Simplifying and Refactoring Introductory Calculus (2018)

Calculus Basic Concepts For High Schools by L. V. Tarasov

https://archive.org/details/LevTarasovCalculusBasicConceptsF...

if their mother language is english. For my kids, mother language German, is much harder to consume such content. Both are doing well in math olympics and similar contests, but i still miss such evolving kind of content like 3Blue1Brown or Brilliant.com in German...

It's frustrating, since it feels like an intelligent enough approach should be able to skip the rote practice, but it turns out that conceptual mastery requires mastery of execution, which requires familiarity, which requires practice.

And I say this as someone with very strong intuition, who was always asking why I needed to practice if I already understood, who often grasped concepts immediately - turns out, I still needed practice to understand thoroughly.

https://arxiv.org/abs/2210.07958

I think the best way to understand this is that division by dx is always allowed. It is genuinely a nonzero quantity. Later, we think more in a more abstract way in a tiny neighborhood around (x,y), considering what happens to dy as dx becomes arbitrarily small, but it never vanishes entirely. That explains why we can say dy = 2x dx or dy/dx = 2x and both are completely true and reasonable. I think the author's argument is that d() is a bit easier to understand because we aren't dividing by dx but it makes no sense (to me) that way. If you cannot divide by dx, a nonzero number, then why not? And if you can, why doesn't dy/dx involve zero division, which is clearly not well-defined? I think answering those questions makes calculus a lot easier to understand and that they are in a sense the hardest questions. The notation this author uses doesn't really illuminate those points and the fact that the author realizes that he is basically teaching the students to accept zero division for most of the year suggests he is basically saying we should go back to a Leibniz-era approach to calculus. I would rather make rigorous what is meant by dx/dy and what exactly dx and dy are.

-- George Berkeley, namesake of UC Berkeley, in 1734, critiquing infitesimal approaches to calculus.

Math uses limits because "dx" as a concept is hard to define and relies on faith that such an object can exist. It behaves as zero when convenient yet is non-zero when that breaks math. Limits have a more rigorous footing.

I have been on the lookout for a specific book I was suggested as a kid. This was when our high school Physics was traveling a few paces ahead of our mathematics curriculum.

All I remember is that the Indian paperback edition had a blue cover. It was very helpful.

I find it hard to understand the persistent calculus hate that I see on HN. For us it was a very enjoyable experience.

We learned it through two courses that sort of raced each other at a tepid pace -- high school Physics (especially dynamics) and high school mathematics.

Garudalife have a store on Abebooks, so available in the UK at under £12 per volume (each circa 340 pages) but inevitably £18 postage. These look interesting and one day we'll have print on demand in each territory!

https://www.amazon.com/Calculus-Ground-Jonathan-Laine-Bartle...

It's interesting that you chose to make that point in a thread about calculus specifically. It had pretty shaky foundations for most of its history, and even today, there's a pretty significant disconnect between the mechanics of epsilon-delta and the meaning we assign to the result.

> Math is not supposed to be easy/simple, but it supposed to be a great way to understand systems based thinking.

Math is a means to an end. Making the tool easy to use is a desirable property. I've heard "it's not supposed to be easy" applied to many disciplines, from film photography to software engineering, and I think it's mostly gatekeeping.

By being a little handwavy at the beginning, you can then tell Dorothy that she's had her ruby slippers with her all along, and by this time they recognize the power and importance of the concept.

In some ways, it is good to be able to explain, from the ground up, why each piece is in place. But, sometimes, understanding how to build a student requires knowing when we need to temporarily handwave something away so that it is actually meaningful when they get it. And then, doing that so they don't maintain a false conception is also important, which is why handwaviness is often helpful ("this is kind of like dividing by zero, but kind of not, and we will get to the distinctions later so just trust us for the moment").

There's also really no excuse not to use it anymore since category theory has provided some of the missing rigor. I think there's a reason that Leibniz et al essentially started with this basis.

In fact, in light of this, I'm starting to wish logicians would stop promoting SIA/SDG.

If you have a correct calculation in SIA involving infinitesimals squaring to zero, it can easily be translated into the classical setting using functions that have their magnitude bounded above by a quadratic in a neighborhood of zero, but with the advantage that if you try it on a non-differentiable function, you'll merely fail to prove the quadratic upper bound instead of getting nonsensical results.

Such a "quadratic bound" approach (in brief f'(x) exists if there exist a constant C and a neighborhood of zero where for all h in the neighborhood |f(x + h) - f(x) - f'(x)h| ≤ Ch²) could actually be adopted over the typical limit of (f(x + h) - f(x))/h without abandoning classical logic, but considering the difficulty of proving the bound for many functions of interest, it might not make much of a difference over just assuming differentiability either way.

Here is an article about the interesting house he designed:

https://torontolife.com/real-estate/look-inside-integral-hou...

(Pointing out the unrelated absurdity of this title being given out to people often not actually present at a company’s founding)

Strongly disagree.

Sequences(discrete) and Convergence are vital to understanding Calculus. Only then the idea of converging to a limit from left or right makes intuitive sense. Pair it with a graphical view of secants converging to a tangent(continuous) and you get the idea of instantaneous change however infinitesimal it might be.

You need both discrete and continuous ideas to build intuition before you introduce limits of functions and continuity.

Some books that i have found useful - https://news.ycombinator.com/item?id=49308281

We spend a lot of time and effort teaching kids how to do algebra, and how to manipulate equations algebraically. By the time they get anywhere near calculus, they know how to do this. Thus, rather than trying to build a whole mathematical world from scratch, the idea is to *build on what they are already practicing* rather than try to drop them in a wild, uninhabited country and say "good luck".

What's funny is the number of adult parents of students who tell me they took four semesters of calculus in college and *never understood what it was about*. This is the real crisis I'm trying to solve. We are teaching. People are learning just enough to pass tests, but aren't internalizing any of it.

I work with engineers on a daily basis. Many never fully grasped what calculus was trying to teach. But they are extremely fluent in algebra. The reason for this disconnect is that no one bothered connecting them strongly.

Generally most of these 'handwavy' notations are rigidly provable, but only under general assumptions, that might not be true in special cases.

If you're looking for a more formal approach, ie the infinitesimal analogue to the usual real analysis, it's called nonstandard analysis and you could probably start with the original, eponymous book written by the creator, Abraham Robinson, for which I unfortunately don't have a link.

If this stuff interests you btw I would also check out Knuth's book on surreal numbers[3], which I believe, in some sense, are the fullest possible extension of what we think of as numbers? But it's been a while since I read into those.

[0] https://www.bravernewmath.com/ [1] https://intellectualmathematics.com/calculus/ [2] https://people.math.wisc.edu/~hkeisler/keislercalc-06-03-26.... [3] https://people.math.harvard.edu/~knill/teaching/mathe320_201...