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Calculus For The People

geogebra.org

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Re: Calculus For The People

#41

Post by dustfinger »

oneepic wrote: post #36
dustfinger wrote: post #32

Earlier quoted context omitted.

One possible consequence of a trend towards less formal ways of presenting mathematics is that the ease of entry to the informal might not motivate beginners to learn the formal. Consider that amateur mathematicians from the past, particularly those whom made great contributions to science, might not have been motivated to teach themselves the rigorous formal notation if everything they read was explained in layman's…

This comment keeps saying "[rigorous] formal notation" -- are you implying that new mathematicians might lack rigorous mathematical technique, or just that they won't be motivated to communicate their results in a formal way that other mathematicians will generally understand?

My intention was to pose the following questions for arguments sake:

In a world with solid coverage of mathematical theory presented informally to appeal to a larger laymen audience, will the world produce more amateur mathematicians that make great contributions to science or less? Will would-be amateur mathematicians be less motivated to learn formal notation given that they can more easily read and understand the same material presented in an informal manner? If they did pick up a base level of formal notation, would they be less motivated to learn how to understand or write rigorous proofs using formal notation for the same reason?

I don't know what the answer is. Maybe there are studies out there that attempt to answer these questions.

Re: Calculus For The People

#42

Post by tacomonstrous »

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…

As a working mathematician who has rarely managed to solve an IMO problem, I have to say this isn't the best advice for everyone, though I agree that focusing on linear algebra and combinatorics is probably a better use of one's time.

Re: Calculus For The People

#43

Post by cgriswald »

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.

Re: Calculus For The People

#44

Post by ksdale »

dustfinger wrote: post #41
oneepic wrote: post #36

Earlier quoted context omitted.

This comment keeps saying "[rigorous] formal notation" -- are you implying that new mathematicians might lack rigorous mathematical technique, or just that they won't be motivated to communicate their results in a formal way that other mathematicians will generally understand?

My intention was to pose the following questions for arguments sake: In a world with solid coverage of mathematical theory presented informally to appeal to a larger laymen audience, will the world produce more amateur mathematicians that make great contributions to science or less? Will would-be amateur mathematicians be less motivated to learn formal notation given that they can more easily read and understand the…

It seems like there could hardly be fewer amateur mathematicians making great contributions to science.

The number of potentially great mathematicians who stop learning because they're satisfied with the informal coverage has to be absolutely dwarfed by the number of potentially great mathematicians who are scared away from mathematics entirely at an early age by excessive formality.

Re: Calculus For The People

#45

Post by gregpetrics »

ksdale wrote: post #44
dustfinger wrote: post #41

Earlier quoted context omitted.

My intention was to pose the following questions for arguments sake: In a world with solid coverage of mathematical theory presented informally to appeal to a larger laymen audience, will the world produce more amateur mathematicians that make great contributions to science or less? Will would-be amateur mathematicians be less motivated to learn formal notation given that they can more easily read and understand the…

It seems like there could hardly be fewer amateur mathematicians making great contributions to science. The number of potentially great mathematicians who stop learning because they're satisfied with the informal coverage has to be absolutely dwarfed by the number of potentially great mathematicians who are scared away from mathematics entirely at an early age by excessive formality.

This.

I've Seen it time and again: Great mathematicians who are awful mathematics students get turned away. They end up making great full stack engineers.

Re: Calculus For The People

#46

Post by Railsify »

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?

Re: Calculus For The People

#47

Post by gregpetrics »

tacomonstrous wrote: post #42
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…

As a working mathematician who has rarely managed to solve an IMO problem, I have to say this isn't the best advice for everyone, though I agree that focusing on linear algebra and combinatorics is probably a better use of one's time.

I agree Linear and Combi is far more useful once you get going on a technical degree and/or career, but go to a university in the US and check the prerequisites on these courses: CALCULUS.

This was another reason I wrote this book. For people who just need to get through calc, here's some help that you can pick up and read in a couple hours.

Re: Calculus For The People

#48

Post by dustfinger »

gregpetrics wrote: post #39
dustfinger wrote: post #32

Earlier quoted context omitted.

One possible consequence of a trend towards less formal ways of presenting mathematics is that the ease of entry to the informal might not motivate beginners to learn the formal. Consider that amateur mathematicians from the past, particularly those whom made great contributions to science, might not have been motivated to teach themselves the rigorous formal notation if everything they read was explained in layman's…

Author of book here. Nice comment, and very interesting question. I am not worried about using "informality" to get more people studying mathematics. The book is informal, but by the end of the book, the integral that gets presented is the correct definition of the integral. I've just collapsed as much of the technical language at possible and focused on the core idea. My thinking is: if someone is hooked, sure they'…

To be fair, I have not read the article. I am to busy with work today, but I did bookmark it for later. Also, thank you for not being insulted by my question. It wasn't meant as an insult, I was just pondering what the net effect might be if in formalism became dominant:

> why bother going all the way "down there" just to say "good enough"? Why not say "good enough" a lot higher up the ladder closer to where the problem originated.

That is exactly what motivated me to pose the question. Will would-be amateur mathematicians, that might have been great, find themselves lacking the necessary motivation to pursue a deeper understanding of the material? Will they stop climbing the ladder and just say -- good enough?

Re: Calculus For The People

#49

Post by rrss »

gregpetrics wrote: post #39
dustfinger wrote: post #32

Earlier quoted context omitted.

One possible consequence of a trend towards less formal ways of presenting mathematics is that the ease of entry to the informal might not motivate beginners to learn the formal. Consider that amateur mathematicians from the past, particularly those whom made great contributions to science, might not have been motivated to teach themselves the rigorous formal notation if everything they read was explained in layman's…

Author of book here. Nice comment, and very interesting question. I am not worried about using "informality" to get more people studying mathematics. The book is informal, but by the end of the book, the integral that gets presented is the correct definition of the integral. I've just collapsed as much of the technical language at possible and focused on the core idea. My thinking is: if someone is hooked, sure they'…

My pages apparently don't align with yours (I see page 46 has only a single exercise), but I don't see where Rudin says anything is "good enough." He states the definition of convergence, meaning that if a sequence satisfies the property then we choose to call it convergent. There is no question of good enough

I don't see a claim about an "infinite set of inequalities" - I see an infinite set of I equalities that must be satisfied.

> Do if/then statements work this way? How can we be sure we can use the natural numbers this way?

Could you be more specific?

I feel like what you call "fundamental informality" I might call "assumption of mathematical maturity."

Re: Calculus For The People

#50

Post by rrss »

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…

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 applicable if the necessary skills aren't taught in college courses.

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