
On Math Teaching: Lockhart's Lament - orib
http://www.maa.org/devlin/LockhartsLament.pdf
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aaco
This is a great reading.

I didn't read it all so far (but I will), but I think if people were taught
mathematics in a more explanatory way there would be a lot more of them
interested in it, and therefore interested in areas related to it, or at least
we wouldn't have a society so afraid of it.

It's amazing how many people don't know the very basics of it like, for
instance, how multiplication is just a compact way of expressing a repetitive
sum. And things like that happen because they were trying to _remember_
numbers and formulas instead of _understanding_ the concept.

It's just hard to forget about a concept, which gives the basis to
understanding more complex things. I think maths teaching would be more
productive if we spent more time elaborating on why things are done in some
way (or how to come up with a concept just by reasoning about it, without
doing calculations and writing symbols on paper) than trying to teach more
complex concepts for someone who doesn't understand the basics, and therefore
won't grasp the complex ones.

There's clearly something wrong about teaching maths, and very likely other
subjects too, or maybe something wrong about teaching (anything).

~~~
unalone
The problem is that schools don't teach logic. Everything that they teach is
arbitrary.

English class is essentially potpourri. Random order of books + Shakespeare.
No teaching as to what makes writing a joy to read. Only emphasis on the
relevant social issues.

History class is horrible, no explanations needed. It stinks because History
has so much potential. If you have a good teacher, the sort that teaches you
shades of grey, then it's amazing. But that's not the curriculum. Also,
there's no emphasis on what MATTERS, which is current events. Rarely do you
learn anything 80s onward. That means that most high school graduates are
functionally retarded in terms of modern politics and news.

The science classes vary between schools, but it's making certain types of
science MANDATORY that just lack logic. Bio and chem? I get physics, but even
there: most people just don't need to know, and NOBODY learns in a way that
makes them like it.

Electives are universally horrid. Teachers regularly have the attitude of "I
didn't specialize in it, so standards don't matter."

...and there's math, which is explained properly here.

I've speculated before on what I think a logical school would be like, and it
would be something like the following:

-Math and physics form the core learning at a younger age, because with them you can do anything.

-English courses focus not on movements, not even necessarily on "we think these guys are great," but emphasize different attitudes on how to write. Aesthetic, deconstructionist... teaching students to actually look at how writing works. Also, pop literature needs to be examined as well. Not teaching it ignores a field of literature that never gets brought up.

-Philosophy would be a great starter high school course.

-What we call "health/fitness" would focus less on STDs in 7th grade and focus MUCH, MUCH, MUCH more on understanding other people. It took me until about sixteen to learn that people aren't solely irrational beings, and the fact that it took me that long is that school is heavy on mysticism (like, assuming that older ALWAYS means better). That should be fixed.

-Emphasis on current affairs and technologies. Learning about the past should be focused on events rather than sweeping ideologies, and more comprehensive on a few key areas.

-All classes should emphasize writing and conveying ideas clearly. But that means no grading things based on length, either: people good at writing short should.

-No grading at all, actually. Grading has done nobody any good, least of all colleges that assume a high GPA necessarily means a working mind.

~~~
albertcardona
I agree with all, but: grading is currently a horrible means of evaluation,
but still some sort of evaluation. Without evaluation --read, _accountability_
\-- no forced studies can acomplish anything. And currently school is nothing
else than forced studies. Demeaning, even.

On the other hand, if you make school optional and create a culture of liking
or perceiving as essential to attend school for one's well-being, then
motivation is way different, and grading is then really an impediment.

~~~
unalone
I think - this is just me, mind you - that interviews are the best way of
weeding out really bright kids from pretenders. It means you can't hide behind
any words and have to rely on who you actually are. Although, granted, that's
not quite as efficient.

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ars
This lament applies to ALL the subjects in school, not just math. The problem
is you just can't teach the joy of each one of these subjects to each of the
students.

Some students will enjoy certain subjects and just 'get it'. But most won't,
and I do not think there is any to teach it. You just can't.

So you teach what you can: at least they can read, even if they don't
understand literature. Same for math: at least they can manipulate the
symbols.

And the proof of the problem is the "new math". It tries very hard to do what
he is saying and it's been a failure. At the end of it students don't know how
to do math.

Except for those students who excel at math - but they would excel at it no
matter how it was taught.

~~~
Retric
There is a stunning gap in basic capability between the best worst students,
but even within subjects there are large ranges. I found Calculus II in
collage a lot easer than mesmerizing multiplication tables in 2nd grade.

Many people say they _hate_ math but love logic puzzles like SUDOKU which is
math but not the type of thing they learn in school. The important thing is to
not confuse students that learning the basic tools like addition and spelling
have anything to do with the type of work you do as an engineer or writer. The
way the system works you need to have a wide range of basically irrelevant
skills before they are willing to teach real math the gray areas of history.
Grouping writing poetry in the same subject as grammar is probably harmful
because teachers spend so much time on grammar they the skip poetry because
it's to "advanced" and it's not on the tests etc. Yet, they don't build off of
each other and many of the greatest poets lived before grammar or spelling
rules where written down.

PS: I expect most students would pay more attention to the basics if I they
had some use for them.

~~~
donw
I think that 'finding some use for them' is an ultimately doomed approach, if
only because each student is going to have different ways in which they want
to use the mathematical tools with which they are presented.

My father, who is an 'educator', seems to think that the best way to learn
trigonometry is to use it to teach carpentry, but I can't think of anything
that would be more dull, for several reasons. One, because I imagine that well
less than half of the class would have a genuine interest in carpentry; and
Two, because building things with wood requires very little trigonometric
calculation, so nearly all of the examples would be highly contrived.

~~~
Retric
Trig _is_ useless. There is about 2weeks of useful math and the rest is a
waste of time. People like putting trig on tests but the basic 4 year HS math
program should be:

Logic + Algebra I, Geometry with a touch of number theory and topology, Grab
Bag (1 year including the useful parts of Algebra II, Trig, and pre Calculus),
Calculus I.

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rkts
I agree with this to some extent, but I think the author goes too far. The
purpose of math education is not to make theoretical mathematicians. Only a
small number of students in math classes will become mathematicians, and the
rest are acquiring a set of tools that they'll use to solve problems in the
real world. Proofs and conjectures and abstract reasoning, interesting as they
may be, are irrelevant to most people's needs.

Here's a better analogy: a mathematician is a composer, and people who use
math are singers and instrumentalists. There will always be a few people who
love composing, and their work is important. But the majority of people have
neither the ability nor the motivation to write original compositions. They'd
rather just be given music to play.

For the composers among us, this can be hard to stomach. Writing music is so
exhilarating, and playing is so mundane! How can anyone prefer the latter to
the former? Well, the fact is, most people do. The rest of humanity really is
different from us, and we need to accept that.

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donw
This is one of the best pieces I have read on mathematical education in quite
some time.

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greyman
I don't necessarily agree with the article. Math was my main subject at the
University, so while I work as a software developer, on paper I am a
mathematician.

I don't agree with the premise that math is an art, and that "Better to not
have math classes at all than to do what is currently being done." First and
foremost, Math is something which learns one to _think abstractly_ in
patterns. And this I believe is indispensable and useful for almost everyone -
even if the concrete pieces of knowledge will be forgotten later or not
directly applicable to one's work or life. Even the so-called "drill" is
somewhat useful.

Also, during my work in software companies, I was able to notice when someone
didn't receive solid Math at university level - they were lacking in the area
of their ability to think more abstractly and thoroughly about more complex
problems.

Of course, the education could be improved by having better teachers with
passion and ability to explain the ideas, no doubt about that.

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brentr
I was sad to see that not once did the author mention the works of Polya.
Discovering new ideas in mathematics ultimately boils down to solving an
existing problem in a new way or solving a completely new problem.

I believe students should be posed the fundamental questions that forced many
of the most beautiful ideas in math to be discovered. The advantage we have
now is that if the students get stuck, they don't have to wait 2000 years for
someone to discover how to find the area under a curve. We can give them
hints. The teacher can compress time with a well thought out question pointing
the students in the right direction.

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rw
Old news but this is something worth duping.

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msluyter
Imagine an ideal math class as Lockhart might envision: one that focuses on
the art, joy, and beauty of mathematics, in which students genuinely _play_
with mathematical ideas. Then, ask yourself how the progress of these students
could be evaluated via standardized testing. The two mix poorly.

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nazgulnarsil
I think the goal of school is supposed to be to teach children how to evaluate
which actions will be most beneficial to them. What school actually teaches is
how to be dishonest/illogical in all the correct ways for a given society.

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dominik
See also: <http://news.ycombinator.com/item?id=256638>

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anewaccountname
Can anyone link to the original pdf? Scribd always crashes my computer (no
other Flash apps seem to).

~~~
Hexstream
Click on the title outside of [scribd] for the original PDF.

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unalone
God, I remember reading this on reddit. Brilliant article.

