote application of memorized formulae, but will fall over the moment that students are presented with a problem in a novel context, or expected to synthesize multiple component skills into a larger solution.

If you've ever tried to tutor someone who "struggles with word-problems", the majority of the time it's because they don't actually understand the things they've been "learning" at all.

Similarly, this tends to be the difference between people who "get" programming and people who don't. If you ever talked to someone who failed a programming course, it's usually clear from their approach that every new language or API they learned became, to them, another set of rote facts to burn into their mind, never coming together to paint a picture of what the formal act of "programming" is generally.

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

#147

Post by kaitai »

pdeuchler wrote: post #120
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…

Potentially unpopular opinion: Maybe our math teachers just suck? If we all had Marvin as our professor I doubt attrition would be as high, whereas most High School math teachers are simply analytically inclined people who realized they had better job security teaching math than english

Actually, many high school math teachers in the US did not major in math, minor in math, or receive any teaching certification in math.

In high school it's not so bad -- only 28% of high school math didn't major in math, and only 12% of our total math teachers in high school have no qualifications whatsoever. [1] In middle school 55% of math teachers didn't major in math, and about 30% didn't major, minor, or get a certificate in math! [2]

So -- you're right.

[1] https://nces.ed.gov/fastfacts/display.asp?id=58

[2] http://nces.ed.gov/pubs2015/2015815.pdf

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

#148

Post by sedachv »

jrcii wrote: post #109
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.

> 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.

The worst part is web development is a completely arbitrary made-up system without any abstractions you can learn and use in a different field of work. It does not even transfer to something as closely related as desktop/mobile GUI toolkits. I view "knowledge" like that as a waste of time and avoid it whenever possible. I wish there was a word for it, something to imply "garbage knowledge."

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

#149

Post by zodiac »

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…

But I still do all of my math on pen and paper

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

#150

Post by complexmango »

crispyambulance wrote: post #42
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…

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…

Why doesn't anyone ask why kids in China, Korea, India etc. don't 'disengage' from math as easily as American kids?
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