Good stuff, though I’d like to point on some of your reflections.
> Part of it was due to laziness. I was a fast reader and had an excellent memory. This allowed me to excel in most subjects without much work. In contrast, numerate subjects required more dedication and systematic study.
It is important you say “laziness”. Usually laziness is about taking less energy-demanding activity across lots of choices. So it looks like “solving problems” was energy-demanding for you, but other activities were not. Whenever you had to solve problems, it felt “tough”, and coupled with lack of reason, you avoided this activity.
But it is interesting to understand, what’s happening to other children, who actually do math. Suddenly you realize, that “solving problems” for them is less energy demanding, which is awkward! How can it be that same puzzle has different energy-demanding levels for different children?
If you say this is due to difference between children, you are correct, but what exactly is different? Interest itself can’t change energy balance. Ability to read may also be same. As you said, you have good memory, so it also isn’t a factor which increases energy level of a puzzle.
Let’s look next.
> It was programming that opened my eyes. As I started learning Python, I understood the difference between the label and the thing. When coding, one works on two levels: the namespace, which contains the labels for the objects, and the objects themselves.
How do you feel this understanding? What exactly makes you able to “see” namespace and objectspace in distinct? Have you had that ability before? What were subjective “energy levels” before doing programming and after?
> At some point, I realized that doing maths is not so different. You are manipulating names that refer to objects.
You say “manipulating”, but you do this manipulation in brain, right? What allows you to do this “manipulation”? Did it feel energy-demanding before?
> I didn’t give myself the chance to be mistaken. I didn’t have a mission that forced me to either learn or fail. At the end of the day, all I did with most of my knowledge was think about it verbally and sometimes talk about it with other people.
This is good. Which kind of “manipulation” gives you chance to be mistaken? You know, if you do “verbal” talk, it often is generated like GPT-3 one—just created word pattern is predictor for next word pattern. It just can’t feel “wrong”, it is what follows. But “manipulation” isn’t like pattern-after-pattern, it is something different. What is it?
> Later, you could have them simulate the models of physics, chemistry and biology. They could engage in competitive or cooperative games which reward curiosity and stimulate them to think. You could have them design the games themselves, or send them to gather data and test theories. The possibilities are endless.
And here it is important to show, that something is omitted. To be able to simulate physics you first must have physics model based on rules in your head. → Exact this ← part is tough, not the subsequent simulation. If you don’t have rules mindset in your head, simulations just won’t “click”.
Whole programming won’t “click”, if you feel mental rule-based transformations energy-demanding.
So yeah, it is not enough to know what math is good for. It is not enough to teach children programming for them to like to understand world. This never taught stuff I talk about—is a special mindset, which reduces energy-demanding levels for most puzzles, so they no longer feel “tough”, but “interesting”.
What is this mindset? How does it look like? How to pass it to other people?
Thank you for your questions, they’re proving very useful.
But it is interesting to understand, what’s happening to other children, who actually do math. Suddenly you realize, that “solving problems” for them is less energy demanding, which is awkward!
I’m not sure this is the case. We’re humans, maths is hard for everyone. I imagine it’s more about developing an ethics of work early on and being willing to delay gratification and experience unpleasant sensations for the purpose of learning something valuable. Though of course it takes a basic level of intelligence to find motivation in intellectual work. And there needs to be some specific motivation as well, i.e. math is beautiful, or math is useful.
As for the other questions… You may be getting closer than me at hitting the target here. I think the comparison between GPT-3 talk, where nothing is wrong, and “manipulation”, is central.
But “manipulation” isn’t like pattern-after-pattern, it is something different. What is it?
I think the whole thing revolves around mental models. Programming “clicks”when the stuff that you do with the code suddenly turns into a coherent mental model, so that you can even predict the result of an operation that you haven’t tried before. I became better at programming after watching a few theoretical computer science classes, because I was more proficient at building mental models of how the different systems worked. Likewise, maths clicks when you move from applying syntactical rules to building mental models of mathematical objects.
It’s easier to build mental models with programming, because the models that you’re working with are instantiated on a physical support that you can interact with. And because it’s harder to fool yourself and easier to get feedback. If you screw up, the computer will stop working and tell you. If you screw up with pen and paper, you might not even realize it.
This is not the whole story, but it’s a bit closer to what I meant to say.
This is false, there are a few genius mathematician who early in childhood proved it is easy for some humans.
I think the whole thing revolves around mental models
Exactly! There is even more specific concept in programming psychology, it is called “notional machines”. Small little machines in your head which can interpret using rules.
I think those also can transfer to math learning, as after rule-based machines concept is grasped, all the algorithmic, iterative, replacable and transitive concepts from math start making sense.
This is false, there are a few genius mathematician who early in childhood proved it is easy for some humans.
Some outliers are hypernumerate. I’m hyperlexic, so attuned to words that I was able to teach myself to read before my childhood amnesia kicked in, so I never had to learn phonics. This doesn’t mean the vast majority of humans aren’t congenitally literate or numerate. OP’s statement may be nominally false, but the exception proves the rule.
As for teaching the aesthetic beauty of math, I would give each student their own blank copy of the 10x10 multiplication table (with a zeros row and column, making it 11x11) at the start of grade 2, and teach them how to fill it in themselves. After that, they can use it in any math class that semester, but they have to make a new one at the start of each semester after that.
The inherent laziness of humanity will drive them to “cheat” by copying from lines above: filling in half the 4′s from the 2′s, half the 8′s from the 4′s, half the 6′s from the 3′s, and so on. And while they’re doing that, they’re learning in an indelible way.
Good stuff, though I’d like to point on some of your reflections.
> Part of it was due to laziness. I was a fast reader and had an excellent memory. This allowed me to excel in most subjects without much work. In contrast, numerate subjects required more dedication and systematic study.
It is important you say “laziness”. Usually laziness is about taking less energy-demanding activity across lots of choices. So it looks like “solving problems” was energy-demanding for you, but other activities were not. Whenever you had to solve problems, it felt “tough”, and coupled with lack of reason, you avoided this activity.
But it is interesting to understand, what’s happening to other children, who actually do math. Suddenly you realize, that “solving problems” for them is less energy demanding, which is awkward! How can it be that same puzzle has different energy-demanding levels for different children?
If you say this is due to difference between children, you are correct, but what exactly is different? Interest itself can’t change energy balance. Ability to read may also be same. As you said, you have good memory, so it also isn’t a factor which increases energy level of a puzzle.
Let’s look next.
> It was programming that opened my eyes. As I started learning Python, I understood the difference between the label and the thing. When coding, one works on two levels: the namespace, which contains the labels for the objects, and the objects themselves.
How do you feel this understanding? What exactly makes you able to “see” namespace and objectspace in distinct? Have you had that ability before? What were subjective “energy levels” before doing programming and after?
> At some point, I realized that doing maths is not so different. You are manipulating names that refer to objects.
You say “manipulating”, but you do this manipulation in brain, right? What allows you to do this “manipulation”? Did it feel energy-demanding before?
> I didn’t give myself the chance to be mistaken. I didn’t have a mission that forced me to either learn or fail. At the end of the day, all I did with most of my knowledge was think about it verbally and sometimes talk about it with other people.
This is good. Which kind of “manipulation” gives you chance to be mistaken? You know, if you do “verbal” talk, it often is generated like GPT-3 one—just created word pattern is predictor for next word pattern. It just can’t feel “wrong”, it is what follows. But “manipulation” isn’t like pattern-after-pattern, it is something different. What is it?
> Later, you could have them simulate the models of physics, chemistry and biology. They could engage in competitive or cooperative games which reward curiosity and stimulate them to think. You could have them design the games themselves, or send them to gather data and test theories. The possibilities are endless.
And here it is important to show, that something is omitted. To be able to simulate physics you first must have physics model based on rules in your head. → Exact this ← part is tough, not the subsequent simulation. If you don’t have rules mindset in your head, simulations just won’t “click”.
Whole programming won’t “click”, if you feel mental rule-based transformations energy-demanding.
So yeah, it is not enough to know what math is good for. It is not enough to teach children programming for them to like to understand world. This never taught stuff I talk about—is a special mindset, which reduces energy-demanding levels for most puzzles, so they no longer feel “tough”, but “interesting”.
What is this mindset? How does it look like? How to pass it to other people?
Thank you for your questions, they’re proving very useful.
I’m not sure this is the case. We’re humans, maths is hard for everyone. I imagine it’s more about developing an ethics of work early on and being willing to delay gratification and experience unpleasant sensations for the purpose of learning something valuable. Though of course it takes a basic level of intelligence to find motivation in intellectual work. And there needs to be some specific motivation as well, i.e. math is beautiful, or math is useful.
As for the other questions… You may be getting closer than me at hitting the target here. I think the comparison between GPT-3 talk, where nothing is wrong, and “manipulation”, is central.
I think the whole thing revolves around mental models. Programming “clicks” when the stuff that you do with the code suddenly turns into a coherent mental model, so that you can even predict the result of an operation that you haven’t tried before. I became better at programming after watching a few theoretical computer science classes, because I was more proficient at building mental models of how the different systems worked. Likewise, maths clicks when you move from applying syntactical rules to building mental models of mathematical objects.
It’s easier to build mental models with programming, because the models that you’re working with are instantiated on a physical support that you can interact with. And because it’s harder to fool yourself and easier to get feedback. If you screw up, the computer will stop working and tell you. If you screw up with pen and paper, you might not even realize it.
This is not the whole story, but it’s a bit closer to what I meant to say.
This is false, there are a few genius mathematician who early in childhood proved it is easy for some humans.
Exactly! There is even more specific concept in programming psychology, it is called “notional machines”. Small little machines in your head which can interpret using rules.
I think those also can transfer to math learning, as after rule-based machines concept is grasped, all the algorithmic, iterative, replacable and transitive concepts from math start making sense.
Some outliers are hypernumerate. I’m hyperlexic, so attuned to words that I was able to teach myself to read before my childhood amnesia kicked in, so I never had to learn phonics. This doesn’t mean the vast majority of humans aren’t congenitally literate or numerate. OP’s statement may be nominally false, but the exception proves the rule.
As for teaching the aesthetic beauty of math, I would give each student their own blank copy of the 10x10 multiplication table (with a zeros row and column, making it 11x11) at the start of grade 2, and teach them how to fill it in themselves. After that, they can use it in any math class that semester, but they have to make a new one at the start of each semester after that.
The inherent laziness of humanity will drive them to “cheat” by copying from lines above: filling in half the 4′s from the 2′s, half the 8′s from the 4′s, half the 6′s from the 3′s, and so on. And while they’re doing that, they’re learning in an indelible way.