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1970-01-01

Feynman was correct for science in school, however arithmetic is fundamental and maybe one level above the root of all mathematics. Children should be able to do most of it via mental lookup tables and apply that knowledge on paper. For some reason, they can't.

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No, it goes beyond that. There's "arithmetic," the applied usage of addition, multiplication, subtraction, and division to permute numbers, and then there's Arithmetic, the set of theorems and axioms that give rise to that system of applied arithmetic. Memorization only works for the applied part, and children aren't usually taught that there is a system of reasoning behind those rules. Without that, no amount of mathematical dexterity in pushing symbols across a page will help them understand anything past the 100 level, and sometimes not even that.

I also think there's a huge undercurrent of resistance from adults to having children learn that system of reasoning because adults don't understand why it's useful, and in my experience when people don't understand something they dismiss it.

Edit: A nice example of another axiomatic system that might be more approachable is Euclid's Elements, in which five postulates are used to develop a system of geometry using an unmarked straightedge and a collapsible compass that you could, if you were careful, use to build bridges and other large buildings.

The study was limited to cities in India. We shouldn't put much weight into this applying globally.

I mean I remember seeing this first-hand student teaching Mathematics 13 years ago in the US. They got to me having never seen any of that stuff, and the curriculum attempted to provide a good education in mathematics. But the staff and the way the whole system is structured is to skip all of that and memorize the single rule you need to know to get through the test. So it's all done by rote and the only time you find out how you've been cheated is when you try to go through Calculus.

And I remember that was how we learned everything when I was a kid, and the teachers chose not to do anything else. I also remember from my math ed curriculum one of the professors joking about the elementary education students complaining about having to learn middle school math from the college perspective. So I think portions of this apply here.

I've also seen carpenters apply trigonometry very effectively to do things like cuts for roofs and stair jacks, so there's certainly a lot of truth to people learning maths by occupation and not in a formal setting, and I think part of it is the formal setting.

Once I got to calc2 and 3. I was so mad. I realized I had spent nearly a decade memorizing things. When I could use calculus to have a factory that made formulas and the rules were on a whole simpler to remember and apply.

Similar here: There were all sorts of volume and area equations I could never remember, then one slow day at work I decided to try and derive the volume of a spehere using what I'd just learned in calculus. After doing so each part of the equation made sense instead of appearing random, and two decades later I still remember it without having to derive it again.

Did you not get to doing lots of integrals? A whole new field of patterns to memorise, which I hated.

Good point, but there are probably plenty of formulas that can be derived with simple integrals, as a backup for if those formulas are forgotten.

Unfortunately, integration, as opposed to derivation, is made of pattern recognition. It's simply that way, like differential equations.

I had a lucky experience taking HS calculus the semester before as physics. I saw other students torturing themselves memorizing the formulae from the physics text and even then struggling to apply it to novel problems.

For the most part, knowing basic calc, it was possible to just draw a free body diagram and either integrate or take a derivative to get the answer. Didn't memorize much beyond f=ma and v=IR, for better or worse.

I still firmly believe that physics and calculus should be introduced together to provide a tangible and practical base to understand the mathematical theory.

It's common in European universities to not have "service courses" that the math departments provide to other departments. The other departments teach the math that is part of their field!

Non-calculus based physics is violence.

Why should they? We have the tables in our pockets at literally all times; doing arithmetic without it might be useful, or a bit faster sometimes, but is hardly an essential skill.

"Will you have enough fuel to make it over the mountain range?"

"My phone is just rebooting" the pilot replied.

The plane has an onboard calculator for this :)

These are the idiotic contrived examples that made me dismiss every lecture on the value of hard work my K–12 teachers gave.

For catching when you keyed something in incorrectly.

It builds a numeric intuition. When you repeat something enough, it begins to do itself - you gain a subconscious mastery. Think about yourself as you read these words. Imagine if you were looking at the letters and actually trying to sound out each word, consciously thinking about each words meaning, and then finally trying to piece together the meaning. You'd spend 5 minutes reading a sentence or two, and oh God help yo if tere ws a tpyo. Instead it all just flows without you even thinking about it, even when completely butchered.

And that sort of flow is, I think, obtainable for most of anything. But 100% for certain for numbers. Somebody who doesn't gain an intuitive understanding of basic arithmetic will have an extremely uncomfortable relationship with any sort of math, which mostly just means they'll avoid it at all costs, but you can't really. I don't even mean STEM careers, but everything from cooking (especially baking) to construction and generally an overwhelming majority of careers make heavy use of mathematical intuition in ways you might not consider, especially if you're already on good terms with numbers.

that's why montessori math is so impressive. it starts with counting out beads one at a time by the hundreds until they have internalized that. then they get beads on a stiff wire 10 at a time, and repeat the process counting them out up to a 1000. and so forth until eventually they hold in their hands blocks of 1000 beads glued together in a cube, and only after they have internalized that the beads get replaced with more abstract woodblocks and sticks. and all that happens in the first year or so at the age of 3.

It's a bit ironic to be saying this in the context of HN, the tech stack is built up on layers of abstraction that little of us have mastery of.

If we made it so that only people who mastered assembly could be considered "real programmers" we'd get nowhere, certainly not to build modern web applications or video games.

People that have a poor understanding of the stack under them almost invariably produce very poor software.

Video games are an area where in fact good software is still produced, mostly because the people working on the cores of games DO know (at least how to read) assembly language.

Modern web applications on the other hand so basically the same things we were doing with computers 20 years ago but consume 100x the resources to do so.

No "modern web application" comes anywhere near the quality of Word or Excel 2003.

so every time i go shopping i have to type all the prices of what i buy into my phone and also have the calculator connect to my bank account and not only make sure i have enough savings, but also tell me that i am not spending more than my average for weekly groceries? and when doing that i need to make sure to not make any typos because my lack of numeric intuition won't allow me to recognize where i made a mistake. and i also won't be able to tell if an item is overpriced. nor will i recognize a bargain unless it is marked with a big colorful sticker.

I suggest learning to use a slide rule and abacus.

And soon those things will be able to read any text shown to them. And speak it out. So why teach kids to read at all. It can all be done by voice.

It is an esssential skill.

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