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Calculus For The People

geogebra.org

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Re: Calculus For The People

#61

Post by lovehashbrowns »

newprint wrote: post #55
Railsify wrote: post #46

Earlier quoted context omitted.

Can you propose a specific curriculum?

There are numerous books devoted to Olympiad preparation, there are websites completely devoted to Olympiads, there are freely available problems sets (with complete solutions). To name the few: Hungarian Problem Books, Problem-Solving Strategies, Challenging Problems in Algebra, books by Titu Andreescu - a former US IMO coach, who published a lot of books on math and IMO prep, http://www.cut-the-knot.org/

Thank you for your suggestions. I purchased Challenging Problems in Algebra and a text book by Titu Andreescu. Looking at the kinds of problems in Olympiad, it seems like they'd actually be helpful and more fun to solve.

Re: Calculus For The People

#62

Post by pugio »

cgriswald wrote: post #43
pugio wrote: post #16

For a truly "from scratch" and deeply empowering introduction to the basic notions in calculus (and all mathematics), I've found nothing better than Burn Math Class[1] by Jason Wilkes. It assumes nothing but basic arithmetic, and proceeds to guide you through how to invent maths for yourself. [1] https://www.amazon.com/Burn-Math-Class-Reinvent-Mathematics/...

I bought it with high hopes but ended up really disliking it. My main criticism is his decision to invent his own notation. For readers new to the subject, it's a great intro to the concepts, but afterwards they've got to learn the proper notation anyway. Why obfuscate it? For readers not new to the subject, the unfamiliar notation just gets in the way.

I've found the opposite to be the case, in my own experience – by first starting with brand new notation, and only later introducing the "standard" notation, he removes the absolute mystique which often surrounds existing notation, and frees you up to realize that a notation is just _a_ way of expressing an underlying idea.

He also does mention existing notation, and has a discussion on the strengths and weaknesses of various notation forms, e.g.: comparison of dM/dx vs ΔM/Δx vs M′. After reading it, I feel much more comfortable with the soup of new (and re-defined) notation one encounters when reading maths papers. With this presentation, notation becomes just a tool I know how to use, rather than some strange Math fiat delivered from on high.

Re: Calculus For The People

#63

Post by sunstone »

If calculus was taught with the concepts of nonstandard analysis rather than the tiresome and archaic "limit as delta x approaches zero" stuff, the world would be an improved place.

Re: Calculus For The People

#64

Post by sn9 »

Railsify wrote: post #46
newprint wrote: post #40

Can I give a very practical advise to people who are reading this and trying to learn math ? As someone, who received a very strong mathematical training in a former Soviet Union, here is my practical advise: 1. Calculus books, just like this one, are absolutely impractical in real life situation, especially, if your goal is "Industrial Mathematics". All you will learn, are basic calculus notations. You will, at best…

Can you propose a specific curriculum?

The Art of Problem Solving series of books is pretty much exactly what was described, is relatively affordable to an adult with disposable income, and they all come with complete solution manuals: https://artofproblemsolving.com/store/list/aops-curriculum

You could spend a year or two doing ~20 problems per day while scheduling review of definitions you've understood and problems you've solved with spaced-repetition software like Anki.

Afterwards, you'd be prepared for any undergraduate mathematics curriculum in the world.

Re: Calculus For The People

#65

Post by bootlooped »

Retra wrote: post #35
bootlooped wrote: post #31

"The learning objective is high conceptual understanding, and applicable utility." I think the lack of this was a big problem with many of the college courses I took, especially the math courses. I've often wondered if it would be better to have a "cs math concepts" set of courses where you, for example, don't need to memorize how to manually integrate a 5th degree polynomial, but instead just learn the meaning of de…

Learning the meanings of things without learning how to do them leaves you powerless to do anything with that knowledge. It's very easy to walk around saying "we could solve problem X with technique Y", but if you don't actually know how to do Y, then you're just conjecturing fruitlessly. For instance, here you're talking about "memorizing how to manually integrate a 5th degree polynomial" as if that's something anyo…

I'm pretty sure I manually integrated (and differentiated) polynomials that large in my calculus classes.

As for the "you don't want to put effort into things", I did put in the effort, I took the classes in question and graduated. I just happen think that particular effort was a waste of time.

There's always a deeper or shallower understanding to be had of a subject. Finding what is appropriate for a given task is the question.

Re: Calculus For The People

#66

Post by jacobolus »

phaedrus wrote: post #59
btilly wrote: post #52

Earlier quoted context omitted.

This is absolutely wonderful advice for a high school student who wants to get a good foundation in STEM. I'm not sure that it is applicable to an adult who needs a rough and ready understanding of Calculus. I personally taught my brother enough Calculus to take a course that had it as a pre-requisite in under an hour. What did I focus on? 1. The idea of approximations. 2. The tangent line. 3. How the tangent line co…

Counting Differential Equations, I've had 5 semesters of calculus. (Calc I,II,III,IV, Diff Equ.) Until reading your comment, item 7, I had never heard this simple explanation of the meaning of the Fundamental Theorem of calculus ("why areas are the reverse of derivatives"). I was taught the fundamental theorem algebraically, and how to apply it, but none of my professors or textbooks ever explained what it meant .

You never had someone in your introductory course show you some problem where you have e.g. a big water tank filling up, and you relate the flow rate with the volume of water in the tank, describing the relation as either sum(flow) = volume or diff(volume) = flow?

That’s quite depressing, since this is really the whole point of calculus.

* * *

Calculus really should be introduced with as much emphasis on physical modeling as possible. Differential equations are at the heart of the past several centuries of science, and understanding the basic ideas involved is crucial for everyone doing any kind of technical work, if not every citizen.

I like this version, which uses discrete computer simulation to cut down on some of the obscurantism of a formal heavily algebraic treatment, and lets students jump right into ideas which are significantly delayed in typical university math curricula. http://www.math.smith.edu/~callahan/intromine.html

Re: Calculus For The People

#67

Post by gregpetrics »

phaedrus wrote: post #59
btilly wrote: post #52

Earlier quoted context omitted.

This is absolutely wonderful advice for a high school student who wants to get a good foundation in STEM. I'm not sure that it is applicable to an adult who needs a rough and ready understanding of Calculus. I personally taught my brother enough Calculus to take a course that had it as a pre-requisite in under an hour. What did I focus on? 1. The idea of approximations. 2. The tangent line. 3. How the tangent line co…

Counting Differential Equations, I've had 5 semesters of calculus. (Calc I,II,III,IV, Diff Equ.) Until reading your comment, item 7, I had never heard this simple explanation of the meaning of the Fundamental Theorem of calculus ("why areas are the reverse of derivatives"). I was taught the fundamental theorem algebraically, and how to apply it, but none of my professors or textbooks ever explained what it meant .

That's really sad.

You might check out the sequence of activities in the Integral chapter of this book to get another perspective on the FTC.

I totally agree. It's the entire point of calculus, and it's what sets the entire field of Differential Equations in motion!

Re: Calculus For The People

#68

Post by lonelappde »

phaedrus wrote: post #59
btilly wrote: post #52

Earlier quoted context omitted.

This is absolutely wonderful advice for a high school student who wants to get a good foundation in STEM. I'm not sure that it is applicable to an adult who needs a rough and ready understanding of Calculus. I personally taught my brother enough Calculus to take a course that had it as a pre-requisite in under an hour. What did I focus on? 1. The idea of approximations. 2. The tangent line. 3. How the tangent line co…

Counting Differential Equations, I've had 5 semesters of calculus. (Calc I,II,III,IV, Diff Equ.) Until reading your comment, item 7, I had never heard this simple explanation of the meaning of the Fundamental Theorem of calculus ("why areas are the reverse of derivatives"). I was taught the fundamental theorem algebraically, and how to apply it, but none of my professors or textbooks ever explained what it meant .

I guarantee you it was in your textbook and probably in lecture and you just glossed over and forgot. What does "applying it" even mean if not "finding the intrgral of a dervative or the derivative of an integral"? That's the entire point of the theorem, which is given the most powerful name mathematics pedagogy gives to important ideas, which should motivate you to glance at it more than once.

Or do you mean that you never connected "derivative and integral are inverses" with "integrals are areas" and "inverses are opposites"?

Re: Calculus For The People

#69

Post by lonelappde »

newprint wrote: post #53
rrss wrote: post #50

Earlier quoted context omitted.

I'm genuinely curious what field you work in and what problems you work on that IMO level algebra and combinatorics skills are frequently useful (IMO = the olympiad, not my opinion). In the fields I have experience with, the usually approach when faced when something gnarly involves a lot of "to first order," or "assume X is much greater than Y," or simulation. I'm somewhat doubtful that your advice is widely applica…

To answer your question, I work in optimization. IMO (International Mathematical Olympiad) are extremely complicated problems that even professional mathematicians often struggle with them. IMO is a level on its own - Gold Standard. Not all Mathematical Olympiads are of the same level of complexity as IMO. Good example are Olympiad caliber problems that are not overkill - Hungarian Problem Books. MO problems are usef…

But industrial mathematics is numerical methods, a rather different beast from the tricky closed form puzzles of Olympiads. I question your claims of causation vs correlation regarding Olympiad work and industrial math and general effort put into basics of mathematical thinking and general mathematical intelligence.

Re: Calculus For The People

#70

Post by lonelappde »

pugio wrote: post #62
cgriswald wrote: post #43

Earlier quoted context omitted.

I bought it with high hopes but ended up really disliking it. My main criticism is his decision to invent his own notation. For readers new to the subject, it's a great intro to the concepts, but afterwards they've got to learn the proper notation anyway. Why obfuscate it? For readers not new to the subject, the unfamiliar notation just gets in the way.

I've found the opposite to be the case, in my own experience – by first starting with brand new notation, and only later introducing the "standard" notation, he removes the absolute mystique which often surrounds existing notation, and frees you up to realize that a notation is just _a_ way of expressing an underlying idea. He also does mention existing notation, and has a discussion on the strengths and weaknesses o…

How can you learn a pile of complicated mathematics but not be able to learn 5 minutes of notation definitions that were chosen for their usability over centuries?
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