That just sounds so different to how I solve math problems. I was naturally good at maths, but it was always from monologue kind of bruteforcing different solutions until one of them seemed to work. I guess it might be a reason also why I find LLMs really exciting since I feel like if I can do it, LLM should be able to do it. I don't feel like I am doing anything special.
I always had problem with trusting my intuition or gut so I was worse in a lot of other real life things however. But math seemed abstract and solvable by words and brute force.
I wish things just magically came to me, but I think I always have to go through things with my inner monologue.
Like if I was to do multiplication in my head e.g. with same numbers, for example 62 x 62. I would have to go through it as a monologue.
I first remind myself of the strategy to do it, which is first I will do 60 x 60. Then it is 3600, then I add 2 x 60, 3720, and then there is 2 x 62 left, but I have to keep reminding myself occasionally what the last numbers were, that initial multiplication was 62 x 62, then I got 3720, and now I have to add 124... okay lets go 3820, 24 left, now 3844. Of course it is easier to remember as I am typing this, but in my head I have to keep reminding myself. And now I am not sure if I did a mistake so I go verify that on the calculator.
> I first remind myself of the strategy to do it, which is first I will do 60 x 60. Then it is 3600, then I add 2 x 60, 3720, and then there is 2 x 62 left, but I have to keep reminding myself occasionally what the last numbers were, that initial multiplication was 62 x 62, then I got 3720, and now I have to add 124... okay lets go 3820, 24 left, now 3844. Of course it is easier to remember as I am typing this, but in my head I have to keep reminding myself. And now I am not sure if I did a mistake so I go verify that on the calculator.
I do multiplication in my head similarly, just strip out the words, the numbers just flickers through and operations happens by themselves and I'm done in a second or two when I'm not rusty. Now that I'm rusty I do it more like (60x60=3600, 2x2=4, 60x2=120, 3600 + 4 + 120 + 120 = 3844), without doing any words, I just did that in my head right now and I am 100% sure not a single word, just the steps, I do sometimes verbalize the numbers so what I wrote in those parentheses is the most verbal my process for mathing that out gets.
Edit: Looking at that, I think it might be easier for someone to correct your thinking if you think in words, but thinking without words is way faster and way more creative since it removes the restriction of only thinking about concepts you have words for.
Edit2: I think your verbalization there is a ritual for the concepts to get to you. For me all I need to do is see 62x62 and imagine I want to solve it and all those thoughts flow to me automatically, basically a shortcut instead of having a large verbal ritual to piece together the concepts. I never did math verbally the way it is taught, so I am not sure how people think that way, to me this was always automatically this way.
If you can or ever were able to do this in 2 seconds that is unimaginable to me. Not in a doubting way, I just have no idea how I could do it in 2 seconds. It would take me 20 to 30s and at least. But most people wouldn't even try to multiply this top of their head.
Also this multiplying of two digit numbers I was never taught, I just came to a strategy that I explained with my thoughts. And I used to do these for fun as a kid. But I would only get faster than that if it was something I had memorized, but not numbers from scratch. And I had to constantly try to repeat numbers in my head that I had stored for addition down the road.
What is usually tougher top of my head and especially if I am tired at all, is something like 68 x 68. Then here first I have to decide whether I go from 60 x 60 or 70 x 70. Since it's that close to 70, I think 70 is more likely be easier. So I think okay 4900 - that's from memory right. Then I start to think what I have to take off from 4900 and what is 68 x 68 lacking compared to 70 x 70. So I will think that if I add 2 x 68, it will make it 68 x 70 and then I have 2 x 70 missing. So I need to deduct 140 and 136. So now at this point, this is much easier done in writing but here I frequently have to repeat numbers or redo some steps because I am not certain or I forget. But otherwise 4900 - 140, this comes easily instantly 4760 - maybe that's how it is for you with 62 x 62. And then now I take 100 off, it's 4660, and further 36 it would be 4624. Right now testing this in my head it took more than a minute because I wasn't sure whether I could just use the logic that I was thinking out of the box and I'm quite tired.
What if you were to have to do something more difficult e.g. 3 digits multiplication or 4?
I got to 4624 in 8 seconds on that now, and I haven't done significant amounts of mental math in over a decade. The 60x8 etc took a couple of seconds each but other than that it just happens automatically. A decade ago each of those would take a fraction of a second and it takes about the same time to calculate as to write it, I tend to calculate the numbers left to right instead of right to left as you are taught since you write it left to right, also more useful when calculating approximations since I just do what I normally do but stop earlier.
> Not in a doubting way, I just have no idea how I could do it in 2 seconds
Words are just slow. I never did math with words, so arithmetics is fast, and so is all the other things you learn in a math degree. I think word based processing in math is a big reason people have such a tough time with it, it really makes it much harder to think, like you are bogged down in a fog instead of up and free and agile with clear sight of everything.
Maybe it was easier for people to get past this stage when calculators were less prevalent. Calculators lets people get away with keeping their arduous word based math instead of just internalizing the concepts.
> What if you were to have to do something more difficult e.g. 3 digits multiplication or 4?
When on tests I write down some intermediate values. I am not a human calculator, I am just fast at math reasoning and being relatively fast at arithmetic's comes for free from that.
I did some physics tests without a calculator, you can approximate all those special functions using regular math logic you learn in high school. I did still ace the test, it isn't that hard once you have internalized it all.
I read the words and then imagine them until I internalize them. I tend to space out a lot when I do that, so I don't really listen during lessons but I tend to learn most things during the lesson that way.
Makes me really bad at following instructions though, since unlike math or physics you shouldn't internalize them instead you should just execute the words and that is really annoying, my brain really doesn't want to think in words. so try to translate everything...
Anyway, I think internalizing things might be harder or impossible if you haven't already done so with all the previous steps. If all your thinking and knowledge is word based then it is hard to break that, and vice versa, I can't think about math in terms of words really. I can slowly translate some things to words but I can't really drive thought with the words.
I used to do math olympiads in high school and the concepts I saw there were always something I hadn't seen before. I always used inner monologue to figure out solutions though. I had never practiced much for math olympiads, I performed relatively well for my area, not good enough to make it to international levels though.
I can't follow verbal instructions as well though, I couldn't listen to the teacher etc. But I think it's because my inner monologue takes all the focus so I just follow my inner monologue which is probably completely another topic than what the lesson is about. I can't really focus away from my inner monologue. When I try, then I would just have philosophical meta discussion about my inner monologue. But also this meant that I couldn't actually learn the subjects as well during the class. I either had to learn from my free time or not at all.
Also have to watch films with subtitles, because audio language I have troubles focusing on.
I think all I have is my inner monologue, no good visual imagery, or other types of "thinking". I think makes me really poor at navigation as well. I never remember how to get somewhere. And also in general for anything 3d, like 3d games I will perform bad at, awareness wise.
Might sound weird but does the look of the operator matter? Say × vs x vs • vs *, I would assume it doesn't but I'm curious if there's a visual change, for lack of a better term.
No, I translate that to the same concept in my head and then I do it.
Edit: Thinking more about it, in my head the numbers are numbers, but the operators are invisible, not sure what to call it but I don't think of the multiplication operator as anything tangible, I just know it is a multiplication of the two numbers.
I always had problem with trusting my intuition or gut so I was worse in a lot of other real life things however. But math seemed abstract and solvable by words and brute force.
I wish things just magically came to me, but I think I always have to go through things with my inner monologue.
Like if I was to do multiplication in my head e.g. with same numbers, for example 62 x 62. I would have to go through it as a monologue.
I first remind myself of the strategy to do it, which is first I will do 60 x 60. Then it is 3600, then I add 2 x 60, 3720, and then there is 2 x 62 left, but I have to keep reminding myself occasionally what the last numbers were, that initial multiplication was 62 x 62, then I got 3720, and now I have to add 124... okay lets go 3820, 24 left, now 3844. Of course it is easier to remember as I am typing this, but in my head I have to keep reminding myself. And now I am not sure if I did a mistake so I go verify that on the calculator.