
Calculus Learning Guide - fauria
http://betterexplained.com/guides/calculus/
======
nextos
Apart from Keisler, mentioned there, I think there are 2 lovely and rigurous
references especially appropriate for anyone with a CS background:

* Henle & Kleinberg [1]

* Hubbard & Hubbard [2]

Both are breathtakingly beautiful. Henle & Kleinberg is small and neat.
Hubbard & Hubbard integrates calculus and linear algebra. In some ways it
reminds me of SICP, described by Peter Norvig as "[...] a rich framework onto
which you can add new learning over a career". [3]

[1]
[https://books.google.co.uk/books/about/Infinitesimal_Calculu...](https://books.google.co.uk/books/about/Infinitesimal_Calculus.html?id=nalhCn80D08C&redir_esc=y)

[2]
[http://matrixeditions.com/5thUnifiedApproach.html](http://matrixeditions.com/5thUnifiedApproach.html)

[3]
[http://www.amazon.com/review/R403HR4VL71K8](http://www.amazon.com/review/R403HR4VL71K8)

------
alexc05
This is fantastic!

After nearly 20 years as a self-taught programmer I've been _thinking_ about
going back to school to get a CS degree (or master's)

Since I haven't taken linear algebra or calculus those are some pretty big
areas of worry for me.

I've been looking for some good resources on the two. I started watching the
video for year one calculus from MIT open courseware (specifically this one:
[http://ocw.mit.edu/courses/mathematics/18-01-single-
variable...](http://ocw.mit.edu/courses/mathematics/18-01-single-variable-
calculus-fall-2006/video-lectures/lecture-1-derivatives/)) ... and was so lost
I had to back out and google "what is calculus?"

I'm pretty basic here :)

Turns out it's the math of "measuring change"

A colleague pointed me to Kahn academy as well, but this is a welcome
resource. Thanks.

~~~
kalid
Kalid from BetterExplained here, glad you like it!

I had similar confusions; despite years of engineering classes, I didn't build
an intuition for Calculus until a decade later. It felt like such a waste.
(Yes, I could memorize rules; but could I visualize the product and quotient
rule?)

I decided to Elon Musk It™ and reason from first principles
([http://betterexplained.com/articles/honest-
learning/](http://betterexplained.com/articles/honest-learning/)).

What would a learning plan look like, assuming limited motivation, a need for
genuine understanding, and goal of starting with appreciation and moving to
proficiency?

Just like learning music, I'd listen to the song, then clap along to it, then
play a few chords along with the band, then learn a bit of music notation, and
then practice playing it. I wouldn't start with scales (er... limits) for
weeks on end. Not if I was honest about my motivations.

Ultimately, we have to acknowledge the approach that helps us. For me,
investing a few hours of exploration before the formalities is well worth it.

~~~
petra
Kalid, thanks for your amazing content!

Reading this from the link you provided:"In my ideal world, every Wikipedia
topic would have a guide that took you from the 1-minute version to a full
technical understanding. Go as far as you wish, make meaningful progress at
each step, and have fun along the way.".

Have you though how to build/scale it ? Because it would seem like an amazing
site, and maybe a good startup.

~~~
defen
> "In my ideal world, every Wikipedia topic would have a guide that took you
> from the 1-minute version to a full technical understanding. Go as far as
> you wish, make meaningful progress at each step, and have fun along the
> way."

This is my biggest gripe with Wikipedia for mathematical topics - as it
stands, it's only useful if you already know the material, in which case you
probably shouldn't be using wikipedia as a reference / refresher except as a
last resort.

~~~
kalid
Exactly. Wikipedia helps me remember things I've forgotten, but it's slow
going when trying to learn a new topic. The first paragraph includes a set of
links that you recursively follow.

------
dandermotj
The more advanced mathematics I learn, the more I realise that in many cases I
do not have a fundamental understanding of many topics, which leads me back to
the beginning. More often than not, Better Explained is at the end of that
path. If only _this_ was how mathematics was taught!

~~~
k__
I never got calculus.

I mean, I got the ideas behind it and could talk with my professors about it,
but I couldn't solve much of these equations.

It was a strange time at university...

I even helped friends to get these exams done, but I myself always failed.
Luckily we only had one math lecture about calculus, the 4 others were about
different stuff.

And it was always the fundamentals which got me. The whole rearranging an
equation stuff all the school life is about totally eluded me. And this is
what the calculus exams were about. At least the professors saw that I got the
other math exams right, so they knew I wasn't an imbecile, haha

~~~
dandermotj
I am never comfortable with my understanding of a topic until I can teach it
(or at least pass it on). There's just always an annoying, niggling feeling
that if I were to be faced with a student who kept asking "why", I'd be a
bumbling mess. I was the same as you for linear algebra. Everyone was always
so happy to learn the rules, answer the questions like its a script and pass
the exam. This frustrated me immensely. Sat down last year and started from
the beginning, and I'm glad I did. Sometimes the key is to wipe the slate of
existing knowledge clean and start again!

~~~
martylookalike
I definitely had similar issues with learning math when I was in school. You
mention that you sat down last year and started from the beginning... would
you mind sharing a little more about where you started, that is what your
beginning was, and what sorts of resources you used? I'm thinking of doing
something similar and am looking for avenues to take.

~~~
dandermotj
Well funnily enough I started back at what a matrix really is. What does it
mean to multiply them, add them, invert them and so on. The first good
resource I used was Better Explained! I also took out some Linear Algebra text
books from the library. First was Linear Algebra for Dummies (don't knock it!)
and then an undergraduate text book. I've still a ways to go but it was a good
start.

------
wyattpeak
I love attempts to shake up maths education, and I reckon the first passage is
a pretty neat explanation, but it seems to slide quickly downhill from there.

The first three chapters are a bit airy-fairy - I get that that's its schtick,
but they're also pretty light on content.

The fourth reads like it's been taken straight out of a maths textbook by a
"cool" teacher - engaging but not fundamentally simpler than what one learns
in class.

The fifth teaches how to use Wolfram Alpha to perform calculus. I could teach
a dog to use Wolfram Alpha.

It all seems to suffer a bit from the "Dummies guide" syndrome - it's easy to
simplify the first few lessons on a subject, but you pretty soon hit a wall
when you reach the parts that actually require a bit of mental legwork.

Calculus is not a simple subject. By teaching it better, you can probably
improve the situation by, like, 20%, but you can't magically make a
complicated subject trivial. It just doesn't seem like the way forward for
education. We need to make the difficulty more rewarding, not pretend that it
doesn't exist.

~~~
jasode
_> It all seems to suffer a bit from the "Dummies guide" syndrome _

I think your criticism is misguided. His tutorial is obviously meant to make
the _introduction_ to the topic _more accessible_. He does that by using
pictures and diagrams to build intuitions. He can do that because he
extensively uses drawing tools to help the learner "visualize" what Calculus
is about.

Compare that with typical calculus teacher with a whiteboard. Because the
teacher's drawing skills are limited, he/she dives right into the delta-
epsilon limit definition: lim h --> 0 for f(x+h)-f(h)/h

Yeah there's some hand drawing of tangent lines to converge on a
"instantaneous change" but that's not enough for a lot of students to "get"
Calculus. The symbolic _notation_ is demonstrated more often than pictures. It
therefore devolves into mindless "plug & chug" for the rest of the semester.

As for the Wolfram Alpha comment, the idea is about "immersion". There are
other tools besides pencil-&-paper that also "do" Calculus and they can be
used to verify self-study Q&A.

Likewise, if I teach Differential Equations to programmers, it's helpful if I
"immerse" them into other tools to numerically solve them such as MS Excel,
MATLAB, Wolfram Alpha, C/C++ code, etc.

His BetterExplained doesn't replace a rigorous Calculus textbook but it can
help with the "aha!" moment to understand what the dry Calc textbook is trying
to teach.

~~~
wyattpeak
>His tutorial is obviously meant to make the introduction to the topic more
accessible.

And I'm saying he doesn't much succeed at that. After the first few chapters,
the content is either trivial or very similar to what you'd find in a
textbook.

>As for the Wolfram Alpha comment, the idea is about "immersion".

You could be right, but it seems like a generous appraisal. It reads to me
more like an attempt to brush over the fact that the student doesn't actually
know any calculus yet.

If your intent is to teach that calculus can be performed intuitively, teach
that - don't teach how you can phrase your question to get a computer to solve
them.

~~~
jasode
_> It reads to me more like an attempt to brush over the fact that the student
doesn't actually know any calculus yet. ... - don't teach how you can phrase
your question to get a computer to solve them._

No, he's _not_ showing WolframAlpha as a _mental shortcut_ to cover up
deficiencies of not knowing calculus.

The point is to utilize it as a _diagnostic tool_ to check the student's
understanding of previous Ch 1 - 4. If the student understands the previous
chapters' vocabulary and concepts, he "doublechecks" that understanding by
seeing if he can "describe" problems to WolframAlpha. Again, the idea is to
create a feedback loop _for the self-directed learner_.

Instead of using WolframAlpha, one could be shown how to use the TI-83
graphing calculator for the same purpose. However, since the student is likely
reading Betterexplained on the web, he can just conveniently click over to
wolframalpha.com and "check his calculus knowledge" there. As a bonus,
WolframAlpha is easier to use, draws prettier graphs, and doesn't cost $100.

------
wolfgke
To quote this site: "Review the intuitive definition. Rephrase technical
definitions in terms that make sense to you."

I openly admit that the technical definitions are often how I imagined the
material intuitively when the material was taught in school. But first when
studying mathematics these definitions occured. I really wish that calculus
(and the rest of mathematics) would be taught in a much more abstract way in
school - it really makes understanding the material much easier since the
"intuitive" definitions suffer a lot from being leaky abstractions (and
mathematics is all about banging to the boundaries of the definitions until
they move or fall down).

~~~
jawilson2
I'm the opposite (which, I admit, is backwards for a lot of engineers.) I need
to have a high-level view to provide a context for the details. It doesn't
help me to spend a week on defining and proving integrals without first
understanding that the goal is to find the area of a rectangle or circle, or
to track planetary motion. Once I see the problem, I can think about how we
arrived at the solution, rather than present it as a solution in search of a
problem. Newton wanted to describe planetary motion, so he invented calculus
(Leibniz too, etc.) to solve it, not the other way around.

~~~
wolfgke
> It doesn't help me to spend a week on defining and proving integrals without
> first understanding that the goal is to find the area of a rectangle or
> circle

I will only discuss this point (but for your other points similar arguments
can be made). So you want to measure some n-dimensional volume? This is a
highly non-trivial problem, since it is quite possible to find subsets of R^n
where you can't assign volumes in a sensible way:

>
> [https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox](https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox)

If you don't care about this subtlety you end up in such a paradox.

So you think hard about how you can define "measureable" in an accurate sense
and come up with the definition of a sigma algebra and think about how you can
even define measures (say of volume) on a sigma algebra. Because these can be
ugly to describe, you will immediately think about how one can describe the
sigma algebra and the measure in a more easy way. So you come up with
generators and note that the measure is defined uniquely by its value on the
generator.

Doing this process on the real line using the Borel sigma algebra/measure is
not that complicated (so you solved the problem for the real case). From here
it is only a small step to the Lebesgue-Stieltjes integral in R^1 that is
really worth doing.

But now you remember that you wanted to compute higher-dimensional volumes. So
you ask how can you recycle what you've already done? The answer is obvious:
You define product sigma algebras and product measures.

As you have seen (Banach-Tarski_paradox) all this is necessary just to even
define volumes. I really know no easier way that is both useful in practise
(at least be able to obtain the volume of spheres (i.e. not just "simple
objects as simplices")) and avoids the problem that was laid out by Banach and
Tarski-

~~~
yaks_hairbrush
I don't like this attitude at all. Euclid, Archimedes, Newton, Leibniz,
Riemann, and many others did a fine job correctly calculating areas, volumes,
and so on without need of measure theory.

Why is it so important to avoid the problem laid out by Banach-Tarski on the
first go through the concept? We, as a species, did not even realize such a
problem existed until about 1920, and yet we still managed to build stuff
based on what we knew.

~~~
wolfgke
> I don't like this attitude at all. Euclid, Archimedes, Newton, Leibniz,
> Riemann, and many others did a fine job correctly calculating areas,
> volumes, and so on without need of measure theory.

The history of mathematics is a history of ever-increasing standards for
proofs. "Proofs" that, say, Euler wrote down that were perfectly valid for the
standards of their time would probably not accepted in todays standard. I
personally also think it's plausible that today's proof standard in "typical"
math papers will not be acceptable in 100 years anymore, when they perhaps
will additionally have to be computer-checkable (and computer-checked).

~~~
yaks_hairbrush
The issue of proof is very different from the issue of learning. I thought it
was the latter being discussed.

~~~
wolfgke
Mathematics is about proofs. Learning mathematics thus means learning about
proofs.

~~~
yaks_hairbrush
I do not agree with your first statement. Mathematics is about developing and
studying tools to solve problems. Proofs are only one aspect of that.

~~~
wolfgke
> Mathematics is about developing and studying tools to solve problems.

There clearly exist methods for solving problems that have nothing to do with
mathematics. This should falsify your thesis.

~~~
possibility
I think maybe you made a mistake, because the exchange reads like this:

A: Knives are for eating.

B: Eating is only a part of it. Knives are for handling food.

A: You can handle food with a fork. Your claim is false.

Specifically, "math is for solving problems" does not imply "problems are only
solved by math".

------
neovive
Has anyone completed the Calculus courses on Khan Academy (Integral,
Differential and Multivariable)? If so, would you recommend Khan Academy for
Calculus instead of the resources mentioned on Better Explained? I'm working
through the Linear Algebra courses now on KA and they are very good.

~~~
jasode
I haven't seen the Khan Academy Calculus videos that require signup but my
answer assumes it is similar to (or exactly the same as) the youtube videos
presented by Salman Khan.

If you're already making great progress from Khan vids for Linear Algebra,
you'll probably have the same success with Khan's Calculus.

S Khan's approach is friendly but it's also very similar to a traditional
teacher using a whiteboard in a classroom. For example, S Khan dives right
into the formal concepts of "functions" and "limits" and all the associated
notation. This mirrors how a professor would present it. In fact, Salman often
remarks that he has his "college textbook" open as a resource to order the
sequence of topics and demonstrate examples.

Kalid Azad's approach with BetterExplained is to present images, pictures, and
analogies that's not found in typical Calculus textbooks. This helps many
learners get past the "mystery" of what the topic is about. You'll notice that
the first chapters of Azad's approach does not discuss "limits"[1]. If you're
already doing fine with Linear Algebra, you may not need all the pretty
pictures as a gentle introduction.

[1] The idea of "limits" is just one example of a philosophical difference on
how to teach calculus. Isaac Newton didn't use the rigorous concept of
"limits" when he invented calculus. His peers derided his approach as "ghosts
of departed quantities". It was the later mathematicians who refined calculus
to be based on more rigorous foundations (Real Analysis) that "limits" was
retrofitted onto calculus. It's interesting that we now think the best
pedagogy for high school students is to force "limits" on them to learn
calculus because ... It's More Rigorous. Unfortunately, "limits" are not
intuitive.

~~~
btilly
Your history of Calculus is a little..muddled.

Newton's approach was based on the idea that rate of change was a fundamental
quantity. This lead to fluxions. His notation for Calculus was x with a dot
over it. He used his method as a rough notation to figure out what the answers
should be, before coming up with pure geometric constructions to use in the
Principia Mathematica.

Inspired by the Principia, Leibniz reinvented Calculus with the idea of
infinitesmals. Infinitesmals are nonzero numbers that are smaller than any
real number. He published, which started a priority dispute between him and
Newton.

Leibniz gave us our dy/dx notation. In the original, d is an operator meaning
"look an infinitesmal away". Literally d(y) = y(x+dx)-y(x). (This converges
faster if you make it y(x+dx/2)-y(x-dx/2)...) You do the calculation, then
drop any infinitesmals and get your answer. Ever wondered why the second
derivative is d^2y/dx^2? That's because you do d(d(y))/d(x)d(x).

The famous quote of "ghosts of departed quantities" was written by George
Berkeley as a criticism of infinitesmals. Far from being a peer of Newton's,
he was a bishop who was demonstrating that science had leaps of faith as big
as any in religion. His criticisms were accurate.

The fact that math's foundations were crap occasionally inspired effort. But
not much. Lagrange's contribution was to view everything as a power series,
and do formal derivatives that way. To him we owe our f' notation.

Then Joseph Fourier upended the whole apple cart with Fourier series. He took
infinite sums of nice functions like sin and cos, and wound up with a step
function. What happens if your step is an infinitesmal away? This was a
crisis.

Cauchy addressed this by making infinitesmals sequences that go to 0. This
approach failed for the chain rule when the derivative is 0, but that's where
the idea of Cauchy sequences comes from.

Weirstrauss in the 1870s came up with the modern limit approach. Mathematics
was (very literally) rewritten.

Since then limits have proven to be a horrible pedagogical tool, but that is
how we teach. However there are serious pedagogical suggestions ranging from
skipping over the justification for Calculus to using infinitesmals (turns out
you _can_ make precise sense of them with modern set theory) to writing the
tangent line approximation with little-o notation. (The last suggestion is
from Knuth! See [https://micromath.wordpress.com/2008/04/14/donald-knuth-
calc...](https://micromath.wordpress.com/2008/04/14/donald-knuth-calculus-via-
o-notation/) for details.)

Sorry for the history lesson, I love this topic. :-)

~~~
Chris2048
Is there any forum for discussing this type of thing?

I have some questions wrt issue like this, but you ask on a place like
MathOverflow and you often get unintuitive answers, as if for the working
mathematician, rather than explanations.

For example, I've always though of dy as intuitively involving some kinf of,
higher-function over another;

d[y(x), a] = y(x+a) - y(x)

such that;

if I(x) = x so that d[I(x), a] = d[x, a]

and d[x, a] = a

then d[y(x), a]/d[x, a] = d[y(x), d[x, a]]/d[x, a]

so we drop reference to a (or lim as a->0) and = dy(x)/dx = dy/dx

But look, there's an implicit 'a' or 'dx' in that y(x), plus y(x) is often
written just 'y' so that you can always switch around which the free variable
is (an equation just describes the relationship, e.g. y = 2x can generate
either y(x) = 2 * x or x(y) = y/2)

Also, since the imagined HO function d[f, a] actually takes a function for the
first arg, there needs to be a implicit identity function for dx, so really
df(x)/dh(x) means "rate of change of f, for change in x, _relative to_
(normalised) rate of change in h, for change in x". Then the identity function
becomes the default such that dx is short for dI(x).

Maybe this descibes the same idea:

[https://en.wikipedia.org/wiki/Differential_operator](https://en.wikipedia.org/wiki/Differential_operator)
[https://en.wikipedia.org/wiki/Derivative#The_derivative_as_a...](https://en.wikipedia.org/wiki/Derivative#The_derivative_as_a_function)

But its a little undecipherable to me...

~~~
btilly
Not that I know of.

I learned it during grad school because it was more fun than the stuff I was
officially learning.

Your intuitive notion is not that different from the original infinitesmal
notation.

What Wikipedia is describing is that "derivative of" is an operator. Meaning
it takes functions and returns other functions. So D(x^2) = 2x. Which is not
exactly the same thing as what you wanted, but it is related.

When you go on in math, you learn that there are a million ways to think about
things. OK, dozens at least. As an illustration Thurston listed a bunch for
derivative in
[http://www.math.toronto.edu/mccann/199/thurston.pdf](http://www.math.toronto.edu/mccann/199/thurston.pdf).
I strongly suspect that this is a real list he maintained, and he didn't jump
from definition 7 to 37 without filling in 29 definitions in the middle.

But when you listen to mathematicians, the odds are that they aren't picking
any way of thinking about things that you would have thought of. And maybe
they DID try your way, then found it didn't work because X, Y and Z. Or maybe
they did try it, but moved on and everyone has forgotten it. Figuring out
which of these is true can be legitimately hard, and doesn't seem to interest
most mathematicians.

But if you find a good forum, let me know. It might be fun for me to visit.
:-)

------
mathgenius
I recently started reading Amir Alexander's 2014 book "Infinitesimal: How a
Dangerous Mathematical Theory Shaped the Modern World". It's a really great
read, this is pre-Newton, pre-Leibniz. People were doing calculus more than
two hundred years before those guys! They struggled with paradoxes to do with
infinitely many zero length points making up a line, and the Jesuits just
outright banned that stuff. Partly because it was protestant mathematics, but
mostly because they wanted a version of mathematics that was somehow eternally
unchanging and crystal clear (like how God is supposed to be.) Boy did that
not work out. Anyway, these early mathematicians used imagery like the threads
in a cloth to describe how infinitely many lines form a plane surface, or the
pages of a book to help understand how infinitely many planes form a solid. So
cool! In the beginning it was Italians working this stuff out (including
Galileo). But then the Jesuits totally banned it thereby casting a pall over
Italian mathematics for hundreds of years!

For a modern treatment of infinitesimals, check out "smooth infinitesimal
analysis" [1]. The way to avoid all the contradictions is to weaken the
underlying logic: in this case they remove the law of excluded middle, so it's
not the case that something is either true or it is false. I think the Jesuits
would definitely have trouble with that!

[1] [https://xorshammer.com/2008/08/11/smooth-infinitesimal-
analy...](https://xorshammer.com/2008/08/11/smooth-infinitesimal-analysis/)

~~~
erichocean
> _The way to avoid all the contradictions is to weaken the underlying logic_

Pet peeve: I would argue that constructive logic is _stronger_ , in the sense
that it allows more reliable (read: usable in the real world) results than
classic logic allows. But that's a value judgement, and there's no definitive
right answer here.

Totally agree Re: Infinitesimals. Dual numbers[0] are crazy useful in computer
science (c.f. Google's Ceres Solver[1]).

[0]
[https://en.wikipedia.org/wiki/Dual_number](https://en.wikipedia.org/wiki/Dual_number)

[1] [http://ceres-solver.org/](http://ceres-solver.org/)

~~~
mathgenius
I agree with you, there's going to be more shades of grey using intuitionistic
logic. I guess I was taking the point of view of the Jesuits!

So actually I understand about Dual numbers, but from what I see this is not
the same thing as smooth infinitesimal analysis (there's no playing around
with the logic.) I feel like if we scratch this a bit more we are going to
understand what Grothendieck was on about with topos theory [1].

[1]
[http://www.oliviacaramello.com/Unification/Unification.htm](http://www.oliviacaramello.com/Unification/Unification.htm)

------
charlieflowers
I am blown away by the "Calculus in One Minute" explanation. And I came in
with a high bar because I already knew about the good material on
betterexplained.com

------
e0m
I love Better Explained. Drawing a parallel between multiplication/integration
and division/differentiation is extremely powerful and almost never taught! It
suddenly pulls perceptively "complex" topics into the realm of the familiar.

------
ilek
Someone posted an article a while back about working to calculus using
functions, I think it was titled something like "If Socrates knew functions"?
I'm not sure. Does it ring a bell for anyone?

~~~
jnbiche
Looks like "If Archimedes Knew Calculus", about "quantum" calculus. Very
interesting, I'm about half-way through already:

[http://www.math.harvard.edu/~knill/pedagogy/pechakucha/](http://www.math.harvard.edu/~knill/pedagogy/pechakucha/)

~~~
jnbiche
To correct myself, the actual title is: "If Archimedes Knew Functions"

Same link:

[http://www.math.harvard.edu/~knill/pedagogy/pechakucha/](http://www.math.harvard.edu/~knill/pedagogy/pechakucha/)

~~~
ilek
Thank you so much!

------
amelius
I'm not sure if I like the example with the integration of the circle
circumference into the circle area. Yes, it sounds intuitive. But here's
another problem with a wrong solution that sounds intuitive: to compute the
hypotenuse C of a triangle with other (perpendicular) sides A and B, you can
simply replace the hypotenuse by a staircase of infinitesimal steps, and hence
you find that C = A+B. We're overlooking something here, but that is not the
point. The point is that the proof should be constructed so that we cannot
possibly overlook something.

Conclusion: don't rely on intuition with these kind of problems.

------
niedzielski
I haven't finished it but I read most of the late James Stewart's "Calculus:
Early Transcendentals" and found it very instructive. It would make a good
supplement for anyone learning Calculus.

------
Chris2048
Stuff like Khan Academy, or even MIT OCW are great, but there can be holes wrt
more advanced or unusual math. There also seems to be an "established" way of
teaching things (what/how), such that alternatives aren't offered.

Thinks like Wikipedia are quite poor on explaining math, especially when the
set theory is pulled. Set theory almost guarantees no intuition for me...

------
mclovinit
This is absolutely fantastic. Great method for teaching new concepts to those
new to the subject matter. I have an engineering background and have to say
that this approach is refreshing to read. It's like a bag of chocolate chip
cookies for the brain :)

------
NKCSS
Looks nice; didn't know that first example; shall give it a read later,
thanks!

------
baldfat
Children should learn Calculus instead of Algebra 1 & 2.

The BIG Problem is the difference between a Good Calculus teacher and a BAD
Calculus teacher is so HUGE and changes the outcomes of their students.

------
sureshkon
As usual, great stuff from Kalid.

------
known
Brilliant write up;

