Simplifying and Refactoring Introductory Calculus (2018)

Posted by E-Reverance 8 hours ago

Simplifying and Refactoring Introductory Calculus (2018)(arxiv.org)

84 points | 38 comments

jgord 5 hours ago|

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.

the-mitr 1 minute ago||

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...

imperio59 4 hours ago||

Honestly it's hard to beat how good 3Blue1Brown is at this... https://www.youtube.com/watch?v=WUvTyaaNkzM&list=PLZHQObOWTQ...

cool_dude85 6 hours ago||

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?

simonreiff 5 hours ago||

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.

archermarks 4 hours ago||

This is effectively how dual numbers work! https://en.wikipedia.org/wiki/Dual_number

ogogmad 1 hour ago||

Dual numbers are basically big/little O notation. They combine beautifully with Robinson's NSA to give the most 18th century-like approach to deriving integral/derivative formulas that I know. And it's fully rigorous!

jjmarr 3 hours ago|||

> 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.

augustusseizure 3 hours ago||

Infinitesimals don't rely on faith any more than any other mathematical idea. The "normal" calculus sequence being based on limits is solely due to the fact that limits were the first discovered method for rigorously formalizing the subject. Infinitesimals do a better job representing how most people intuitively think about calculus though, as evidenced by being the vehicle through which the entire field was discovered in the first place, and so now that they too rest on a formally rigorous foundation we should probably consider rebuilding calculus education around them.

mkl 5 hours ago|||

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

anthk 2 hours ago||

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).

richard_chase 4 hours ago||

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

imperio59 4 hours ago|

Which book is that exactly? I'm finding lots of calculus book for "Stewart's Calculus" :(

Jtsummers 3 hours ago||

James Stewart is the author, there are 8 or 9 edition of the same book. Any will do.

xiphias2 5 hours ago||

,, 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.

matherial 3 hours ago|

> 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.

ogogmad 1 hour ago||

> even today, there's a pretty significant disconnect between the mechanics of epsilon-delta and the meaning we assign to the result.

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)

A (nonstandard real) quantity is o(δx) when it's infinitesimal relative to δx, i.e. s ∈ o(δx) whenever s/δx is infinitesimal. So δx² ∈ o(δx) but δx ∉ o(δx).

bee_rider 6 hours ago||

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…

nophunphil 5 hours ago|

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)

conorbergin 6 hours ago||

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

anthk 2 hours ago||

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/

rramadass 2 hours ago||

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

ogogmad 1 hour ago||

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.

drunkboxer 15 minutes ago|

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.

E-Reverance 6 hours ago|

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

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