Simplifying and Refactoring Introductory Calculus

EE-Reverance about 4 hours ago 11 commentsRead Article on arxiv.org

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jgord•21 minutes 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.

cool_dude85•about 1 hour 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•16 minutes 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.

mkl•36 minutes 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

bee_rider•about 1 hour 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•36 minutes 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•about 1 hour ago

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

scythe•24 minutes ago

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...

E-Reverance•about 1 hour ago

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

cyberax•about 1 hour ago

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.

light_hue_1•43 minutes ago

> 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.