9 ms·

Calculus For The People

396 points by tannrckb 7y ago

visarga 7y ago
Too little to learn from. I prefer the Chinese AI based curriculum, at least it makes an effort to find my blindspots and focus on them.
kstenerud 7y ago
"You might notice that when h is very close to 0, the slope of the line very closely matches the graph of f(x)" Huh? "and therefore, the slope of the line very closely matches the growth rate of the function as well." Growth rate? What's that? "Notice that as h gets close to zero, the secant line almost perfectly matches the growth of f at point A. " Not sure what this means... "For instance, in this situation we can study the limit of the slope of g when h tends to 0. As we can see, the limit of the slope of g as h tends to 0 is 4." Wait... where is this 4 coming from? "From this, we can conclude that the growth rate of the function f at x=2 is 4." What the hell is growth??? "Sometimes limits are obvious like this one" And now I give up.
- tannrckb 7y ago
  To find the slope of a line tangent to a point (x, f(x)) on a line, you can "draw" a secant line through two points (x, f(x)) and (x+h, f(x+h)). Then, identify the slope of the line passing through these two points. This gives an approximation of the slope of the tangent line passing through (x, f(x)). To get a more and more accurate approximation, you can look at what value the slope tends to as h approaches 0. So, (x+h, f(x+h)) gets closer and closer to (x, f(x)), the slope of the line passing through those two points tends closer to the tangent line passing through (x, f(x)). In other words, we are identifying the limit of the slope as h approaches 0. Based on the points of confusion you mentioned, I recommend a refresher on algebra. I think that will clear up your confusion
  - garmaine 7y ago
    I think he is pointing out that this “for the people” tutorial uses math jargon without introducing it.
- slumenta 7y ago
  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.”
  - kstenerud 7y ago
    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.
    - Sniffnoy 7y ago
      It's using a naive, informal notion of those. If you were to define it formally, well, you'd have the derivative. Which is what he does quite soon after. This is how definitions frequently work in mathematics -- they're meant to take some naive informal notion and formalize it, by coming up with a formal definition that matches how it should work. So, it's assuming you already have some informal notion of growth rate in your head, like being able to talk about the velocity of an object even when that velocity is not constant. (Imagine the x coordinate is time, and the y coordinate is position (we'll work in one spatial dimension here); then the "growth rate" is velocity.) Then it discusses how to define this formally.
      - kstenerud 7y ago
        So you're just drawing a line from start to end and calling that the velocity? That just averages the whole thing out, doesn't it? Unfortunately, I don't have an informal notion of growth rate in my head :/
        mac01021 7y ago
        Yes, that's right. The growth over any finite window, if you partition it into smaller windows, is the sum of the growth within each partition. Draw enough pictures and you'll develop the intuition that, if you keep partitioning smaller and smaller, you'll reach a point where the average growth rate across a partition is never going to change very much by subpartitioning further. If instantaneous growth rate is going to be defined at all, it has to be very close to the average rate over that tiny interval, no?
- gregpetrics 7y ago
  Good point. Thanks. I do think that limit is obvious and equal to 4. The point tends to 4. That said, I also agree the "growth rate" thing is coming in a little too quickly there. It's meant to foreshadow derivatives in the next chapter, but it seems like maybe it's introducing confusion to the reader. I went ahead and made some revisions to try to ease that connection of the "slope of a secant line" as an estimate of "growth rate" of a function. That said, no matter what I do, this is one of those "object equivalencies" in calculus that there's no way to really make for someone. At the end of the day "slope of secant line" and "growth rate" are two different objects that in the context of a mathematical model are equivalent, but in a mathematical vacuum, are not. I write about this a little bit at the end of the book here: https://www.geogebra.org/m/x39ys4d7#material/fxpkwpt7 https://www.geogebra.org/m/x39ys4d7#material/fxpkwpt7 Sadly, the resolution for you isn't really very concrete. To get another person to "learn" an object equivalence is a challenging thing. There's really only two options: 1. tell them. 2. put evidence in front of them and hope they make it themselves. I went for option 1 after sprinkling in a bit of option 2. I've tried to slow it down a bit more, but of course, every learner will be different on when they're ready to make this important connection. So at some point or another, this speed-bump needs to get hit. If you have more thoughts let me know!
dm3 7y ago
My favourite resource for the introduction to Calculus is "Calculus Made Easy" by Silvanus P. Thompson[0]. 0: http://www.gutenberg.org/files/33283/33283-pdf.pdf http://www.gutenberg.org/files/33283/33283-pdf.pdf
kstenerud 7y ago
A common failing of these "for the people" guides is that they fail the most basic UX test: User observation. When you build a UI, at some point you have to test it on actual people, observing them as they try to use it. Without fail, you'll discover a whole bunch of assumptions you'd made without even realizing it. It's only natural, since you've been working on this project for months and have intimate understanding that you've gained during your time of designs and rewrites and refactorings. But your user doesn't have that history, and you can't remember where common knowledge ends and your assumptions begin anymore. So you do observation tests to expose as many of these as you can. If you want to make a "for the people" instructional site, it's imperative that you offer an easy feedback mechanism so that people can instantly tell you when something confuses them. Simply relying on success stories exposes you to survivorship bias. Understanding is a two-way street. Design your medium with that in mind, and do LOTS of iterations with real people.
- maxmunzel 7y ago
  Actually I think that GeoGebra is by far the most intuitive and for-the-people that a math toolbox can be. When I picked it up in Highschool, I didn’t need to google a thing about it, contrary to the CAS Systems I use nowadays... (Maybe not a fair comparison)
  - kstenerud 7y ago
    Except that there's no "I don't understand" button where you can tell the author what you don't understand. So it's basically a one-way textbook with no feedback mechanism for the author to discover his assumptions and clarify them.
    - deleted 7y ago
      [deleted]
    - falcor84 7y ago
      What other popular resource has such a button?
- 7y ago
pugio 7y ago
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/dp/0465053734 https://www.amazon.com/Burn-Math-Class-Reinvent-Mathematics/...
- garmain 7y ago
  I agree
- rhizome31 7y ago
  Thank you for the recommendation. As an adult learning mathematics I'm half way through "Mathematics Rebooted" by Lara Alcock and I really like it so far. It's a good read to complement school books, video lectures and Khan Academy. Burn Math Class seems like a great candidate to be next on my math reading list.
- cgriswald 7y ago
  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.
  - pugio 7y ago
    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.
master_yoda_1 7y ago
For which people? Who does not go to school?
- vibrio 7y ago
  Do you believe school is sufficient?
  - master_yoda_1 7y ago
    Yes if you focus and don't waste time. Your whole life won't be sufficient if you can't focus and keep wasting your time in unrelated stuff like this shallow article.
    - vibrio 7y ago
      Serially commenting on a shallow article is sort of doubling down on wasted time, no?
- strikelaserclaw 7y ago
  " It was written for people who think they can't understand calculus." Even many people who went to school or are going to school think they can't understand calculus. I think most math from undergrad is understandable by a lot of people provided they get rid of "math fear" and put in the work.
exabrial 7y ago
Just curious, I really like the template used to build the site. Any idea what it is?
- gregpetrics 7y ago
  It's a "Geogebra Book." Get a free Geogebra account at geogebra.org, and get started writing.
  - HuangYuSan 7y ago
    Östareich oida
oneepic 7y ago
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, this is cool.
- dustfinger 7y ago
  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 terms. Similarly, those that would not be motivated to teach themselves the formal notation may read plenty of laymen explanations about mathematical theory, but also never be motivated to learn the formal notation. Consequently, a trend towards "presenting material in a more down-to-earth way" may might lead to a global average decline of amateur mathematicians with knowledge of formal notation. 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?
  - oneepic 7y ago
    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?
    - 7y ago
aj7 7y ago
“Most people think calculus is absolutely impossible no matter how hard they think.“ This is how people who KNOW calculus think. In fact, it is INDEED impossible for people weak in algebra and trigonometry, which is virtually the ENTIRE set of people who wash out of calculus.
bootlooped 7y ago
"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 derivatives and integrals.
- Retra 7y ago
  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 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.
  - bootlooped 7y ago
    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.
    - Retra 7y ago
      You don't have to talk about integrating higher degree polynomials, because a polynomial is just a sum of monomials, and you just end up integrating monomials over and over again. If you can integrate a single monomial, you can integrate all polynomials. That makes your objection seem weak, as though you don't know how to integrate, and you're just imagining that it is hard. That undermines your point because there are actually things in math that are hard, but you'd be woefully unprepared to understand them if you don't see the value in memorizing the absolutely trivial stuff. It's like a child saying "I don't want to be forced to memorize the shapes of the letters, because that's not what makes a good writer." Does a good writer sit around looking up letter shapes in a diagram all day because they can't be bothered to remember them? Either way, the optimal solution is the same: do the task over and over again until you are so familiar with it that you can recall it from memory. Memorization is a necessary part of learning.
newprint 7y ago
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 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.
- tacomonstrous 7y ago
  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.
  - gregpetrics 7y ago
    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.
sunstone 7y ago
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.
norswap 7y ago
Rifled through it, seems to be just the standard stuff. I was particularly disappointed by: > The bad news is that this is a little harder than using the Monkey Rules to calculate derivatives. In some sense the Monkey Rules, particularly the Quotient Rule and the Chain Rule, "blow functions up" when they systematically calculate derivatives. In order to go backwards, and undo the Monkey Rules to find antiderivatives, you need to think a bit like a forensic analyst who studies the site of an explosion to see what sort of bomb was used. We'll discuss this analogy more later when we practice finding antiderivatives. ("Monkey rules" are the derivation rules, this kind of cuteness is a big part of the purported dumbing down) Anyway, systematically calculating derivatives was always a big sticking point for, as indeed you often need to use multiple rules and it's not quite obvious which chaining of rules will get you there. I was hoping the authors could introduce a systematic algorithm (which no doubts exists but I never bothered looking up - I don't do much integrals day to day) or at least some strong form of intuition that goes beyond "if we did this we'd have something on which we could apply that rule".
- gregpetrics 7y ago
  Good feedback. I struggled with deciding if I should write activities that illustrate the full algorithm for derivatives and antiderivatives. At this time I left it out, but I do have the materials... The book was written with a bit of a promise to keep the algebra out, and overdoing it on Monkey Rules (derivatives) and Lucifer's Rules (antiderivatives) breaks that promise. That said, calculating derivatives and antiderivatives is the fundamental algebraic task of a calculus student. I'm thinking about your feedback right now... and will likely make adjustments in the near future to introduce optional tracks for extra practice on this.
  - norswap 7y ago
    That's great to hear :)

396 points by tannrckb 7y ago

Too little to learn from. I prefer the Chinese AI based curriculum, at least it makes an effort to find my blindspots and focus on them.

kstenerud 7y ago

"You might notice that when h is very close to 0, the slope of the line very closely matches the graph of f(x)" Huh? "and therefore, the slope of the line very closely matches the growth rate of the function as well." Growth rate? What's that? "Notice that as h gets close to zero, the secant line almost perfectly matches the growth of f at point A. " Not sure what this means... "For instance, in this situation we can study the limit of the slope of g when h tends to 0. As we can see, the limit of the slope of g as h tends to 0 is 4." Wait... where is this 4 coming from? "From this, we can conclude that the growth rate of the function f at x=2 is 4." What the hell is growth??? "Sometimes limits are obvious like this one" And now I give up.

tannrckb 7y ago
To find the slope of a line tangent to a point (x, f(x)) on a line, you can "draw" a secant line through two points (x, f(x)) and (x+h, f(x+h)). Then, identify the slope of the line passing through these two points. This gives an approximation of the slope of the tangent line passing through (x, f(x)). To get a more and more accurate approximation, you can look at what value the slope tends to as h approaches 0. So, (x+h, f(x+h)) gets closer and closer to (x, f(x)), the slope of the line passing through those two points tends closer to the tangent line passing through (x, f(x)). In other words, we are identifying the limit of the slope as h approaches 0. Based on the points of confusion you mentioned, I recommend a refresher on algebra. I think that will clear up your confusion
- garmaine 7y ago
  I think he is pointing out that this “for the people” tutorial uses math jargon without introducing it.
slumenta 7y ago
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.”
- kstenerud 7y ago
  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.
  - Sniffnoy 7y ago
    It's using a naive, informal notion of those. If you were to define it formally, well, you'd have the derivative. Which is what he does quite soon after. This is how definitions frequently work in mathematics -- they're meant to take some naive informal notion and formalize it, by coming up with a formal definition that matches how it should work. So, it's assuming you already have some informal notion of growth rate in your head, like being able to talk about the velocity of an object even when that velocity is not constant. (Imagine the x coordinate is time, and the y coordinate is position (we'll work in one spatial dimension here); then the "growth rate" is velocity.) Then it discusses how to define this formally.
    - kstenerud 7y ago
      So you're just drawing a line from start to end and calling that the velocity? That just averages the whole thing out, doesn't it? Unfortunately, I don't have an informal notion of growth rate in my head :/
      - mac01021 7y ago
        Yes, that's right. The growth over any finite window, if you partition it into smaller windows, is the sum of the growth within each partition. Draw enough pictures and you'll develop the intuition that, if you keep partitioning smaller and smaller, you'll reach a point where the average growth rate across a partition is never going to change very much by subpartitioning further. If instantaneous growth rate is going to be defined at all, it has to be very close to the average rate over that tiny interval, no?
gregpetrics 7y ago
Good point. Thanks. I do think that limit is obvious and equal to 4. The point tends to 4. That said, I also agree the "growth rate" thing is coming in a little too quickly there. It's meant to foreshadow derivatives in the next chapter, but it seems like maybe it's introducing confusion to the reader. I went ahead and made some revisions to try to ease that connection of the "slope of a secant line" as an estimate of "growth rate" of a function. That said, no matter what I do, this is one of those "object equivalencies" in calculus that there's no way to really make for someone. At the end of the day "slope of secant line" and "growth rate" are two different objects that in the context of a mathematical model are equivalent, but in a mathematical vacuum, are not. I write about this a little bit at the end of the book here: https://www.geogebra.org/m/x39ys4d7#material/fxpkwpt7 https://www.geogebra.org/m/x39ys4d7#material/fxpkwpt7 Sadly, the resolution for you isn't really very concrete. To get another person to "learn" an object equivalence is a challenging thing. There's really only two options: 1. tell them. 2. put evidence in front of them and hope they make it themselves. I went for option 1 after sprinkling in a bit of option 2. I've tried to slow it down a bit more, but of course, every learner will be different on when they're ready to make this important connection. So at some point or another, this speed-bump needs to get hit. If you have more thoughts let me know!

dm3 7y ago

My favourite resource for the introduction to Calculus is "Calculus Made Easy" by Silvanus P. Thompson[0]. 0: http://www.gutenberg.org/files/33283/33283-pdf.pdf http://www.gutenberg.org/files/33283/33283-pdf.pdf

kstenerud 7y ago

A common failing of these "for the people" guides is that they fail the most basic UX test: User observation. When you build a UI, at some point you have to test it on actual people, observing them as they try to use it. Without fail, you'll discover a whole bunch of assumptions you'd made without even realizing it. It's only natural, since you've been working on this project for months and have intimate understanding that you've gained during your time of designs and rewrites and refactorings. But your user doesn't have that history, and you can't remember where common knowledge ends and your assumptions begin anymore. So you do observation tests to expose as many of these as you can. If you want to make a "for the people" instructional site, it's imperative that you offer an easy feedback mechanism so that people can instantly tell you when something confuses them. Simply relying on success stories exposes you to survivorship bias. Understanding is a two-way street. Design your medium with that in mind, and do LOTS of iterations with real people.

maxmunzel 7y ago
Actually I think that GeoGebra is by far the most intuitive and for-the-people that a math toolbox can be. When I picked it up in Highschool, I didn’t need to google a thing about it, contrary to the CAS Systems I use nowadays... (Maybe not a fair comparison)
- kstenerud 7y ago
  Except that there's no "I don't understand" button where you can tell the author what you don't understand. So it's basically a one-way textbook with no feedback mechanism for the author to discover his assumptions and clarify them.
  - deleted 7y ago
    [deleted]
  - falcor84 7y ago
    What other popular resource has such a button?
7y ago

pugio 7y ago

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/dp/0465053734 https://www.amazon.com/Burn-Math-Class-Reinvent-Mathematics/...

garmain 7y ago
I agree
rhizome31 7y ago
Thank you for the recommendation. As an adult learning mathematics I'm half way through "Mathematics Rebooted" by Lara Alcock and I really like it so far. It's a good read to complement school books, video lectures and Khan Academy. Burn Math Class seems like a great candidate to be next on my math reading list.
cgriswald 7y ago
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.
- pugio 7y ago
  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.

master_yoda_1 7y ago

For which people? Who does not go to school?

vibrio 7y ago
Do you believe school is sufficient?
- master_yoda_1 7y ago
  Yes if you focus and don't waste time. Your whole life won't be sufficient if you can't focus and keep wasting your time in unrelated stuff like this shallow article.
  - vibrio 7y ago
    Serially commenting on a shallow article is sort of doubling down on wasted time, no?
strikelaserclaw 7y ago
" It was written for people who think they can't understand calculus." Even many people who went to school or are going to school think they can't understand calculus. I think most math from undergrad is understandable by a lot of people provided they get rid of "math fear" and put in the work.

exabrial 7y ago

Just curious, I really like the template used to build the site. Any idea what it is?

gregpetrics 7y ago
It's a "Geogebra Book." Get a free Geogebra account at geogebra.org, and get started writing.
- HuangYuSan 7y ago
  Östareich oida

oneepic 7y ago

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, this is cool.

dustfinger 7y ago
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 terms. Similarly, those that would not be motivated to teach themselves the formal notation may read plenty of laymen explanations about mathematical theory, but also never be motivated to learn the formal notation. Consequently, a trend towards "presenting material in a more down-to-earth way" may might lead to a global average decline of amateur mathematicians with knowledge of formal notation. 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?
- oneepic 7y ago
  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?
  - 7y ago

aj7 7y ago

“Most people think calculus is absolutely impossible no matter how hard they think.“ This is how people who KNOW calculus think. In fact, it is INDEED impossible for people weak in algebra and trigonometry, which is virtually the ENTIRE set of people who wash out of calculus.

bootlooped 7y ago

"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 derivatives and integrals.

Retra 7y ago
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 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.
- bootlooped 7y ago
  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.
  - Retra 7y ago
    You don't have to talk about integrating higher degree polynomials, because a polynomial is just a sum of monomials, and you just end up integrating monomials over and over again. If you can integrate a single monomial, you can integrate all polynomials. That makes your objection seem weak, as though you don't know how to integrate, and you're just imagining that it is hard. That undermines your point because there are actually things in math that are hard, but you'd be woefully unprepared to understand them if you don't see the value in memorizing the absolutely trivial stuff. It's like a child saying "I don't want to be forced to memorize the shapes of the letters, because that's not what makes a good writer." Does a good writer sit around looking up letter shapes in a diagram all day because they can't be bothered to remember them? Either way, the optimal solution is the same: do the task over and over again until you are so familiar with it that you can recall it from memory. Memorization is a necessary part of learning.

newprint 7y ago

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 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.

tacomonstrous 7y ago
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.
- gregpetrics 7y ago
  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.

sunstone 7y ago

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.

norswap 7y ago

Rifled through it, seems to be just the standard stuff. I was particularly disappointed by: > The bad news is that this is a little harder than using the Monkey Rules to calculate derivatives. In some sense the Monkey Rules, particularly the Quotient Rule and the Chain Rule, "blow functions up" when they systematically calculate derivatives. In order to go backwards, and undo the Monkey Rules to find antiderivatives, you need to think a bit like a forensic analyst who studies the site of an explosion to see what sort of bomb was used. We'll discuss this analogy more later when we practice finding antiderivatives. ("Monkey rules" are the derivation rules, this kind of cuteness is a big part of the purported dumbing down) Anyway, systematically calculating derivatives was always a big sticking point for, as indeed you often need to use multiple rules and it's not quite obvious which chaining of rules will get you there. I was hoping the authors could introduce a systematic algorithm (which no doubts exists but I never bothered looking up - I don't do much integrals day to day) or at least some strong form of intuition that goes beyond "if we did this we'd have something on which we could apply that rule".

gregpetrics 7y ago
Good feedback. I struggled with deciding if I should write activities that illustrate the full algorithm for derivatives and antiderivatives. At this time I left it out, but I do have the materials... The book was written with a bit of a promise to keep the algebra out, and overdoing it on Monkey Rules (derivatives) and Lucifer's Rules (antiderivatives) breaks that promise. That said, calculating derivatives and antiderivatives is the fundamental algebraic task of a calculus student. I'm thinking about your feedback right now... and will likely make adjustments in the near future to introduce optional tracks for extra practice on this.
- norswap 7y ago
  That's great to hear :)