The way calculus is usually taught is a mess. A mix of epsilon-delta formalism, without adequate motivation, differentials and excessive focus on computations.<p>For young students, a great introductory textbook is <i>Calculus Made Easy</i>. 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.<p>For a more mature audience, I like <i>Infinitesimal Calculus</i> by Henle & Kleinberg.
I don't buy it.<p>AoPS is a great organization, but the focus is on pure, theoretical mathematics.<p>Understanding calculus is key to understanding many beautiful areas of applied mathematics: image processing, signal processing, control systems, electronics, etc. I consider them more elegant than theoretical mathematics.<p>Now, for that, you don't need all the messy manipulation (integration-by-parts and similar), but you do need the basics of area-under-the-curve, of derivative-as-slope, and similar, as well as some of the theory.<p>But that's not too hard to learn.<p>My own opinion is that the basics of calculus should be taught alongside the basics of algebra in elementary school. Plenty of people have had success doing both.
A couple of good past discussions:<p>2014: <a href="https://news.ycombinator.com/item?id=7207495" rel="nofollow">https://news.ycombinator.com/item?id=7207495</a><p>2009: <a href="https://news.ycombinator.com/item?id=717982" rel="nofollow">https://news.ycombinator.com/item?id=717982</a>
What does HN think of Khan Academy's treatmemt of calculus? <a href="https://www.khanacademy.org/math/calculus-all-old" rel="nofollow">https://www.khanacademy.org/math/calculus-all-old</a><p>Ideally, by an expert in calculus, who has done the whole KA course on it (though why an expert would do that, I don't know...)
> That student who breezed unchallenged through algebra, geometry, and trigonometry, will breeze through calculus, too.<p>I took calculus last year (AB Calc BC, 10th grade), and I can say that my experience was certainly a counterexample. I did OK, but it was definitely a marked difference from "breezing through" algebra.
> "For an avid student with great skill in mathematics, rushing through the standard curriculum is not the best answer. That student who breezed unchallenged through algebra, geometry, and trigonometry, will breeze through calculus, too."<p>That was me. I was great at calculus type things, but Matrix Theory hit me like a ton of bricks. I still have that text book, sitting on my other desk, staring menacingly at me from across the room; Matrix Analysis, Horn and Johnson. Geometry in High School gave me a taste, but would have been nice had we had available another proof based class in the math curriculum; Formal Logic or Discrete Maths at a high school level. Maybe even Linear Algebra?
This article is primarily a marketing piece for their $500 courses, which I'm sure are good but if you don't have money or accessible local math student clubs and want to like the article says 'explore math' try these math foundations playlists for free <a href="https://www.youtube.com/user/njwildberger/playlists" rel="nofollow">https://www.youtube.com/user/njwildberger/playlists</a> Wildberger starts from the very beginning, proving laws of each ring/field with basic arithmetic. He also has an algebraic trig method that a primary school kid could do and an interesting discrete algebraic calculus method, plus plenty of abstract algebra content.
I think this is a side-effect of the way our educational system is structured. We rely on big, standard tests to evaluate schools and their students. It would be too expensive, and somewhat subjective, to evaluate students' reasoning abilities, their conceptual knowledge. Can you imagine if these tests were evaluated by rooms full of people reading proofs, rather than by bubble-sheet scanners? So we test what we can test, which is procedural knowledge, and we teach that at the expense of deeper understanding. And because the best way of measuring procedural knowledge is to measure how many procedures you know, we race students forward to as much calculus as they can memorize.
<i>If ever you are by far the best, or the most interested, student in a classroom, then you should find another classroom.</i><p>It is unfortunate that the prevailing educational trends are to get rid of tracks, lanes, and advanced classes, and dumping all the kids into mixed ability classes.