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Calculus they won't teach you [video]

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Re: Calculus they won't teach you [video]

#51

Post by mhuffman »

jwsteigerwalt wrote: post #42

Integrals -> Derivatives -> Series -> Limits? That’s not a progression I am familiar with. In trigonometry and precalculus I learned Series then Limits, then in calculus derivative first then integrals. There are definitely new directions to go with Series and Limits after learning integrals, but the concepts are prerequisites to calculus (and arguably trigonometry).

This is the same progression that I learned, but I am not sure it is the best one. I wonder if there has been any rigorous scientific study on the pedagogical benefits one way or another?

Re: Calculus they won't teach you [video]

#52

Post by michael_j_ward »

dotancohen wrote: post #31

This is exactly the calculus that I was taught in school - right down to the pizza slices analogy.

If you don't mind sharing, what grade-level, decade, location were you taught this?

The key part about the pizza example is the demonstration of mathematical thinking completely disparate from what I experienced at a Catholic high-school in Chicago during the 2000s.

Re: Calculus they won't teach you [video]

#54

Post by hirvi74 »

jwsteigerwalt wrote: post #42

Integrals -> Derivatives -> Series -> Limits? That’s not a progression I am familiar with. In trigonometry and precalculus I learned Series then Limits, then in calculus derivative first then integrals. There are definitely new directions to go with Series and Limits after learning integrals, but the concepts are prerequisites to calculus (and arguably trigonometry).

I remember when I took calculus we studied those topics in the order of:

Limits -> Derivatives -> Integrals -> Series.

Re: Calculus they won't teach you [video]

#55

Post by r_strongbow »

amluto wrote: post #26
alexb_ wrote: post #13

Earlier quoted context omitted.

I disagree. The biggest thing that can help you learn something complex is by introducing it as it was discovered, and making someone feel like they could have come up with it on their own.

I don’t see why the history of discovery of something should be a particularly good way to rediscover it. As a concrete example, if someone asks “how do we know the Earth is round” or “how do we know the planets orbit the sun”, one could go through ways to figure this out with math and minimal technology. But one could also point out that we have spaceships and probes and have actual pictures! You can see the Earth f…

I've been interested for awhile in a distinction between math and philosophy here. Just descriptively speaking, we don't tend to teach math from original source (at least, not well-established math like you'd see in high school and early college). We don't teach calculus out of Newton/Leibniz/Cauchy/etc, we don't teach Fourier analysis out of Fourier, etc. Maybe geometry students will work from the Elements, but I certainly didn't.

But in philosophy, students do read everyone from Plato to Rawls in the original. There are (lots of) supplementary texts, of course, but they're companions.

I think that shows an interesting divide that the way we actually think about philosophy is not just a catalogue of arguments and positions one could take, but that the famous works are things worth reading unto themselves. There are a few math papers like that ("God Created the Integers" tries to anthologize them) but they are the exception rather than the rule.

But maybe there's another explanation?

Re: Calculus they won't teach you [video]

#56

Post by hammock »

mhuffman wrote: post #51
jwsteigerwalt wrote: post #42

Integrals -> Derivatives -> Series -> Limits? That’s not a progression I am familiar with. In trigonometry and precalculus I learned Series then Limits, then in calculus derivative first then integrals. There are definitely new directions to go with Series and Limits after learning integrals, but the concepts are prerequisites to calculus (and arguably trigonometry).

This is the same progression that I learned, but I am not sure it is the best one. I wonder if there has been any rigorous scientific study on the pedagogical benefits one way or another?

Yeah why don’t we do more a/b testing or controlled testing in education?

Re: Calculus they won't teach you [video]

#57

Post by yamtaddle »

ultra_nick wrote: post #23
pryelluw wrote: post #11

Earlier quoted context omitted.

Yea, it has gotten worse in the US thanks to the common core bullshit. My daughter’s math education has been eroded with shitty ed tech apps like the ones from MHM[0]. I’ve been able to support her but there’s many kids that are not so fortunate. [0] https://www.hmhco.com/

What's wrong with common core?

It could be worse but one thing that constantly bothers me is the stupid new vocabulary. One of the worst examples: "number sentence". It means what you, an adult, might guess it does (well, it means one of the things you might guess it means) but why use it at all? They introduce it before the kids have a decent idea of what a normal sentence (what, a "word sentence", I suppose?) is. Many of the symbols in them aren't numbers (+, -, =). They don't really act like sentences or serve the same purpose. It's misleading as hell and the kids don't even have the context that might, even hypothetically, make it useful for scaffolding, when they introduce it in kindergarten.

They do this for lots of stuff. There's a whole Common Core Math jargon the motivation for which I just can't grasp.

Also, much of the approach is effectively what you do with kids who are struggling in math—you shower them with techniques and explanations for the same thing, hoping one will stick. The "show 5 different ways how you could have solved this" worksheets drive gifted kids, at least, batty, and that's most of the work. "I fucking solved it, I've already shown you a hundred times I understand this other technique (which I'll never use because I also know several better ones), why are you still bothering me?" They'll have them do two problems with all that extra stuff, in the time and space the kids could have practiced ten problems. It's making my kids hate math in early elementary. Wonderful. Just wonderful.

[EDIT] To illustrate by example why I find "number sentence" in particular so deeply stupid: would anyone think a good approach to introducing sentences to very young children, in language classes, might be to label them "word equations" or "word formulas" or "word algorithms" or anything like that? Would anyone think that would be an improvement? God no, it's plainly a bad idea—but that's exactly what someone, somewhere, decided to do, in reverse.

Re: Calculus they won't teach you [video]

#58

Post by tzs »

jwsteigerwalt wrote: post #42

Integrals -> Derivatives -> Series -> Limits? That’s not a progression I am familiar with. In trigonometry and precalculus I learned Series then Limits, then in calculus derivative first then integrals. There are definitely new directions to go with Series and Limits after learning integrals, but the concepts are prerequisites to calculus (and arguably trigonometry).

Integral calculus before differential calculus is how it is done in Apostol's "Calculus". Here are the first several chapter titles to give an idea of what is covered when.

  Introduction
  The Concepts of Integral Calculus
  Some Applications of Integration
  Continuous Functions
  Differential Calculus
  The Relation Between Integration and Differentiation
  The Logarithm, The Exponential, and The Inverse Trigonometric Functions
  Polynomial Approximations to Functions
  Introduction to Differential Equations
  Complex Numbers
  Sequences Infinite Series, Improper Integrals
  Sequences and Series of Functions
(the rest of volume I is vector algebra and calculus and linear algebra).

That book was widely used at Caltech, MIT, and several other top schools for decades.

Re: Calculus they won't teach you [video]

#59

Post by diordiderot »

r_strongbow wrote: post #55
amluto wrote: post #26

Earlier quoted context omitted.

I don’t see why the history of discovery of something should be a particularly good way to rediscover it. As a concrete example, if someone asks “how do we know the Earth is round” or “how do we know the planets orbit the sun”, one could go through ways to figure this out with math and minimal technology. But one could also point out that we have spaceships and probes and have actual pictures! You can see the Earth f…

I've been interested for awhile in a distinction between math and philosophy here. Just descriptively speaking, we don't tend to teach math from original source (at least, not well-established math like you'd see in high school and early college). We don't teach calculus out of Newton/Leibniz/Cauchy/etc, we don't teach Fourier analysis out of Fourier, etc. Maybe geometry students will work from the Elements, but I ce…

Two contributers I can think of

1. Being able to tease apart language from ideas can't be 'taught' you have to do it

2. People disagree much more in philosophy than math

Re: Calculus they won't teach you [video]

#60

Post by dotancohen »

michael_j_ward wrote: post #52
dotancohen wrote: post #31

This is exactly the calculus that I was taught in school - right down to the pizza slices analogy.

If you don't mind sharing, what grade-level, decade, location were you taught this? The key part about the pizza example is the demonstration of mathematical thinking completely disparate from what I experienced at a Catholic high-school in Chicago during the 2000s.

In Israel it's been taught this way for decades.
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