Marvin Minsky: What makes mathematics hard to learn? (2008) • Hacker News Board

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

#41

Post by Jtsummers » Tue, Jan 26, 2016, 2:01 PM UTC

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…

Re edit: Put a space between the * and the characters.

  5 * N

versus

5*N

Pairs of asterisks adjacent to non-space characters become markers indicating italicized text.

  5*N is easier to remember than 16*N

5N is easier to remember than 16N

Versus

  5 * N is easier to remember than 16 * N

5 * N is easier to remember than 16 * N

And I think the word you want is exponential, not logarithmic. Related to each other, since exponential expressions become linear on a logarithmic scale, but exponential describes the growth of 2^n better.

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

#42

Post by crispyambulance » Tue, Jan 26, 2016, 2:01 PM UTC

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, however, "rediscover" math when they get to their first class that involves theorems and proofs (often in high-school geometry) and when it starts being used in science coursework. It is then that they realize that math is a way of thinking and this can transform their opinion and motivation for math.

The best teachers find ways to relate mathematics to real life and real purpose. Yeah, it is always going to be challenging, but having a purpose creates motivation to get through the tedium.

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

#43

Post by artursapek » Tue, Jan 26, 2016, 2:08 PM UTC

> Fascinated by electronics and science, the young Mr. Minsky attended the Ethical Culture School in Manhattan, a progressive private school from which J. Robert Oppenheimer, who oversaw the creation of the first atomic bomb, had graduated.

Crazy to read this sentence while I am halfway through Cat's Cradle.

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

#44

Post by vdnkh » Tue, Jan 26, 2016, 2:08 PM UTC

"Why do some children find Math hard to learn? I suspect that this is often caused by starting with the practice and drill of a bunch of skills called Arithmetic—and instead of promoting inventiveness, we focus on preventing mistakes. I suspect that this negative emphasis leads many children not only to dislike Arithmetic, but also later to become averse to everything else that smells of technology. It might even lead to a long-term distaste for the use of symbolic representations."

Substitute "Arithmetic" for "Interview Problems"

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

#45

Post by ashark » Tue, Jan 26, 2016, 2:09 PM UTC

kazinator wrote: post #36

qb45 wrote: post #3

Earlier quoted context omitted.

As much as I regret that humans don't have 8 fingers at each hand and count in base 16, I think it's more likely that she simply is/knows a computer nerd.

Maybe the child has simply played a doubling game before, along these lines, and memorized it: What's 1 and 1? 2. What's 2 and 2? 4. What's 4 and 4? 8. What s 8 and 8? 16. What's 16 and 16? 32. That's just five facts. The game could occur socially between kids. It is natural to ask a question, then take the answer and "up the ante" by re-formulating the answer into a harder question, back into the other child's face!…

I used to run through this (and on up to higher powers of 2) in my head as a kid in idle moments, well before I knew it had any relevance to... anything, really.

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

#46

Post by Phithagoras » Tue, Jan 26, 2016, 2:12 PM UTC

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?

It may be that the girl happened to know 16+16=32 because she had answered the question before, and it stuck in her memory.

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

#47

Post by marknutter » Tue, Jan 26, 2016, 2:14 PM UTC

What irritates me about the way math is taught, in fact, the way most subjects are taught, is that the ultimate goal is to become better at the subject matter rather than applying the subject matter to some ultimate purpose. Math is mental masturbation in the same way that Sudoku is. Some people love solving little puzzles for the sake of solving little puzzles, but without the necessity of applying those lessons to actual problems that exist in the real, physical world, there's very little value in it. I would really like to see the whole concept of breaking down education into neat little subdivisions go away and instead focus on what it takes to solve larger problems, whether that be creating great new works of art, advancing technology, curing a disease, doing social good, or even fixing cars. The curriculum for each of those myriad problems will be different but perfectly mapped out. There's very little calculus an auto mechanic or a dance instructor needs to know to excel at their chosen profession, for instance.

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

#48

Post by Jtsummers » Tue, Jan 26, 2016, 2:15 PM UTC

kazinator wrote: post #36

qb45 wrote: post #3

Earlier quoted context omitted.

As much as I regret that humans don't have 8 fingers at each hand and count in base 16, I think it's more likely that she simply is/knows a computer nerd.

Maybe the child has simply played a doubling game before, along these lines, and memorized it: What's 1 and 1? 2. What's 2 and 2? 4. What's 4 and 4? 8. What s 8 and 8? 16. What's 16 and 16? 32. That's just five facts. The game could occur socially between kids. It is natural to ask a question, then take the answer and "up the ante" by re-formulating the answer into a harder question, back into the other child's face!…

It's also doubling. Like ashark, I tended to play with numbers like that as a kid. Plus, not sure how old I was but I think I was 7, perhaps 8, I was introduced to the question:

  If I agreed to pay you a dollar today, and double your pay
  each day. How much would you get paid after a week? After
  30 days?

By a teacher (presented differently, but that's the question). The power of doubling just stuck with me (how quickly it grew), long before I actually knew the concepts of linear versus exponential growth. This resulted in certain arithmetic facts sticking with me better than other, perhaps more logically obvious facts. Random things like this will stick with kids when they can start to see patterns or features within the structure of it.

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

#49

Post by bluedino » Tue, Jan 26, 2016, 2:15 PM UTC

It's not that it's hard to learn, it's just not easy. You can't take shortcuts (you can at first but it bites you later), you must do things in the correct order, you must be exact. There are lots of little things to learn and remember.

I think some peoples brains are just wired to 'get' math just as some people 'get' programming. And some people just 'get' art. Certain people just 'see' certain things and you can't really teach that.

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

#50

Post by fmap » Tue, Jan 26, 2016, 2:18 PM UTC

This is a wonderful series of articles and I find myself nodding along with most of it. However, these lines really made me cringe:

  A child was sent to me for tutoring because of failing a geometry class, and gave this excuse: " I must have been absent on the day when they explained how to prove a theorem." 

  No wonder this child was confused—and seemed both amazed and relieved when I explained that there was no standard way to make proofs—and that “you have to figure it out for yourself”.  One could say that this child simply wasn’t told the rules of the game he was asked to play.  However, this is a very peculiar case in which the ‘rule’ is that there are no rules! (In fact, automatic theorem-provers do exist, but I would not recommend their use.)

I think interactive theorem provers would go a long way towards making children understand symbolic reasoning. The way these programs work is that you have feedback available at every step of a proof. Children can learn basic causal relationships by looking at the world around them. By visualizing the basic relationships between logical formulae you can similarly learn to inuit the effects of reasoning steps. Interactive theorem provers provide the visualization.

The game has rules, and you can learn them.