Notesale: Turn your study into money

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I have a love/hate relationship with calculus: it demonstrates the beauty
of math and the agony of math education
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My closest
analogy is Darwin‟s Theory of Evolution: once understood, you start
seeing Nature in terms of survival
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You know why sugar and fat
taste sweet (encourage consumption of high-calorie foods in times of
scarcity)
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Calculus is similarly enlightening
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But most of us learn these formulas independently
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Unfortunately, calculus can epitomize what’s wrong with
math education
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It really shouldn‟t be this way
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Specifically, staying encouraged despite
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Teachers focused more on publishing/perishing than teaching

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Self-fulfilling prophecies that math is difficult, boring, unpopular
or “not your subject”
Textbooks and curriculums more concerned with profits and test
results than insight

„A Mathematician‟s Lament‟ [pdf] is an excellent essay on this issue
that resonated withmany people:
“…if I had to design a mechanism for the express purpose of destroying a
child’s natural curiosity and love of pattern-making, I couldn’t possibly do as
good a job as is currently being done — I simply wouldn’t have the
imagination to come up with the kind of senseless, soul-crushing ideas that
constitute contemporary mathematics education
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Instead, let‟s study paint chemistry, the physics of light,
and the anatomy of the eye
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After all, they have the “rigorous, testable” fundamentals to
start appreciating art
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Imagine studying this quote (formula):
“This above all else: to thine own self be true, and it must follow, as night
follows day, thou canst not then be false to any man
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But if this were math class, we‟d be
counting the syllables, analyzing the iambic pentameter, and mapping
out the subject, verb and object
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Don’t
confuse the finger for the moon
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We‟ve forgotten that math is about ideas, not robotically manipulating
the formulas that express them
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If you need answers right away for that big test, there‟s
plenty of websites, class videos and 20-minute sprints to help you out
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Equations
aren‟t enough — I want the “aha!” moments that make everything click
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Diagrams, animations, and just plain talkin‟ can often provide more
insight than a page full of proofs
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We don‟t need
to be writers to enjoy Shakespeare
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Not long ago, reading and writing were the work of trained
scribes
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Why?
Because we expect it
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So expect that calculus is just another subject
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But the rest of us can still
admire what‟s happening, and expand our brain along the way
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I‟d love for everyone to understand
the core concepts of calculus and say “whoa”
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It‟s correct, but not helpful for beginners
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Arithmetic is about manipulating numbers (addition, multiplication,
etc
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Algebra finds entire
sets of numbers — if you know a and b, you can find c
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Using calculus, we can ask all sorts of questions:
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How does an equation grow and shrink? Accumulate over time?
When does it reach its highest/lowest point?
How do we use variables that are constantly changing? (Heat,
motion, populations, …)
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Like evolution, calculus
expands your understanding of how Nature works
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Suppose we know the equation for circumference (2
* pi * r) and want to find area
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Here are two ways to draw a disc:
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Make a circle and fill it in
Draw a bunch of rings with a thick marker

The amount of “space” (area) should be the same in each case, right?
And how much space does a ring use?
Well, the very largest ring has radius “r” and a circumference 2 * pi * r
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The final ring is more like a pinpoint,
with no circumference at all
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Let’s unroll those rings and
line them up
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But if we take
thinner rings, that triangle becomes less jagged (more on this in
future articles)
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For each possible
radius (0 to r), we just place the unrolled ring at that location
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Calculus showed us that a disc and ring are intimately related: a disc is
really just a bunch of rings
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And sometimes the little things are easier to work with
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That‟s great, but it can be
hard to relate: honestly, how often do you know the equation for
velocity for an object? Less than once a week, if that
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That ring/circle thing we made? You could build it out of
several pipe cleaners, separate them, and straighten them into a crude
triangle to see if the math really works
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A note on rigor (for the math geeks)
I can feel the math pedants firing up their keyboards
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Did you know we don‟t learn calculus the way Newton and Leibniz
discovered it? They used intuitive ideas of “fluxions” and
“infinitesimals” which were replaced with limits because“Sure, it
works in practice
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We‟ve created complex mechanical constructs to “rigorously” prove
calculus, but have lost our intuition in the process
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Eat it
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Would it be so bad if everyone understood calculus to the “nonrigorous” level that Newton did? That it changed how they saw the
world, as it did for him?
A premature focus on rigor dissuades students and makes math hard to
learn
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The natural log can be seen as an
integral, or the time needed to grow
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Happy
math

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