
In Math You Have to Remember, In Other Subjects You Can Think About It - tokenadult
http://www.maa.org/devlin/devlin_06_10.html
======
tomkin
> Numerous studies over the past thirty years have shown that when people of
> any age and any ability level are faced with mathematical challenges that
> arise naturally in a real-world context that has meaning for them, and where
> the outcome directly matters to them, they rapidly achieve a high level of
> competence. How high? Typically 98 percent, that's how high. I describe some
> of those studies in my book The Math Gene (Basic Books, 2000). I also
> provide an explanation of why those same people, when presented with the
> very same mathematical challenges in a traditional paper-and-pencil
> classroom fashion, perform at a lowly 37 percent level.

This. I love that this argument is being made. There is this notion among
academics that the student should bend to the (fabricated?) stringent
regulations on how math is taught or expressed. As a society, we accept the
fact that there are those who grasp concepts, learn and develop sensibilities
of material in various different ways.

For me personally, I found the way that math was taught in school to be
completely disconnected with its purpose. There would be times where material
was applied, but more often not. No other subject can get away with this.
Music - instruments, scale and training. Art - painting, modelling and theory.
English – writing, reading & comprehension. Science - hypothesis, experiments,
conclusion. Most subjects have an _execution_ factor. How far Jamie has to
walk to get 3 bags of milk is _not_ execution, it's _practise_. This is a
gross simplification of a beautiful subject - but this is the point where many
get lost: Purpose.

For those who absorb material differently, this is where the conversation
needs to start.

~~~
tgrass
Painting, modelling and theory; writing, reading & comprehension. These are
practice. Getting your painting into a gallery or your book into the hands of
an unobligated reader - that is execution.

In Civil Engineering, we ran problem after problem in school. Nothing I do now
professionally in civil design resembles the practice I did in school.

Which is just to say, execution should be the goal of ALL programs, but none
seem to come close.

[edit] it's Friday morning and I'm done; any suggestions for a synonym for
'unobligated'?

~~~
tomkin
I agree that they are practice in some sense, but the end result is valuable
immediately. When I was a young student, I would take my painting, science
experiment or short story home to show my mom. When I did well in math I
showed my grade, not the actual _work_. Here is where we find _perception of
activity_ vs. _successful comprehension_.

Math is not valuable unless it has purpose and if some are not given purpose
they will stop investigating it. It's easy to categorize those who have that
viewpoint. As someone who _found_ math later in life on my own terms, I can
say that it was much more enjoyable than what was being taught as a student.

This was foretold in each math class I had in the schools I attended. There
was always a "Math is necessary for..." poster on the wall in the classroom,
and yet, no other class had a "Art is necessary for..." or "History is
necessary for..." poster. We've identified the problem. Now what do we do
about it?

Language Studies (aside from immersion) also could benefit from being more
execution-oriented. For me, 4 years of French class was easily lapped by a
month in Quebec.

~~~
tgrass
I completely agree. I found math later too, at 29, and had it been taught
better it might have clicked with me earlier.

The only academic program I've experienced that got right to the crux of its
execution was History: but only because I feel its only real purpose is self
reflection.

------
CurtMonash
I was generally regarded as math prodigy. I got my first algebra book when I
was 4 years old, and mastered the material quickly. I eventually got a
mathematics PhD from Harvard when I was 19 years old.

Traditional mathematics classrooms contributed very little to that learning,
except for the portion of a university education directed toward mathematics
major. And I even have my doubts about that part.

Meanwhile, the best bit of math teaching I've ever done may have come when I
was helping my stepdaughter with her algebra homework. I noticed her applying
on over-specific rule to a class of formula simplification problems. So I
deliberately changed the problem she was working on to one where the bogus
rule didn't apply ... :)

I've come to believe that for students of all levels, the most important part
of mathematics education is when you coach them through problem-solving. Yes,
the formalities are wonderfully powerful and fun, at least to my taste. But
they should be a last resort, whether for solving a problem or checking the
soundness of a result. Formalities-first is exactly the wrong way to approach
things, for math users and math teachers alike.

~~~
kenjackson
I'm interested in how you mastered algebra at 4. Were you already strong at
arithmetic when you got the algebra book (and algebra was just a continuation
of your already acceleratd progress) or was algebra just very intuitive and
the start of a great math career?

~~~
yaks_hairbrush
Can't speak for OP, but I started learning algebra at 5. It happened that I
really liked playing with calculators at age 2-3, and that helped give me an
extremely solid mental arithmetic foundation.

Essentially, the special thing about me probably wasn't some super-human
neural circuitry for understanding math. It was the fact that I was wired,
somehow, to enjoy playing with a calculator. I had more arithmetic practice by
age 4 than most kids have by age 11.

~~~
prezjordan
I feel like I can relate to this (a little). I'm no prodigy, but even as a
young kid (6 or 7) I would write down algebra problems all day and solve them.
I did a ton of mental math, too, writing down several dozen 2-digit numbers
and adding them all in my head.

Haven't really thought about that in a long time. Playing with a calculator
also got me into programming (TI-Basic on the TI-83 in middle school).

------
jere
>In Math You Have to Remember, In Other Subjects You Can Think About It

This title/quotation is quite depressing, since the student has it totally
backwards. Math and any field with mathematical roots are the only things you
can _think_ about and figure out. I can't count the number of times I walked
into an EE test unprepared and was able to work out some formula, which I
hadn't memorized, from other principles.

Most of my history/english/social studies consisted of rote memorization of
facts, dates, and obscure words. I'm still flabbergasted on why the SAT has
middle school level math problems while also testing you on an intractable
memorization problem: vocabulary.

[update: apparently, analogies have been removed from the SAT. thanks,
nickbarnwell]

~~~
nickbarnwell
> Most of my history/english/social studies consisted of rote memorization of
> facts, dates, and obscure words. I'm still flabbergasted on why the SAT has
> middle school level math problems while also testing you on an intractable
> memorization problem: vocabulary.

As someone who still has the SATs fresh in mind, vocabulary is actually a very
small, and very easy, portion of the exam. It's a subcomponent of the Critical
Reading section, which largely concerns itself with drawing conclusions from a
given text. Anyone even moderately well-read should be capable of acing CR,
and it was a not unusual occurence at my gymnasium (~5 out of 15 students in
my year managed 800s, IIRC)

~~~
jere
It's been about a decade since I took mine. Apparently, they got rid of
analogies, which is what I was thinking of:

>In 2005, the test was changed again, largely in response to criticism by the
University of California system.[32] Because of issues concerning ambiguous
questions, especially analogies, certain types of questions were eliminated
(the analogies from the verbal and quantitative comparisons from the Math
section). <http://en.wikipedia.org/wiki/SAT#2005_changes>

That sounds like a great change. Each analogy required you to know 4 words (at
least one of which would be rather obscure). I read quite a bit in grade
school (still proud of myself for seemingly being 1 out of only 3 students who
read the required 750 page _John Adams_ over a summer), yet suffered miserably
during this section.

~~~
cchurch
I'm (un)fortunate enough to have taken the SAT long enough ago to have had
analogies on it. I suffered through them, scoring disproportionately poorly on
them as compared to my math score. It was frustrating since I'm a voracious
reader. I did not understand why such an important test would require
memorizing so many obscure words to perform well, especially since most of the
math section is about solving puzzles and trick questions. It seemed a
completely useless skill.

It was years later that my (bilingual) wife explained her view on the
analogies. She saw them as logical puzzles to figure out. If a student
understands word roots and has some concept of foreign (romantic) languages or
latin, it is simple for them to apply this knowledge to remove possible
answers, figure out properties of the words, and solve the analogy.

My high school (even with 3 years of mostly worthless spanish) never presented
me this toolbox. We spent 2 years in english classes with SAT prep vocabulary
tests. We memorized random words and their definitions. It never occurred to
me that this was contrary to the purpose of the verbal section of the test.

It is sad that people seem to feel the same way about the math section as I
felt about the analogies section. "There's no way I can memorize every
possible answer."

------
micro_cam
As a mathematician who made the jump to software I feel that, while there is
certainly room for improvement in math education, methods based too much on
intuition and feel are a hugh step backwards.

Math is about thinking abstractly. To do it at the level required by modern
science, data analysis and engineering you need to be able to focus on the
abstract symbols, equations and rules that govern them without relying on an
intuition for underling objects. For example, no one has valid intuition for
fluid turbulence, n dimensional manifold theory or complicated probability
distributions, so great leeps in understanding these are made by people who
have an intuition for how equations behave and rigorously show that it is
valid. Real world applications are often only found after the fact.

I think that they key to a good math education is not just showing students
real world application but teaching them to find beauty and pleasure in
abstract symbolic reasoning and the rigors of proof.

------
vlad
My favorite part about the benefits of learning to solve problems rather than
memorizing skills for a particular section of a book at a time:

> When they had been at school, their social class, as determined by their
> parents' jobs, were the same at both schools. But eight years later, the
> young adults from Phoenix Park were working in more highly skilled or
> professional jobs than the Amber Hill adults... 65 percent of the Phoenix
> Park adults were in jobs more professional than their parents, compared with
> 23 percent of Amber Hill adults. In fact, _52 percent of Amber Hill adults
> were in less professional jobs than their parents_ , compared with only 15
> percent of the Phoenix Park graduates...

Students at the school that that taught problem solving rather than
memorization continued math education in college because they enjoyed it,
leading 65% of them to get better jobs than their parents at only 24 years
old. On the other hand, the majority of students at the traditional high
school avoided math in college, and chose "less professional" jobs than their
parents at 24. Over their lifespans, I can only imagine the disparity
increased...

> Which statement would give you more pleasure? "Because of good teaching, my
> child scored 79% on her last math test," or "Because of good teaching, my
> child has a much better job and leads a far more interesting and rewarding
> life than me."

The author mentions that even the best students at traditional schools had
trouble on tests since they practiced the skill-of-the-day on assignments,
leading me to believe they would show remarkable improvement on standardized
testing in a new curriculum. Maybe the author believes that testing students
after-the-fact is not as good a use of taxpayer money as providing resources
and education for teachers to change the curriculum would be.

~~~
barrkel
For context here: Phoenix Park was predominantly working class, while Amber
Hill was predominantly "affluent".

In this context, the results sound a lot like reversion the mean.

------
Lockyy
I experienced the strict setting of sitting in a classroom, facing forwards,
listening to the teacher give demonstrations and then answering tens of
questions that followed the same pattern. This was from 16-18 and was only a
small change from what I had experienced previously. Before that we had at
least been allowed to talk to each other, I think because the teacher
recognised how needed that is.

Once I was studying for my A-levels however I was in a group of people who
wanted to learn maths, we had chosen the subject after all. But we were given
a teacher who demanded that we were silent, demanded that when we had a
problem we asked her and not each other. Except for some students she came to
the conclusion that their not understanding was their fault and would just
tell them that they should have listened originally, for others it would just
take too long to get an answer because the class was overcrowded with
students. Twenty-five students at that level is far too many for one teacher.

Some students fell behind, and when they decided that instead of doing the
parts of the subject they had no chance of understanding they would work on
things they did understand to get up to speed, they had less advanced mock
papers taken from them and told to do what they were told to do.

Students dropped maths, very able students dropped the subject. A lot of
people's marks were hurt drastically due to just coming to hate the subject.
We knew that discussing things with each other would help us a lot, it was
good that I could turn to someone and ask them to explain something right then
without having to wait for the single teacher to deal with the five other
people who needed help before me. Hence, we did it anyway because it worked
and we weren't stupid enough to keep silent just because the teacher told us
to, we had to do this to learn, and we had voluntarily chosen the subject to
learn.

So yes, I completely agree with the argument that the traditional approach is
broken and that nobody wins. It's the only thing at that school that I have
bad memories of, every other department, even other teachers within the maths
department, knew the value of group based learning. A bad teacher using a bad
method can ruin peoples view of an entire subject extremely quickly which can
lead to disastrous results.

------
danibx
Where can I learn about math notation and how to read math symbols
expressions? Many times I find some articule that explain some idea using math
symbols. And I can understand nothing. This week I was looking for a method to
calculate the distance between two geolocation coordinates. Át. First the
article used math symbols to explain and I could understand nothing. I didnt
even know how to Google that "enigma". But a few lines below there was the
same algorithm, but This time it was written in Javascript. And it was very
easy and simple to understand. How to learn "math syntax and grammar"?

~~~
roryokane
Good question. Some resources I found:

<http://en.wikipedia.org/wiki/Table_of_mathematical_symbols>. It is more of a
reference than a tutorial. It also includes only symbols, not other features
of notation such as division lines and superscript/subscript.

[http://mathoverflow.net/questions/33152/is-there-a-
reference...](http://mathoverflow.net/questions/33152/is-there-a-reference-
containing-standard-mathematical-notations). Someone with a similar, more
specific question, looking for a dictionary of what to name variables. Nobody
found any such dictionary.

<http://www.alcyone.com/max/reference/maths/notation.html>. A decently large
list of notations. But it doesn’t have, for instance, the notation of
calculus, and you must run a web search yourself to learn more about what a
listed notation means.

Hopefully in conjunction these resources will help. But it’s too bad that
there isn’t anything quite like what you’re looking for; I would be curious to
read such a tutorial, too.

------
pessimizer
I'm all about physical metaphors being used to teach math - instead of handing
someone an equation and saying that the equation is what the concept being
taught _is,_ it's easier to remember the physical metaphor that the concept
_is like_ , and then to see how the equation describes both the concept and
the metaphor.

More of this: <https://en.wikipedia.org/wiki/Where_Mathematics_Comes_From>

~~~
RockofStrength
I'm a big fan of that approach. Betterexplained.com uses the method.

------
learc83
I was helping my 9th grade step brother with his math homework, and I've
noticed his math classes are no longer divided into subjects they way they
where when I was there 10 years ago (algebra, geometry, algebra II,
trig/precalc, calculus, discrete math)

His math book jumped around so much, that one week he was working on
probability, the next basic geometry, and the next simple factoring.

It looked exactly like someone wrote a whirlwind study guide that covered just
enough to pass some very specific standardized test.

~~~
zmoazeni
Personally, I would have preferred a breadth-first approach to mathematics
when I was in school. All of those concepts are related, but it's very
difficult to make those connections when you are "only learning
[algebra|geometry|etc]" in isolation.

Edit: related [http://steve-yegge.blogspot.com/2006/03/math-for-
programmers...](http://steve-yegge.blogspot.com/2006/03/math-for-
programmers.html)

~~~
learc83
This definitely isn't some kind of holistic approach to math. It's basically
just here memorize this probability forumla, and the algorithm for multiplying
two brackets together, because that's what you're getting tested on at the end
of the year.

I could understand if it was something like showing how bracket multplication
works by adding in an example from geometry like this
[http://math1sfun.wordpress.com/category/algebra/expanding-
br...](http://math1sfun.wordpress.com/category/algebra/expanding-brackets/)

But it's nothing like that at all.

------
OrionTheDog
Well I was not happy with the plug and chug approach to calculus so I wrote a
book that uses a more intuitive geometric approach (at last it is intuitive to
me) and put it all online for free at <http://www.thegistofcalculus.com> but
nobody cares to read it so I guess people are actually fine with just
memorizing everything.

It is only about 50 pages and explains the meat & potatoes of the math class.

------
jwingy
As someone who was taught the 'traditional' way of mathematics, can someone
give a few pointers of de-programming myself from the traditional way that I
was taught? (Although maybe it won't be so hard since I feel like I've
forgotten quite a bit)

Someone already recommended this for calculus which looks pretty promising:
<http://www.gutenberg.org/files/33283/33283-pdf.pdf>

~~~
fferen
If you can code, Project Euler is really good. Particularly the problems past
#100; those tend to concentrate more on mathematical insights than programming
ones. It's what kindled my interest in math in high school.

~~~
eru
Yes, but Project Euler merely gives you a stimulus to go out and research, and
a test to apply your new skills. They are not so much a tool on their to
acquire the skills.

E.g. learning about dynamic programming will simplify at least half of all
Project Euler problems. But it would be rather harder than necessary to try
and come up with all the generalities of dynamic programming just from
personal attempts at solving Project Euler problems.

------
jere
>In an international survey conducted in 2003, students from forty countries
were asked whether they agreed or disagreed with the statement: "When I study
math, I try to learn the answers to the problems off by heart."

Is "off by heart" really a well known phrase in 40 countries? I'm from the US
and this question really confused me (even though I'm aware of the phrase "by
heart").

~~~
xyzzyz
They obviously didn't ask this question in English in all these countries.

------
Mon_Ouie
This reminds me of one of the math teachers I had, who used a technique I
found quite interesting and effective compared to how other teachers taught us
about new concepts.

He would rarely come up with some theorem, ask us to solve a few exercises,
and go on to the next chapter. When he wanted to introduce a new concept, he'd
usually ask us to solve an equation or a more concrete probleme (I can
remember something like the fence problem mentioned in the article, which was
used to introduce us to finding the maximal output of a 2nd degree
expression). After that, we would either find a solution through some struggle
— and then try to figure out a more efficient approach — or we'd stumble on a
problem by trying to use approaches we'd been taught before — and therefore
try to find a better way to represent the problem.

On an amusing sidenote, while we were each working individually, he'd walk
around in the class, looking at what we were doing, and crossing out with a
red marker pen mistakes we'd made.

------
jasim
This captures the problem with mathematical education well enough:

> When Boaler would visit a class being taught in a Railside-like fashion and
> ask students what they were working on, they would describe the problem and
> how they were trying to solve it. When she asked the same questions of
> students being taught the traditional way, they would generally tell them
> what page of the book they were on. When she asked them, "But what are you
> actually doing?" they would answer "Oh, I'm doing number 3." [p.98]

The typical math education relies a lot on rote learning (just take a look at
the cheatsheets here: <http://tutorial.math.lamar.edu/cheat_table.aspx>).
There are a lot of equations you just have to remember by heart to be able to
be productive in solving the typical textbook problems. For a high-school
student who does not have any insight on the beauty of mathematics, this is a
huge turn off. They leave school with the impression that this is a cold hard
subject with perfect proofs that you have to learn by rote, and nothing more.
No one talks about why Math is beautiful. The most positive thing about Math
I've heard from people is that it is the best subject to get a perfect score.
The arts are subjective and there is no perfectly right answer, but Math, if
you know how to solve these types of problems without missing a sign or a
bracket here or there, you get a 100/100.

The problem, I think, is because we teach the results of hundreds of years of
evolution of Maths. For example, most courses on Calculus start teaching it by
talking about Limits, and from there moves onto Differentiation. Integration
is considered to be the 'advanced' part of Calculus. It was very recently that
I discovered Apostle's textbook on Calculus (students of universities who use
that text are lucky) where he treats Integration first - because that is the
right historical order in which Calculus evolved.

I could appreciate it a lot more when I understood what kind of problems were
Newton and Leibniz trying to solve when they came up with the formalized
notion of Calculus. Calculus was described by them using the concept of
'infinitesimals', not through Limits. Limits was a clever abstraction that was
evolved later to better explain Calculus and keep it consistent. But when we
start teaching students Calculus with Limits, show them the perfect way where
Limits can be used to find the differentials of trigonometric functions, they
do not know this background. For them, there is no moment of 'awe'. They are
not even shown a glimpse of the amazing intellectual pursuit that was behind
this fantastic subject. All you see are a bunch of equations, some proofs that
are mathematically perfect, and you just learn them by rote.

The typical Math education needs to focus more on the evolution of the
subject, the pains faced by mathematicians (or physicists!) to which they came
up with these solutions. The logical gaps in new ideas and how they were
filled later. Let the students understand that this is not a 'perfect'
subject. There were human beings who faced real problems who came up with
these solutions. Even better, let them understand that some of the things they
learn was the result an intellectual pastime for these mathematicians. It was
imperfect, and there was joy when mathematicians brought it closer to
perfection.

With the advent of computers, most of the evaluation criteria used in High
School Math is becoming redundant. Moving from one step to another without
making careless mistakes is priority number one now. If we reduce the
importance on that manual aspect of the typical Math problem solving, and
instead focus on teaching the more interesting, insightful things about Math,
the students will go away with a totally different idea about Maths. Like
programming, it becomes a universe of abstractions where your curiosity drives
you to learn more.

~~~
tgrass
After Vector Calc, I wanted to go back to the fundamentals, to understand
instead of remembering.

I came across Silvanus Thompson's 1910 reprinted textbook Calculus Made Easy
[1], and it was hands down the best primer on any topic I've delved into.

1\. [http://www.amazon.co.uk/Calculus-Made-Easy-Very-Simplest-
Int...](http://www.amazon.co.uk/Calculus-Made-Easy-Very-Simplest-
Introduction/dp/0312185480)

~~~
jasim
I've Silvanus sitting in my shelf, but am yet to look into it yet.

A couple of recommendations (not specific to just Calculus):

\- What is Mathematics? (Courant [http://www.amazon.com/Mathematics-
Elementary-Approach-Ideas-...](http://www.amazon.com/Mathematics-Elementary-
Approach-Ideas-Methods/dp/0195105192))

\- Calculus (Apostle [http://www.amazon.com/Calculus-Vol-One-Variable-
Introduction...](http://www.amazon.com/Calculus-Vol-One-Variable-Introduction-
Algebra/dp/0471000051)).

\- Mathematics from the Birth of Numbers ([http://www.amazon.com/Mathematics-
Birth-Numbers-Jan-Gullberg...](http://www.amazon.com/Mathematics-Birth-
Numbers-Jan-Gullberg/dp/039304002X)) This book was written by a Swedish
surgeon without any background in Mathematics. He started working on this when
his son started attending university. A recommended read.

\- The Calculus Lifesaver (Adrian Banner). This book is supposed to be a guide
for students to crack their exams. But I found the book surprisingly
informative. <http://press.princeton.edu/titles/8351.html>

\- Godel Escher Bach. I've read only the first couple of chapters. My interest
in mathematics was rekindled to a great degree by Godel and the Incompleteness
Theorem.
([http://en.wikipedia.org/wiki/Kurt_G%C3%B6del#The_Incompleten...](http://en.wikipedia.org/wiki/Kurt_G%C3%B6del#The_Incompleteness_Theorem))

\- <http://us.metamath.org/>. The concept alone makes me happy! Metamath is a
collection of machine verifiable proofs. It uses ZFG to use prove complicated
proofs by breaking it down to the most basic axioms. The fundamental idea is
substitution - take a complicated proof, substitute it with valid expressions
from a lower level and keep at it. It introduced me to ZFG and after wondering
why 'Sets' were being taught repeatedly over the course of years when the only
useful thing I found was Venn diagrams and calculating intersection and union
counts, I finally understood that Set theory underpins Mathematical logic and
vaguely how.

\- The Philosophy of Mathematics. From the wiki: studies the philosophical
assumptions, foundations, and implications of mathematics. It helped me
understand how Mathematics is a science of abstractions. It finally validated
the science as something that could be interesting and creative.
<http://plato.stanford.edu/entries/philosophy-mathematics/>

I think the Philosophy of Mathematics should be taught during undergraduate
courses that has Maths. It helps the students understand the nature of
mathematics (at least the debates about it), which is usually pretty fuzzy for
everyone.

~~~
rsanchez1
If you've only read the first few chapters of Godel Escher Bach, you should
really set a goal to continue reading. The book is filled with so much good
information presented in a digestible format. Topics are slowly revealed
throughout the book until you just get it. It's a great experience.

------
TreyS
"As someone who was taught the 'traditional' way of mathematics, can someone
give a few pointers of de-programming myself from the traditional way that I
was taught."

I tutor high school students in math and breaking the memorization, process
based approach to math is usually pretty difficult. The biggest thing is to
absolutely avoid memorizing or even looking at formulas. Instead, try and look
at things graphically. A lot of formulas seem confusing and unintuitive but if
you look at it on a graph, it makes a lot more sense.

A basic example is something like the distance formulas. Students often say
"Gah I can never remember the distance formula. I need to memorize it before
the test" because

d = sqrt( (x1 - x2)^2 + (y1 - y2)^2 ) seems complex and confusing.

But if you draw a right triangle, it's apparent that the distance formula is
just finding the hypotenuse.

In a nutshell, focus on the why's and not the how's.

------
atpaino
As an undergrad math major, I've just recently run into this problem as I
begin to take graduate level math courses. In the past (i.e. through high
school math and basic calculus) I have always had a deeper understanding of
the math going on such that I didn't really need to memorize many formulas or
rules because I could always "reinvent" them (at least the simpler ones such
as cos^2(x) + sin^2(x)=1) if I needed them.

I have found this level of understanding far more difficult to attain
recently. For example, this past semester I took a linear algebra course.
Whenever I am given a definition or property, I typically try to prove/justify
it to myself or at least figure out why it is interesting. I found in my
linear algebra course that this was very difficult to do, since there were so
many definitions that just seemed to be "dropped in", unexplained by the
professor. Thus, I found myself memorizing formulas and theorems instead of
having the deep understanding of them I was used to.

I may have gone off on a little bit of a tangent, seeing as this article
relates more to basic mathematics, but I think the underlying problem is the
same. I'm not really sure what the solution is in my case, but I would guess
that if my professor were to devote more time to explaining the usefulness of
some of the more abstract concepts (such as eigenvalues) I would probably feel
more comfortable with the subject.

------
angdis
I recently picked up my neice's high school alegbra book and the "mile wide
inch deep" problem is plainly evident: way too much clutter and convolving of
topics. When I helped her with some problems sets, it became apparent that the
teacher had been drilling the students with rigid mechanized approaches to
solving different types of problems. Actual THINKING about the problem at hand
and gaining intuitive grasp of graphing and relationships seems to be
completely ignored in the teaching. Very sad indeed.

------
jkn
_Parents whose own math education was more traditional believe [...] that the
presence of weaker students [in mixed ability groups] will drag down the
better ones._

These parents are so wrong. Nothing makes you better at math than to explain
what you have understood, well what you think you have understood, to other
students. It's a challenging exercise that will benefit the brightest (except
for pretentious asses I guess).

------
prezjordan
Is this satire? I think this is completely backwards. You can intuit math, you
can't intuit the king of England during the 1400s, or what the word
"defenestrate" means (well I guess you can if you know a bit of French or
Latin).

Everybody forms their own way of doing mathematics through the mind, but
vocabulary and history are simply memorization.

~~~
dllthomas
The title is a (depressing) quote from a student who apparently didn't
understand enough math to understand the problem with their position.

------
gavanwoolery
I did very poorly in math throughout high school (and mediocre in college). I
still am not that great, but I feel I have a far deeper understanding than
your average college or high school "A" student. It has nothing to do with how
smart I am. It has everything to do with the fact that I have been using math
in "real world" (ok, virtual world) problems for the last decade.

You absolutely cannot "teach" math (beyond the elementary level) and expect
students to fully understand how it works. Memorizing equations is not
understanding.

You must learn math, the hard way. Give a student a relevant problem, and tell
them to solve it. Don't tell them how to solve it, let them derive the
solution on their own. It might take weeks in some cases, but I guarantee they
will have a much better understanding and increase their general problem-
solving ability in all of mathematics.

------
krakensden
> The other, less common scene appears much more chaotic. Groups of students
> sit around circular tables discussing how to solve a particular problem, or
> standing at the whiteboard arguing about the best way to proceed. The
> teacher moves around the room talking with the different groups in turn,
> making suggestions as to how to proceed, or pointing out possible errors in
> a particular line of reasoning the students are following. Occasionally, the
> teacher will call the entire group to order and ask one group to explain
> their solution to the rest of the class, or to give a short, mini-lecture
> about a particular concept or method

I had some classes that did this. In was invariably even less useful than
"please memorize this formula", simply because of the social pressures.

------
tokenadult
Thanks for the many interesting comments. I'll reply jointly here to a few of
the comments, and invite further discussion on some points made in the
submitted article that I'm curious about.

From a top-level comment:

<http://news.ycombinator.com/item?id=4084559>

 _As someone who was taught the 'traditional' way of mathematics, can someone
give a few pointers of de-programming myself from the traditional way that I
was taught? (Although maybe it won't be so hard since I feel like I've
forgotten quite a bit)_

From a second-level comment, which has already received some helpful replies:

<http://news.ycombinator.com/item?id=4084426>

 _After Vector Calc, I wanted to go back to the fundamentals, to understand
instead of remembering._

There is a FAQ page on the Epsilon Camp site

<http://www.epsiloncamp.org/FAQ.php>

that includes some Frequently Asked Questions articles about learning
mathematics for deeper understanding. The FAQ article "Problems versus
Exercises"

<http://www.epsiloncamp.org/faq/faq_1.php>

relates to what kind of work to set for yourself to build deeper
understanding, and the FAQ article "Learning Mathematics"

<http://www.epsiloncamp.org/faq/faq_3.php>

points to writings by various mathematicians, including the book Numbers and
Geometry by John Stillwell, about how to appreciate mathematics as a deep,
connected subject.

The submitted article mentioned "Numerous studies over the past thirty years
have shown that when people of any age and any ability level are faced with
mathematical challenges that arise naturally in a real-world context that has
meaning for them, and where the outcome directly matters to them, they rapidly
achieve a high level of competence. How high? Typically 98 percent, that's how
high. I describe some of those studies in my book The Math Gene (Basic Books,
2000)." The most striking example of this that I remember from a news report
was a Wall Street Journal series in the 1990s that followed two young men in
an inner city ghetto, one who was a good high school student and the other who
was a street criminal. The street criminal usually skipped high school, but
happened to show up the day students could take one of the major standardized
tests (probably the PSAT, if I remember correctly). The street criminal, who
sold illegal drugs among other activities, scored just as well on the test as
the more regularly attending student who had learned most of his mathematics
from school lessons. That's a rather stark illustration of what's missing in
school lessons for children who don't have an outside-of-school environment
for learning mathematics.

<http://www.ams.org/notices/200502/fea-kenschaft.pdf>

The article also says that many students say, "You have to be willing to
accept that sometimes things don't look like - they don't see that you should
do them. Like they have a point. But you have to accept them." I wonder how
that relates to the quotation attributed to John von Neumann,

<http://en.wikiquote.org/wiki/John_von_Neumann>

"Young man, in mathematics you don't understand things. You just get used to
them."

And from a third-level comment:

<http://news.ycombinator.com/item?id=4084865>

 _I experience math (and programming) quite differently than learning a
language or painting: Once I grasp a concept, I can use it. Before that, it's
mostly useless to me._

I ask, because when I studied mathematics in school, I had a drive to
understand the general principles first before I launched into working on my
homework, while some of my classmates were successful--at least in the context
of school--by working on the homework and DEVELOPING some level of
understanding as they tried to figure out answers for the homework. (I was in
a "tracked" mathematics class, taking algebra in eighth grade in an era when
most Americans took algebra in tenth grade, if at all, and most of my
classmates had parents who were engineers or medical doctors and could ask
their parents for help at home if the school lessons were confusing, as they
often were.) I also have a very strongly visual approach to grappling with
mathematical problems. So when I first learned algebra, which was presented to
me as a bunch of "Do this to the equation, and then do this" with little
rationale, I found that very dissatisfying. Later in the school year, we
learned about coordinate graphing of systems of equations in the Cartesian
plane, and I remember thinking, "Why didn't you tell me this in the first
place?" For historical reasons, and perhaps for reasons of what most learners
consider most easy, usually purely procedural algebra for solving systems of
two equations in two unknowns has been taught in school before graphing
systems of equations in the coordinate plane. But for some learners, it would
be easier and more accessible to reverse that order. What do you think about
the issue of students working first according to instructions, to DEVELOP
understanding a la the von Neumann quotation, versus getting the "big
picture," perhaps explicitly visually, before working on problems.

I'll comment also that the approach taken to learning mathematics in school in
most of the newly industrialized countries of east Asia and southeast Asia is
plainly superior to the United States approach for at least two reasons:

1) the school textbooks in those countries explicitly encourage students to
THINK about why a procedure will or will not work, and about how many
different ways there might be to solve a problem, and

2) the school textbooks show multiple representations of most mathematical
concepts, building from "concrete to pictorial to abstract" as in the
Singapore Primary Mathematics series

[http://www.singaporemath.com/Primary_Mathematics_US_Ed_s/39....](http://www.singaporemath.com/Primary_Mathematics_US_Ed_s/39.htm)

and the follow-up New Elementary Mathematics series

<http://www.singaporemath.com/New_Elementary_Math_s/47.htm>

which interleave arithmetic, number theory, geometry, and algebra in
increasing depth and interconnection throughout all grade levels.

~~~
btilly
A few amusing anecdotes.

I had a lot of trouble learning to solve 2 equations in 2 variables because I
did not see the point. Given any word problem that you were supposed to solve
that way, I could solve it in my head. It wasn't until I was shown that I
couldn't solve 3 equations in 3 variables that I realized that I needed to
learn the boring way.

Secondly the single most useful thing that I did in school was try to generate
a table of how likely it was to get various dice rolls when you rolled 4
6-sided dice and took the top 3. I learned a lot from that, and that sparked
my interest in math.

Thirdly my biggest complaint about the way we teach stuff is that we present
matrices and matrix multiplication with no context. It makes no sense to
people. But if you know what a linear function, and realize that a matrix is
just a way to write one down, then matrix multiplication turns out to be just
function composition.

Just think how surprising the associative law is for matrix multiplication. I
remember sitting there thinking, "How on Earth did anyone think it up, and see
the associative law?" It becomes something you memorize because it makes no
sense.

But the associative law always holds for function composition. Given three
functions f, g, h and a thing they act on v, then by definition:

((f o g) o h)(v) = (f o g)(h(v)) = f(g(h(v))) = f((g o h)(v)) = (f o (g o
h))(v)

Since matrix multiplication is just a way to write out function composition
for a certain class of functions, it likewise must follow the associative law.
THAT is how they thought it up!

I can hear the complaints already. "Oh, but this is too abstract for college
students, they can't possibly understand this approach!" Bull. They can, and
they do if you have the guts to present it this way. I've done it, with
success.

~~~
roryokane
“… try to generate a table of how likely it was to get various dice rolls when
you rolled 4 6-sided dice and took the top 3.”

I felt the urge to code this. Here is the result:
<https://gist.github.com/2899137>. It doesn’t tell you “likelihood of various
dice rolls”; it can either print out the rolls for each trial or tell you how
common each of the 6 numbers were in all the rolls.

~~~
ihodes
That's simulating the answer; you can get an exact answer without using a
computer.

~~~
roryokane
That’s true. But I thought doing it by hand would require writing a tediously
large table because you have 6^3 possible roll results to give the probability
of, if you were actually going to write the “likelihood of various dice
rolls”. I suppose the appropriate compromise is a symbolic manipulation
program like Mathematica, which can work with exact numbers easily while
automating the creation of the table. (If anyone can explain the problem, it
would be great if they could link to a document demonstrating the solution on
somewhere like <http://www.mathics.net/> .) Or is there an easier, simpler way
to solve this by hand?

~~~
btilly
There are 6^4 possible rolls. But you don't need to list them all. You can
take shortcuts.

After doing it by hand, I became interested in how I could do it by
computer... :D

------
mistercow
The substance of the article seems good, but why was it necessary to mention
that it was a _female_ high school student? Is her lack of a penis somehow
involved with this discussion?

~~~
jere
Uh... perhaps since we have huge gender gaps in STEM fields, such a quotation
by a female high school student is especially alarming?

~~~
amcintyre
As mistercow says, it would have been a good idea to explain that. That detail
just hanging there with no explanation caused me to pause as well, because
I've encountered some mathematicians (not people I've worked with, just
visiting professors and such) that felt the need to tell stories about person
X, who was in some way not as clueful as they'd expected, "and by the way she
was a woman."

I'm not in any way suggesting that the article author (Devlin) is like that,
just that I would personally try not to make a statement like that without
proactively making it clear that I'm not "that guy."

------
mahmud
(Shrink the size of math text books. No reason why an algebra or trig text
needs to be 900 pages, each page in primary colors)

------
lurker14
" In Math You Have to Remember, In Other Subjects You Can Think About It"

Most math geniuses I know (including however much of a genius I am) know the
exact opposite! Other subjects are memorizing facts; you can noodle through
math problems using logic. In fact, much of our success in other subjects is
in weaseling through multiple-guess tests using mathematical logic!

~~~
colanderman
You should read the article instead of responding to the title, which is in
fact a quote by a student about the current state of education in the US.

