

The case against teaching math - petewarden
http://www.ordinary-gentlemen.com/2010/03/the-chloroformed-mind-the-case-against-teaching-math/

======
jerf
Bear with me here, I have to lay down some foundation to get to my point.

 _Bias_ has a specialized definition in machine learning; in a (really thin)
nutshell, it represents the set of concepts a given technique can represent.
If your banana vs. ball classifier can only represent "roundness" and
"yellowness", the bias will prevent it from seeing a yellow cube as anything
but a banana. (If you really dig in, I find this concept is really a superset
of the traditional meaning of "bias" and find it a lot more edifying, but
that's beyond this post.)

A learning agent, even a human, can only learn what their biases permit. Much
of intellectually growing up is the process of learning better biases, for
instance putting away "magic" and replacing it with "chemistry". If you've
ever _felt_ your brain stretching as you learn something (a foreign language,
Haskell, etc), that's your biases growing. Characterizing growing biases
mathematically has been a great challenge for machine learning, but it clearly
happens to humans.

Alright, getting up to the payload here: I have actually had similar concerns
about our educational patterns before because you can't effectively teach
something until you have the necessary biases in place. Unfortunately, you
can't stop learning, you are always learning. So what happens when you "teach"
something to students that lack the ability to apply the correct biases? You
get the square yellow cube effect above... one way or another it _will_ fit
the biases you have (give or take growth, which takes time). The result is
semantic gibberish. Now, by itself it is inevitable that you will go thorough
quite a lot of gibberish as you grow up (see also "Kids Say the Darndest
Things"), but why are we using precious school time to do that for math?

Furthermore, it is transparently obvious that things can be taught too soon.
My 18-month-old is frolicking around my feet now, and he can't add. There's
not much I can do to correct that right now. At some point he will, but this
suffices to prove the point that there is a "too soon". So, when is it no
longer "too soon"? Are we really sure that our traditional answer is anything
more than traditional, that it has any actual truth?

Teaching things early is not harmless, either. Think about it; how many adults
have downright childish issues with math? Childish misunderstandings, childish
opinions, childish beliefs? Coincidence? Bad motivation? Or a schooling system
that jammed it into their brains before they were ready. And those that don't
have childish problems... is it merely because we were ready soon enough,
rather that necessarily any actual unique skill? Some people never recover.

It is at least a question worth research and thought before we knee-jerk an
answer of "if it was good enough for my grandpappy it's good enough for you".

(Getting a little more controversial, this is why I don't support really early
sex education (i.e., elementary school). You will accomplish nothing except
hilarious misunderstandings on a very important topic. This can have real,
negative consequences, and the intentions count for nothing.)

~~~
epochwolf
_Think about it; how many adults have downright childish issues with math?
Childish misunderstandings, childish opinions, childish beliefs? Coincidence?
Bad motivation?_

Right here. I taught myself programming and I got pointers (references, same
difference) within the first week I started reading about php. I enjoy
learning new concepts except for math. I really want to learn math but I am so
afraid of just opening a math book and not understanding anything in it.
(Posttraumathematic stress disorder?)

~~~
rw
Are you in school? Get an independent study going with a professor on a topic
she likes and you wouldn't mind learning. Emphasize that you want to get your
training wheels for autodidactic math education.

~~~
epochwolf
Sadly, I'm graduating in 6 weeks. I think I'll have to figure it out on my
own.

~~~
buckler
May be you've seen these HN threads about self-learning maths already... if
not, hopefully you'll find these textbooks useful. Here's a little HN link
dump, in the order of time at which the discussions took place (from the
oldest to the most recent):

\- <http://news.ycombinator.com/item?id=108723>

\- <http://news.ycombinator.com/item?id=201913>

\- <http://news.ycombinator.com/item?id=458926>

\- <http://news.ycombinator.com/item?id=755043>

\- <http://news.ycombinator.com/item?id=1058359>

\- <http://news.ycombinator.com/item?id=1193352>

One caveat is that many of these textbooks do not come with a solutions
manual. So, in terms of practice, I think that's where the dearth of uploaded
lecture materials could come in. Sometimes, there might be no
e-learning/opencourseware material for the topic you're interested in. In such
cases, I found that if you look through universities' Websites closely enough,
professors upload a lot of course materials on their personal pages. You can
do this by finding a random university's math dept. page, then searching from
course/faculty listings.

Last of all, I'm pretty certain you've heard of this site already, as it's
being circulated quite a lot lately, but many have professed that videos from
khanacademy.org to be quite helpful when grasping basic mathematical concepts.

~~~
epochwolf
Thanks for the list. Saves me from cluttering HN with a another similar
thread. :)

------
teilo
This story is dead on. I know this from personal experience and from teaching
my own children. I could not add until I was 10. It was painful. I still
remember it vividly. Suddenly everything clicked, and I rocketed through all
the material I was behind on.

I have six children, five of which are old enough to be in school. We have
been home schooling them, I believe I have some creds on this.

Two of them (both girls) took right to math early on. No problem there. Two of
them (both boys) couldn't be bothered, no matter what method we tried.

One of the girls was doing math before she could read, because she was a late
reader. Couldn't read until she was 9. But suddenly she got it, and took off,
and is reading voraciously today. Her younger sister was reading at 4.

Both boys are now doing pre-algebra. One is 13, the other 11.

We decided not to sweat it when thy were way behind. We just kept getting them
to do what they could. If they weren't getting some concept, we went slower,
or went to other topics they could understand. Eventually everything started
clicking.

Cookie-cutter education could never do this. I agree that national education
standards that fit all children into specific requirements by age, are foolish
and harmful. As guidelines they are fine, but if the government is behind them
they are never just guidelines. They won't fix the problem of bad and lazy
teachers (in fact, they make that problem worse), and they don't help good
teachers teach better.

~~~
abstractbill
Just adding my own experience to this: Until the age of 15, maths was
considered a real "problem subject" for me. At every parent-teacher conference
it was generally accepted that I would never be good at it (and I had no
problem with that at all - I found the whole subject very boring and
difficult).

Then one day, at age 15, a classmate showed me how to do quadratic equations.
I have no idea why, but that day a really bright light suddenly switched on
for me. I quickly caught up to, and then passed the abilities of everyone else
in the class, and then started teaching myself calculus in my spare time. I
went on to get a PhD in Pure Mathematics.

I do think I just _wasn't ready_ for a long time.

~~~
bokonist
I'm curious - before math clicked for you, were you good at other subjects and
generally one of the brightest in your class? IE - did your teachers think you
were dumb overall, or just that you were a smart kid who for some reason
didn't get math?

~~~
abstractbill
I was always good in the sciences, especially physics, and also in music. My
teachers seemed to think I was smart, but they definitely thought I would
never get maths.

------
shadowsun7
I disagree with this.

Sure, kids don't like math. I myself didn't like math when I was a kid. Why?
Well, because math takes work. Sometimes lots of work.

I believe a solution to this has to do with _how_ math is taught, as opposed
to just abolishing the subject altogether. I don't hate math any longer - I
can sometimes see the beauty in it, and I sure as hell know that it comes in
handy, especially when working with computers.

But even if it isn't taught properly - there is something to be said for doing
math at an elementary level. Working on math problems tend to give you a
certain kind of mental tenacity, that would come in handy later. The question,
really, is how to make math seem fun/relevant - especially when it gets more
abstract.

Steven Strogatz has some cool ideas:
[http://opinionator.blogs.nytimes.com/category/steven-
strogat...](http://opinionator.blogs.nytimes.com/category/steven-strogatz/)

~~~
barrkel
I used to skive off school and go down the woods and do maths problems to
entertain myself, so that I could avoid Irish lessons.

------
greenlblue
Completely False. I started to learn mathematics starting in preschool and
when I arrived in the US I was shocked by what passed as mathematics education
in the US. The stuff American students where doing in 6th grade I had already
done in 3rd grade. It might be that starting later is a good idea but the
exact opposite was true for me. If I had been forced to learn mathematics the
way American students learn it, well then I would have been bored out of my
mind and would have given up learning it.

~~~
Qz
The American education system is constantly changing. I don't even know how
they teach math to kids these days but I would probably think it was
completely foolish. Apparently they changed the way they teach spelling and
vocabulary and I think the new method is complete nonsense.

But the point of the article is that the current mantra seems to be that if
'kids aren't good at math in grade X' then 'teach even more math at grade
X-1', which has diminishing returns when you start getting down to
kindergarten and such.

------
Tichy
I learned maths AND was able to solve "story problems" in elementary school.

Looking back, the maths taught in elementary school is very basic, too. Just
multiplication and division of integers, not even fractions yet.

To be honest, I don't care that much anymore if somebody doesn't want to learn
maths. It's just their loss. But maths was one of the few things I enjoyed at
school, so I absolutely see no point in removing it from the curriculum.

There are some school forms (like Montessori I think, quite popular here in
Germany), where pupils can apparently choose for themselves what they want to
learn at any particular moment. Not sure how exactly it works, but might be a
good idea.

~~~
lotharbot
In Montessori, there is some emphasis on choosing what you're interested in,
but I think the more important part (when it comes to math) is that it's
typically taught using strong visuals, manipulation of number blocks/cubes,
etc. Instead of focusing on rote memorization of algorithms, the focus is on
patterns and sensory experience. Kids "get" patterns; what they struggle with
is algorithms applied to symbols.

I went through "normal" school and was always way ahead in math. My
Montessori-educated sister, 3 years younger than me and not particularly
mathematically inclined, could intelligently converse with me about my algebra
homework when I was in sixth grade and she was in third. She wasn't as
familiar with the symbols as I was, but she recognized many of the patterns
(like difference of squares) in physical form.

The Montessori math approach is outlined at
<http://www.infomontessori.com/mathematics/introduction.htm>

------
tokenadult
This is blogspam. The original article from Psychology Today was already
submitted to HN.

<http://news.ycombinator.com/item?id=1211198>

But I must acknowledge that some searches for submissions of the original
article that I just tried on Google failed, although SearchYC works with
enough keywords.

~~~
epochwolf
I don't think this sinks to the level of blogspam. The OP made a summary of
the original and added his own comments. I looked at the first few entries on
the home page and most of them seemed to be in this format. There is original
content here and the guy writes well. To call it blogspam is (in my opinion)
wrong.

Note: Wikipedia defines blogspam as "the post of a blogger who creates no-
value added posts to submit them to other sites".

~~~
bmj
I agree. And the folks at that blog don't typically do the blogspam thing, at
least in my experience.

------
metamemetics
No math until after elementary school? what? I remember doing multiplication
timed tests in 2nd grade, waiting until 6th grade would seem to be catering to
the lowest common denominator and doing a great disservice to many bright
students.

------
cabalamat
Can someone tell me WTF "pre-calculus" is meant to mean? AFAICT, it's just a
fancy word for algebra.

~~~
euroclydon
If I remember, since geometry is such a large topic, pre-calculus also teaches
the parts of geometry necessary for traditional calculus study. That way, the
geometry course can be made more simple.

~~~
cabalamat
You don't need any geometry to understand algebra. Though graphs (and
therefore cartesian co-ordinates) certainly help).

~~~
euroclydon
To understand calculus, not algebra.

~~~
cabalamat
Well corrected. |You don't need geometry to understand calculus, either.

------
hkuo
This is silly. He's on his way to make a good point, but fails when his main
points become 1) no math 2) focus on stories and language early on.

The good point he could have arrived at is to put less emphasis on things a
child is not immediately acquiring. It would be like trying to force-feed
mashed potatoes down a drinking straw. Perhaps sometimes it's math. Perhaps
sometimes it's music.

Instead, he just says, stories and language are the important part! Leave math
for later!

What a buffoon. He started with a bigger picture that ended with a close-
minded detail.

To the original author, remind yourself that "not everyone is you." Kids can
grow up to be scientists, artists, doctors, lawyers, even _gasp_
mathematicians! And we all take different paths, even at a young age.

------
derefr
Just a slight quibble:

> R–recitation. He wrote that by "recitation" he meant, "speaking the English
> language." He did "not mean giving back, verbatim, the words of the teacher
> or the textbook." The children would be asked to talk about topics that
> interested them–experiences they had had, movies they had seen, or anything
> that would lead to genuine, lively communication and discussion. This, he
> thought, would help them develop the capacity to reason and communicate
> logically.

He means "rhetoric." That's what the _second_ "R" of the "Three Rs" (a.k.a.
the Trivium) is supposed to mean.

------
derefr
This occurred to me when I took my first college psych course, and we were
talking about "operational stages." The whole lecture was about how kids can't
work with disconnected mental abstractions until they reach a certain level of
cognitive development (the Formal Operational Stage.) It was also enforced
that moving between stages of cognitive development just takes _time_ , not
more or better prerequisite education. In that light, it's kind of obvious
that young children shouldn't be being taught math, any more than they should
be being taught first-order logic.

------
arvinjoar
I think that intrinsic motivation is key, without it, it will be impossible to
really learn. If you don't have intrinsic motivation to learn something, you
can always go fotr the monkey approach, where you ape after your instructor to
finish a task, or solve a problem. That is by no means "learning" though.

------
WingForward
You lost me at "Psychology Today".

