I have strong opinions on how Calc should be introduced - visually.

I think we don't cover basics like the distributive rule in school very well, and that it should be a much more nuts n bolts visual / measuring / counting experience.

Ive attempted to outline how I think this stuff should be taught, by making a video tour of the concepts - from Counting, to Distributive rule / algebra, to Quadratics then the Derivative, here :

https://www.youtube.com/playlist?list=PLEInJ-Z4qBKYxbK1Mm13g...

All of these things are covered in some great books :

  W W Sawyer Vision in Elementary Mathematics
  Algebra by Gelfand
  Calculus by Thomas

We have superb resources now like 3Blue1Brown, KhanAcademy and ArtOfProblemsolving.com / BeastAcademy .. so you _can_ get your kids a superb math education, even as many schools seemingly give up on teaching Algebra and Calculus.

Got to the place where he says "As you can see, this is identical to the d/dx() operation except that the result is not divided by dx."

What does it mean with his d() operator to "divide by dx"? All of a sudden it seems like he has changed dy/dx from unfortunate notation that sort of looks like a division into something that actually is dividing two meaningful things, dy and dx? And so what the hell are dy and dx?

I think Stewart's Calculus is excellent and it is rightfully the standard textbook. No modifications needed in my opinion.

,, additionally, moving limits to the end of a first-year course allows students to develop intuitions around the derivative first before seeing the formal proof of their validity’’

Waiting a year to get from intuition to theorems is a perfect way to ruin math.

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

At the same time there could be more examples taught on why these building blocks were historically needed and what they are used for solving nowadays.

Looks like this came out nearly 8 years ago, so… how’d it work out? Given the way job titles work these days I guess we could have some Senior Engineers here who learned calculus from this paper…

This guy has an interesting publication history, programming books and what looks like evolutionary biology from a creationist perspective.

Does anyone know how to simplify the actually hard part of calculus: solving integrals? I refer to the million different substitutions and trig/hyperbolic formulas, along with the endless amount of other heuristics. I wonder if there's a way to bypass or simplify most of that.

That's an exercise under SICP (an infamous Scheme course) it works best with either Racket with

       #lang sicp 

at the top of the SCM file, or with Chicken Scheme 5 once you run these commands in a terminal:

      chicken-install srfi-203
      chicken-install srfi-216

Then set this ~/.csirc file:

   (import scheme)
   (import (srfi 203))
   (import (srfi 216))

Try it, because under SICP you will learn Calculus by literally learning the rules of derivation, integration and squared and cubic roots as an example of recursion.

Online, interactive SICP in the browser, you don't need to install anything:

https://iain-s.github.io/isicp/

Some good books for introductory Calculus;

1) Calculus: Basic Concepts for High Schools by Lev Tarasov. Soviet-era book written as a dialogue between the author and reader. Absolutely fantastic (also see his other books on Probability etc.) - https://mirtitles.org/2018/09/04/calculus-basic-concepts-for...

2) Calculus: An Intuitive and Physical Approach by Morris Kline. A classic; any book by Morris Kline is a must-have - https://store.doverpublications.com/products/9780486404530

3) Calculus: The Princess of Mathematics by H.C.Verma et al. A two-vol must-have affordable set. The author is a well-known Indian Physics professor and this is written specifically for students to "understand" calculus i.e. it is not a typical textbook. - https://garudalife.in/calculus-the-princess-of-mathematics-v...

4) How to Think about Analysis by Lara Alcock. Provides conceptual insight like the Tarasov book above. Checkout the author's other books too. - https://global.oup.com/academic/product/how-to-think-about-a...

I believe we need to study Calculus alongside Probability/Statistics nowadays due to their pervasive use in ML/AI/etc. To that end;

a) Methods of Mathematics Applied to Calculus, Probability, and Statistics by Richard Hamming. It is by Hamming so one of the best. - https://store.doverpublications.com/products/9780486439457?_...

b) Calculus and Statistics by Michael Gemignani. Similar to the above - https://store.doverpublications.com/products/9780486449937

where I heard of this from : https://youtu.be/4ZB2PNUYR1Y

I’ve always been curious about differentials and how to build a rigorous theory of what the fuck dx, dy, dy/dx, etc. are. For example, if you study Tao’s Analysis and Analysis 2, you will not see anything at all about differentials, and I think that maybe you won’t see the dy/dx notation at all. So, can anyone recommend a textbook about differentials?

My only experience is as a physics TA and teaching X-ray techs, so take this with a grain of salt. I think the author is trying to address a real problem, but he's not working on the right parts.

First, limits are harder than derivatives. Historically, humans figured out the derivative in the late 1600s, but the modern rigorous definition of the limit didn't exist until the 1800s. Slow-walking the definition of the derivative doesn't fix the problem of understanding limits.

The limit of a function f at a point x is defined as the value y, if it exists, such that for all d > 0 there exists an e > 0 such that for all x' in [x - e, x + e] we have |y - f(x')| < d. That's an earful. But for essentially all limits in introductory calculus we evaluate using two rules: the limit of a continuous function f at a point x is f(x), and the squeeze theorem. So my suggestion is to elevate these to the status of axioms. Introducing another number system does not help when students will not do anything nontrivial with it anyway.

The second problem is that "introductory" calculus includes too much material and is consequently pushed too late in the curriculum and seen as a weed-out course. Students spend too much time on "preparation" that doesn't prepare them for calculus. Studying logarithms and trigonometry is orthogonal, so basically all of "precalculus" is not actually pre-calculus. To me a four-year high school curriculum could be written up just fine with two years of algebra and geometry (not separated), one year of calculus and then statistics, which provides an ideal application for the theory of derivatives when you learn regression. But the author has included multivariable calculus and fiddly techniques for taking derivatives of ugly functions into "introductory" calculus. I think this is a step in the wrong direction. Laborious algebra calculations can be moved into an optional methods course for engineering students; we should be ensuring the core ideas are as accessible as possible so that doctors don't write papers about the trapezoid rule anymore:

https://diabetesjournals.org/care/article/17/2/152/17985/A-M...

Eh. I think that the standard calculus approach is mostly fine, but just needs tweaking.

The only major change that I'd like to make is the removal of sequences and all the associated theorems from the introductory calculus. Instead, start with limits of functions and the notion of continuity.

It immediately leads to the notion of the derivative. And after that, it's just a lot of building blocks.

> Again, by using differentials instead of derivatives, we have transformed a number of processes that students find unintuitive into a single process where the intuition is supplied by the student’s knowledge of algebra.

Ah yes. Algebra. The subject all students love dearly. If only we could get students to love and appreciate calculus as much they love algebra!

I have a hard time even imagining an article that is more disconnected from the reality of teaching calculus to tiny humans.

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