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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)

#21

Post by qb45 »

gambiting wrote: post #16

" This child imagined ‘Math’ to be a continuous string of mechanical tasks—an unending prospect of practice and drill. It was hard to convince him that there would not be any more tables in subsequent years" Well, I enjoy maths, but for me my entire mathematical education was always about memorizing tables. It was made worse by the fact that during Polish exams you cannot use advanced calculators(only ones that can d…

> I had to remember loads of different formulas, trigonometric values for common angles etc

Real math is about being able to derive this stuff when woken up at 5AM.

What you get in school is drilling kids to appear as if they understood math so that adults who don't remember this stuff anymore feel intimidated and leave with impression that their children are learning some rocket science and the "education system" deserves their money.

No teacher literally reasons like that and no one will tell you that, but that's exactly what they do when they are overwhelmed by a horde of barely interested kids who somehow have to pass the final exams. The other part of this insanity being politicians who set up those exams.

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

#22

Post by Dowwie »

"we need to provide our children with better cognitive maps of the subjects we want them to learn"

This style of learning works for everyone -- including children. There is a growing community of people sharing educational, animated gifs on Reddit that recognizes the value of visual learning.

Popular subreddits include:

- /r/educationalgifs - /r/mechanical_gifs/ - /r/GifRecipes/

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

#23

Post by qb45 »

746F7475 wrote: post #20
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…

I don't know, I always remembered things like 6+7 = 13 and if numbers I was adding were close to 6 and 7 I knew instantly how to proceed. Of course there's other tricks like, subtracting the 1 out of 6 to get 5 and then adding it to 7 making 8 then removing the 5 to get 10 with the other 5 and you are left with 10 and 3 which is easy, I guess. At least for me it was just silly shortcuts that stuck with me like 6+7=13…

I think that's what you get with unsupervised learning.

Tricks only get you so far, you can't do 6+7 on fingers or 7·7 as (4+3)(4+3) (I had a huge problem with this one, too many numbers to track). So you fail. And when you fail, you memorize the problematic cases and go on with tricks based on those.

And yes, it's completely incomprehensible for people who learned the pencil-and-paper algorithms and think that those are the end of the world.

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

#24

Post by oftenwrong »

Dowwie wrote: post #22

"we need to provide our children with better cognitive maps of the subjects we want them to learn" This style of learning works for everyone -- including children. There is a growing community of people sharing educational, animated gifs on Reddit that recognizes the value of visual learning. Popular subreddits include: - /r/educationalgifs - /r/mechanical_gifs/ - /r/GifRecipes/

Also http://setosa.io/ev/

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

#25

Post by qb45 »

ghubbard wrote: post #10
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?

There is a popular [citation needed] children's song called Inchworm[0] that has a chorus that lists powers of two. Two and two are four Four and four are eight Eight and eight are sixteen Sixteen and sixteen are thirty-two Everyone familiar with this song would know that 16 and 16 are 32. [0] https://en.wikipedia.org/wiki/Inchworm_%28song%29

Complete lyrics (is DMCA coming at me?):

  Inchworm, inchworm, measuring the marigolds
  You and your arithmetic, you'll probably go far.
  Inchworm, inchworm, measuring the marigolds
  Seems to me you'd stop and see how beautiful they are.
  
  (Kids Singing: 2 & 2 are 4, 4 & 4 are 8, 8 & 8 are 16, 16 & 16 are 32)
Nothing to do with imperial units at all.

Even if she didn't know the song, she might have learned 16+16 from those.

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

#26

Post by pas »

adrianN wrote: post #13
libeclipse wrote: post #8

He makes a really good point about traditional teaching methods. What's the point with the obsession over 'the correct method'? If the answer is correct, and the method in question can reliably reach the correct answer, in my opinion, the correct method is that one, and it differs from person to person.

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.

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

#27

Post by aout »

There is something quite interesting about the form of the article. While Mr Minsky talks about teaching using the "deck of cards method" and why it might not be the best approach if the child wants to advance to more complex subjects he also wants us to learn about his experiments (which are quite good!).

Having read the article I find it well written but WHAT THE HELL?! Why doesn't he use pictures? simple diagrams? "sexy" stuff?

It seems that I'm having the same problem as the child is having: the form does suck and it discourages me from reading a boatload of articles like this one. If the presentation of the subject was better I could read 10 more and maybe, finally, take an interest in "teaching methods" that would lead me to being curious and accept any "bad form" of content :)

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

#28

Post by sergiosgc »

aout wrote: post #27

There is something quite interesting about the form of the article. While Mr Minsky talks about teaching using the "deck of cards method" and why it might not be the best approach if the child wants to advance to more complex subjects he also wants us to learn about his experiments (which are quite good!). Having read the article I find it well written but WHAT THE HELL?! Why doesn't he use pictures? simple diagrams?…

He doesn't use pictures because neither his target audience nor the subject matter require pictures.

His target audience is composed of people who are interested in mathematics teaching (parents, teachers and such). He expects (correctly in my opinion) that these have handled and can handle long texts.

The subject matter is an opinion piece without much underlying data, so graphical representation is not required.

The writing style must be adapted to circumstances. Increases in picture density do not monotonically increase text appeal.

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

#29

Post by ideonexus »

gambiting wrote: post #16

" This child imagined ‘Math’ to be a continuous string of mechanical tasks—an unending prospect of practice and drill. It was hard to convince him that there would not be any more tables in subsequent years" Well, I enjoy maths, but for me my entire mathematical education was always about memorizing tables. It was made worse by the fact that during Polish exams you cannot use advanced calculators(only ones that can d…

> I'm not saying he was wrong, but in a lot of educational systems maths is 90% about remembering stuff.

It is/was, but the Common Core Math standards are trying to change that. Children are now being taught how to look at addition, subtraction, multiplication, division, fractions, and other concepts from a variety of perspectives. Predictably, it's driving their parents insane because they were taught the "one true algorithm" for each of these tasks and can't help their kids when confronted with alien algorithms that change from semester-to-semester.

http://www.corestandards.org/Math/

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

#30

Post by sago »

qb45 wrote: post #21
gambiting wrote: post #16

" This child imagined ‘Math’ to be a continuous string of mechanical tasks—an unending prospect of practice and drill. It was hard to convince him that there would not be any more tables in subsequent years" Well, I enjoy maths, but for me my entire mathematical education was always about memorizing tables. It was made worse by the fact that during Polish exams you cannot use advanced calculators(only ones that can d…

> I had to remember loads of different formulas, trigonometric values for common angles etc Real math is about being able to derive this stuff when woken up at 5AM. What you get in school is drilling kids to appear as if they understood math so that adults who don't remember this stuff anymore feel intimidated and leave with impression that their children are learning some rocket science and the "education system" de…

It's cargo cult mathematics.

People who understand math can do X, Y, and Z.

Therefore we should make sure children can do X, Y, and Z.

Then they'll understand math too!

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