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The way calculus is usually taught is a mess. A mix of epsilon-delta formalism, without adequate motivation, differentials and excessive focus on computations.

For young students, a great introductory textbook is Calculus Made Easy. It is around 100 years old, and develops all the material using infinitesimals. Which is essentially modern non-standard analysis, minus rigor. It is also the way Newton and Leibniz thought about calculus, and the way most physicists intuitively think about problems.

For a more mature audience, I like Infinitesimal Calculus by Henle & Kleinberg.

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No way high school or college level classes are teaching intro calc with delta-epsilon proofs, it just won't make any sense. It's generally introduced in Real Analysis, which is generally a sophomore level class.

Develop the intuition, then crystallize and formalize the idea with a proof. Otherwise, it's just not going to make any sense.

My last year high school math class had this approach, and it was a regular math course taken by every student.

Same thing at university, in a CS degree. Freshman Calculus taught at a level somewhere between Spivak and Rudin. In fact, both books where on the official course bibliography.

All this took place in a fairly big EU country, ~10 years ago.

I was taught the Epsilon-Delta definition of the limit in Calc 1, right off the bat in first year (University of Waterloo). Real Analysis here is a third year course run by the Pure Math department and from what I've heard it's one of the hardest undergrad courses the university offers, period. I'm nearing the end of my second year as a math student, finishing up calc 3 and linear algebra 2. When I take a look at a few of the questions on assignment 1 from a past real analysis course I have no idea how to do any of it, it's way out of my league.

Real analysis is not actually that hard, you do however have to learn a bunch of notation, concepts and processes before you can grasp things. The people who struggled with it at my course (different country, similar content I suspect) were, entirely unsurprisingly, the people who didn't attend all the lectures.

I don't think the difficulty is due to anything inherent to real analysis. I think it's due to the fact that most of the professors in the pure math department participate in the development of the math contests for CEMC [1]. Thus they tend to love creating extremely complicated and challenging homework problems.

If you have time, click around on that page and check out some past Euclid (grade 12) math contests. Even if you've come through real analysis I bet some of the later problems in those Euclids will give you trouble.

[1] https://www.cemc.uwaterloo.ca/contests/contests.html

I took Calc I at a community college last fall and epsilon-delta was about halfway through.

Huh? I learned differentiation with delta-epsilon proofs in the 11th grade. It was pretty standard at my public high school. I'm sure I didn't understand it very well at the time, but it was good to have seen it when I took real analysis.

Had to look it up to be sure, since I didn't recognize the name, but it looks like this refers to using limits to "invent" differentiation?

If so, yeah, we did it that way too, in 11th or 12th grade. (Had the same teacher both years, can't remember when exactly Calc was)

I remember it involving a lot of drawings of graphs so we could understand what each term referred to, and can't really imagine an easier way to learn it.

Yeah, exactly. I honestly don't know how else you'd teach the derivative either, but I guess it's done.

The limit definition of differentiation is definitely intuitive, but epsilon-delta refers to the most common definition of what it means to take a limit. It's also simple, because when you say it in words it makes perfect sense, but it's not at all obvious why someone would define limits in such a way, or what the alternatives are.

Epsilon-delta is the formalization introduced by Cauchy, Bolzano and Weirstrass during the 19th Century.

It is not the way calculus was invented by Newton and Leibniz. They thought in terms of infinitesimals. However, this approach is relatively hard to formalize. It was only done by Robinson in the 1960s.

To see why Cauchy et al had to work on epsilon-delta, take a look at [1], an excellent book.

[1] https://www.macalester.edu/aratra/

Same here. That's how I learned it as well. That said, I didn't really "get it" until I was well into the class. I still remember that "aha moment" very vividly, even though it was in 1992 or so. I was sitting at a table in the school library working on a problem about a pool being emptied while it was raining and my brain said "Holy crap! This is just about how fast things are changing!". With all of the proofs, graphs, etc. things were never put into such a simple statement.

The end result of months of flailing followed by the "aha" was a grade of C for the class AND being the only one to get a 5 on the AP test.

EDIT: As a fun corollary, I really hated Calculus class in high school. Ironically, I ended up getting a Masters in EE, so I ended up spending the next several years doing Calculus in > 50% of my courses.

EDIT EDIT: Also fun to note was that Calculus was only offered to Seniors at the time and required 4 years of math beforehand: Algebra, Geometry, Advance Math, and Trigonometry. That meant you had to double-up on math in either Sophomore or Junior years, which basically limited class size to a small handful of people.

Yes, _Calculus Made Easy_ is a classic. Here's link to Project Gutenberg pdf: https://www.gutenberg.org/files/33283/33283-pdf.pdf

I don’t know about the rest of the world but in Quebec calculus is not taught with any epsilon-delta formalism. You don’t encounter that until mathematical analysis, which is in the curriculum of a very few majors in uni.

This differs throughout the USA.

"Best" schools (<1%) will have what the article is describing (strong student mathematical problem solving communities lead by teachers with strong mathematical training). Typically very expensive, either directly through tuition or indirectly because they are public schools but only serve students in very high net worth zip codes.

"Very Good" schools (<10%) will include some of the epsilon-delta formalism but might have some curricular problems and a "gifted just means moving through the terrible not-real-math curriculum faster" problem that the article talks about.

"Decent" schools will teach how to compute derivative/integrals by rote and maybe talk a bit about the physical intuitions in a very hand-wavy way. Calculus is very much a continuation of Algebra or Trig where you learn some rules and how to pattern match and don't ask too many questions about why.

The other half of USA high schools? They don't even offer a Calculus course of any kind [1]. Which... if you don't teach it at all, you can't teach it wrong...

As an aside, AP CS has the same problem as the problem identified with AP Calc. AP CS is the epitome of a "Java School" course.

[1] https://www.theatlantic.com/education/archive/2016/06/where-...

>The other half of USA high schools? They don't even offer a Calculus course of any kind [1]

That's a bit misleading. There are many districts where one school serves as a magnet school where anyone interested in taking more advanced classes can go. So yes while it may be true that half of all schools don't offer calculus--far less than half of all students can't readily take calculus.

Not to mention that many schools allow dual-enrollment, where one can take courses at a local community college to make up for the dearth of courses at a high school.

> AP CS is the epitome of a "Java School" course.

As a high school student who took AP CS last year, I agree, but in reality the course isn't even about Java. Code is written by hand on the free response portion and Java syntax is mentioned but not really emphasized (things like missing brackets and semicolons are okay on the FRQs). I finished all the questions in about 25 minutes, with over an hour left to twiddle my thumbs.

Even questions about algorithmic analysis (in APCS's case, just sorting algorithms) are done without proper big-O notations, which is just plain stupid IMO.

For a student that has done CS on their own for years now, APCS feels like the fake, industrialized, watered-down version.

> watered-down version

All of the AP tests are roughly comparable in my opinion. Just not too many high school kids are as overprepared for the tests in chemistry, physics, statistics, economics, etc.

I read that "Java School" bit as being in the Joel Spolsky sense. As opposed to a more SICP approach.

I would be curious to hear your thoughts on the AP CS AB version's material, specifically whether you think it would have been of any use to you. They killed it some years ago, there's just the A now, so you'll have to search around for an old practice exam to look at. I took it in 2007 since I was ahead of the class enough to continue studying on my own. I don't remember exactly what was covered anymore but I do remember enjoying the studying and writing the exercise programs more, it was worth the lower score. I doubt the A has absorbed anything from it.

If you keep up your self-study, expect further disappointment in most undergrad data structures courses. =P

AP CS AB looks amazing, it has everything from binary trees to proper data structures, and even Big-O notation![1]

It's sad they killed it, it looks so fun.

[1]: http://mchs.virtualbeach.com/cs/documents/AP%20Computer%20Sc...

> Typically very expensive, either directly through tuition or indirectly because they are public schools but only serve students in very high net worth zip codes

It's news to me that Stuyvesant considers zip code for admissions. I thought it was a standardized test open to all students in the system.

It is. Students citywide match in SHSAT score order from the top down.

Let me recommend Callahan & al.’s Calculus in Context, http://www.math.smith.edu/~callahan/intromine.html

Which focuses on what might be called the calculus approach to mathematical modeling in science/engineering, and uses computer simulations.

I suggest don't let the K-12 and college educational systems make calculus a mess.

Instead, if have some algebra and trigonometry from high school and want to learn calculus, then just get one or a few good freshman COLLEGE calculus books and work through them -- at each lesson or section, read and think about the material and then work all the more challenging exercises. Check answers in the back of the book, a copy of the Instructor's Guide, on the Internet, etc.

For the books, get mostly old ones known for decades to be good. Get just good, used copies -- the subject hasn't changed much in decades. For the books, DO get ones that are good on (A) the completeness property of the real numbers ("Calculus is the elementary consequences of the completeness property of the real numbers."), (B) limits, (C) the epsilon-delta definition of limits, (D) the epsilon-delta, limit definitions of the derivative and the (Riemann) integral, (E) applications. Get more than one such book, use the one that looks the best as your primary source and use the others for alternate explanations and more exercises.

That's what I did: I got a good book and worked through about half of it. Then for calculus in college, I asked to skip freshman calculus and start on sophomore calculus, the rest of the book. A prof gave me a little oral exam, define the derivative, with some TeX notation

f'(x) = d/dx f(x) = lim_{h --> 0} (f(x + h) - f(x))/h

and I was in. I did well, made As both semesters. Went on to advanced calculus, ordinary differential equations, advanced calculus for applications, real analysis, functional analysis, real applications, peer-reviewed publications, teaching calculus, etc. E.g., for an application I derived and used

y'(t) = k y(t) (b - y(t))

to please the BoD at FedEx, keep a crucial investor from leaving, and save FedEx from going out of business.

From what I've seen of high school materials for calculus, I'd advise trying hard to avoid them -- again, just start with one of the best college texts. The one I used for the actual course was Johnson and Kiokmeister, then also used at Harvard, now ancient but still fine.

This is off-topic to the article.

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