Calculus For The People
geogebra.org
geogebra.org
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.
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.
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.
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.
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 useful, because they make you think out of box, they often times involve several branches of math in one problem, such as number theory problems go in hand with combinatorics; they don't require complex mathematical machinery, and technique of solving problems, directly translates to solving complex problems in analysis/abstract algebra.
I frequently see topics similar to this one popping-up on HM and a lot of people interested in learning or re-learning math as an adults. Excellent ! So let them learn discreet mathematics by solving problems in algebra, combinatorics and number theory. Once they are comfortable, they can move on to more abstract subjects. My view is very similar to Concrete Mathematics, by Knuth, Patashnik, Graham. Their "concreteness" is very down to earth: combinatorics, number theory and few other things in the mix.
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 connected to approximations.
4. The derivative.
5. The easiest formulas for differentiation and why they are true. (All handwavy, heuristic big-O arguments.)
6. That all possible max/min points can be found at the boundaries, or by finding where the derivative is 0 or non-existent.
7. The Fundamental Theorem of Calculus aka why areas are the reverse of derivatives.
8. The advice that if he had to actually calculate a derivative or integral, he should use a program like MAPLE.
Did he have to review his notes a bunch of times so it stuck? Of course!
But he went on to ace the course. And my guess is that he understood what makes Calculus tick better than most who took the course. (Sanity check. If you do not understand why the tangent line and derivative are connected, then you do not understand Calculus.)
How you did that introduction is very important. Every time you tell something new, it better be really small - or explained quickly and well, so the concept would stick before brain would get tired.
Thank you for listing your points.
The result is that I was able to break my exposition into one piece at a time, starting with ideas that were already accessible. More specifically I started with the idea that f(x) = approx(x) + error(x) where we want approx simple and error small. This motivates a language for describing what "small error" means, which motivates little-o for polynomials.
That's enough to do tangent lines and answer questions like, "Given the equation for position vs time, how fast is the rock going when it hits the bottom?"
Only after he could calculate tangent lines did I introduce the derivative.
Contrast to the usual approach where limits are an abstract concept with no obvious application, and then the derivative is introduced. It actually combines several ideas jumbled together. That's a big mental knot that almost nobody gets. And explaining it more carefully doesn't help because people keep getting to holding too many unintegrated ideas in their head at once.
And seriously, heuristic arguments that you can reproduce whenever you forget them are good. Take the product rule.
f(x0+h) = f(x0) + f'(x0) h + o(h)
g(x0+h) = g(x0) + g'(x0) h + o(h)
(f*g)(x0+h)
= (f(x0) + f'(x0) h + o(h)) * (g(x0) + g'(x0) h + o(h))
= f(x0) * g(x0) + f(x0) * g'(x0) h + f(x0) * o(h) +
f'(x0) h * g(x0) + f'(x0) h * g'(x0) h + f'(x0) h * o(h) +
o(h) * g(x0) + o(h) * g'(x0) h + o(h) * o(h)
= f(x0) * g(x0) + f(x0) * g'(x0) h + o(h) +
f'(x0) * g(x0) h + o(h) + o(h) +
o(h) + o(h) + o(h)
= f(x0) * g(x0) + (f(x0)*g'(x0) + f'(x0)*g(x0)) h + o(h)
I had him do that calculation with only minimal prompting. And I think that this is a calculation that I'd like every Calculus student to be able to do on demand. If you forget the rule and you know that argument, you can figure it out again.
I wish it surprised me. But I suspect that your experience is more normal than not.
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
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!
Or do you mean that you never connected "derivative and integral are inverses" with "integrals are areas" and "inverses are opposites"?
[1] https://www.amazon.com/Burn-Math-Class-Reinvent-Mathematics/...
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.
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.
In the meantime, I'm thinking of how I might hack it with what I have now by just using URLs attached to text like "Help me! I don't understand!" to out-of-book auxiliary resources to provide another perspective on some of the slipperiest topics (like "growth rate" vis-a-vis derivatives).
I really would like to know where people stumble with these activities... and then fix it. So that feature would be really helpful. Or if someone wants to copy this and change it themselves, go for it! I put this up on Geogebra because it is a fully open platform.
Which brings me to another reason for going with Geogebra: your point.
Geogebra books are a nice balance of free hosting, wide distribution, and a reasonably friendly UX format. That said, it's hardly perfect. But it offers me what I think is my best chance to fail fast, get feedback, and make changes with the approach to teaching calculus that I present in this book. But I want to fully acknowledge your point regarding UX. I absolutely agree. The "Geogebra Book" format has its limitations and introduced new assumptions along the way that make it harder for users. If nothing else, it's A BOOK. There is an inherent "one-way-ness" to it. I don't see that being as much of an issue as you, but I agree some of it needs to be broken down. How else will revisions make it into the book? That's the whole point of it being open.
As a sidenote: I also want to point out that I have tested these activities as best I can in a variety of LMSs and in a variety of classrooms from Ivy League students gunning for top marks to adult learners who "can't do math" and gave up on Calculus 20 years ago, but due to some reason or another now need to know it. This book is sort of the "least common denominator" of all these testings, and is itself a testing.
So... you're right! Thanks. Want to work together to test it more?
I've already changed it just from reading these comments, and I'd be really interested in continued revisions!
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.
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?
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.
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.
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.
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.
> 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?
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."
I know it seems formal because it adheres to a certain structure, but even this is informal at a fundamental level.
Specifically, how can we be sure we can perform a countably infinite number of distance measurements in the metric space to be sure the sequence stays close to p? (this is the infinite stack of inequalities I alluded to)
He doesn't say. Implicitly, Rudin is saying here that this definition of "converges" is good enough. And he's not wrong. It is a very good definition. To me at least this is Rudin, the towering statue of formality, being informal.
He could/should have actually gone down to a more fundamental level and whipped out mathematical induction as an axiom to assure us that we can do such things, but then that would have taken him off his narrative goal, and also probably lost even more readers. Furthermore, even if he did so, an axiom is an assertion that "you just have to trust me on this one."
Now look, I'm not bashing formality. I'm a huge fan of it, and teach upper level math classes formally. But it has it's place and it is NOT in Calculus 1. Furthermore, I think folks need to realize that even the most formal of treatises have informalities buried in them at the very least in the form of stated axioms.
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.
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
“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.”
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.
Unfortunately, I don't have an informal notion of growth rate in my head :/
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?
Common person or engineer only needs to accept/trust that point in a curve has well defined 'slope' and it's not an approximation.
No. We're talking about instantaneous velocity. You know, the thing the speedometer displays. How fast is the car moving at any given moment? Like, a car doesn't need to be moving at constant speed for a speedometer to give meaningful information, right? Sometimes it is moving faster and sometimes it is moving slower. Sometimes it is moving at a rate such that if it stayed at that rate it would go 60 miles in an hour, and sometimes it is moving at a rate such that if it stayed at that rate it would go 30 miles in an hour. This is the informal notion of instantaneous velocity you should already have. Now the question becomes, how do we formalize this? Which is what the page is trying to answer.
Actually, what does growth rate even mean?
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.
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
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!
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".
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.
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.
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.
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.
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.