
You never did math in high school - xSwag
http://j2kun.svbtle.com/you-never-did-math-in-high-school/?_nospa=true
======
sfrank2147
I'm a former math teacher, now a programmer. I think he leaves out a few
considerations:

1) You need to be able to do basic calculations before you can do advanced
proofs. I taught a lot of high school seniors, and I had a ton of students who
were smart enough to handle abstract concepts, but couldn't follow along when
I showed them cool proofs because they got caught up on the basic calculations
(because they hadn't learned them well in middle/high school).

2) Good high school teachers DO do a lot of pattern recognition/abstract
reasoning. That's the entire idea behind a discovery lesson and constructivist
teaching - having students learn formulas by discovering patterns and
reasoning about them.

3) Again, as he points out, American high schools do do proofs in Geometry. He
thinks they're really pedantic, but there are good reasons why 2-column proofs
are so tedious. For one, students seeing proofs for the first time freak out,
so giving them structure helps. For another, if the students write out every
single step, it's easier to identify who really knows his/her stuff and who's
BSing.

~~~
vkjv
1\. This. And many of the smarter students also gravitate towards these
things. 2\. It's easier for many students to grasp things when they are not
abstract. 3\. Yes, it still rattles my brain that Algebra teachers force
students to memorize the quadratic formula. It's ridiculous. The method of
completing the square is straight forward, more applicable in other
situations, and can even derive that verbose formula. It only fosters the,
"memorize every possible form of the question that could be on the exam" type
of learning.

~~~
Grue3
The formula for quadratic equation solutions is not that hard to memorise. It
is useful to remember since it makes it possible to get the solutions
instantly, whether numerically or algebraically. It also exposes discriminant
of the equation, another useful concept (to instantly determine the number
real solutions). Of course, the derivation of these formulas should also be
taught, but it is really inefficient to derive things from scratch every time.

~~~
eli_gottlieb
I think I've actually used the quadratic formula maybe _twice_ outside of an
academic math class in which the quiz question was _designed_ to invoke the
quadratic formula.

I'm a computer scientist, one of the _more_ mathematical occupations.

I've _never_ been in a life-or-death situation where I thought, "aha, thank
God I memorized that formula in high school!". Also, I don't actually remember
it anymore, other than "negative b plus or minus the square root of b squared
minus four-a-c, all over two-a".

Well _huh_. I guess I _do_ remember it, but apparently at a verbal-auditory
level rather than visual-symbolic. As usual for me.

Still: I've never had occasion to _use_ that recitation I just made.

~~~
colomon
1\. I'm implemented the quadratic formula in software more than twice over the
years, I think. In, you know, real programs I was paid money to write.

2\. Have you ever been in a life-or-death situation where _anything_ you
learned in class in high school was useful to you? "Well, I would have died,
but then I remembered that Julius Caesar's last three words were not actually
'Et tu, Brute!' according to Shakespeare..."

------
matthewmacleod
Hmm. I'm not sure I agree.

It seems that "solving algebra problems and doing two-column geometry proofs"
is a necessary step on the road to "generating your own questions about
whatever interests you and trying to answer them". That is, an understanding
of the concepts and established mechanisms for dealing with abstract reasoning
and patterns is required in order to have any hope of moving further in
mathematics.

Contrary to the point made, we _do_ teach students music in school by
explaining and using the established tools we use to create music. We teach
notation, rhythm, keys, harmonies… we then exploit that to compose, perform or
understand music.

Mathematics has always seemed the same to me. I don't really use much of it
day-to-day, but occasionally I'll come across a geometry problem or something
when I'm building software; maybe I end up doodling triangles, and using basic
trig and algebraic manipulation to understand more or solve my problem.

Much of our teaching processes focus on skills, rather than a more abstract
notion of "education." There's been much said about why this is a bad thing;
I'm rather ambivalent on it myself, seeing from casual observation how much
benefit skill-focused education can offer to those who would otherwise simply
learn nothing. Of course, this works better where self-motivated students are
not stymied by too-strict adherence to curricula. IOW, perhaps we don't teach
math, but we do teach the skills that are required to "experience" math at a
later date.

So maybe I've convinced myself of the validity of the title, if not the
individual arguments.

~~~
6d0debc071
> It seems that "solving algebra problems and doing two-column geometry
> proofs" is a necessary step on the road to "generating your own questions
> about whatever interests you and trying to answer them". That is, an
> understanding of the concepts and established mechanisms for dealing with
> abstract reasoning and patterns is required in order to have any hope of
> moving further in mathematics.

\---

I'd agree with you, if what we taught was an understanding of the concepts and
established mechanisms. However, it seems to me that, most of what I saw in
schools was just symbol manipulation.

For example, people didn't actually seem to understand that to get the area of
a circle you took the radius, multiplied it by the ratio of the diameter to
the circumference and squared it. They understood that you took the radius,
multiplied it by a magic number, and for unknown reasons squared that.

The mapping of the symbols onto reality was often missing. It wasn't problem
solving beyond the level of having a lookup table in your head that said 'When
calculating an area do this, then this, then this.'

All that said, there are things it makes sense to memorise after you
understand them - low level components where the speed gained in doing so
allows you to use them in higher level abstractions. My point isn't that it
doesn't make sense to teach people tools. But that to just give them the tools
without the understanding of how they function seems harmful to their ability
to create and adapt their own tools down the line.

~~~
chowells

        The mapping of the symbols onto reality was often missing.
    

Perhaps because it's not a part of math? Math is an art. It's totally
unconcerned with things like "reality". If you're so concerned with reality,
you've probably never done math.

~~~
eli_gottlieb
This is what mathematicians _actually believe_.

~~~
nbouscal
Some do, others don't. There is an interesting dichotomy: much of mathematics
does appear to be completely dissociated from reality, and yet in the other
direction, reality appears to be entirely mathematics. I think to reach a
unification, it is absolutely essential that we have people working in each
direction. You aren't likely to develop a theory of (∞,n)-categories by
working backward from reality, and yet it turns out to be useful to have done
so once you start exploring e.g. topological quantum field theories.

------
macspoofing
>There’s one kind of student I routinely encounter, usually in a freshman
calculus course, that really boils my blood: the failing student who “has
always been good at math.”

I understand the larger point about the difference between high-school math
and high-level mathematics, but come on, don't be so pedantic! Everyone calls
whatever it is you study in High-School & Elementary school - Math.

On a related matter, physicists and other scientists have done a pretty good
job of communicating to the general public what it is they do. On the other
hand, very few people actually know what professional Mathematicians actually
do - something I realized when I struggled to explain it to my dad the other
day.

~~~
colomon
Also, put it into the terms he'd like to put it into: His failing student has
always been good at symbol manipulation, but now that he's gotten to this
teacher's more advanced symbol manipulation, he's failing. Suddenly it sounds
like the kid has a perfectly valid concern, quite possibly with this guy's
teaching style, and the teacher's response is to make fun of the kid for using
the word "math" in the fashion 99% of the American population uses it.

------
eagsalazar2
99% of kids who were "always good at math" will continue to be good at math in
college. So the _entire_ article is a rant against a straw man to make a case
for his beliefs on how math should be taught. Not that I disagree that math
education is stupid, but just saying this rant is has no foundation.

The 1% of kids who did well in high school and then fail in college because
they are so attached to their rote memorization of techniques have a
profoundly broken approach to problem solving that is bigger than the
education they received. I've tutored many kids exactly like that and it is
very hard to pry them free of that mentality. It is part of their personality.
Also, those kids were never really good at math in high school either and were
battling (using tutors for help frequently) uphill to get through their entire
primary curriculum.

The much bigger and _real_ tragedy of math education in the US is the very
large percentage of kids who have been labeled as "not good at math". __Those
__kids 99% of the time are actually plenty good at math but have fallen out of
the system because of frustration and a poor fit for their learning style.
Those kids don 't end up in universities trying to take calc for science
majors at all because they believe they aren't capable and that is a crime.

~~~
eric_h
A friend of mine who does math tutoring for high school seniors and college
freshman told me that many of her students who did well on their AP Calculus
exams did not actually understand what a derivative/integral was. They could
procedurally find the derivative/integral for a given function, but they
didn't know what it meant.

Admittedly, this is anecdotal, but it does seem to support the argument/rant
in the article.

I feel that I, personally, was very lucky in my high school mathematical
education in that my teachers exposed us to the concepts/meanings of all of
the operations we were doing long before they exposed us to the procedural
trivia of finding integrals/derivatives and the like. It is unfortunate that
not all schools are like this.

~~~
tunap
Just another anecdote... when I passed Calc 225 my 1st semester my counselor
informed me my math prerequisites were fulfilled(counting college prep in HS).
She did not encourage me to consider pursuing further, just that I didn't need
to take any more and focus on "business". Working PT/3rd shift & taking 19+
credits a semester I figured my path was paved in business so I took her
advice...Until 2nd year, taking motherload of hard classes, I was informed she
was incorrect & I had to take probability. I missed first the first week of
class, assuming cakewalk, and never caught up. The dominoes fell from there,
my GPA suffered & after 2 years of midnight merchandising I realized I did NOT
want to work sales or anything remotely retail. After grad, I washed out of
corp sales job & Merrill job in under 8 months. Went back to construction and
the picture hasn't been especially rosy ever since. Cest la vie.

TLDR: Take the maths for the sake of knowledge, even if you don't 'have to'.
It can't hurt.

------
tokenadult
I try to teach genuine mathematics to elementary school pupils in
supplementary classes during the weekends. Because what I do in my classes is
intentionally quite different from school lessons in mathematics in most
elementary schools, I have to explain my approach to new clients. My FAQ
"Problems versus Exercises"[1] is the first in a series of four FAQ documents
about how genuine mathematics involves problem-solving, and sometimes doing
something that looks frighteningly hard at first. I think this kind of
approach to mathematics can be helpful to hackers and to their children.

[1]
[http://www.epsiloncamp.org/ProblemsversusExercises.php](http://www.epsiloncamp.org/ProblemsversusExercises.php)

------
Typhon
I've already read that somewhere...

[http://mysite.science.uottawa.ca/mnewman/LockhartsLament.pdf](http://mysite.science.uottawa.ca/mnewman/LockhartsLament.pdf)

EDIT : Oops, I hadn't seen that he already linked to this text. That'll learn
me to post too quickly. Oh well.

~~~
ww2
Even the music metaphor is the same. And the mention of geometry proof. The
blogger is so uncreative.

~~~
sp332
Creative isn't the point. It's still a problem, so it's still worth talking
about!

~~~
brockhaywood
Agreed but not even a passing reference to the clear basis for his argument
makes me question what else he is regurgitating.

~~~
sp332
_(Fair disclaimer, this analogy is borrowed from a much better writer, Paul
Lockhart)._ And the words "Paul Lockhart" are hyperlinked to
[http://worrydream.com/refs/Lockhart-
MathematiciansLament.pdf](http://worrydream.com/refs/Lockhart-
MathematiciansLament.pdf) which Typhon already pointed out several comments
up-thread! Your reading comprehension needs a tune-up.

------
bitL
There is a huge disconnect between intuitive mathematics and the formalized
one taught at universities since the middle of the 20th century due to
Bourbaki's group. For many people this emptied mathematics and made it
inaccessible to a large portion of population, making them 2nd class citizens
of the future, group which would be otherwise capable of mastering it with a
proper pedagogical style.

IMO this is a pedagogical insanity, flooding young kids with formalisms that
took centuries to emerge without any explanation about their background and
enforcing form over content, which is what cuts many super talented people and
forces them to focus at different fields.

There are many problems with contemporary math that are conveniently avoided
(binary logic for example - most of the population doesn't believe it has any
connection to thinking due to weirdness of material implication and teacher's
insistence that this is the right way to think, never mentioning that its
distant father Aristotle was so discontent with it that he immediately
developed a first proto-modal logic), etc. If some constructionists and
intuitionists weren't going against the scientific current, we wouldn't have
had computers for a long time.

~~~
peterjancelis
I always really liked math in school, cause it was just logic and that appeals
to my lazy side. I was good at it too.

The last 2 years of high school, however, I had picked the 8 hour math options
(25% of total course time) and the fun was quickly beaten out of it by having
to learn formal ways to write a proof. Saying the same thing in plain language
was 'invalid'.

From that point on math felt more like learning a foreign language than about
doing logic.

~~~
axm
Math is a formalism of everyday logical reasoning. It's the difference between
a formal language and English.

Learning rigorous mathematics is frustrating at first because the veracity of
a statement can seem intuitively obvious but difficult to prove. However it is
a key stepping stone to modern mathematics and will considerably sharpen your
intuition after you've gone through the process.

------
bluedevil2k
I think a certain amount of blame has to go to the teachers though as well. My
personal anecdote, I did extremely well in Calc AB in high school, aced the AP
exam, aced first semester Calc III in college, then had a professor in linear
algebra who, in hindsight many years later, was a terrible teacher. He zoomed
through everything, didn't explain, and just _presented_ rather than taught. I
still remember his comment to help us understand - "if you're having trouble
picturing 11 dimensions, picture a 3D picture, but in 11 dimensions." Thanks!
My last math course, Partial Diffs I did well again. To a certain degree I
feel "math is math" but how it's taught is different from prof to prof.

~~~
jimktrains2
Are you me?

Halfway through Calc III it was interesting to watch the visual students, who
normally would do very well, have a rough time, while other students, who just
treated it as an abstract system had more overhead/trouble/were slower when
learning initially, but it paid off when getting to un-visualizable systems.

~~~
Steuard
In my experience, virtually all of Calc III is "visualizable" (even somewhat
esoteric stuff like Lagrange multipliers[1]), because it's mostly about
vectors (which have a natural geometric interpretation). My claim would be
that to be _really_ good at math, you need to be skilled at both visualization
(and other intuitions) and abstract systems. They complement each other well.

[1] E.g.
[http://www.slimy.com/~steuard/teaching/tutorials/Lagrange.ht...](http://www.slimy.com/~steuard/teaching/tutorials/Lagrange.html)

~~~
j2kun
Lagrange multipliers are esoteric? Oh man, I was under the impression that
they formed the basis for most many useful optimization techniques :)

------
Jtsummers
Past discussion with lots of comments, circa 12 February 2014:

[https://news.ycombinator.com/item?id=7221713](https://news.ycombinator.com/item?id=7221713)

~~~
tomaskazemekas
I wonder how did the submitter this time come up with a tag '/?_nospa=true' at
the end of the original url. ( [http://j2kun.svbtle.com/you-never-did-math-in-
high-school/?_...](http://j2kun.svbtle.com/you-never-did-math-in-high-
school/?_nospa=true)). Is it a way to resubmit an old item to HN again?

~~~
Jtsummers
Hadn't noticed that, but yes. Putting in content like that can get around the
duplicate filters.

------
jiaweihli
I was decent at math pre-college - placed in top 5 in several state-level
competitions, and enjoyed my fair share of more obscure branches that weren't
traditionally taught in school (number theory, combinatorics).

I agree with the majority of the author's points, but I despise his quick
judgement on freshman students complaining about calculus.

I also said the same 'ironically stupid thing' in my freshman year, but that's
because I _dreaded_ doing calculus as it's traditionally taught. It's much
harder to find elegance in calculus than it is in say, algebra or geometry.
(Mostly because the 'grunt' work behind it is so much more tedious.) Those are
similar to programming in the sense that coding has elegant, extensible
solutions and quick, dirty hacks. With calculus, I always felt like I was a
inadequate human version of Mathematica.

More simply put, I could always solve problems using shortcuts in high school
both to save time and to give myself more of a mental challenge. In intro
calculus classes, there is no such thing.

~~~
jimmaswell
I'd say derivatives/integrals being applied to velocity/acceleration/etc, and
using integrals to find volumes of revolution and such, are pretty elegant.

~~~
jiaweihli
I agree with you here, but I meant more from a process rather than application
perspective.

By analogy, it's like choosing to use options / maybe monad instead of null in
a service that takes input. The end service's functionality is the same
regardless! (maintainability might be a different matter though, hehe)

------
mephi5t0
I was in Math high school and in Math class. We had 12 of mixed math per week:
algebra, geometry, linear algebra, probability and what not. Too bad in
college we couldn't pick classes like in USA, so we had to do some of them
over again in 1st and 2nd semester. Our math teacher was a very old, smart and
animated Jewish guy. He received money from Soros fund for being awesome, and
also survived Nazi blockade of Leningrad. In his 80s he still went there EVERY
year to walk across frozen lake in memory of that event. For 4 years he taught
us he always asked after explaining the task: so, Joe, what do you think we
should do here? And we can go on and on about how we should find a common
denominator or perhaps build an additional triangle and... he just smile and
say "we need to think". And then he will tell us step by step his thought
process. By the end of the 12th grade we all knew answer to that question.

------
noonat
In my years of mentoring and tutoring, I've run into a number of students of
programming (and, indeed, teachers) with the same problem.

Programming is about problem solving first, and teaching programming is about
teaching someone to look at a problem analytically, and how to use an abstract
flow of logic to solve a problem, and how to diagnose issues with your own
logic when things go wrong. It's about critical thinking. It's about attention
to detail.

It's about all of these things first, and about slapping keys second. But too
many people see learning programming as learning the act of typing code, and
focus too much on rote memorization of syntax, or teaching tools instead of
thinking. Someone should come out of a course saying that they learned how to
think in this new way as programmer, not that they learned a new programming
language.

~~~
j2kun
The worst is when I spend five minutes helping a student understand what the
problem is, and then he concludes by asking me what he should type.

------
golergka
By the way, there are schools where you actually do math. I was lucky to get
into one of these on the second try, after 7 series of exams and interviews.
The math lessons (apart from algebra and geometry, which we had to learn too,
of course), were set up pretty simple: you were given a single sheet of paper
with some axioms and definitions about the topic at hand and a list of lemmas
and theorems that you had to prove. When you thought you could prove on of
them, you called on of the teachers (there were about 5 per class), sat down
with them, and tried to defend your proof. No homework, nothing else but this
sheet.

I didn't pursue a career in mathematics, like a lot of my classmates, but
these lessons gave me more anything else I did in all years spend on
'education'.

~~~
SamuelMulder
Do you mind sharing more information? What school was this?

~~~
golergka
Here's the book on math teaching lessons, and although it's in Russian, you
can get an overall idea of the level of math involved by skipping through the
pages: [http://www.mccme.ru/free-
books/57/davidovich.pdf](http://www.mccme.ru/free-books/57/davidovich.pdf)

It describes a 4-year program, from 8 to 11 grade of russian school, for kids
from 13-14 to 16-17 years old respectively.

~~~
SamuelMulder
Thanks! I'm very interested in this approach and will take a look. Wish I
could find something like it in English, as I don't speak Russian :)

------
paul_f
Other things you probably never did in High School: Science

~~~
graylights
Yes, a lot of high school subjects focus on the results rather then the
process. Community colleges are also guilty. It's not just contained to math
and science but even the arts and philosophy.

It's a part of the prevailing attitude "Everyone should go to college." High
schools focus on rote learning instead of critical thinking to improve their
chances of admission to a good college. Then those same colleges frown on
those mechanical methods. The worst part is we're training people to be
spoonfed knowledge rather then seek it.

~~~
raverbashing
The problem is of course in SAT and other admission tests that expect the rote
learning

~~~
jiaweihli
Ha, this problem is nowhere near as bad as it is in China, where your
admissions are nearly (outside of connections) solely based on your college
admission test results.

At least in the US, the most selective universities look at other aspects of
your application.

That's not to say there aren't other problems in the States, such as negative
discrimination / racial quotas.

~~~
thaumasiotes
I've been talking to a few chinese college students. I'd heard that the gaokao
was everything in college admissions, and I'm prepared to believe it plays a
strong role, but I was a little disoriented by the results of asking four
students how they got into their particular college:

复旦 student #1: My gaokao score wasn't high enough for 北大, but it was high
enough for 复旦, so I came here.

复旦 student #2: My high school recommended me to 复旦, they gave me an interview,
and then they offered me a place in the school of social science. Because of
the offer, I didn't need a very high gaokao score.

财大 student #1: I took 财大's own entrance exam and qualified for admission, so I
didn't put in any effort on the gaokao and got a low score. That meant my
gaokao score wouldn't get me in to any other universities, so I came to 财大.

财大 student #2: I took the gaokao and my score was high enough for 财大, but not
for 复旦, so now I'm at 财大.

So half of everyone I've asked fits the mold, but that seems like a low figure
to me. The sample size here is only four; have I stumbled upon a wildly
unrepresentative group?

~~~
jiaweihli
In the case of the second student, I feel there's some missing context there.
My guess is it was an alumni recommendation, which would explain the
circumstances.

For student #3 - I'm unaware of how popular the practice of having a test per
university is (or how practical it is to take them as a prospective
applicant). Maybe this is restricted to certain universities or majors, or
works similarly to EA or ED in the US?

------
vkjv
The author seems to be building a false dichotomy. Math is the combination of
two. To continue the use of the music analogy, if you can sing and write songs
but do not know any theory and cannot express your ideas to anyone but
yourself, is it truly music? As an amateur musician, I can say it is
absolutely frustrating to work with people who cannot transcribe or manipulate
their work in any meaningful way--even a simple tab would do.

Mathematics are the same way. Yes, you need to solve problems, but you also
need to solve problems in ways that can build on your past knowledge and be
shared with others.

(FYI, I'm a lifelong amateur musician, programmer, and data analyst. My formal
education consisted of a double electrical engineering / mathematics major.)

------
iopq
Some students are taught this way:

x + 5 = 10 the equals sign is a magical mirror so when you take operations
across it it changes them to the opposite of what they were

so adding five becomes subtracting five, multiplying by two becomes dividing
by two, etc.

x = 10 - 5 by way of magical mirror

~~~
jiaweihli
This is how I learned it - in hindsight, I feel this is easier to reason
about, as it forces you to apply operations synchronously. (As opposed to,
say, subtract 5 from both sides, then add 8 => which we can reduce to add 3.
But this reduction might just be noise to your brain.)

~~~
iopq
This way you get "proofs" like:
[http://www.math.hmc.edu/funfacts/ffiles/10001.1-8.shtml](http://www.math.hmc.edu/funfacts/ffiles/10001.1-8.shtml)

If you have an intuitive understanding like "both sides represent a number, as
long as I manipulate it the same way the equation stays true" then you
wouldn't have problems with logic when dividing by zero

~~~
jiaweihli
Yes, I understand the concept that you quoted. Theoretically, it's more sound.

However, I want to point out that the purpose of the 'magic mirror' method is
to ease mental math. It frees your brain from having to spend time / space on
the extra step of applying the same operation to both sides.

Putting this in the context of 'showing your work':

Normal: x / 3 = 4 => 3 * x / 3 = 4 * 3 => x = 12

Mirror: x / 3 = 4 => x = 4 * 3 = 12

It's a bit complex to describe - I feel how you do basic math such as this is
hardwired into your brain at a very young age.

------
cmollis
I find it interesting that the author states that essentially mathematics is
formalized 'pattern matching', yet rails against the insidious imposition of
these very patterns in the pedagogy (in the form of rote exercises).

Isn't naive pattern recognition the basis of deeper dimensional understanding
(ya know, the 'theory')? Isn't this how intelligence is built?

It seems pretty easy to make rag on the lack of 'true understanding', when
you've spent 25+ years recognizing the patterns.

------
dahart
I'd use different terminology.

Arithmetic, two column geometry proofs, and even transposing a sheet of music
to another key, are all math.

I think the author is talking about education, not really math. Jeremy wants
kids learning math to think about, be interested in, and search for meaning in
math. He's right that teaching rote mechanics doesn't lead to curiosity for
most people. But there are no high school classes that reliably lead to
curiosity.

------
graycat
> You never did math in high school

Yes I did. Your claim is BS.

Your claim is based on no knowledge of me and what math I learned in high
school and, thus, is incompetent.

Your claim is an insult to me and the math I did learn in high school.

In the article, you imply that a student's claim "I was always good at math"
is poorly founded. But for some students, that student claim is correct. Your
implication is based on no knowledge of me and is incompetent. Moreover, for
me you claim is flatly wrong -- In high school I always was good at math.
E.g., my plane geometry teacher was severe in the extreme, likely the most
competent in the city, and I commonly toasted her. Your implication that the
student's claim is poorly founded is an insult to my abilities at math.

The claim in your title is guaranteed to be wrong for some thousands or tens
of thousands or more good US high school math students present and past.

From research in applied math, I hold a Ph.D. degree from one of the world's
best research universities and have published peer-reviewed original research
in applied math and, thus, know what the heck I'm talking about.

You owe many thousands of good math students a profound apology.

------
girvo
I did. But only because of doing the elective advanced mathematics in high
school, which required me to learn proofs, work things out from first
principles, and basically do everything I'd end up doing again in first and
second year university (BAppSci in Mathematics). The thing is, it had an 80%
failure rate, because the rest of what we'd learnt was exactly how the OP
described. Such a shame.

------
geebee
This is an important topic, and I'm glad that it gets a lot of attention on
HN. I hope this spreads.

I've participated in a few of these discussions so far (apologies if you're
tired of hearing me repeat myself), and I truly believe that almost all these
issues are downstream of a single and very fundamental problem: math teachers
are rarely drawn from top math students.

Here in the US, we love a plan, and we have a potentially harmful concept of a
career ladder. A classroom teacher is on the bottom run, and it's considered
career progress to set the curriculum for all teachers.

My take on it is this - if the US drew it's math teachers from the top 10% of
math graduates, the "plan" wouldn't be nearly as important. Yes, it's a good
idea to have a general standard for where students should be, it's a good idea
to have some kind of training for math teachers, and it's a good idea to check
in every now and then. But think about the relative importance of "a great
curriculum" vs "teachers drawn form the top 10% of math graduates".

We could change up the curriculum to reflect the problems Mr Kun has
identified, but without an armada of top math teachers, it would make
absolutely no difference. If we drew our math teachers from the top ranks of
math students and allowed considerably autonomy (along with general
guidelines), I suspect many of the improvements Mr Kun talks about (along with
so many other complaints about math instruction) would happen on their own.

So... how do we get very strong math majors who are inclined to teach into the
profession, and how do we keep them there? To me, this is the upstream bug fix
that will be referenced (perhaps in one line) when all these other bugs are
closed.

~~~
misingnoglic
Definitely what needs to be done is that teachers need to be given more
respect for the work that they do. This includes not paying them below
anything else a mathematician could make elsewhere, and in general making it a
better environment (who cares if your math teacher is the best at math if s/he
has a bunch of bullshit standards to get through). I'm a college student who
would love to be a math/cs teacher but I don't see it as anything sustainable
or worth it.

~~~
geebee
Yeah, I see the same two issues: money and autonomy.

I don't think that teaching necessarily needs to pay as much as what strong
math majors can earn elsewhere, but it does need to clear the "comfortable
middle class" barrier. Here's the thing - the comfortable middle class barrier
includes owning a pleasant house in a safe neighborhood, travel, the ability
to afford child care, and so forth... in SF, that probably takes $200k or more
a year. Honestly, I'm not sure $200k even covers it. So this is a tall order
in some places. And while SF is insanely expensive, this is still true of a
lot of urban areas (los angeles, boston, certainly new york...)

Autonomy is also critical. Strong math students will not be willing to engage
in this profession if they are excessively constrained by a bureaucratic
curriculum that prevents them from applying their knowledge, skill, and
passion for teaching. They'll leave.

Now, combine the two - low pay and low autonomy, and there's no way.

I'm not sure people understand just how much money and autonomy it would take
to get really strong math and related students to go into teaching.
Incremental changes are welcome, but not close. This is an order of magnitude
difference.

------
upofadown
I started Electrical Engineering when I was 35 so I got to observe the high
school to university math transition from a semi-unique position. Yes, as the
author states, there were a lot of students with good high school math marks
that didn't really know any of the sort of math taught in university. But that
math turned out to be continuous functions (linear mostly), limits, and a
whole lot of memorized integrals and derivatives of those continuous
functions. This was in an era where people were carrying around lap tops.

The one token "numerical methods" class was all about solving ... wait for it
... continuous functions.

The problem is that to fix this weird situation we have to start teaching
iterative methods of solving problems at all levels of the educational system.
I have the distinct impression that people at those various levels tend to
assign blame to those at other levels (this article could be an example of
that).

So does it really matter that high school students don't learn the wrong math?
There is a much bigger question here.

------
tjr
The (seemingly intended to sound ridiculous) description of teaching music
matches pretty well with what I remember from my occasional K-12 music
classes. There was also an element of listening to music, but it wasn't
particularly interesting. My first exposure to what I see as a serious study
of music was in college.

But why single out math in particular? I look back on pretty much all of my
K-12 education as fairly trite and superficial, in terms of "doing real work
in the subject". My experience with, say, college-level history was much more
intense (and seemingly more true to the field of history) than anything in
high school. On the other hand, I'm not sure I would be able to do well at
"real math" like calculus or graph theory or whatever you wish to deem "real
math" if I was struggling with adding numbers and solving equations, and I
attribute getting past such struggles to doing bountiful rote exercises...

------
Jugurtha
In my experience, and opinion, going somewhere for learning purposes should be
a humbling experience. College should be a place where a student realizes how
little he knows.

I say this because in my experience, the first symptom of ignorance is a
feeling you know a lot.

I'm depressed since childhood because since a very young age, I had read about
great minds. How could I ever feel I'm "good at maths" after reading about
Gauss, or Galois?

It made me feel like the lowest form of life.

That is akin to the way Military Generals feel towards Alexander the Great:
You can be a great General, but you probably will never be Alexander the
Great.

Maybe this should be done freshman year: Before even a single "maths" course
is dispensed, a session on the achievements of Gauss and Galois, at age 17 or
19.

Maybe a brief discussion on who Lagrange was, and what he did in his teens.

This should take out any feeling of being "good at maths", and make students
shut their mouth and open their ears.

------
Bahamut
I had an interesting transition from high school to college as well - I aced
math classes beforehand, doing well in courses such as college Calculus III,
Linear Alegbra, and Differential Equations, but ended up failing an intro
proof class my first semester of college.

After failing that class, I still took enough from it to pass subsequent
proof-based math classes with good grades. Ultimately I left a top PhD program
in math after 4 years and did well for myself since that failure, but it's
interesting to note that transition from easy & menial calculations to full-on
hard logic that challenges you at the highest level mentally. Our education
system does a poor job of preparing us for it.

Edit: For further context, I was a top math & science student in NY, having
placed top 50 in competitions nationwide and similarly competitive state and
region-wide

------
buyx
In South Africa, with its bottom of the world rankings in school mathematics,
this problem is particularly acute. With the government-set matric (school-
leaving) exams it is sufficient to work through past exam papers and memorise
the answer patterns, to be guaranteed good marks. This isn't a recent
phenomenon, but has been the case for many years, although it seems to have
gotten worse in recent years. Although there is a more realistic matric exam
used by most private schools (Independent Examination Board), they will
inevitably have to lower their standards as well to remain competitive with
the government-set exams.

I am ashamed to admit that even when I got to university, I preferred the
handful of maths and physics lecturers who followed a similar approach - work
through the homework, memorise the answers, and pass.

~~~
chii
>I am ashamed... I preferred the handful of maths and physics lecturers who
followed a similar approach

no shame in admitting that you've been thru a broken system. Tertiary
education isn't about setting a bar, it's about learning and discovery -
setting a bar should be left for vocational training institutions, where you
get certified that you are capable of doing such and such. Universities
_should_ ostensibly be about personal learning and inherent motivation. I
would garner that you shouldn't even be rewarded with any sort of formal
certification from a university. Those who need such a formal cert ought to
take an exam from a vocational training and certification institution.

------
mitosis
To continue the author's music analogy, what do you think music class would be
like if it were taught by teachers who have never listened to an orchestra in
their life, are barely aware of the existence of different music styles, and
cannot play any instrument? That's right, it would consist of drawing treble
clefs, memorizing note positions, and transposing music.

Many high school math teachers actually have _no idea_ what math is about. It
is illogical to expect them to teach what they don't know.

There is no incentive for people who would make great math teachers to go into
teaching in America. No social recognition. Ridiculously low salaries.
Internal pressure to conform and avoid making bad teachers look bad. Who in
their right mind would willingly do that, when they could be doing
mathematics?

------
mcguire
The author has a strong point. There is a serious disconnect between "math" as
it is taught up through high school and "math" as it is taught in college (and
how it really is, as a subject).

"Math" in high school is about calculation. _Math_ is about useful
abstraction. Students are expected to jump that gap on their own, without any
outside help. At the same time as being expected to learn some concepts that
are actually fairly difficult in their own right.

God help any students that have a full math professor teaching freshman
calculus---the lectures will be about proofs while the homework and tests will
be about calculated answers.

------
JetSpiegel
So, who complies a list of subjects worth of being called Mathematics?

Is arithmetic not maths?

~~~
ColinWright
Arithmetic is to mathematics as planing a piece of wood is to building a piece
of furniture, or as typing is to programming. It's a critical, underlying
skill, but it's not the whole thing.

The question of what mathematics might actually be is like the question of
what pornography is. I can't define it, but I know it when I see it.

~~~
beejiu
From my experience of professional mathematicians at university, arithmetic
was usually their worst skill. They'd quite often struggle (for a short
moment) to multiply, say, 12 and 16 together. The public perception of a
mathematician is somebody who remembers Pi to a 1,000 digits. A real
mathematician doesn't care less.

~~~
lostcolony
This. So much this. Arithmetic is bookkeeping, it's turning an elegant
abstract process into an ugly solved instance through rote mechanical labor.

In most of my college math classes, I largely 'got' it, but what killed me,
and many other students, was the arithmetic. We could take multiple integrals,
but dammit, 2*3 /= 5. Thank God for partial credit.

------
mjhoy
The method of teaching might be the problem, but I think it's the wrong thing
to focus on. A teacher who loves math, who is inspired by it, will teach it
"right" (I think) even if ultimately they need to teach all of the boring
routine steps to solving some particular kind of problem. At least that is
what worked for me. Watching a teacher work with problems in class who loved
math and loved patterns made me aware there was a "there" there.

------
darkxanthos
This is timely for me as I've just started a course in Abstract Algebra after
taking a class on proofs. There doesn't seem to be a great way to get through
this course without full understanding of the concepts since it is so proof
laden.

The idea struck me yesterday that there's no real reason one couldn't teach
this in elementary school or Jr. High and maybe that would be amazing for
students? These kind of courses shape the way you reason.

------
websitescenes
Right on! Believe it or not, I failed every single math class I ever took
multiple times. Just so boring and pointless. It wasn't hard, just pointless.
I would like to see the way it's taught updated. It shouldn't be about the
semantics, it should be about critical thinking. That's interesting.

------
venomsnake
As a person from Eastern Europe (where hard sciences were very important
subjects) studied in a special school for math talented children and got
dragged screaming and unwilling to math competitions - I definitely did math
in high school.

Also some of the crazy stuff we did for physics/chemistry required a lot of
math.

------
taksintik
Must admit. I hated math because it always felt impractical. I think I was
misguided and really what I needed was a better teacher /well rounded
understanding of math /problem solving

------
Datsundere
Every kid should do calculus in high school. period

If not going into the nitty gritty detail, but an overview of how
derivatives/integrals are so applicable in the real world.

~~~
btilly
My experience when I taught math at a college level said that the ones who had
Calculus in high school were worse prepared. _shrug_

------
NDizzle
Okay. I only have a high school education. Where do I go from here? (not being
snarky - honest question. Khan academy?!)

------
graycat
"You never did math in high school"

Yes I did. The claim is false.

"I was always good at math."

From the ninth grade on, yes. The main means of measurement were standardized
tests of math ability and/or knowledge.

I'll compare 'math' aptitude, knowledge, and accomplishments with you any
time, any day, for money, marbles, or chalk. I'll give you a head start and
big odds, and I will totally blow you away.

F'get about my opinion. Instead, (1) in the ninth grade I was sent to a math
tournament, (2) twice I was sent to NSF summer programs in math, (3) I was a
math major in college and got 'Honors in Math' with a paper on group
representation theory, (4) my MATH SAT score was over 750 both times (strong
evidence of being "good at math"), (5) my CEEB math score was over 650, (6) I
never took freshman calculus, taught it to myself, alone, started in college
with sophomore calculus and made As, (7) got 800 on my Math GRE knowledge test
(means I knew some math), (8) used the differential equation

y' = k y (b - y)

to save FedEx (a viral growth equation for revenue projections that pleased
the Board and saved the company), i.e., an original application of math, (9)
used the statistics of power spectral estimation of stochastic processes to
'educate' some customers and win a competitive software development contract,
(10) did some original work in stochastic processes to answer a question for
the US Navy on global nuclear war limited to sea, (11) studied solid geometry
in high school and later used it, the law of cosines for spherical triangles,
to find great circle distances in a program, I designed and wrote, to schedule
the fleet at FedEx, a program that pleased the Board, enabled funding, and
saved the company, (12) my Ph.D. research was in stochastic optimal control,
complete with measurable selection, that is, 'math'.

The claim is false, badly false.

Finally, as you hint, we will end with original work done, and I will pull out
two of my peer-reviewed published papers and my Ph.D. dissertation, and with
what you wrote you won't have the prerequisites to read any of them. Then, you
lose the bet.

I feel sorry for your students. Go back to teaching the quadratic equation and
binomial coefficients and f'get about your broad views of 'math'.

------
the_watcher
I get what the author is frustrated with, although I don't necessarily agree
with his conclusion that you aren't doing math - you are doing math, but you
are not learning to use math the way it should be used (in his music analogy -
you are learning music, but you are not learning to use music the way it
should be used, which requires actually playing). It's a common, and valid
complaint about American education - we are very good at stripping out all
practical applications of a subject and simply teaching a method for solving a
problem that is completely unrelated to anything in reality. In one of Richard
Feynman's books, he writes about his experience editing textbooks. He notices
this, and in his typically fantastic way, rips the textbooks to shreds. One of
his examples went something like this:

Some problem, while accurate in that the process to get to the answer worked,
was a word problem. Part of this problem read "The Earth has two suns. One is
blue, the other is green." He stops this early in the problem, since even
here, they have made the problem about something that is completely unrelated
to reality. Think back to high school - you have not lived long enough (in
most cases) to be able to mentally reapply a process to a problem you have had
in your own life, so education should be going far out of its way to present
problems in a way that the kids encountering them can understand. The Wire had
a great example - using gambling to teach probability (there's a moral
argument there that I won't touch - the point is the kids could understand why
they were learning math, since it solved a problem or gave them an advantage
that they could immediately relate to). Programming could help with this (I
haven't taken a math class since my senior year of high school, and always did
well in all of them, but I feel like I understand why I would use algebra
better now than ever, since I can actually relate to the concept of variables
in reality, rather than "make the numbers into letters"). Baseball has taught
me more about statistics and data analysis than any class I ever took.

I remember learning about finding the slope of an equation in 7th grade. I had
a huge argument with my teacher (I was a bit of a handful back then), because
she actually could not give me a single example of why I would ever need to
care about the slope of a line beyond future math classes. That's a problem.

TL;DR - The author is likely more frustrated with American education's
tendency to remove all relatability from a subject (and then not arming
teacher's with good examples of how to reapply the methods they learned to a
problem that encourages more investigation) than he is strictly accurate about
students not actually doing math. And I agree.

~~~
pdevr
It is interesting that the author had to portray a hypothetical scenario of
learning music theory and never practising to get his point (which I more or
less agree with) across.

Sometimes, it is necessary to create unrealistic scenarios and problems
because real world problems may be too complicated for beginners. So you end
up oversimplifying. In this process, sometimes the essence of the material is
lost.

A solution is to pay equal attention to the problem sets and examples as the
rest of the course content.

------
mnemonicsloth
Does anyone else think this is a profoundly ignorant and destructive article
whose anodyne plausibility underscores the absence of a consistent model of
education in the cultural lexicon?

I'm not trying to be impolite here; I just want to know how far my views are
from the mainstream.

------
XorNot
Is this really an important question? We still suck at teaching whatever it
is, and that's a problem. What you call it is irrelevant.

~~~
wycx
A good first step is to come up with some possible reasons for why we suck.

I know I felt that mathematics was completely devoid of context during my high
school and undergraduate years.Now that I have problems to solve, I have a
real purpose when I go back and relearn what I was taught years ago.

Perhaps we should not have mathematics classes at all. Instead, we should just
expect to encounter mathematics every subject, and the mathematics is taught
where appropriate.

~~~
calibraxis
Also, I'd use the natural explanation, which the evidence points to: schools
are reasonably successful at inculcating conformity and ignorance.
([http://www.youtube.com/watch?v=pFf6_0T2ZoI](http://www.youtube.com/watch?v=pFf6_0T2ZoI))
If they didn't fulfill this social function, they'd be dismantled or fixed.

(Thinking otherwise is like thinking that invading Iraq was about WMDs and the
US gov't was noble-but-fumbling. Contorting ourselves into logical pretzels to
preserve an illusion.)

I wish people like Edward Frenkel well. (Author of "Love and Math: The Heart
of Hidden Reality", who tries to undo the damage done by math education.) But
they're fighting an educational system which is inherently opposed to
supporting critical thought, which fires effective teachers who refuse to work
in the correct ideological framework.

