
A Mathematician’s Lament (2002) [pdf]  - gshrikant
https://www.maa.org/external_archive/devlin/LockhartsLament.pdf
======
mrcactu5
This is how people think of Math class:

"Music class is where we take out our staff paper, our teacher puts some notes
on the board, and we copy them or transpose them into a different key. We have
to make sure to get the clefs and key signatures right, and our teacher is
very picky about making sure we fill in our quarter-notes completely. One time
we had a chromatic scale problem and I did it right, but the teacher gave me
no credit because I had the stems pointing the wrong way."

~~~
nextos
That's very sad and very true.

For some reason, I stumbled upon Hubbard & Hubbard Vector Calculus a few days
after first reading this essay and it stroke me as the opposite to this wrong
math teaching.

------
jnbiche
I actually disagree pretty strongly with this essay. I think Lockhart is
advocating for an educational approach that would cater strongly (and
exclusively) to a particular learning style, specifically, those learners who
thrive when going from abstract to specific (deductive learning).

For those of us who do best with _inductive_ learning, the type of education
he proposes would bring even greater misery to our grade school education. It
was not until I took statistics and probability (for science and engineer
majors) in college that I truly began to enjoy math once again. I was able to
start with concrete ideas and applications, and then work my way back to the
theory behind them.

I'm now reading "Concrete Mathematics" and really enjoying it. Knuth's ideas
on math education are pretty diametrically opposed to those of Lockhart, as
far as I can tell, and they give rise to something very close to my ideal
learning environment for math.

Why not allow children to follow which ever of the two math paths that is best
suited to them, instead of forcing concrete thinkers into an abstract world,
and abstract thinkers into a concrete world?

Edit: I should probably add that I agree with Lockhart that there's a problem
in the way math is taught, but I disagree with him on the solution.

~~~
jacobolus
Endless drilling, memorization, and focus on pushing numbers through
“formulas” is not the same as “inductive learning”. It’s entirely possible to
do lots of pattern matching and bottom-up “inductive” problem solving in a way
consistent with Lockhart’s recommendations.

Here’s a great book chapter wherein a mathematician teaches some 6-year-olds
in what might be called an “inductive” way:
[http://www.ams.org/bookstore/pspdf/mcl-5-prev.pdf](http://www.ams.org/bookstore/pspdf/mcl-5-prev.pdf)
( _so much better_ than a standard first grade mathematics curriculum)

Edit: to clarify, I basically think you’re reading something into Lockhart’s
essay that isn’t there.

~~~
jnbiche
Agreed, which is why I included the note (preceding your comment) adding that
I agree with Lockhart that there's a problem in the way math is taught.

Edit: Can you expand on your idea? I did just re-read the essay (for the 3rd
time) and Lockhart seems pretty absolute in his belief that math should be
taught as a playful, creative abstraction. And he seems to disdain practical
applications of math. So how would you teach math in an inductive manner that
maintains the level of abstraction and "playfulness" he's advocating for?

By the way, love the book chapter! That's _exactly_ how I'm trying to teach my
kids. Not always easy, but they respond well to it.

And it's entirely possible I'm reading something into the essay that isn't
there. But I'm not yet convinced.

~~~
jacobolus
I’ll let Lockhart speak for himself:

> _Now let me be clear about what I’m objecting to. It’s not about formulas,
> or memorizing interesting facts. That’s fine in context, and has its place
> just as learning a vocabulary does— it helps you to create richer, more
> nuanced works of art. But it’s not the fact that triangles take up half
> their box that matters. What matters is the beautiful idea of chopping it
> with the line, and how that might inspire other beautiful ideas and lead to
> creative breakthroughs in other problems— something a mere statement of fact
> can never give you.

> By removing the creative process and leaving only the results of that
> process, you virtually guarantee that no one will have any real engagement
> with the subject. It is like saying that Michelangelo created a beautiful
> sculpture, without letting me see it. How am I supposed to be inspired by
> that? (And of course it’s actually much worse than this— at least it’s
> understood that there is an art of sculpture that I am being prevented from
> appreciating).

> By concentrating on what, and leaving out why, mathematics is reduced to an
> empty shell. The art is not in the “truth” but in the explanation, the
> argument. It is the argument itself which gives the truth its context, and
> determines what is really being said and meant. Mathematics is the art of
> explanation. If you deny students the opportunity to engage in this
> activity— to pose their own problems, make their own conjectures and
> discoveries, to be wrong, to be creatively frustrated, to have an
> inspiration, and to cobble together their own explanations and proofs— you
> deny them mathematics itself. So no, I’m not complaining about the presence
> of facts and formulas in our mathematics classes, I’m complaining about the
> lack of mathematics in our mathematics classes._

------
drcomputer
I loved mathematics throughout school. I don't know about anyone else, but I
can see quite clearly how mathematics has shaped the way I think and form
concepts, ideas, and understandings of my perception of the world. I would say
that mathematics is foundational to my perception, if there exists anything
that serves as the base way I interpret information.

That said, I paint and I hated painting classes. I hated almost every art
class I took. I paint okay, but painting is more about getting rid of negative
emotions for me, than anything. I never really liked piano lessons either, I
prefer to gain a small ability and spend years perfecting it with a
combination of the few I've learned and perfected, into various impromptu
permutations. I guess some people call this jazz, but all the stuff I've
studied makes it sound like classical music does to me.

With math, I don't really care about creating it. I just want all of the math
in my head, with the right understanding of it, because I think that makes me
a better computer scientist and software developer. I don't know if that's
irrational reasoning, but I know that understanding math correctly is hard,
and writing code is easy.

------
jackmaney
A beautifully written and tragic essay.

(Note: What follows is US-centric.)

After nine years of teaching mathematics courses (one semester as an
undergraduate, 4.5 years as a graduate student, and 4 years as an assistant
professor) and navigating university politics, I'm convinced that this is, at
its heart, a cultural issue.

There's a hatred of mathematics in mainstream American culture that runs very,
very deep. And it will probably take generations to change that (if changing
it is even possible at this point).

~~~
dxbydt
>There's a hatred of mathematics in mainstream American culture that runs
very, very deep

Completely agreed. As an immigrant, I can say it is very much a US thing.
Haven't seen this much math-hate, but more importantly, math-utilitarianism,
as in the US. In Asian countries & in Europe ( UK, France especially), people
don't constantly fixate on stupid questions like "what is it good for ? ",
which is ultimately a proxy for "how do I make money with this thing ?". But
when I taught math here in the US as a graduate student TA, the majority of
questions focussed on this single metric - usefulness.

So math texts here are forced to invent bogus problems like "You want to house
pigs with 500 feet of fencing. What dimensions of your rectangular pen will
house the most hogs ?". Then the American kid says, Ah! Now I see the point of
all this! Let l be the length of my pen and b its breadth. You want me to
maximize the area of my pen l _b subject to 2l+2b=500 so I can house the most
pigs! Ok so I see that b = 250-l, so l_ b is 250l - l^2, so I take its
derivative & equate to zero & l=125, b=125, and that's the biggest pen that
can house the most pigs. Very nice!

In other countries, you simple wouldn't come up with all these sort of bogus
utilitarian problems in animal husbandry. Students here learn exponentials &
Taylor expansion as part of "how do I compute compound interest on my bank
account", because that's supposedly the only legitimate use of e^x !

I honestly found teaching pre-calc, calc-1 & calc-2 a complete travesty,
because the theorems & the entire courseware was essentially perverted - it
was all in service of how to make use of the math for some bogus application,
rather than learn it for its own good. The worst was when I had to teach how
the horizontal range of parabolic trajectories varied - the textbook had
examples of the US bombing Japan, & the students went to work computing the
best possible angle for firing the missile, so that it would fly across in a
parabolic trajectory and land the farthest thus maximizing its horizontal
range & kill the most number of Japanese! There was no thought given to how
violent & nasty this was.

I actually have very radical ideas about how things should be taught - like
you must learn Rolle's theorem before learning shit like pre-calc. Learn as
much of undergrad real analysis before you get into application oriented shit
like calc-1, calc-2 etc. Don't bring in garbage like LCR circuitry into pde's,
even though yes, you can use a third order differential equation to compute
current through an LCR circuit.

Applications have their place, but such an overemphasis on application is
simply not healthy. It actively distorts the culture & the body politic. Note
that American students don'r ask "what use is rock and roll ? otr what use is
hbo ? or what use is literature ?" all those things are given a free pass. But
when it comes to math, suddenly use becomes the primary criterion. Read pages
6-7, & especially 12 of Lockhart's lament, where he chooses to parody this
point of view via Simplicio.

~~~
jinfiesto
I'm also a math teacher, and agree that there's something about American
culture that's really messing with mathematics education. As a student, I was
always annoyed by applications questions. Not because I have anything against
applications, but because they were usually contrived, and often outside of my
field of study. There were all sorts of physics questions that would pop up in
the lower math classes (Never took physics) that I could do the math for, but
had no reasonable physical intuition about how the system in question should
actually behave. I was mostly just crunching numbers.

------
ColinDabritz
Such a wonderful essay. It's very applicable to the teaching of computer
science/software engineering as well. So much of the problem is the
misunderstanding people have about the field. It's a creative, constructive
discipline, and so much of the instruction is consumption, mimicry, and
repetition.

Solving well defined problems is relatively easy. Our real problem is that
real problems are not well defined.

~~~
theoh
I think I agree but isn't your last line a bit like saying "hammering in a
nail is easy. The problem is that these screws aren't nails."

Chuck Close: "I think while appropriation has produced some interesting work …
for me, the most interesting thing is to back yourself into your own corner
where no one else’s answers will fit. You will somehow have to come up with
your own personal solutions to this problem that you have set for yourself
because no one else’s answers are applicable." ... "See, I think our whole
society is much too problem-solving oriented. It is far more interesting to
[participate in] ‘problem creation’ … You know, ask yourself an interesting
enough question and your attempt to find a tailor-made solution to that
question will push you to a place where, pretty soon, you’ll find yourself all
by your lonesome — which I think is a more interesting place to be."
[http://www.brainpickings.org/2012/12/27/chuck-close-on-
creat...](http://www.brainpickings.org/2012/12/27/chuck-close-on-creativity/)

------
zodiac
FWIW one of our professors here at uwaterloo taught a first year abstract
algebra / number theory class in a very Lockhart-esque way (Math 145; he even
quoted Lockhart on one of the assignments). I learned a lot of math and
enjoyed myself, but the main problem I observed was figuring out how to fairly
grade students, and the fact that the homework took a lot more time than a
class taught normally.

------
mathattack
I have a lot of sympathy for his point of view. I loved Math growing up. High
school drove the interest out of me, and I didn't get it back until senior
Calculus, when I started doing well again. Then I learned to appreciate CS
theory, economic theory, etc. Trying to figure out how to break the cycle for
my kids: Stats for practical work, and math for curiosity.

------
4ad
A very similar piece by V.I. Arnold[1]: [http://pauli.uni-
muenster.de/~munsteg/arnold.html](http://pauli.uni-
muenster.de/~munsteg/arnold.html)

[1]:
[http://en.wikipedia.org/wiki/Vladimir_Arnold](http://en.wikipedia.org/wiki/Vladimir_Arnold)

------
samwilkinson
As a Physicist, I feel obliged to mention that black holes were first
hypothesised by Physicists (contrary to the essay), albeit through
Mathematical enquiry.

~~~
tokai
I would assume that he means that mathematicians worked with singularities,
before we knew, or hypothesised, that singularities arises in the real world.

------
WhitneyLand
I often hear the excuse that mainstream courses like Algebra and Calculus are
taught first and in a boring way because you have to learn mechanics before
getting to the good stuff.

However I don't see why they couldn't start a Calc course with one of those
cool documentaries on Newton. For me it was incredibly motivating to hear the
questions that drove the theory.

Beyond that it seems certain classes like discrete math or combinatorics might
allow more creativity and experimentation in secondary school without
requiring a ton of foundation.

Geometry, if I recall correctly, was one of the exceptions in early math where
you are allowed to veer off the path a bit. Is everything else algorithmic
until college?

------
MaxScheiber
[https://news.ycombinator.com/item?id=6187014](https://news.ycombinator.com/item?id=6187014)

With that being said, I've always loved this essay. As of recently, I've
viewed it as relevant to the recent argument that programming should be a
requirement in American public schools, either as a tool in math and science
classes or a free-standing course. This kind of mathematical reform might
actually be a prerequisite for programming and computer science, given that it
would develop mathematical maturity much more effectively than the current
system does.

~~~
jnbiche
Yes, but Lockhart is arguing _against_ any practical applications of math in
the early years. I'm almost certain he would oppose any attempt to connect
math to programming, for the reasons outlined in his essay (roughly, "math
should be about imagination and playing, not applications"). In fact, I
disagree with this essay, for reasons outlined in [1].

1\.
[https://news.ycombinator.com/item?id=8847132](https://news.ycombinator.com/item?id=8847132)

~~~
birdsareweird
Who do you think mathematical programming has to be practical?

~~~
saraid216
You're taking away the wrong thing from the grandparent post.

If you teach math as a prerequisite for something else, you're naturally going
to be undervaluing it. Math should be taught for the sake of learning math,
not because it's the gateway to something else. It is, yes, but that's _not
the point_.

~~~
birdsareweird
You know the problem with that sentiment? Most people do not find abstract
patterns and rote symbolic manipulation to be interesting. They simply don't,
and if you teach math class mistakenly assuming they do, 90% tunes out.

If you can show them how those abstract patterns reflect things in nature, and
how those symbolic manipulations represent ideas and algorithms and systems,
and can explain complicated things, that's something else.

e.g. You don't need computer graphics to teach linear algebra, but how many
people who 'know' matrix algebra know that the columns of a matrix are the
basis vectors for the principal axes, and that matrix multiplication is the
(affine) transform tool from photoshop?

I would also suggest that if you want to get kids interested in trigonometry
and you fail to mention that every single thing in a 3D video game is made out
of triangles, you are a bad teacher and you should feel bad.

~~~
saraid216
I'm not interested in teaching math to people who don't want to learn math,
honestly. It baffles me why so many people think that if someone is
uninterested in math, they must be tricked into being good at it anyways.

Teaching shouldn't be so dishonest.

~~~
birdsareweird
Why do you think creating interest from real life is going to make them not
"want to learn math"? That makes no sense.

Teaching is all about benevolent dishonesty. The first thing a teacher does is
dumb themselves down to their students' level.

~~~
saraid216
> Why do you think creating interest from real life is going to make them not
> "want to learn math"? That makes no sense.

I don't. I think it's entirely possible to cram knowledge into people's minds
unwillingly.

But you're making an unnecessary assumption: that you _have_ to teach everyone
math. Why is that?

~~~
birdsareweird
Why do we teach everyone history? Because it's valuable to know where we came
from, and to realize that whatever's going on today, it's probably happened
before.

Why do we teach everyone math? Because it creates quantitative literacy,
because it helps people deal with systems and complexity, to break down
problems logically, and to not let biases get in the way of the truth. Or at
least that's what I wish it would be focused on, instead of producing droids
who know how to execute symbolic algorithms.

~~~
saraid216
> Because it creates quantitative literacy, because it helps people deal with
> systems and complexity, to break down problems logically, and to not let
> biases get in the way of the truth.

None of these things are math. These things can take advantage of math, yes,
but they have as much to do with math as figuring out how to use Microsoft
Word does.

> Or at least that's what I wish it would be focused on, instead of producing
> droids who know how to execute symbolic algorithms.

You want to know how to get them to focus on it? Ask them to.

Stop asking people to teach math. Ask them to teach quantitative literacy.
Recognize that this isn't necessarily math. Ask them to teach systems theory
and complexity theory. Recognize that math is not the best vehicle for
understanding those things, especially for grade schoolers. Ask them to teach
logic. Recognize that set theory isn't covered in grade school at the moment,
and that learning logic isn't going to happen through math. Ask them to teach
ways of discerning truth despite bias. That means covering the scientific
method, covering statistics, covering research strategies, covering fact-
checking.

Be. Honest. With. Your. Goals.

Your goal isn't "Students should know how to derive polynomial expressions."
You've stated your goals. Recognize them for what they are. Stop asking math
teachers to carry all that weight for you. Stop hoping that students will
magically gain "quantitative literacy" from geometry proofs about angles.

You get "droids" because you've asked for "droids".

------
blahblahblah3
I think for most people, math is best learned in the context of some
application that they care about (the last four words are very important). Few
people appreciate the beauty of the abstract game itself.

For example, most people who play poker online quickly learn about expected
value, probability, and variance.

~~~
eruditely
Precisely. Poker is one of the most advanced games that legitimately teaches
you what you need to know! The language used by poker players should be
adapted wholesale.

------
ashark
HN is big on curated lists lately. Is there one for resources to aid in
teaching mathematics the way Lockhart would prefer?

I've seen this posted so many places so many times that surely there's a
market for materials and support for it. Where are they?

~~~
GregBuchholz
With a quick search, here's the closest thing I could find:

[http://mathoverflow.net/questions/5074/are-there-
elementary-...](http://mathoverflow.net/questions/5074/are-there-elementary-
school-curricula-that-capture-the-joy-of-mathematics)

...but I too would like to see what you are looking for, at an elementary
school age level.

~~~
ashark
Now that I've thought about it a bit more, what I'd really love is a directory
of sensibly ordered learning resources (free online stuff, videos, links to
books on Amazon, whatever), headed by one or two mission-statement type papers
(like Lockhart here) for the pedagogical strategy they generally follow, for a
whole bunch of topics with several strategies per topic. That way you could
pick a topic, read the statements for the various strategies, select one or
more that you like and just start working down from the top.

... and I'd also like a pony :-)

------
prestonbriggs
Lockhart has a book, "Measurement", that elaborates in the same vein. Quite
nice.

~~~
mushishi
I personally found it inspiring and beautifully written.

------
drcomputer
I loved mathematics throughout school. I don't know about anyone else, but I
can see quite clearly how mathematics has shaped the way I think and form
concepts, ideas, and understandings of my perception of the world. I would say
that mathematics is foundational to my perception, if there exists anything
that serves as the base way I interpret information.

That said, I paint and I hated painting classes. I hated almost every art
class I took. I paint okay, but painting is more about getting rid of negative
emotions for me, than anything. I never really liked piano lessons either, I
prefer to gain a small ability and spend years perfecting it with a
combination of the few I've learned and perfected, into various impromptu
permutations. I guess some people call this jazz, but all the stuff I've
studied makes it sound like classical music does to me.

With math, I don't really care about creating it. I just want all of the math
in my head, with the right understanding of it, because I think that makes me
a better computer scientist and software developer. I don't know if that's
irrational reasoning, but I know that understanding math correctly is hard,
and writing buggy programs is easy.

------
drcomputer
I loved mathematics throughout school. I don't know about anyone else, but I
can see quite clearly how mathematics has shaped the way I think and form
concepts, ideas, and understandings of my perception of the world. I would say
that mathematics is foundational to my perception, if there exists anything
that serves as the base way I interpret information.

That said, I paint and I hated painting classes. I hated almost every art
class I took. I paint okay, but painting is more about getting rid of negative
emotions for me, than anything. I never really liked piano lessons either, I
prefer to gain a small ability and spend years perfecting it with a
combination of the few I've learned and perfected, into various impromptu
permutations. I guess some people call this jazz, but all the stuff I've
studied makes it sound like classical music does to me.

With math, I don't really care about creating it. I just want all of the math
in my head, with the right understanding of it, because I think that makes me
a better computer scientist and software developer. I don't know if that's
irrational reasoning, but I know that understanding math correctly is hard,
and writing code is easy.

------
krazydad
Love the bit about the misconception that Mathematics is mainly about utility.
I remember reading something about G.H. Hardy (which I can no longer find) in
which he said he would get a little bit disappointed if he found that one of
his results ended up finding a practical use.

~~~
Retra
That's not a misconception.

It would be great if all mathematics had obvious utility, but that's demanding
perfection. It doesn't mean mathematics would be worth doing if it had no
utility. In fact, we often try to do mathematics in a way that guarantees some
amount of utility; that's why we use proof-based methods in leu of wishful
thinking.

Should I write a mathematics paper concluding "true = false" and argue that
mathematics is not about utility? No. Such a thing is utterly useless, and it
would be ridiculous to propose that I'm doing mathematics without taking the
necessary care to ensure my work is useful.

~~~
saraid216
I'd be more concerned that such a paper would be a lie. When did utility
become more important than truth?

~~~
Retra
Truth is important because it has utility. It enables consistency and
reproducibility in a way that nothing else does.

~~~
saraid216
Really? What consistency and reproducibility does "An individual is a
worthwhile human being" have?

------
aaronem
Vital context: _The Underground History of American Education_ , free online:
[http://mhkeehn.tripod.com/ughoae.pdf](http://mhkeehn.tripod.com/ughoae.pdf)

~~~
GregBuchholz
Wow. It certainly has an excellent hook in the very first sentence:

"Our problem in understanding forced schooling stems from an inconvenient
fact: that the wrong it does from a human perspective is right from a systems
perspective."

~~~
aaronem
It gets better from there. Keep reading.

------
WhitneyLand
I often hear the excuse that mainstream courses like Algebra and Calculus are
taught first and in a boring way because you have to learn mechanics before
getting to the good stuff.

However I don't see why they couldn't start a Calc course with one of those
cool documentaries on Newton. For me it was incredibly motivating to hear the
questions that drove the theory.

Beyond that it seems certain classes like discrete math or combinatorics might
allow more creativity and experimentation in secondary school without
requiring a ton of foundation.

------
WhitneyLand
I often hear the excuse that mainstream courses like Algebra and Calculus are
taught first and in a boring way because you have to learn mechanics before
getting to the good stuff.

However I don't see why they couldn't start a Calc course with one of those
cool documentaries on Newton. For me it was incredibly motivating to hear the
questions that drove the theory.

Beyond that it seems certain classes like discrete math or combinatorics might
allow more creativity and experimentation in secondary school without
requiring a ton of foundation.

------
drvortex
If the musician woke up in the first sentence, how is he still dreaming in the
next one?

------
sarciszewski
My favorite part is "The Standard School Mathematics Curriculum".

------
eximius
Always a good read.

