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Simplifying and Refactoring Introductory Calculus (2018)

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Re: Simplifying and Refactoring Introductory Calculus (2018)

#41

Post by drunkboxer »

ogogmad wrote: post #39

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.

Re: Simplifying and Refactoring Introductory Calculus (2018)

#42

Post by the-mitr »

jgord wrote: post #13

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 Derivati…

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...

Re: Simplifying and Refactoring Introductory Calculus (2018)

#43

Post by gloveone »

jjmarr wrote: post #22
cool_dude85 wrote: post #7

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?

> 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 defin…

Your comment was correct 100 years ago. But today it is highly inaccurate. Initially Calculus was developed using the ideas of infinitesimals throughout, although this was not yet fully formalized. The first to ground with mathematical rigor was O believe Cauchy with the epsilon-delta definition of limits. For historical reasons this caught on and is the standard way we introduce students to the subject till today. But since then we have already discovered fully rigorous and zero faith ways to define and work with limits: Robinson's non standard analysis and using nilpotent infinitesimals a la synthetic differential geometry. These provide completely rigorous way to view all the classical intuitions that initially develop the subject and are much easier to work with than the current standard epsilon-delta gymnastics. Unfortunately, mathematicians are very conservative and we tend to stick with the conventional way of doing things way more than we should. In fact a huge part of mathematical community have not fully engaged with the beautiful way of defining and using infinitesimals for calculus, even though it would greatly aid the students learning and intuition and solve the disconnect when working with physics using infinitesimals.

Re: Simplifying and Refactoring Introductory Calculus (2018)

#44

Post by u1hcw9nx »

simonreiff wrote: post #14
cool_dude85 wrote: post #7

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 h…

Hyperreal numbers make Leibniz notation literal and algebraically consistent and rigorous, rather than a convenient shorthand for limits.

Author suggests using them. Resistance to Hyperreals seems to come from era before there were rigorous definitions for them.

Re: Simplifying and Refactoring Introductory Calculus (2018)

#45

Post by pelasaco »

jgord wrote: post #13

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 Derivati…

> 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...

Re: Simplifying and Refactoring Introductory Calculus (2018)

#46

Post by nextaccountic »

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...

Re: Simplifying and Refactoring Introductory Calculus (2018)

#47

Post by srean »

ogogmad wrote: post #37
archermarks wrote: post #17

Earlier quoted context omitted.

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

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!

True. Knuth had a paper where he suggests using (a slightly modified) O notation for teaching calculus.

https://www-cs-faculty.stanford.edu/~knuth/calc

From the Arxiv paper:

> Students have to memorize a diversity of processes for essentially performing the same task.

Is that true for differentiation ? I don't recall having to memorize many things, just how differentiation composes over +,-,×,÷, function composition and the differential of a few standard forms.

Symbolic integration, on the other hand, is a whole can of worms.

Re: Simplifying and Refactoring Introductory Calculus (2018)

#48

Post by tgv »

jgord wrote: post #13

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 Derivati…

Missing are exercises, feedback and motivation. Without those, the content won't stick.

Re: Simplifying and Refactoring Introductory Calculus (2018)

#49

Post by srean »

ogogmad wrote: post #39

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.

https://rulebasedintegration.org/

Re: Simplifying and Refactoring Introductory Calculus (2018)

#50

Post by voidhorse »

nextaccountic wrote: post #46

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 th…

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.

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