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The Calculus Trap (2005)

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Re: The Calculus Trap (2005)

#31

Post by Hasz »

nextos wrote: post #8

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 thou…

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.

Re: The Calculus Trap (2005)

#32

Post by dTal »

tomrod wrote: post #21
morpheuskafka wrote: post #5

> That student who breezed unchallenged through algebra, geometry, and trigonometry, will breeze through calculus, too. 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.

I expect it was more of a teacher issue then. A solid math teacher is worth their weight in platinum.

I had an excellent Calculus teacher in high school, and I did very well on the AP exam. It was still vastly more challenging than the trigonometry class that preceded it. Even today, I use trigonometry all the time almost without thinking, but I can't for the life of me remember how to long-divide polynomials (or picture a scenario where I'd need to).

There's just so friggin much deeply abstract symbol manipulation in calculus class (which in school also covers essentially "advanced algebra"). It's a different ball game.

Re: The Calculus Trap (2005)

#33

Post by User23 »

throwawayjava wrote: post #13
jefft255 wrote: post #9

Earlier quoted context omitted.

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 ( "Very Good" schools ( "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 e…

> 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.

Re: The Calculus Trap (2005)

#34

Post by hackermailman »

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 https://www.youtube.com/user/njwildberger/playlists 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.

Re: The Calculus Trap (2005)

#35

Post by Jach »

applecrazy wrote: post #25
throwawayjava wrote: post #13

Earlier quoted context omitted.

This differs throughout the USA. "Best" schools ( "Very Good" schools ( "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 e…

> 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 m…

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

Re: The Calculus Trap (2005)

#36

Post by Jach »

morpheuskafka wrote: post #5

> That student who breezed unchallenged through algebra, geometry, and trigonometry, will breeze through calculus, too. 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.

I got As in my pre-college math courses but I don't think I breezed through them. My schooling had no separate trig course, it went algebra, geometry, intermediate algebra, and calculus. Various pre- or honors versions. Since the BC exam covers quite a bit more material than the AB, my school teacher required those interested in it to take pre-calculus Honors first which was mostly about shoring up any gaps in previous studies while getting a foothold on Calculus AB material near the end. I think that extra prep made a huge difference in my performance, along with the teacher himself and an insistence on memorizing a lot of things to not be dependent on a rule sheet and getting an intuition for solving problems that even the TI-89 wouldn't help much with.

Re: The Calculus Trap (2005)

#37

Post by nextos »

Hasz wrote: post #31
nextos wrote: post #8

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 thou…

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.

Re: The Calculus Trap (2005)

#38

Post by HiroshiSan »

hyperpallium wrote: post #28

What does HN think of Khan Academy's treatmemt of calculus? https://www.khanacademy.org/math/calculus-all-old 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...)

As someone who tried teaching themselves math with Khan Academy, I feel like Khan Academy suffers from the same problem most sites do, in that they tell you the information before you can discover it for yourself. This is different than how AoPS teaches math, where they ask you questions which then guide you towards self discovery, and as a result, you start asking your own questions which create paths for you to go on and explore.

Re: The Calculus Trap (2005)

#39

Post by HiroshiSan »

hackermailman wrote: post #34

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 https://www.youtube.com/user/njwildberger/playlists Wildberger starts from the very beginning, proving laws of each ring/field with basic arithmetic. He also has an a…

Not sure how you got this impression, the concluding paragraph is "However, we are not the only other option. Other options students have are to become involved in extracurricular programs, such as math teams. Math contests should be selected with some care: those that encourage mass memorization or just test standard curricular tools tend to exacerbate the ills of the calculus trap rather than enhance problem-solving ability. Students can also pursue independent study if they are able to find mentors. University professors are occasionally willing to fill this role to some degree. There are also many summer programs and good books for extracurricular study, and some communities have developed grassroots programs to provide opportunities for eager students. These options are usually not as easy as “enroll in the next course,” but they will be far more rewarding than settling into the calculus trap."

Also these articles are normally targeted to those already invested in the AoPS ecosystem, whether it be books, courses, or their forums.

Re: The Calculus Trap (2005)

#40

Post by redler »

User23 wrote: post #33
throwawayjava wrote: post #13

Earlier quoted context omitted.

This differs throughout the USA. "Best" schools ( "Very Good" schools ( "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 e…

> 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.
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