
Does math have big scary teeth or something? - carterschonwald
http://ask.metafilter.com/135720/Does-math-have-big-scary-teeth-or-something#1939302
======
parse_tree
The comment that the link leads directly to is devastatingly good. If you
don't want to do a lot of reading, just skip the main article and read that
comment.

~~~
amalcon
He does have some good points, but they're drowned in nonsense. For example,
he suggests that memorization is taught in lieu of problem solving because
this is what society thinks is a smart person. Memorization _is_ taught in
lieu of problem solving, but it's because we haven't yet figured out a good
way to teach problem solving. It's not some vast conspiracy; the outcome we
actually want is just a really hard one.

~~~
amackera
I don't think that the author was implying the existence of a vast conspiracy
to keep people dumb; rather, the author was suggesting that society is
operating in a manner which he believes is suboptimal.

Let's say memorisation is taught in lieu of problem solving for the reasons
you stated, then consequently the people who _are_ good at memorisation become
thought of as smart, since they do well academically. This, in turn,
influences the education system to focus more on memorisation (though, of
course, this is all hypothetical).

~~~
amalcon
Interestingly, "smart" is only equated with memorization in artificial
situations (like academics and trivia shows). Case in point: if you ask people
who's presently the smartest person in America, they might say the jeopardy
guy (I actually suspect a lot of them would say Hawking). If you ask who's the
smartest person in American history, the overwhelming response will probably
be Einstein (except a few who will say Hawking, or -- very rarely -- Tesla).
Einstein was a notorious problem-solver. Certainly, if you asked someone if
they'd prefer their child to grow up to be like Einstein or like the Jeopardy
guy, they'd probably pick Einstein.

Likewise, I don't think the author was implying any sort of conspiracy. I
probably used the wrong word. I should have said that there's no real effort
on anyone's part to promote memorization over problem-solving. Memorization is
more prominent in these artificial situations because it's easier to train and
test, rather than because people think it's superior.

~~~
brutimus
FYI: Hawking is British and teaches at University of Cambridge in England. :-)

~~~
amalcon
Right. Silly me. s/U.S./world/g, or go back in time 30 years or so and use
Feynman.

------
10ren
Counter to the linked-comment: _maths_ is not pure problem solving.

There are a lot of facts to be learned in mathematics; it is more like a
language, with a huge vocabulary. And, like language, very little of
mathematics is actively justified as to whether it is or is not the best way
to go about solving problems. People just build on the existing mathematics as
a basis... in a similar way to how people write novels in a language.

The attitude of mathematicians is as an imperial, aristocratic attitude to
courtly protocol. The _true problem-solving mathematician_ is more like the
wild barbarians, who break all the protocols to achieve actual outcomes. One
might say: empirical not imperial.

But the aristicratic attitude makes a lot of sense, because facts and
protocols are things that one can learn and advance with, whereas a problem
solver is only as good as their last solution. It's a bit magical. While one
can practice problem solving, the return on that investment is no where near
as reliable as the return on investing in learning established, conventional,
standard _facts_. It's a good investment; it gives you a sustainable
competitive advantage.

However, there is something massively cool about being one of the people on
the frontier, who germinate and disseminate facts, rather than passively
receive them.

~~~
scotty79
Math requires a lot of memorized knowledge to solve anything interesting.

Without knowing the right theorems to use in the proof you are trying to
compose ... all your efforts might be futile despite having excellent problem
solving skills.

Math uses mountains of knowledge derived over many years by hundreds of the
smartest people in history.

You just can't physically repeat their work for the purpose of making your
proof. You just have to know what is known so far.

For me one of the most fun parts of educations was non-organic chemistry in
primary school. So little to learn and you could attempt to solve problems
that twisted your brain like a pretzel (and succeed). I could solve problems
my teacher couldn't because I was probably smarter then him and we both knew
the same about the problem domain because it was all that was to know.

------
Mz
I memorized my way through all kinds of math in high school. I was in my
thirties before I knew some of those formulas had practical applications. I
knew my oldest son couldn't memorize his way through math. So I taught a
conceptual approach. It was the math concepts I retained anyway. When I went
back to school, knowing the concepts stood me in good stead. Looking up what
all the little squiggles meant and refamiliarizing myself with some of the
formulas quickly got me up to speed. I was waivered into my statistics class
based on my 17 year old SAT scores because I had never taken college algebra.
I ended up with the highest grade in the class and explained a lot to my
classmates. So I made sure my son knew the concepts, even though he is
terrible at dealing with numbers. It worked beautifully.

How it's taught can make a big difference.

------
ubernostrum
After reading a lot of comments in a lot of threads like this one on a lot of
sites, I've noticed a particular factor which sticks out: people who direct
this sort of commentary at education in the US almost all seem to have
attended very large universities.

I did not attend such a university, and my experience of "academics" was
rather different from what's described in the linked comment; there were some
things that I know were unusual about the way my college education was
structured, and that not even all of the students there were getting the same
level of or approach to education, but I suspect that there's a fundamental
difference between large and small schools. The larger the institution, the
more "efficient" it has to be with the resources at hand; this seems to lead
naturally to the sort of bulk-processed "memorize this and regurgitate it on
an exam" approach being complained about here. A smaller institution,
meanwhile, isn't under the same pressure to move large volumes of students
through its curriculum, and can afford to do things differently. I'm well
aware that mine did, and thankful every day for that fact.

------
ahpeeyem
The comment is quite similar to the idea of "mappers and packers:"
[http://the-programmers-stone.com/the-original-talks/day-1-th...](http://the-
programmers-stone.com/the-original-talks/day-1-thinking-about-thinking/)

This theory goes that some people (mappers) create a mental map of how the
world works, and any new knowledge has to be integrated into the map - or if
it doesn't fit, the entire map has to be rearranged. Packers just store little
packets of knowledge for each situation, without connecting them together.

The mappers therefore can do problem solving by connecting the different parts
of their map when they come across something they haven't seen before, whereas
packers are will try and apply one of their existing knowledge packets to a
new situation, and if none of them fit they are lost.

I think the analogy works, certainly some people I've helped with using their
computer don't seem to be able to apply knowledge learned in one program to
another one, and need to learn everything by rote instead of figuring it out
as they go. So either they don't have enough experience yet to build a meta
map/model that would let them figure it out on their own, or they don't even
try to.

~~~
sfnhltb
I would think this is more like what happens in each individual on a given
topic - you start off with little packets and eventually you integrate them
into a map. How well you are at doing this depends on how good you are at
linking the packets together, how you are taught or learn them, and whether
you have constructed a similar enough map before to construct the new one more
easily.

------
brutimus
The highlighted response was bullseye hit of what I noticed going through
college. The couple classes I remember it being most evident in were the very
low level programming classes: Intro to Java/Javascript and a PHP class. These
were the classes that pretty much everyone in the business school had to take.

People would work through problems assigned out of the book, but if they ever
ran into an issue not specifically outlined for them, they were usually stuck.
I spent many nights in the computer lab helping classmates through problems
which just required stepping back and taking it one small step at a time. And
even if they couldn't work through it on their own, they had no idea where to
start on researching a solution -- they didn't know how to ask questions, etc.
(Disclaimer: I nearly failed every damn accounting class I had to take, 5 in
total I believe.)

I find myself being the exact opposite of what the linked reply states as
being the 'norm'. I can't remember simple facts to save my life. I love
playing guitar, but I can't memorize notes/chords. To this day I can't
honestly tell you what a noun/pronoun/verb/adverb, etc are. Forget about
people's names, it ain't happening. I find no real joy in reading fiction, I
forget it all anyways. I couldn't ever remember the bajillion accounting terms
I had to deal with in business school, but I loves me some calculus.

Also, he seems to point out "America" quite a bit in his response. I have a
hard time believing this trend only applies to Americans. I'd like to hear
either a non-American or someone with a little international time chime in on
this.

~~~
Tichy
"I find no real joy in reading fiction, I forget it all anyways"

But that is good: you can read lord of the rings over and over again as if it
was the first time.

------
cowmoo
Rote memorization is a problem in programming, too. Not to rag on PHP or
Visual Basic, but people could get by with just memorizing certain keywords -
or not solve hard problems by using a third-party plugin or googling for code
snippets.

I hated my Automata and Functional Programming Language classes because I was
forced to write out formal proofs to prove really obvious programs to be
correct, or to demonstrate something arcane as the Turing machine. But it
forced me to think about programming in a different light and find my own way
to find the solution.

This reminds me to always to look under the hood of all of the web frameworks,
data ORM's that I'm using and to improve my 3rd party libraries, in
programming. Also in music, not to blindly play the scales or the tabs of
popular music, but learn the different patterns, what makes a song tick, and
composition. Also in sports, not just play or practice according to drills,
but to analyze post-game what went right, what went wrong, and to apply it in
future matches.

~~~
nl
"Not to rag on PHP or Visual Basic, but people could get by with just
memorizing certain keywords"

That's bullshit. Yes, PHP/VB programming may be easier, and yes, they may not
be tackling hard problems, but "problem solving" approach revolves around
analysis and how to address different situations.

That is best expressed by the "if.. then.." construct.

Look - I'm not going to claim the ability to program in VB makes you a good
problem solver. However, I am claiming that it is a difference of skill level,
not skill set. Memorization is a different skill to problem solving.

~~~
billswift
Memorization, at least in its meaning of remembering facts, is a necessary
prerequisite of understanding, which is needed for effective problem solving.
[http://williambswift.blogspot.com/2009/03/learning-
journal-a...](http://williambswift.blogspot.com/2009/03/learning-journal-and-
record.html)

------
presidentender
It's pretty sad how strongly memorization, as opposed to critical thinking, is
encouraged and rewarded. This seems (based on my University education) to be
increasingly true even in disciplines like Math and CS.

Beginning in grade school, the "smart kids" are those who memorize things, not
those who solve problems. I'm better at rote memorization, but I pride myself
on solving problems.

------
swolchok
I run screaming from courses that cover mathematical background I don't have
(e.g., machine learning because of linear algebra) because it's impossible to
keep up without having terminology and definitions memorized.

Recently got annoyed with the overuse of math in
[http://www.eecs.umich.edu/courses/eecs461/lecture/Lecture8.p...](http://www.eecs.umich.edu/courses/eecs461/lecture/Lecture8.pdf)
. The lecture covers numerical integration in an embedded system. If you know
dx/dt = f(x,u) (where u is the input), the completely obvious approximation by
computer is to execute x += dt * f(x, u) at intervals of dt seconds. Said
lecture takes ages to get to this point, unnecessarily puts things in
matrices, and never puts the point so concisely. IMO, it's easy to use linear
algebra to belabor simple points.

~~~
btilly
You know the missing mathematical background isn't actually that hard to
learn. And once you learn it, you may notice important details you're
currently missing.

For instance the lecture you got annoyed with was covering how to correctly
model a _second degree_ differential equation. So you know that d^2x/dx^2 =
f(x, u). This is a different and more complicated problem than a first order
differential equation. They handled it by turning it into a system of first
order differential equations. And then demonstrated how the obvious way to do
it lead to numerical instability, creating a physically incorrect result. And
then they had to do more complex stuff to avoid the numerical instability.

The underlying concepts share a lot with a simple first order equation. But
the generality is letting you tackle and deal with issues that a simple model
can't. Plus the technique described can be expanded in a straightforward way
to model a combination of interacting things.

But you froze up at the idea of linear algebra and didn't even realize that
they were dealing with a much more complicated problem than you thought.

~~~
scotty79
It could be easier to understand if instead using z-Transform they shown that
you could do better numerical second order integration if you take few
previous points into account instead of just one as in naive method of
integrating twice with forward Euler integration.

Of course purpose of this lecture was probably not to make you understand what
you are actually doing while numerically integrating, just to teach you how to
put this stuff into simulink that has z-Transform as one of it's primitive
building blocks.

------
grumblebee
Hi. I'm the author of that post. Wow. I wrote my opinion on a message board
and now it's everywhere. On some sites they are referring to it as an
"article." Ah, the risks you take when you post stuff on the web...

In any case, I think some of you make good points below about weaknesses in my
argument (which, if I WAS writing an article, would have been more rigorously
researched and reasoned), but I stand behind the majority of what I wrote.

Feel free to ask questions, challenge me or call me a narcissist. It's all
good.

------
silkodyssey
I think all jobs require problem solving to some degree just that some jobs
are easier than others and others while not necessarily easy are sufficiently
well defined to enable a worker to develop expertise in that area. And with
expertise in a field problems are often solved intuitively without going
through all the "problem solving mechanics".

Getting to this point where problems can be solved intuitively can sometimes
take years of study and practice. We can look at this as an investment. A
person invests time and energy in wiring his or her brain to solve a
particular set of problems and after achieving this goal the person is left
with little desire to repeat the process for another problem domain. A doctor
may apply problem solving skills to diagnose and treat a rare disease but may
not be able muster the effort required to learn photoshop. It's not due to a
lack of problem solving abilities but more of a desire to step not out of
their problem solving comfort zone.

------
rabidsnail
The main purpose of the educational system, up to and including graduate
school, is to produce obedient workers. We're fortunate to work in a field too
young to have developed an orthodoxy. That's how we can get away with thinking
so much. Most people are not so fortunate.

------
tesseract
See also <http://www.metafilter.com/70699> (and the recently published printed
version, although IMO the material added beyond the original article is not
worth getting the book for, which is a shame).

------
gchpaco
I was trying to explain the difference between hackers, if you will, and
"programmers" just going through the motions. I likened it to the difference
between someone who spoke a language frequently and someone who was working
out of a conversational phrasebook. The thing is one cannot help but learn a
language as a human being, but you can sure go through life without problem
solving skills!

------
rick_2047
Well all my school life I have had some great English/Hindi literature
teachers. I may be safe in saying that I have learned problem solving from
them. They were not of the types which tell you to learn the poem or prose by
heart.Give prepared notes on certain lines and then tell you to rote learn
them.They told us how to get to heart of a story.On of my most experienced
teacher(our batch was her last,she retired at the age of 60 I think),told us
that just reading a story is a disgrace in itself to the author [when you are
in a Literature class].Every story was a part of his/her life when he/she
wrote it.Every poem was a real emotion when it was laid out on paper.She used
to throw random books at us (on me sometime literally), told us to
"understand" them. Then we had this huge discussion over the plot and how this
can be interpreted or how that is misinterpreted by most people.Those were one
of the best days of my school life.

Now this may seem not a particular type of "problem solving" to some people.
But it is when you take it seriously.You get the same sensation when you make
a perfect connection between an event in the authors life and how he
interpreted it in his story,that you get when you solve a mathematical
problem.Because these two have essentially the same procedure.1)Finding the
roots of the problem ,2)Understanding the roots of the problem,3)Interpreting
the roots of the problem,4)Using those to solve the problem. Perhaps that's
why one of my Maths professor used to say "Look at the problem,its screaming
its solution to you".

