Данило Глинський

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We're humans, maths is hard for everyone

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.

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?

One of latest games I am really-really fond of, is QED (though it was just me, my friend didn't enjoy it). From OPs list I've played Portal and Braid, and while those are visually interesting, they weren't enough hard to make me happy. But QED had.

> Who developed it/What other things they developed
It was written by Terrence Tao, a brilliant modern mathematician, exactly to explain math logic to layman.

> What platforms it is on/When it came out
Purely web-based, completely free.

> If there are (good) Sequels/Prequels/DLC
It had one sequel (FOL and predicates), which is now combined with main game.

> If you believe you finished it
Hell yes! I finished it, and I liked that nowhere I had to lookup for solutions.

> How long is the game (how long it took you, NOT the number of puzzles)
It took me two weeks, and quite a bit of thought.

> How difficult is the game, in general (you can compare to other games)
It is of mixed difficulty. Overall it is hard, but doable. If you are a math student and it seems easy, then you may try to find optimal solutions. I think some optimal solutions were obtained with computer brute-force.

> How the puzzles fit into the Deduction/Efficiency/Technical/Linchpin categories
All of these! The efficiency though is optional, personally I stopped searching optimal solutions after some time.

I’d say, it is very strange how different people understand same words differently. Originally I thought that those 2 activities are in same category, but now that I read your explanations, shouldn’t I adjust my “categorization” heuristics? Who’s wrong here?

This issue seems small compared to original topic, but how can we improve anything, if we don’t speak same language and don’t know what’s right and who’s wrong?

Yet in general, if people are asked about the relative number of restaurants in various fast-food chains, their estimates generally bear a close relation to the truth.

The link is broken. Is it this article https://psycnet.apa.org/record/1992-18641-001 ?