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
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 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.
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.
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.
I think this is practically fixed by Robinson's NSA when it's combined with big/little O notation:
δy = f'(x) δx + o(δx)
(Pointing out the unrelated absurdity of this title being given out to people often not actually present at a company’s founding)
#lang sicp
chicken-install srfi-203
chicken-install srfi-216
(import scheme)
(import (srfi 203))
(import (srfi 216))
Online, interactive SICP in the browser, you don't need to install anything:
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
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...
Generally most of these 'handwavy' notations are rigidly provable, but only under general assumptions, that might not be true in special cases.
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...
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.
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
But they are not immediately needed to understand the limits.
Try to see how far you can get just with the epsilon-delta formulation of limits of functions.
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.