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Comment by karmakurtisaani | Hacker News Reader

I don't have hard scientific evidence, but memory absolutely plays a role as far as I can tell. Even for simple deductions we rely constantly on toy examples or similar scenarios to that we're currently thinking, which can then guide our thinking further.

For example (inspired by my kid's struggles to learn multiplication), if you're calculating 7x6, it helps to remember that 6x6 = 36, and then just add another 6. Much easier than calculating 6+6+...+6.

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There is a lot of cognitive science research confirming the point you're making. Cognitive scientists draw a major distinction between working memory and long-term memory (there's also a third "sensory memory" that appears in the literature but isn't all that germane to learning math). In cognitive load theory, managing working memory is important to getting things into long-term memory, so if you have to do 6+6+...+6, that's going to eat up your working memory and you're less likely to be able to commit the more important thing you're trying to learn to long-term memory. There's a lot of published research on this.

It is appealing to some educators to lean on 6+6+...+6 because it shows conceptual understanding, but counterintuitively students who don't make the jump to memorizing 6x7 or something close like 6*6+6 (cognitive scientists call this "fluency") end up having difficulty learning higher-order concepts because of the working memory bottleneck.

This is interesting. Does that make you smarter?

Wouldn't having a total grasp of adding 6 seven times be considered smarter than using that shortcut?

And wouldn't the logical evolution of that mental shortcut be to simply use the tool in your pocket instead?

I too have similar tricks for math, but there must be plenty I'm not aware of because I just use my phone, and because there is no clear incentive nor reward to learning these tricks.

Is it smarter to memorize math shortcuts that you don't actually need, or is it smarter to circumvent them and employ your attention fruitfully elsewhere?

I also know a man who was born without a left hand. He said he never needed it anyway and wondered how people found that annoying thing useful.

Did that man get a bionic replacement that was stronger and faster than his flesh?

> This is interesting. Does that make you smarter?

I'd rather not think in terms of "smarter" or "intelligent", since then you open the discussion to nitpicking what we mean by those anyway. I'm more interested in how to think reasonably fast and accurately, i.e., how schools typically measure smarts. There these shortcuts absolutely help.

> Wouldn't having a total grasp of adding 6 seven times be considered smarter than using that shortcut?

Not necessarily, since by having to do more calculations exposes you to the risk of errors, and being overall slower. The user AlanYx had a very nice response to my previous comment on this.

> And wouldn't the logical evolution of that mental shortcut be to simply use the tool in your pocket instead?

Well.. Depends. If every time you see a multiplication, your immediate reaction is to go to your pocket and start typing numbers, you have a lot less feeling for the numbers themselves, and your flow of thinking is interrupted by thinking about the phone.

> Is it smarter to memorize math shortcuts that you don't actually need, or is it smarter to circumvent them and employ your attention fruitfully elsewhere?

Ideally, sure. However, in practice what tends to happen is that you do the calculation with your phone, get your answer, look at the phone in your hand and see all the social media notifications. The attention then goes to checking the messages, reacting to a funny meme and scrolling the latest social media outrages. You can't really think of the phone as a microchip in your brain that instantly provides you with the correct answer, it comes with its usual drawbacks too.

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