
The Math I Learned After I Thought Had Already Learned Math - snake117
http://blog.mrmeyer.com/2015/the-math-i-learned-after-i-thought-had-already-learned-math/
======
rsy96
The revelation he has had looks trivial to me. Graphs as equations,
equivalence of expressions or matrix as linear operators are all something I
learned and understood from the very beginning.

I wonder if it has something to do with the textbooks. As a Chinese I often
found American textbooks on mathematics so softcore. They have so many
analogies, so many "real world" examples that masquerade the true mathematical
meaning of the concept. Many of Chinese people argue that these are the reason
that Americans are more creative, but I cannot help but wonder maybe the lack
of rigor underlies some of problems with American math education.

Or maybe I am just the exception. Maybe other Chinese struggle with math just
the same.

~~~
zintinio3
Americans generally struggle with mathematics, and not just advanced concepts.
The problem is cultural, because it is acceptable to be poor at math, it's
viewed as wholly unnecessary. Some examples are A&W's Third Pounder vs
McDonald's Quarter Pounder (3 < 4 so quarter pounder is bigger) [1], how long
does it take to go 80 miles if you're driving 80 miles per hour [2], and
Verizon Math [3].

As for Chinese struggling with math, I can give you firsthand experience at my
_American_ University, where the foreign (Chinese and Indian) students are
known for rampant cheating, in both the Undergraduate and Graduate levels.

We have a serious problem with logic, math, and science. People who are good
at the three are ridiculed and alienated rather than celebrated, although it's
not a hard rule, just something I've noticed. Reading for pleasure is the
exception rather than something normal. We're very anti-intellectual once you
get out of the big cities.

[1] [http://www.nytimes.com/2014/07/27/magazine/why-do-
americans-...](http://www.nytimes.com/2014/07/27/magazine/why-do-americans-
stink-at-math.html?_r=0)

[2]
[https://www.youtube.com/watch?v=m2eyq9qTOQY](https://www.youtube.com/watch?v=m2eyq9qTOQY)

[3]
[https://www.youtube.com/watch?v=MShv_74FNWU](https://www.youtube.com/watch?v=MShv_74FNWU)

~~~
wfunction
The Verizon Math was gold.

------
trengrj
I think that maths at school it still too focused on creating engineers (the
non software kind) and much of the material is aimed at producing people who
can do calculus.

I'd prefer for there to be more focus on proofs, logic, geometry, number
theory, graph theory, and cryptography etc. I think these concepts would give
people a more rounded understanding of mathematics and prevent their
experience of mathematics being one of pain.

~~~
hugh4
I don't know if there's too much calculus, but there's definitely too much of
the wrong type of calculus.

Congratulations, you have just learned to integrate. Let us now spend the next
three years running through several thousand different examples of analytic
integration of one-dimensional continuous functions.

More dimensions? Path integrals? Numerical integration methods that you'll
almost certainly have to apply in the real world because most functions don't
have analytic integrals? Differential equations? Never heard of them. Now
let's all integrate (x^2 - sin(x))/coth(x).

~~~
Gibbon1
I think there is too much focus on the mechanics and not enough on the big
picture behind what you're trying to do. Mostly it's a lot of. Step one
recognize the problem. Step two apply the solutionator+ and turn zee crank.
Step three receive kibble.

+Solutionator: Bunch of mechanical steps that the student understands in the
exact same way a trained monkey understands an organ grinder.

~~~
gh02t
It's kind of funny, most people who major specifically in math or applied math
will take something equivalent to a course in Real Analysis. RA is basically
starting over from scratch and reteaching you the exact same material as
calculus, but in a much more formal and rigorous manner. Mostly it focuses on
eliminating the not-so-rigorous concept of an infintesimal quantity and
formalizing the notion of limits and convergence (albeit, there is a way to
rigorously treat infintesimals called nonstandard analysis, but I digress). It
involves a lot more open-ended thought than the endless memorization of
earlier calculus classes. It's often dreaded by undergraduates as being very
proof-heavy, but it was one of my favorites. It always felt like learning the
"essence" of calculus, at least for me (who sucks at/dreads rote
memorization).

~~~
eli_gottlieb
My real analysis textbook says on the back:

 _Was plane geometry your favorite math course in high school? Did you like
proving theorems? Are you sick of memorizing integrals? If so, real analysis
could be your cup of tea. In contrast to calculus and elementary algebra, it
involves neither formula manipulation nor applications to other fields of
science. None. It is Pure Mathematics, and it is sure to appeal to the budding
pure mathematician._

~~~
eru
Complex analysis is even neater. (But that's almost algebraic in its
niceness.)

~~~
gh02t
I really enjoyed measure-theoretic probability theory myself.

~~~
eru
We did that as a precursor before probability theory. Really puts the latter
on a sound footing.

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toothbrush
Cool, i also had an "aha" moment while reading:

“It was only after grad school that I learned (from Lockhart’s book
Mathematician’s Lament) to consider natural numbers as stones that can be
arranged in various patterns that illustrate the different properties of a
number. For example, evens are piles of stones that can be arranged into two
equal rows, and square numbers have just the right number of stones to make a
square! It’s really fun thinking about various operations in this way, and
there are some beautiful proofs based on this technique. For example, why the
sum of the odd numbers 1 + 3 + 5… Is always a square.”

I should hang out with maths teachers more often!

~~~
roymurdock
This is a really helpful way to think about natural numbers. I like to remind
myself that we can gain efficiency through abstraction, but true, intuitive
understanding comes through concretization/deconstruction.

Interestingly enough, I use round stones to represent ideas when I meditate.
As the ideas come to me, I pick them up, examine them, and weigh them. If the
idea is pressing, I delve into it and think it through. If not, I put the
stone down and wait until my mind picks up the next.

Mental stones: some of modern life's most useful tools :)

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jwmerrill
Two good ways to represent plane curves are parametric form:

(x(t), y(t))

or implicit form:

f(x, y) = 0

Parametric form is naturally associated with one point of view of what a plane
curve is: the set of points traced out as a parameter is swept over its
domain. Implicit form is naturally associated with a different point of view:
the set of points that satisfy a certain relation.

Functions of the form

y = f(x)

can be easily re-expressed in parametric form:

(t, f(t))

or implicit form

y - f(x) = 0

so both the parametric viewpoint and the implicit viewpoint are equally valid
and useful ways of understanding the graph of a function. You could rephrase
the author's insight as saying that he had always understood graphs of
functions parametrically, but later learned to also understand them
implicitly.

Depending on the application, it may be more convenient to have a parametric
representation of a curve, or an implicit representation of a curve. For
example, it's easy to find a point on a parametric curve, but hard to test if
a point is on a parametric curve; on the contrary, it is hard to find a point
on an implicit curve, and easy to test if a point is on an implicit curve. If
your curve is the graph of a function, it is easy to convert back and forth
between these forms, but in general, converting from one form to the other may
be quite hard.

For me, the relationship between implicit and parametric representations is a
piece of math that I didn't really learn until long after I thought I had
already learned math.

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SteveBash
I somehow understood the same thing(a graph is a collection of points that
makes a function true..) a year ago through Gilbert Strang Big picture of
derivatives: [https://www.youtube.com/watch?v=T_I-
CUOc_bk](https://www.youtube.com/watch?v=T_I-CUOc_bk)

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jwmerrill
It looks like the blog is down at the moment. Here's google's cached version
of the post in the meantime:

[http://webcache.googleusercontent.com/search?q=cache:7SWRaHW...](http://webcache.googleusercontent.com/search?q=cache:7SWRaHWxfiEJ:blog.mrmeyer.com/2015/the-
math-i-learned-after-i-thought-had-already-learned-
math/+&cd=1&hl=en&ct=clnk&gl=us)

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xpinguin
I really didn't get the article. I mean, I've got that we have some mapping f:
R -> R, or equivalently f: (x, y) \in RxR, so we could represent it by drawing
on 2d plane, but no further...

Author claimed, that he had been missing something essential regarding that
stuff, but I wonder what exactly does he talk about?

~~~
ColinWright
People think of "drawing a graph of a function" as drawing a line, so if
_y=sin(x)_ they end up with a wriggly line going infinitely far left and
right.

What they don't think of is that the line is actually the collection of points
(x0,y0) such that the equation _y0=sin(x0)_ is true. Kids in high school don't
think of the graph of a function as being a subset of points of the plane,
being:

    
    
      { (x,y) : y=sin(x) is true }
    

Realising that opens the doors to equivalences between different ways of
thinking. We can think of a permutation of objects as both the act of
permuting them, and as the result of applying that permutation to the default
initial position. We can think of a vector (4,6,9) as a location in space, and
as the movement to get from (x,y,z) to (x+4,y+6,z+9). We can think of "3" as a
location on the number line, or as the action of adding 3 to something, or as
the action of multiplying 3 my something, and so on.

We can think of the graph you draw as a line, or as a subset of the plane, and
we shift effortlessly between them, deliberately blurring the distinction, and
from that blurring can come power.

Does that help?

~~~
sago
This hurts when they deal with equations like

x^2 + y^2 = k

and can't even begin to understand how you could _graph_ something like that.

It took me a while to grok it, so I second his experience of this not being
well stressed (30 years ago, admittedly). I did further maths A-level (and got
an A), but didn't grok it until uni.

~~~
bitwize
Having long been fascinated with computer graphics this bit was easy for me to
get. I imagined a computer raster-scanning the plane and plotting a point when
the condition in the equation was true. I just had to scale it up in my head
from screen resolutions to infinite resolution and infinite extent and... oh
boy, this is already starting to make my head hurt...

But the _gist_ of it I got. And I admit, I'm unusual.

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ginko
Is "secondary math" university mathematics? How can anyone think he's
"learned" that?

~~~
titanomachy
No, it's high school math. The kind you learn before university. secondary
school == high school in many places.

