
A Gentle Introduction To Learning Calculus - mqt
http://betterexplained.com/articles/a-gentle-introduction-to-learning-calculus/
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michael_dorfman
It's unfortunate that most of the comments so far only engage either with the
title of the piece, or with it's (acknowledged) debt to "A Mathematician's
Lament".

I think the author's main point-- that most teaching of mathematics tends to
undervalue the "aha!" types of insights that connect various pieces-- is a
good one, and it looks like he has some concrete ideas about how to go about
it.

I, for one, look forward to seeing the author follow through with additional
concrete examples.

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kalid
Thanks :). I don't know what it is, but for a field concerned with abstraction
people miss the forest for the trees.

I try to catalog insights right as they happen, and try to capture the before-
and-after: what, exactly, was the turning point that made it click?
Unfortunately that makes articles slow coming, but I think they can be more
genuine than explaining something after you're already an expert. Most of us
can't remember what it was like trying to learn to read, for example -- it's
too hard to relate to that problem.

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bridgetroll
The author is obviously under the delusion that an introduction to calculus
should be gentle and soothing, like slipping into a warm bubble bath. It
should not. It should be swift and shocking, like jumping into a river in
early Spring.

Mathematics requires an exact and expert hand. They are EMTs. This of your
Calc 1 mid-term: "Quick, you have two minutes, resuscitate this equation!"

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vlad
Rarely do math and science instructors teach material the way it is
discovered. Also, math and science courses are typically taught the way
history or English is taught--you have some topics, you have things to
memorize, you're given questions that are very similar to those you've seen in
school or on your homework. I can imagine that only top schools teach this
material in terms of tools and knowledge, versus learning and writing down the
right equations on a test.

~~~
kalid
That's exactly it. We show math as this complete and perfect system, whereas
it was discovered, often times, through trial-and-error. Retracing the steps
can help show why certain decisions were made.

For example, why does C have a separate short and long data types, but
javascript just has "var"? Because when C was made, memory was at a premium.
When javascript was invented, everything could be a 4-byte value.

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a-priori
Actually, the reason was more that C was designed for systems programming, and
that sort of code often has to interface with hardware devices through memory-
mapped registers. To do this, you need to be able to write code that uses
variables of the correct width. This is why C has all the various sized
variables in signed and unsigned variants.

~~~
kalid
Good point -- C was designed to interface with hardware ("high level
assembly") whereas javascript was designed to interface with web pages &
browsers. Either way, the history helps understand the design decisions.

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nazgulnarsil
this is copying the message and style of A Mathematician's Lament

~~~
kalid
I was definitely enjoyed that essay and want to spread its message; it's why I
linked to it and quoted from it :).

However, I've been writing about these concepts in this way since 2001, way
before I ever heard of the essay (here's my earlier website from school, which
was the the precursor to this blog:
<http://www.cs.princeton.edu/~kazad/resources.htm>).

~~~
nazgulnarsil
oh, thought you were just another me too blog.

good on you then. :)

~~~
kalid
Hey, no prob -- I dislike me-too posts as much as the next guy :)

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dnaquin
I stopped reading here: _How does an equation grow and shrink? Accumulate over
time?_

Equations, don't grow or shrink. Maybe he was confusing equations with
functions.

~~~
kalid
I used the term loosely. But to most people, I think y = x^2 is an "equation".

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abless
True and it's good if people understand it better that way. Nonetheless, I'd
advice on introducing a correct grammar as soon as possible.

