
You never did math in high school - DavidChouinard
http://j2kun.svbtle.com/you-never-did-math-in-high-school
======
cperciva
Yes I did. Also:

 _when a freshman college student tells me that he was always good at math, it
translates to “I was very good at following obscure steps to manipulate
mysterious symbols, without any real understanding of what they mean.”_

The most common complaint I hear about undergraduate students is that they
_aren 't able to blindly manipulate symbols_, and consequently get hopelessly
confused when they have to deal with "unintuitive" concepts.

~~~
baddox
It's even worse than that. It's more like "I was very good at doing enough of
the practice/homework problems to memorize all the possible combinations of
problems that will end up on the test."

~~~
Pxtl
Exactly. You have to remember who the loudest complainers in a school are -
not the nerds, not the losers, but the _keeners_. The kids who play the game
of earning marks instead of learning to think.

That's who the current system panders to, because they're the ones who scream
loudest when it doesn't. And they find it morally offensive if you ever ask
them to solve a problem that isn't _extremely_ similar to an example they've
already seen.

And they're the ones who think they're "good at math". That's who highschool
caters to because that's what standardized testing caters to and that's what
the noisiest parents and students want... and teachers that have higher ideals
are swimming up-stream if they want to actually challenge the kids.

------
ruswick
The OP comes across as very fond of deriding the US schooling system, but
proposes nothing beyond an egotistical assurance that _he_ can teach math well
even though no one else can.

Math education is _not_ homogenous. People are not homogenous. Everyone
experiences math in different ways both in school and in everyday life. Some
people will simply memorize the steps necessary to solve problems. Some will
actually grasp the theory behind the problem solving. Some will experience
very little of what is considered "classic" high school math. (At my school,
those who are not tracked into calculus or statistics as Seniors take a course
that deals with things like formal logic. It actually seems really
interesting.) The OP's blanket assertions are simply wrong.

Moreover, math is a massive field. The OP's claim that calculus and algebra
are not math but that "pattern recognition" is is just absurd.

~~~
m_ke
Jeremy is a phd student at UIC right now and I'm pretty sure he taught a few
classes there.

As someone who moved to the states at 13 I have to agree with him. Most
teachers in America are forced to prep their students for exams instead of
teaching them how to think critically. Instead of building an intuition,
students are forced to memorize rules and equations so that they can finish
their standardized exams in time. The problems done in class and for hw are of
the same format as the exams, with the numbers used being the only difference.
I've seen plenty of college students get offended when a professor puts a
problem on an exam that requires some critical thinking.

I did my math degree at a school that had a special program for future math
teachers and they were consistently the worst performing students in my
classes. Abstract Algebra was a disaster for them, most couldn't put together
even the simplest proofs. With exams approaching, they all begged the
professor to tell them which proofs they were required to memorize. Somehow I
got by with perfect scores by understanding the content while they had a tough
time pulling off anything above 20%.

~~~
yummyfajitas
The problem isn't prepping for exams, the problem is that the exams cover only
mechanical transformation.

Put real problem solving on the exam and this goes away. Of course, then
teachers who can't problem solve (most of them) will be up in arms...

~~~
m_ke
That's exactly what I was getting at.

------
thrush
I think that you're getting at a great point, but another contributing factor
may be that high school has a different scheduling structure.

    
    
      High School:
        Classes/Week   = 5
        Time/Class     = 45 min
        HW Frequency   = Every Day or Every few Days
        Exam Frequency = Every Week or Every Couple Weeks
    
      University:
        Classes/Week   = 2 or 3 + Discussion
        Time/Class     = 90 or 120 min
        HW Frequency   = Every Week or Every Couple Weeks
        Exam Frequency = Every Month or 3 per Semester
    

All I am trying to say is that difficulty of material is not always the
problem. Sometimes the system can be the problem. It's clear that university
expects you to spend more time doing independent study given there is more
time between sessions, yet many new students aren't accustomed to spending
their free time this way.

~~~
adamnemecek
Indeed. For me, the worst part about college was how everything was spaced
out. Once you got into the zone, you didn't have that much time before you had
to go to another class or something.

------
pbiggar
See Lockhart's Lament for an in-depth view of the same argument:
[http://www.maa.org/sites/default/files/pdf/devlin/LockhartsL...](http://www.maa.org/sites/default/files/pdf/devlin/LockhartsLament.pdf)

~~~
Zarel
He actually summarizes and links to Lockhart's Lament in the first six
paragraphs.

More interesting is his link near the end, about how he teaches graph theory
to college students: [http://jeremykun.com/2011/06/26/teaching-mathematics-
graph-t...](http://jeremykun.com/2011/06/26/teaching-mathematics-graph-
theory/)

~~~
pbiggar
Whoops, I read to the end but missed that link!

------
shalmanese
I remember my dad giving me the following problem on a long distance car ride:

An A4 sheet of paper, when folded in half, becomes an A3 sheet of paper with
the exact same ratio of long side to short side. Given this information, can
you figure out the what the ratio is?

I must of spent 3 hours of that car ride doodling on a piece of paper trying
to nut out that problem with no additional information or tools, ultimately
unsuccessfully.

To this day, I think that was when the light switch went off for me about math
as a creative problem solving endeavor and not just a rote series of
calculations.

~~~
potash
Interesting, I'd never thought about this before. In case you (or anyone else)
are still wondering...

If an A3 sheet is X units by Y units (with X the long side) and A4 is 2Y units
by X units then their ratios are X/Y and 2Y/X, respectively. If these are
equal we have

2Y/X = X/Y

cross multiplying gives

2Y^2 = X^2

rearranging and taking a square root (X and Y are positive) gives

X/Y = sqrt(2).

------
mamcx
I have tough that education must be more integrated. Take for example, the
Pythagoras theorem. Is a boring fact, with null use...

Wait!

If somebody tell me before that I can use it for get out of a jungle... I
could have listen better.

The class could have started like this:

"You were traveling in plane, when suddenly, it crashed. Nobody else survived.
You don't know where you are. It look like a jungle. Not civilization around,
no cellphone, nothing. You will die in 1 week if not reach civilization. How
can you escape?"

And if in history (at the same time that in maths) we talk about the man
(Pythagoras), and about the compass. Then in social about the problems in
traveling in the ancient cultures. Geography about maps. In artist class ( _"
Art" class was more about technical diagrams for me, we never ever do oleo or
similar stuff_), how draw maps and in spanish build histories about it. Then
all the clases related to each other. Probably the math class must be the last
of it, to make this build-up effective.

In short, all the classes connected around the _theme of the week_ or what are
we doing at the moment.

------
mullingitover
This just made me remember the whole class struglling with Riemann sums in
Calc II, and Russian professor yelling "This is grade school calculus!" at us.
Apparently Russians do Calculus in grade school.

~~~
Scene_Cast2
Maybe not grade school, but calculus comes in tenth grade or so, if my memory
is right. Quadratic equations are 8th grade, compared to 11th in North
America.

It's actually telling of the quality of education, given that I did better in
Math in college than in high school (in North America), and I still found it
not anywhere rigorous enough.

~~~
michael_h
There is no 'North America' when talking about education. There isn't even a
'United States'. The high school experience differs widely.

I went to public school in southeast Ohio. Quadratic equations (for me) were
8th grade. Calculus started midway through 10th, but didn't really get moving
until 11th grade.

~~~
amichal
My US public high school did not offer calculus at all during my tenure.
Resulted in very interesting conversations with university admissions
departments.

------
hawkharris
I remember reading a fun fact in an MIT faculty newsletter a few years ago:
the amount of money that American students spend failing calculus — paying for
the course and tutoring, then redoing it — is greater than the budget of
Avatar (over $300 million).

Every year, our students spend the budget of a James Cameron film on failing
to understand calculus...

------
analog31
I taught a college pre-calc course for a short time. We had "show your work"
exams, so I got to see how students solved problems. I think that many of them
had been taught "test taking skills" in high school, including a method of
finding answers to math problems by guess-and-try or process of elimination.
Basically, their teachers had hacked the standardized testing process.
(Probably including the AP exam).

Honestly, college math wasn't much better. Rather than confronting them with
critical thought, we simply replaced the old hack with a new one:

1\. Recognize the "form" of the problem, corresponding to a section in the
textbook.

2\. Plug the parameters of the problem into an equation solvable by the method
of that section.

3\. Apply the method and write the answer.

When I realized this, I told my students about it.

There was a chapter on maxima and minima. But the only function they had
learned was the quadratic, so all optimization problems boiled down to
arranging things into a quadratic. The text had them graphing each quadratic.
I decided it would be more interesting for the kids to see how we make our own
formulas, so we derived one for the optimum of a quadratic function, and
memorized it.

Now is this math or not? Well, I think there's a place in math that involves
classifying the forms of expressions and equations, sort of like taxonomy in
biology. But it shouldn't be the only thing.

------
6d0debc071
Any good books, or online resources, people can recommend on _actually_ doing
maths for people from a non-maths background?

I mean this is an old refrain, and not one I disagree with, but it's missing
the positive side of the argument.

~~~
trentmb
BOOK OF PROOF:
[http://www.people.vcu.edu/~rhammack/BookOfProof/index.html](http://www.people.vcu.edu/~rhammack/BookOfProof/index.html)

Foundations of Higher Mathematics: [http://www.amazon.com/Foundations-Higher-
Mathematics-Peter-F...](http://www.amazon.com/Foundations-Higher-Mathematics-
Peter-Fletcher/dp/053495166X)

How to Solve It:
[http://en.wikipedia.org/wiki/How_to_Solve_It](http://en.wikipedia.org/wiki/How_to_Solve_It)

EDIT:

Mathematical Reasoning: Writing and Proof:
[https://sites.google.com/site/mathematicalreasoning3ed/](https://sites.google.com/site/mathematicalreasoning3ed/)

~~~
aet
I used Foundations as a text in college, but I think it may be out of print
now. At more than $200 there may be more reasonable choices. Might be an
option from Dover that covers similar material.

------
Pxtl
I have to disagree with this.

For example, he attacks geometric proofs. Obviously, geometric proofs are
useless and pointless, except that they represent a very simple and intuitive
sandbox for learning proofs. Which is fantastic. How better would you teach
kids to understand what math _means_ beyond arithmetic? You give them a bunch
of simple rules that make obvious sense, and then show them how you can use
those to prove non-obvious things.

Our curriculum has all but dropped geometric proofs and the kids are poorer
for it.

I think our curriculum here in North America (I'm in Ontario, but I imagine
Americans have very similar classes) is fine - the problem is that people are
afraid to make the problems really hard and force the students to really
stretch their skills beyond just "same as the example but change the numbers".
That's what's missing, not some esoteric subject matter, just taking the
existing tools in the existing toolbox and pushing the kids to build _higher_
instead of _more_.

~~~
j2kun
It's the way geometry is taught, not the content, that is the problem. This is
essentially the argument for all types of math; they aren't "math" because
they're taught in a way that removes all the math and leaves... what exactly?

------
primitivesuave
Jeremy has put together some of the best graph theory educational notes, and I
refer to his website a lot when I'm trying to show teachers how to infuse
something abstract and meaningful into their traditional mathematics
education. I highly recommend checking out his work, he really exemplifies
modern mathematical education.

------
gboudrias
Well, I sure wish this article had more explaining of what we should actually
teach, instead of "go read this other article". I was waiting for the point
but it wasn't there.

------
graycat
Nonsense. I did four years of math in high school, and it really was 'math'.
It wasn't all research or 'critical thinking', but I didn't memorize "steps"
either -- I have an awful 'rote' memory, at least without some oral hints.

And I did fine with math later on, in including my Ph.D. in applied math
(stochastic optimal control) and peer reviewed publications in applied math --
with theorems and proofs.

The OP has a small point but takes it way, way too far.

------
asperous
This may be true but personally I have found that the mathematics I was taught
was good enough for most purposes.

People learn how to drive a car by doing it. They aren't encouraged to
discover the laws of combustion or shown the magic of kinematics. They just
learn how to press the petals.

~~~
jimmaswell
Well yes, but understanding the math is the important part for most people
unless they use it daily, while in driving a car it doesn't matter much as
long as you can drive it.

------
baddox
Interestingly, I observed that the inverse seemed to be true in college (I was
a computer science major). People (mostly math majors) who were _very good_
math students, meaning they breezed through the college calc courses,
struggled when we met again in more CS-relevant classes like Discrete
Mathematics and Algebraic Structures. It should be troubling to any educator
to see people excel at Calc 2 integrations, yet struggle to accomplish much
simpler (at least to me) tasks like proving that the product of two even
integers is an even integer.

~~~
cowsandmilk
> struggle to accomplish much simpler (at least to me) tasks like proving that
> the product of two even integers is an even integer.

I always see articles that claim I didn't do math in high school, and then
people talk about things like this. I did this in high school. In the US. In
2001.

Literally, one of our test questions was: (a) prove the product of two even
numbers is even (b) prove the product of two odd numbers is odd (c) prove the
product of an odd and even number is even

And this wasn't regurgitating a proof we learned in class or had on homework,
it was a question we had not seen before.

Blanket statements about what people see and don't see in high school are
worthless.

~~~
baddox
I didn't make any blanket statements. I simply didn't encounter any proofs in
high school math courses, other than "geometric proofs" which weren't formal
proofs.

------
joshontheweb
I definitely felt like my grade school education did me a disservice in the
math department. I liken it to learning grammar for 12 years without ever
writing an essay.

This is why I think programming should be taught starting in elementary
school. It gives you an application for the math you need to learn. I never
cared about learning algebra, calculus, or trigonometry until I started
programming. Now I wish I had paid attention and now I am going back and self
educating on the subjects in order to empower my programming.

------
brg
Other's have pointed this out already, but there is a need for everyone to
learn symbol manipulation, geometric representation, and 5 line arguments
before mathematics mastered. Saying that this is not part of a complete
mathematics education is saying that grammar and vocabulary are not part of an
English education.

And yes, my school did mathematics in high school. We went through much of
Euclid's Elements.

------
Imagenuity
My college experience was that teachers taught how to pass exams, as described
in the article. But when asking the teacher _how_ the math worked, they were
unable to _because they learned and were trained through the same system._
They didn't 'do' math or know math. They didn't possess the deeper
understanding I was looking for someone to explain.

------
nayefc
The assumption that's explaining why high schools in America SUCK: "freshman
calculus course".

In my school and country, we take calculus 1 & 2 in grade 11; that's two years
before we go to college. In grade 12, we take differential equations.

~~~
RogerL
Calculus is taught in high school. However, it is not mandatory/standard - you
have to be on the college track, and it is skippable. (these are all general
statements, different school districts make different choices).

For example, in my high school, in a fairly small rural school, we were taught
calculus in senior year. I was bored, and did it sooner, but that was the
general rule. We also had things like advanced placement biology, physics, and
so on. Generally speaking, at the end of those courses you can take the
optional AP tests, which allow you to skip the Physics I, Calc I&II, etc.,
that are the normal fare for freshmen.

I can't recall anyone who had diff eq taught in high school, though we really
did end up using it anyway in advanced physics. To roughly sketch the limits
of what we did, I remember needing Green's theorem, and having to prove things
using elliptic integrals, but nothing much beyond that.

~~~
nayefc
Interesting. When I came to college here in the US (CS, engineering
department), I was rather shocked that most, if not all, engineering students
came in with zero calculus or advanced physics.

------
skywhopper
He's spent _20 hours_ doing one-off lectures to high school math students, and
he believes he's an expert teacher? He seems to have an interesting approach
and can get the students intrigued, but giving an interesting hour-long
presentation and teaching a high school class for a full year are very
different things.

I'm not trying to be cynical or discouraging, but dial back the certainty in
your genius methods a couple of notches until you've done more than just
gotten kids interested for an hour.

------
xupybd
I love this article. I hated "Math" in school, loved in at university. Why
don't they teach real Math at school?

------
andrewcooke
wish i'd known about this earlier. when i didn't get something i assumed i had
to work harder. didn't realise i was supposed to blame the teacher. damn.

------
xname
I don't like this kind of claim. Even calculation of 1+1 is math. To be
constructive, instead of claiming what students learnt is not math (which
actually is math), please articulate what you think needs to be added to high
school math. However, please be aware that not everyone is expected to learn
calculus after high school.

~~~
DerpDerpDerp
A lot of people separate "math" (the process of reasoning about structures or
relations) from "arithmetic" (the process of calculating with numbers).

I think this is reasonable, as "math" is essentially just another name for
"meta-arithmetic", and we draw similar boundaries between science and
engineering, for example.

------
benihana
I wish I would have known this in high school and college. I was convinced I
sucked at math because I did poorly on math tests. It wasn't until I started
programming that I realized I _was_ good at math; I was just bad at the
chickenshit busy work they made us do in high school algebra and geometry.

