
Calculus For The People - tannrckb
https://www.geogebra.org/m/x39ys4d7#material/phuyhqtw
======
newprint
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.

~~~
btilly
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 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 master the subject? Heck no!

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.)

~~~
phaedrus
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_.

~~~
jacobolus
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](http://www.math.smith.edu/~callahan/intromine.html)

------
pugio
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/...](https://www.amazon.com/Burn-Math-Class-Reinvent-
Mathematics/dp/0465053734)

~~~
cgriswald
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
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.

~~~
lonelappde
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?

------
kstenerud
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
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
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.

~~~
falcor84
What other popular resource has such a button?

------
oneepic
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
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?

~~~
gregpetrics
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'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.

~~~
rrss
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."

~~~
gregpetrics
Whoops. Page 47. I paraphrase: Sequences "converge" if there exists an N such
that the sequence stays within epsilon of p (the mark) for all indexes larger
than N.

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.

~~~
lonelappde
Most US schools teach formality in Geometry with 2-column mechanical proofs,
following Euclid. Formal and informal belong side by side throughout the
curriculum.

------
kstenerud
"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.

~~~
slumenta
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
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
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
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 :/

~~~
Sniffnoy
> 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?

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.

------
dm3
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)

------
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.

------
norswap
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
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
That's great to hear :)

------
aj7
“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
"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
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
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
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.

------
visarga
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.

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

~~~
gregpetrics
It's a "Geogebra Book." Get a free Geogebra account at geogebra.org, and get
started writing.

~~~
HuangYuSan
Östareich oida

------
master_yoda_1
For which people? Who does not go to school?

~~~
vibrio
Do you believe school is sufficient?

~~~
master_yoda_1
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
Serially commenting on a shallow article is sort of doubling down on wasted
time, no?

