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The Calculus Trap (2005)

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Re: The Calculus Trap (2005)

#41

Post by chongli »

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.

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.

Re: The Calculus Trap (2005)

#42

Post by rhizome »

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.

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

Re: The Calculus Trap (2005)

#43

Post by applecrazy »

Jach wrote: post #35
applecrazy wrote: post #25

Earlier quoted context omitted.

> 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 rem…

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...

Re: The Calculus Trap (2005)

#45

Post by barry-cotter »

HiroshiSan wrote: post #38
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…

You, and AoPS’ target audience of current and potential math nerds are very far from the norm. Inquiry based learning is actively harmful to learning for many students, and much less efficient for more or less everyone.

> Why minimal guidance during instruction does not work: An analysis of the failure of constructivist, discovery, problem-based, experiential, and inquiry-based teaching

> Evidence for the superiority of guided instruction is explained in the context of our knowledge of human cognitive architecture, expert–novice differences, and cognitive load. Although unguided or minimally guided instructional approaches are very popular and intuitively appealing, the point is made that these approaches ignore both the structures that constitute human cognitive architecture and evidence from empirical studies over the past half-century that consistently indicate that minimally guided instruction is less effective and less efficient than instructional approaches that place a strong emphasis on guidance of the student learning process. The advantage of guidance begins to recede only when learners have sufficiently high prior knowledge to provide "internal" guidance. Recent developments in instructional research and instructional design models that support guidance during instruction are briefly described.

https://www.tandfonline.com/doi/pdf/10.1207/s15326985ep4102_...

Re: The Calculus Trap (2005)

#46

Post by threatofrain »

WillPostForFood wrote: post #29

If ever you are by far the best, or the most interested, student in a classroom, then you should find another classroom. 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.

I believe it's been roughly found that tracking helps smart kids moderately and hurts everyone else with the brain drain. It's a question of to what degree we are individuals, and to what degree we are a community.

I think it's a question that cuts right down the middle of many people's value systems.

Re: The Calculus Trap (2005)

#47

Post by Mvandenbergh »

threatofrain wrote: post #46
WillPostForFood wrote: post #29

If ever you are by far the best, or the most interested, student in a classroom, then you should find another classroom. 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.

I believe it's been roughly found that tracking helps smart kids moderately and hurts everyone else with the brain drain. It's a question of to what degree we are individuals, and to what degree we are a community. I think it's a question that cuts right down the middle of many people's value systems.

There's also the hybrid view: it's important to ensure that the most talented kids are supported properly because they are our future leaders, inventors, artists, etc.

Re: The Calculus Trap (2005)

#48

Post by Ntrails »

chongli wrote: post #41
Hasz wrote: post #31

Earlier quoted context omitted.

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.

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 f…

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.

Re: The Calculus Trap (2005)

#49

Post by sevensor »

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.

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.

Re: The Calculus Trap (2005)

#50

Post by Izkata »

sevensor wrote: post #49
Hasz wrote: post #31

Earlier quoted context omitted.

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.

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.

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