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Calculus they won't teach you [video]

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

#101

Post by abdullahkhalids »

Maursault wrote: post #65

Let me know, not if anyone here can do Calculus (that is ordinary; who can't? ), but rather whether you actually personally use Calculus, and please describe generally an example of needing Calculus to figure something out in the last 5 years. I think my point is that the history of mathematics is exponentially more interesting than mathematics. We like mathematics, but we like stories far more than we like whatever…

In my personal life, if I talk about the economy with someone, or listen to an expert talk about it, I use the conceptual framework of calculus to evaluate the relations between quantities. For instance, understanding changes in demand of various products in the face of inflation based on the price sensitivity of those products.

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

#103

Post by dragonwriter »

yamtaddle wrote: post #57
ultra_nick wrote: post #23

Earlier quoted context omitted.

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 are…

> 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

Specifically, it means “equation or inequality”.

> They don't really act like sentences or serve the same purpose.

They act exactly like sentences, and serve exactly the same purpose, because they are declarative sentences about the relationships between numbers.

> It's misleading as hell land the kids don't even have the context that might, even hypothetically, make it useful for scaffolding, when they introduce it in kindergarten.

The idea is that they are mutually reinforcing, not sequential, concepts.

It also, as a sibling comment notes, has nothing to do with Common Core, its an orthogonal development in pedagogical approach to the Common Core standards.

> The "show 5 different ways how you could have solved this" worksheets

... also have less to do with the Common Core standards thablazy development of the particular materials (demonstrating proficiency with different articulations of operations and ways of solving problems is part of Common Core, to an extent, but even where it is doing it by using them all on the same problem, in the same assignment, and doing that for a substantial number of problems on one assignment, is just bad exercise design.)

> To illustrate by example

What you offer there is an analogy, not an example. Analogies don't really act like examples and don’t serve the same purpose.

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

#104

Post by watwut »

dvt wrote: post #63
tzs wrote: post #58

Earlier quoted context omitted.

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…

Starting out with integration makes so much more sense, and it's way more graspable from a geometric standpoint (what's the area under this line?) as opposed to the weirdness (and relative abstraction) of the derivative.

Derivatives are massively easier. Starting with easy concepts makes more sense

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

#105

Post by watwut »

Maursault wrote: post #65

Let me know, not if anyone here can do Calculus (that is ordinary; who can't? ), but rather whether you actually personally use Calculus, and please describe generally an example of needing Calculus to figure something out in the last 5 years. I think my point is that the history of mathematics is exponentially more interesting than mathematics. We like mathematics, but we like stories far more than we like whatever…

I did a module for calculating financial figures and values in work. Knowing calculus was very useful - both to understand what I am doing and to figure simplifications where possible.

Imo, history of mathematics is not interesting unless you are interested in mathematics itself too. It is sort of stuff that some people good in math and interested in care about, but is otherwise completely uninteresting for anybody else.

Unless you turn it into history tabloid. Which you can do in pretty much any field, but also it will loose any meaning. At that point, you can do paris Hilton or Kayne West.

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

#106

Post by zozbot234 »

makr17 wrote: post #89
pjlegato wrote: post #62

Earlier quoted context omitted.

There was another step: Leibniz' pre-limits formulation of calculus used hyperreal numbers: infinite and infinitesimals (small-delta) numbers. This version is conceptually simpler and for many people more intuitive than the limits-based explanation. It was the standard approach for many years. Infinitesimal calculus was discarded in late 19th century pedagogy in favor of limits as no rigorous proofs yet existed of it…

For the formal underpinnings of the infinitesimal approach, the phrase to look for is "Nonstandard Analysis": https://en.wikipedia.org/wiki/Nonstandard_analysis Fun stuff.

Edward Nelson's Internal Set Theory is a somewhat cleaner and more intuitive foundation that's practically equivalent to non-standard analysis.

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

#107

Post by hansvm »

drewrv wrote: post #94
makr17 wrote: post #89

Earlier quoted context omitted.

For the formal underpinnings of the infinitesimal approach, the phrase to look for is "Nonstandard Analysis": https://en.wikipedia.org/wiki/Nonstandard_analysis Fun stuff.

I understand that nonstandard analysis is a firm footing on which to build calculus, but is it totally equivalent to real analysis? I loved real analysis in college, for example I found cantors set kind of crazy. Does that "exist" in nonstandard analysis? Or are there other mind-bending implications?

Standard math is a subset of nonstandard, and you can prove the existence of NSA within ZFC, so they are exaxtly equivalent.

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

#108

Post by noobermin »

Maursault wrote: post #70
cycomanic wrote: post #66

Earlier quoted context omitted.

I do calculations on Maxwells equations all the time, that's s calculus. Doing CAD, or any finite element calculations - > calculus. Modern engineering would not be possible without calculus. Just because software engineers mostly use linear algebra, doesn't mean that most other technical disciplines are the same.

> Modern engineering would not be possible without calculus. Was not my question. Nearly everyone uses computers. Computers would not be possible without electronics. How many computer users work with Maxwell's Equations or have even heard of Kirchhoff's circuit laws? A very tiny percentage, undoubtedly. The reason I asked was the Penn State professor's epiphany that no engineer among the many he asked actually used…

How many programmers use linked lists? I just use the list type in python, it's just "list" not a "linked list". I don't have to deal with individual nodes or pointers.

I don't have learn sorting algorithms, I just use list.sort or sorted.

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

#109

Post by zozbot234 »

yamtaddle wrote: post #57
ultra_nick wrote: post #23

Earlier quoted context omitted.

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 are…

Well, if you take Montague semantics seriously then a "word sentence" is precisely a very broad generalization of a "number sentence", mostly differing by modality (accounting for "possible worlds", time, epistemic state, indexicals, attitudes etc. etc.) The basic "algorithmic" approach to language and words was already known to the ancient Indian grammarian Pānini. It's incredibly sad how few in the supposedly advanced West are aware of the countless innovations India brought to so many fields of science, art and culture.

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

#110

Post by noobermin »

Different strokes for different folks and all that but the concept of infinity doesn't really bother me that much because it doesn't really exist to me as a number more than it is a fact of the unbounded-ness of some set with a total ordering. It is more a statement of potential rather than actual existence, that given an infinite set, you can always find a "larger" element, which to me is just a result of the way it is defined.

Infinitesimals to me are a bit more difficult because physics (the real world) doesn't actually need real continuity, otherwise physical models won't work because many continuous quantities in physics (fluid densities, temperatures, etc) aren't really continuous on small scales compared to the problem size, but that isn't needed to model such systems well. It seems like there are only a few pathological examples mathematicians come up with which is interesting and fascinating that require actual continuity, but most physical systems do not need real, actual continuity.

On the same token, I guess (countable) infinite sets aren't really needed either.

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