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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
My experience was that the visuals were a cool distraction from familiarity with the rules/identities. People always want a shortcut to comfort, and they often want it to be that a thorough exploration of motivating examples, often with graphics, grants understanding and comfort before facility, but I've always found that a relatively cursory description of the motivating example is more than enough, and after that I just need well-organized reference materials, worked examples, and rote practice.
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.
Of possible interest, presents basic notions of calculus as a dialogue
Calculus Basic Concepts For High Schools by L. V. Tarasov
https://archive.org/details/LevTarasovCalculusBasicConceptsF...
Missing are exercises, feedback and motivation. Without those, the content won't stick.
> so you _can_ get your kids a superb math education
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...
Honestly it's hard to beat how good 3Blue1Brown is at this... https://www.youtube.com/watch?v=WUvTyaaNkzM&list=PLZHQObOWTQ...
Thanks for the shout-out! I teach calculus to homeschool co-op students regularly, and I got tired of the existing books on the market (I used to teach from Saxon Calculus) so I wrote my own, "Calculus from the Ground Up". This paper represents the principles I used for writing the book.
Interestingly, my reformulation of the Leibniz notation for the second derivative was actually the result of writing the book. I was wanting to write a piece on why the second derivative looks the way it does. I had about 10 different calculus books I was looking through trying to find a solid answer and there was none. So, I eventually decided to try and derive the formula myself. I was quite surprised when I was able to derive a formula, but it was different. I tried to figure out for a while how to get from my formula to the standard one, until I eventually realized that the standard formulation was itself problematic.
It's in the "Calculus from the Ground Up" book as "Appendix B", but I don't use it in the main text so as not to confuse students who take further calculus courses. I found a middle ground for the book which neither forces the new notation nor commits the mistakes of the previous one. The book is not heavy in higher-order derivatives anyway, so the usage is minimal.
On the second derivative side, a fuller treatment (including applying the approach to partial differentials) is given in the paper "Total and Partial Differentials as Algebraically Manipulable Entities".
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?
Actually Leibniz invented the modern dy and dx notation and did view the differentials as genuinely nonzero, which is generally speaking a safe assumption. In other words, dy/dx really was a quotient, albeit of really tiny values (at least we assume dx can become arbitrarily small while remaining nonzero). The calculation Leibniz would do looked something like this. First he would consider an equation y = x^2. Then he would consider a nonzero difference so something like y + dy = (x + dx)^2 = x^2 + 2x dx + (dx)^2. At this point he would use his starting equation to subtract y from the LHS and x^2 from the RHS, leaving: dy = 2x dx + (dx)^2. Then he would divide by dx leaving dy/dx = 2x + dx and since dx is infintisimal, he would just lop it off. Suffice it to say, just ignoring the nonzero dx on the RHS, or casting it to 0 while conveniently ignoring the division by 0 on the LHS, was rather disturbing to many critics. A lot of work had to be done by Riemann, Cauchy, and Weierstrass over the following century after Newton and Leibniz invented calculus to answer the question you are asking.
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.
They are differentials. https://en.wikipedia.org/wiki/Differential_(mathematics) has some info but is not great as a beginning introduction. dx is an infinitesimal bit of x, and dy is an infinitesimal bit of y. dx here is the same dx as in an integral, which you can think of as the width of one of the infinite infinitesimally thin rectangles whose areas are being added up to find the area under the curve: https://en.wikipedia.org/wiki/Riemann_integral
> And what are these Fluxions? The Velocities of evanescent Increments? And what are these same evanescent Increments? They are neither finite Quantities nor Quantities infinitely small, nor yet nothing. May we not call them the ghosts of departed quantities?
-- 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.
dx and dy are the smallest change ever in a function. Think about the speed of something
(that's the reason of derivatives, you can't trace the speed of a stopped car in a track, you can't divide by zero, of course, there's no motion).
I think Stewart's Calculus is excellent and it is rightfully the standard textbook. No modifications needed in my opinion.
James Stewart made a lot of money off his textbook.
Here is an article about the interesting house he designed:
https://torontolife.com/real-estate/look-inside-integral-hou...
Which book is that exactly? I'm finding lots of calculus book for "Stewart's Calculus" :(
It's a bit frustrating that smooth infinitesimal analysis isn't mentioned even once, even as the author lists other systems that add infinitesimals to real numbers. It's a much better system to develop calculus on. It features infinitesimals as normal everyday mathematical objects, rather than the hack that hyperreal numbers are.
Of course, I know that something like SIA would never be adopted. The main problem is that it is based on intuitionistic logic rather than classical logic. As such, it requires new intuitions that may not be appropriate to develop while studying calculus (it would work if it were a middle school topic). This is unfortunate, because those intuitions would make calculus much simpler and remove a large number of edge cases (a great deal of quirks with calculus are actually quirks in classical logic in disguise)
However, it was not even cited! And it was not cited most likely because the author never heard about it (even though he hedged with "and other systems"), even though the author spent a great deal to explain how teaching calculus with infinitesimals (that's what differentials are) is much simpler and easier to understand than epsilon-gama limits.
Anyway let me drop some links
An one-page motivation (explains what it is all about) https://publish.uwo.ca/~jbell/invitation%20to%20SIA.pdf
A 14 page exposition https://arxiv.org/abs/0805.3307
Wikipedia article https://en.wikipedia.org/wiki/Smooth_infinitesimal_analysis
A book on SIA, that not only develop multivariate calculus but also builds classical mechanics using the same infinitesimal arguments of Newton and Leibniz, but within a rigorous mathematical setting (well that's just a free sample containing the table of contents, but the book itself is available elsewhere) https://api.pageplace.de/preview/DT0400.9780511368400_A23677...
The smooth infinitesimal analysis assumption that all functions are infinitely differentiable indeed removes a lot of edge cases, but only in the way that the C assumption that all memory accesses are within bounds removes a lot of edge cases. Just as a C programmer may have difficulty telling a valid C program from a meaningless sequence of symbols, so does SIA make it difficult to tell a valid assumption from a paradoxical statement that would allow defining a non-differentiable function.
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.
I think "Smooth Infinitesimal Analysis" is actually a bad idea because I think that the combination of Robinson's Nonstandard Analysis with little/big O notation gives you nilpotent infinitesimals in a more rigorous and elementary way.
In fact, in light of this, I'm starting to wish logicians would stop promoting SIA/SDG.
I completely agree. I have Bell's book on the infinitesimal approach and it is infinitely (hah) more intuitive (hah again) than epsilon-delta limit foundations. It trades a heady second order logical statement for simple algebra.
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.
Double limits understood visually
https://projecteuclid.org/journals/bulletin-of-the-american-...
,, 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.
I'm curious if you've taught calculus? Do your students remember limits by the end of calculus? Most studies show that students DO NOT RETAIN limit concepts (ESPECIALLY epsilon-delta ones). It is used as a crutch and then largely discarded before anyone is actually comfortable/familiar with them.
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").
> Waiting a year to get from intuition to theorems is a perfect way to ruin math.
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.
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…
At the very least, Founding Engineers!
(Pointing out the unrelated absurdity of this title being given out to people often not actually present at a company’s founding)
Not really. I teach high school calculus to homeschool co-ops. My earliest students are just a few years out of college. I usually only teach 2-10 students per class, and I don't think I managed to get any school to adopt my "Calculus from the Ground Up" book.
This guy has an interesting publication history, programming books and what looks like evolutionary biology from a creationist perspective.
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
Thanks for the list.
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.
I hope you check out my book, "Calculus from the Ground Up":
https://www.amazon.com/Calculus-Ground-Jonathan-Laine-Bartle...
> Calculus: The Princess of Mathematics by H.C.Verma et al.
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!
Calculus Made Easy, but renewed by Martin Gardner. Yes, that one.
where I heard of this from : https://youtu.be/4ZB2PNUYR1Y
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.
It's not really possible. It's akin to saying simplify multiplying large numbers or long division. It can be sometimes done by having a heap of tricks up your sleeve, by practicing a bunch you might get better at guessing which trick to use when. The usefulness of knowing these tricks and recognising when to use them is entirely dependent on your motivations.
That's an exercise under SICP (an infamous Scheme course) it works best with either
Racket with
#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:
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...
SICP teaches you calculus in a really easy way, you are basically teaching the computer how to derivate and integrate in Lisp which a much easier notation. The functions almost define themselves.
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?
Depends what you're looking for. Full Frontal Calculus[0], Intuitive Infinitesimal Calculus[1], and Elementary Calculus[2] are all textbooks on the calculus sequence using an infinitesimal pov. The basic approach is to extend the Real numbers to include infinitesimals (greater than zero but smaller than every positive real number) and transfinites (greater than every positive real number), collectively called the Hyperreals.
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...
Try https://en.wikipedia.org/wiki/Elementary_Calculus:_An_Infini...
Generally most of these 'handwavy' notations are rigidly provable, but only under general assumptions, that might not be true in special cases.
Manfredo P. do Carmo: Differential Forms and Applications. Short and incredibly beautiful.
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.