In the long run, the mechanics of math (how to do long division, or differentiate an expression, or expand a geometric series) are important only insofar as to help us model and predict and analyze the world, intuitively. However, students who do not intuitively see the power of mathematical intuition as a tool for understanding and modeling the world better, think that they are taught the mechanics of math just for the sake of "no phone/calculator/computer available" circumstances.

Mathematics is about examining things and understanding their essence - what statements can we make about all such things? How can we prove that? How can we use that to figure out other things?

But nobody ever tells kids that - they think they’re just in ‘advanced counting’ lessons.

I wonder whether a split in curriculum could help - similar to English language/literature. Mathematics needs a ‘numeracy’ program that starts in kindergarten and covers the mechanical ‘how numbers work’ stuff… then a separate mathematics program that starts in middle school and teaches reasoning and proof. Start with geometry.

The more abstract it gets, the more it warrants an introduction with "here's how it's used in real life".

I can recall the point when it started becoming mechanical - it was when I started doing Derivatives and Integrations. I was 'just solving puzzles' until I hit a chapter, tucked away way back in the syllabus, almost at the end of the 2 year arc - a chapter about Applied Differential Calculus. I still remember the feeling of "Oh... I get it now" euphoria, with a tinge of sadness - why wasn't this covered early on?

Same thing would have happened in Engineering as well, but at the time my teacher was good. They started by explaining the applications of Fourier Transform before we actually got to learn the mechanics of it.

I have the same thing with learning languages: I despise focusing on the rules of conjugation and word order and rote memorizing rules, exceptions to them and exceptions to exceptions. I much prefer to bootstrap with some vocab and then acquire by immersion, even if that means I get the conjugations wrong in the beginning. But I also find myself in the minority by far, to the extent that pretty much all language teachers I ever had (except my high school French teacher - she was awesome letting me translate MC Solaar lyrics for grades) didn't even really seem to believe that there are other ways of learning other than studying rules.

> outside of certain career choices.

As someone who did a lot of calculus in university, and definitely hit a eureka moment while integrating over vector fields that helped me conceptualise some general day-to-day stuff better in my head, would most people generally benefit from that same conceptual "grokking": of course. Would it be worth the time & effort investment for them if they're not using it in their career: no.

Abstract thinking is really useful during arguments, even about politics, religion, etc.

The "mental block" you're referring to here is a combination of individual aptitude, capacity, access to educational resources & time-/effort-budgeting over a lifetime. This isn't some free casual object people are refusing to collect on their travels, it requires investment.

Yes, calculating the right amount of fence is useful, but not only (as someone pointed out) you don't need a lot of math for that, people just take take a ruler on the plan, or count steps in field, going across the entire length, then estimate, and it works.

> estimation of 500km route arrival time while taking traffic statistics into the account

Who does this? How would you convince a friend to learn math in order to do this? What people do is they remember how long a 200 km route took and so they estimate the 2.5× longer path will take 2.5× more time.

We need examples of real world math applications, because such examples are scarce across the Internet.

Who does it? Navigation app in your phone. They modeled this problem using math. While you could probably model it somehow without math, but why would you want to do it other way, when you have reliable, mature and flexible tool

I didnt write that you shouldnt leverage tech.

Do it, yet be aware how it works and why. You can leverage math in other custom scenarios

Let's be honest: advanced math is useful in programming contexts. Everywhere else, in 2023 you're using computer programs that will do the hard math for you.

Math is not about calculators or computers.

It is about modeling (e.g real world)

It is useful whenever non-standard, already modeled in your calculator problem apprears.

The real problem is you can cannot find a language which you don't speak, useful.

The only reason why things like Addition, Subtraction, Multiplication, Division come easily even to people without any education is because it has to become a part of their everyday vocabulary if you have to as little as start a lemonade stand. If Im not wrong Arabs arrived at algebra trying to workout inheritance laws from the Quran. Want to build a home, you can only brute force that much, eventually you are likely to invent the Pythagoras theorem, measurement techniques, myriad other geometrical methods.

All this is possible because if you have to do a thing, anything at all, well, you have to arrive at the most logically consistent formulation of the problem. That helps you to not only do it well, but do it right.

The real question there fore is introducing Math in a way it becomes a part of people vocabulary and general language.

Perhaps he didn’t and isn’t it a part of the problem to apply math?

>Then you basically not only get better tools to operate (model) in real world, day2day life, but also it opens you doors to highly paid careers.

can you elaborate on how “it opens you doors to highly paid careers”?

Depends on the level of proficiency

Examples can be engineering or fancy financial related companies like Jane Street

> Since math modeling is everywhere in "modeling" industries like engineerings, financial-ish jobs and othet

In other words, certain career choices, as the OP says.

(Also, while it might not be the tools needed for the average homeowner, there are plenty of optimisation problems similar to "how much fence do I need" which are most easily solved by solving the Euler-Lagrange equations)

While it's always better to work out the numbers if possible, sometimes it isn't possible and people get trapped trying.

I was in a shop this weekend where the price per 250g of coffee was displayed, but the woman in front of me only had a 175g container. Neither her, the person serving her, or the other person working in the café knew how much to charge her. It's 175/250 * £PRICE_PER_250.

In supermarkets, prices of items are displayed beside each other - a sharing bar of chocolate is £x/100g, but the multipack is £y/item, and each item is z grams. Which is better value?

Cooking - I have a recipe in a book that serves 2 people, but I'm cooking for 3. How much X do I need?

I forgot all my calculus after high school, had to relearn it in uni and then I promptly forgot again.

Exactly as the article says, it was more about proving we had the capacity (proxy iq test?) to learn it.

You don't need calculus in real life and I think the focus on calculus is ridiculous when we could explore other more practical areas, like category theory (which only my lucky friends who did advanced math got to play with)

I love the wikipedia intro: Calculus is the mathematical study of continous change, the same way geometry is the study of shape and algebra... that's it perfectly. And the most basic application is in everyone's life and also one of the basic physics thing: The relationship between location, speed and acceleration. I find this very essential, vs category theory at least..

Category theory is about connecting the dots between different areas of maths. The "general application" is to allow you to reason over the structure of a problem you're interested in, while throwing away all the superfluous details. It arose when geometers and topologists realised they were working on the same problems, dressed up in different ways. I think the utility for technical people, from this perspective, is pretty clear.

As for the general working person? I think it's just an exercise in learning to do abstractions correctly, which is valuable in any line of work.

There are actually people who advocate that we should base maths education on category theory much earlier (much as New Math was interested in teaching set theory early on, as a foundational topic). CT is an unreasonably effective tool in a large section of pure maths, so this doesn't sound unreasonable to me; it wouldn't be nearly so scary if it were introduced gently much earlier on (in the same way we start to learn about things like induction in the UK in secondary school, long before formalities like ordinals are introduced at uni). Currently only a very specific, highly-specialised section of the population learn CT, but if something like this were to happen, I'm sure we'd see lots of benefits which are hard to identify at the moment.

I guess to try and mirror your calculus example, I'd try and motivate why someone should care about abstraction itself, perhaps with examples like 'calculating my taxes each year is exactly the same problem, except the raw numbers have changed'.

Alternatively it might go over better to say something like: "Imagine you have a map with a bunch of points, and paths which you can walk between them. CT is the study of the paths themselves, the impact of walking down them in various routes: for mathematicians, this means looking at things like turning sentences such as 'think of a number, add 4 to it then divide by 2 then add 6 then subtract 1' into 'think of a number and add 7'. Once you've spotted this shortcut on this silly toy map, you'll recognise the same paths and the same shortcut when you see on your tax form 'take your income, add £400 to it, divide by 2, add £600 and subtract £100"

I've only read pieces of it, but I think this moves in the right direction towards making category theory useful to day-to-day life in non-trivial ways.

I don't even know what "calculus" is really.

I have had plenty of math classes both in high school and later at university but I don't recall any significant distinction that would leave me with some concrete idea of "algebra" vs. "calculus" vs. "whatever" years later.

It's the continuity/limits/integrals stuff.

Also it was never optional. Either in high school or CS at uni.

"Calculus" is the application of that theory without argument. It's an advanced high school class or an early college one. There you'll integrate or differentiate real valued functions for use in optimisation problems or for determining qualitative features of such a function (e.g. where is it flat, where is it defined, etc).

In the US, you can probably pass calculus without writing a proof, but you can't pass mathematical analysis without at least understanding epsilon/delta proofs.

Hey now that I think of it, "mathematical analysis" had continuity, limits, some integrals. And then every mathematically inclined uni specialization had "integral and differential calculations (let's shorten it to calculus)" which was more advanced use of integrals :)

A rose by any other name would smell as sweet, but it may be called a thorn in a different locale.

In Poland we do those in high school. :)