31–40 of 78 posts
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 derivatives and integrals.
A lot of people are just linking out to their favorite calc intro instead of commenting on this one. I like it so far, but I already did 2 years of calculus a few years ago, so I'm not learning much. I do appreciate the growing trend of presenting material in a more down-to-earth way, maybe with less-formal language and showing the reader it's not as scary as they might've thought previously. Kudos to the author, thi…
It seems great on the surface. More people might read texts about mathematics, but if the trend were taken past some threshold, then there might be a global consequence as well.
Obviously this is just speculation. Another possability is it will simply result in a change in the personality type of those ammeture mathemeticians whom make contributions to science. I suspect there will be some sort of net effect, but it might not be what we expect.
Is there anyone here that was inspired by layman articles on mathematical theory and later went on to learn rigorous formal notation?
Earlier quoted context omitted.
Quoting the prerequisites: “On the matter of prerequisites, this book assumes you are competent, if not a Jedi, at basic algebra and arithmetic. Specifically, an understanding of lines, their equations, slope, y-intercepts, x-intercepts, and so on is more or less assumed. I think this is reasonable.”
And I do understand those. What I don't understand is growth rate and the slope of a function that is a curve, not a line, therefore not a slope.
Think of your morning drive to work. Your distance from the office is always changing as you accelerate and decelerate along the way. It may even stop changing at various times (when you're parked, at stop signs or red lights). If you were to record your distance over time, it would be a curve that goes down (and up when you go in reverse), and sometimes remains level.
The slope of that curve at a given point, that is the derivative, corresponds to the speed displayed by your speedometer at that particular moment in time.
This is differential calculus in a nutshell: given a curve representing your distance over the entire trip, find the speed at any instant in time. We can then relate this to integral calculus in the following way: given a way to record your speed at every moment in time (speedometer), determine the total distance you travel. If it sounds like two sides of the same coin, well it is! This is the brilliant discovery of the fundamental theorem of calculus.
"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…
For instance, here you're talking about "memorizing how to manually integrate a 5th degree polynomial" as if that's something anyone who knows calculus actually does. What it really sounds like is that you don't want to put effort into things. Giving you easier classes isn't going to solve your problem.
Granted, you'll want a broader understanding of things, but the best way to get that is often to actually learn as many details as possible over the long term, not by watching teaser trailers and being told that's the whole plot.
A lot of people are just linking out to their favorite calc intro instead of commenting on this one. I like it so far, but I already did 2 years of calculus a few years ago, so I'm not learning much. I do appreciate the growing trend of presenting material in a more down-to-earth way, maybe with less-formal language and showing the reader it's not as scary as they might've thought previously. Kudos to the author, thi…
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…
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/...
A lot of people are just linking out to their favorite calc intro instead of commenting on this one. I like it so far, but I already did 2 years of calculus a few years ago, so I'm not learning much. I do appreciate the growing trend of presenting material in a more down-to-earth way, maybe with less-formal language and showing the reader it's not as scary as they might've thought previously. Kudos to the author, thi…
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…
A lot of people are just linking out to their favorite calc intro instead of commenting on this one. I like it so far, but I already did 2 years of calculus a few years ago, so I'm not learning much. I do appreciate the growing trend of presenting material in a more down-to-earth way, maybe with less-formal language and showing the reader it's not as scary as they might've thought previously. Kudos to the author, thi…
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…
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'll run up against walls if they try to use my book and only my book, but that would be the time to turn to Stewart (famous Calc text) or comparable. My thinking it that at that point the student is ready for "rigor" and "formality", and they won't even think twice about. They might even appreciate it. I've seen it happen over a decade of calculus teaching. It happens more than you think.
But to take this a little further, I believe the "formality" you mention actually hides a fundamental and insidious truth about mathematics: Mathematics fundamentally is informal. Burrow down deep enough into the epsilon/delta of limit definitions, and you'll see at the bottom is what amounts to an informal "this is good enough I guess".
For instance, at the bottom of epsilon/delta definition of what it means to converge in Baby Rudin (pg. 46), he essentially says "if you can get sequence within epsilon of the target anywhere past N" that's good enough. But why?! There is no more unpacking or additional fundamentalism at that point. How can we be sure we can make a claim about an infinite set of inequalities? Do if/then statements work this way? How can we be sure we can use the natural numbers this way? That fundamental informality then persists throughout the text. It's fine of course, and this is the agreed upon way to do mathematical calculus, but it's also a fundamental informality.
From my point of view (and this is part of what got me writing this book in the first place): 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.
I'm hardly the final arbiter on this matter. But that's my opinion.
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 be able to solve very basic toy problems. 2. Instead, learn basic algebra and combinatorics on extremely proficient level. This is what often is missing in US education.
In order to get to 2. 3. Learn how to do a. complex algebraic manipulations, b. solve complex algebraic inequalities, c. basics of number theory, d. combinatorics. Notice, nothing going beyond Real Numbers and I'm not even including Euclidean geometry.
4. Best sources for that are Math Olympiad problems and technique to solve them. You will learn how to crack extremely complicated algebraic expression, how to factor them and represent them in different forms, how to do tricky substitutions. Same technique is applicable in working with complicated integrals/diff. There is an entire layer of mathematics that devoted to inequalities and they are very applicable in solving calculus problems. Most of the technique and materials to solve those problems aren't taught in high schools and even college course.
Being able to solve moderately complex algebraic problems is must before learning calculus and analysis. Crush your ego, google/amazon for books and materials on how to solve (basic) Olympic problems that are intended for HS 9-12 graders and see what you can do.