
A Mathematician’s Lament (2002) [pdf] - kjhughes
http://www.maa.org/sites/default/files/pdf/devlin/LockhartsLament.pdf
======
nsphere
"There is such breathtaking depth and heartbreaking beauty in this ancient art
form. How ironic that people dismiss mathematics as the antithesis of
creativity. They are missing out on an art form older than any book, more
profound than any poem, and more abstract than any abstract. And it is school
that has done this! What a sad endless cycle of innocent teachers inflicting
damage upon innocent students. We could all be having so much more fun."

As a Mathematician I've always identified as a creative person, yet struggle
to convince "artsy" persons that what I do is creative. As soon as I utter the
word math they convulse and shutter as if they were afflicted with a PTSD
flashback. I try explaining what the idea involves and draw a diagram or
three, but all they do is just nod and say "yeah, yup, okay.." Are the ideas
of math that inaccessible to the general population when compared to a work of
Art?

~~~
john_b
I think that language (at least English) fails us here. The word "creative"
can be used to mean both "relating to or involving the imagination or original
ideas, especially in the production of an artistic work" [1] or "resulting
from originality of thought, expression, etc.; imaginative" [2]. It seems that
most people identify the word more strongly with the artistic sense of the
word, in which creativity is a proxy for a kind of self-expression that is not
bound by any rules, logic, or structure. In doing so, they seem to mistake one
of the more visible manifestations of creativity with its essence, which
really lies in the "originality of thought" and "imagination" of the creative
person.

If we had a specific, unique, and widely used word for the "artsy" free-
expression type of creativity, I think the confusion many people express when
you try to convince them that mathematics is a creative endeavour would be
greatly diminished.

[1]
[http://oxforddictionaries.com/us/definition/american_english...](http://oxforddictionaries.com/us/definition/american_english/creative)

[2]
[http://dictionary.reference.com/browse/creative](http://dictionary.reference.com/browse/creative)

~~~
dwc
> creativity is a proxy for a kind of self-expression that is not bound by any
> rules, logic, or structure

I do not find this to be the case. Art is full of structure, rules and logic.
When artists "break" the rules, it usually means they been able to operate
using _underlying_ rules, and understand well the rules they break.

~~~
saraid216
Or they're creating new rules.

~~~
dwc
Yes, indeed. For instance, the creation of bebop in jazz. Of course Bird could
only do this by a comprehensive understanding of the existing rules and seeing
deeper.

------
softbuilder
_" but later in college when they finally get to hear all this stuff, they’ll
really appreciate all the work they did in high school.”_

So painfully spot-on. My mathematical education was horrible. Meanwhile I had
been writing code since I was a little kid. It wasn't until I was an adult
that I realized how much math I had been learning while programming. And
worse, that I had been completely miseducated about what math actually is.

~~~
gxs
Interestingly, the converse is also true.

I was always into computers and tech but never really programming.

I was a math major in college, however, and after graduating I started
programming. The transition was almost seamless, I picked up programming
really quickly, it was surprising to me how much the "ways of thinking" are
alike.

~~~
saraid216
One of the key mistakes of educational strategy is dividing up knowledge into
a bunch of subjects like "Language", "Math", and so on. You can find linkages
in just about everything; this siloing really only serves to make people think
they're particularly good at one thing and not another, when really it's a
preference for one perspective over another.

~~~
marcusf
So the pragmatists question is then naturally, how do you teach it? How do you
shard the curriculum if not by subject? It seems a bit overbearing to ask each
teacher to have the proper renaissance man's education of knowing a bit of
everything.

~~~
saraid216
To me, a teacher's proper role isn't having knowledge and dumping it into
someone's brain. It's knowing _where to find_ that knowledge. My three Rs are
"reading, research, and reflection": a teacher's job is to (1) provide useful
material to consume, via lecture or homework or whatnot, (2) point towards
larger resources for further exploration, and (3) guide thought processes to
make useful conclusions.

A teacher's job is not to teach. It's to provide a space in which a student
can learn. A focus is useful for this, but the focus doesn't need to be an
abstract subject. It's a MacGuffin; it can be anything.

------
bridgeyman
I read this a couple weeks ago and decided to try and find a math book that
incorporates some history in it. I found _Journey through Genius: The Great
Theorems of Mathematics_ with some great reviews.

I am working on an iPad app that pairs up people to mentor each other through
books like this. It has video chat and a shared whiteboard, so it is ideally
suited for discussing math. If anyone is interested in reading the book with
someone, email me at bridger@understudyapp.com. I could really use some beta
testers for the app!

If you have already read it, you could still mentor someone in it to review
the material again.

~~~
codyb
To add to the suggestions here, "Number" by Tobias Dantzig is absolutely a
wonderful history of mathematics. I mean, shoot, it's actually got a quote
from Einstein:

"This is beyond doubt the most interesting book on the evolution of
mathematics which has ever fallen into my hands. If people know how to
treasure the truly good, this book will attain a lasting place in the
literature of the world. The evolution of mathematical thought from the
earliest times to the latest constructions is presented here with admirable
consistency and originality and in a wonderfully lively style."

—Albert Einstein

[http://www.amazon.com/Number-Language-Science-Tobias-
Dantzig...](http://www.amazon.com/Number-Language-Science-Tobias-
Dantzig/dp/0452288118)

~~~
auggierose
if you are in a hurry:

[http://www.engineering108.com/Data/Engineering/Maths/Number_...](http://www.engineering108.com/Data/Engineering/Maths/Number_the_language_of_science_by_Joseph-
Mazur_and_Barry-Mazur.pdf)

------
dsego
Still haven't finished reading this and all the painful memories came back
from middle school. Not math (although it too wasn't a fun experience) but
art. For four years we had a teacher that thought music and visual arts by
dictating. For 45 minutes we would write down everything word for word. Then
after a few weeks she would ask us broad questions and we'd have to recite
everything back to her. I still remember (I'm almost 30 now) rote learning
from my notebook, walking around the house, repeating those stupid sentences
in my head, memorising descriptions of works of art I've never seen, painting
techniques I've never witnessed, music I've never heard. Wow, just talking
about it makes me angry, makes me want to to choke the bitch's neck. Sadly,
that's the state of schools in Croatia, people equate schools with rote
learning and memorising. You have to do it to get into a good university and
get a good job. Who will create those jobs? No wonder the country is in the
shitter.

~~~
dsego
oops, meant taught instead of thought.

------
jobenjo
Love this essay. I read it years ago when my brother was working with Paul
Lockhart, who deeply influenced him as a math teacher.

My brother and his wife have since started an organization called Math For
Love ([http://www.mathforlove.com](http://www.mathforlove.com)) focused on
changing the way math is taught. They run workshops for teachers and provide
great material for students.

If you're in Seattle and interesting in pedagogy and math, you should check
them out.

------
ctl
I love this article, but: what can a practicing math teacher take away from
it? How can you apply this stuff if you still have to teach a standard
curriculum?

I'm really asking -- my friend is about to start as a high-school math
teacher.

I guess the first recommendation would be: motivate every new technique by
starting with one or more problems that the technique helps to solve. (Here
"problems" is meant in the Lockhart sense -- real puzzles, not exercises.)

But how often are "techniques" actually taught in high school math, especially
algebra and precalculus? A lot of high school math consists of digesting new
definitions, or the generalization of old definitions. A fair amount of it
consists of learning theorems that go unproven, or that are proven (by the
teacher) too quickly for students to understand where they come from -- and in
general it isn't satisfying to solve a puzzle with a theorem that one doesn't
actually understand.

On top of that... students have to spend time with problems before they become
genuinely interested in their solutions, so progress would be slower with this
method. It's not clear that you could teach a whole year's curriculum in one
year like this. (And if you fail to do that you'll eventually get fired.)

Any insight? I believe that it's possible to teach math, even standard high
school curriculum, in such a way that students are at all times intrinsically
interested in what's presented. But it would be awfully hard to do at scale,
at the standard pace, as a high school teacher would have to. How might a
teacher start in that direction?

~~~
jfarmer
I co-founded Dev Bootcamp and while I was still there one of my not-so-secret
missions was to make mathematics less alienating. I only say that because _it
was incredibly difficult_ , even in an environment where I had complete
autonomy and authority to make whatever curricular and pedagogical decisions I
wanted. The problem becomes combinatorially more complex in a public school
where teachers have much less autonomy, have to teach to a common set of
state-wide standards, and have students of varying levels of interest.

Here are my scattered thoughts, though. I'm going to try to not suggest a pie-
in-the-sky solution like "new curriculum!"

First, I majored in mathematics at the University of Chicago, but I hate,
hate, _hated_ mathematics in high school. Take something you'd see in Algebra
II like matrix multiplication, matrix inverses, and solving systems of linear
equations. You're presented with these things called matrices and taught a
bunch of rules. Where did these rules come from? Why are we calling this
"multiplication" when it doesn't look or act anything like multiplication?

And sure, I see that when I go through the steps you tell me to go through
like a monkey I get an answer that works, but how do we know there aren't more
correct answers? How did anyone even come up with these steps in the first
place? It's not like someone sat down and tried a trillion random combinations
of symbols and steps until one of them happened to work.

Augh. In that world the only recourse for students is to memorize, usually
just enough to do the homework or pass the test, and then promptly forget. The
only experience they associate with math is the utterly humiliating feeling of
being terrible at it.

So, I think that's one of the root problems. People remember what they feel
and most people remember feeling stupid, humiliated, and possibly ashamed when
it comes to mathematics. It's only a matter of time before that becomes part
of their identity. "Oh, I'm terrible at math. Oh, I'm not smart enough to do
math." and so on.

If I were a HS math teacher my top priority would be to watch out for when
those counterproductive, self-defeating beliefs were forming and do whatever I
could to preempt them.

Second, I think the way math is taught is overly symbolic. What most non-
mathematicians don't realize is that when most mathematicians look at a set of
abstract symbols they don't "see" the symbols _per se_ , they see what those
symbols are meant to represent. They freely move between a geometric and
algebraic picture of the world, but the algebraic picture is usually
incredibly compressed.

I think the key thing is not to pick a side -- algebra vs. geometry -- but to
show the relationship between the two. Geometric objects admit a symbolic
representation and vice versa.

Third, students have this idea that math is all about being "right" or
"wrong", that it's "black" or "white", that there's some universe of Proper
Math that is insisting on certain rules for no rhyme or reason

Here's a silly but illustrative example that I think students would cover in
6th or 7th grade: order of operations.

Hey class! Look at this expression: 4 _5+6. What does it equal?

A bad teacher says "It's 26 and any other answer is wrong." An ok teacher
says, "Remember the order of operations. If we apply those rules we get 26, so
that's the right answer."

A great teacher shows their students that some things are necessarily true and
other things are definitionally (or conventionally) true. This teacher would
do something more like...

Who got 26? Who got 44? Students who said the answer was 26, how did the
students who got 44 arrive at their answer? Students who said the answer was
44, how did the students who said 26 arrive at their answer? Neither of you
are wrong _per se _. We could have chosen to live in either world, but we have
to choose one consistent set of rules.

These rules lead us to 26. If we chose the other set of rules, we'd get at 44.
We only do this because we don't want to have to write down parentheses all
the time, but without them it's unclear what order we're supposed to apply +
and _. So we need to agree on a set of rules so that two people looking at the
same expression both understand how to make sense of it.

It's like traffic laws. There's nothing stopping people from driving on the
left side of the road. In fact, there are countries where everyone does drive
on the left side of the road. The important thing is that everyone agrees on a
convention -- left-side or right-side. It works as long as everyone agrees and
breaks if people don't.

I could go on, but I'll stop here. Like I said, these are my scattered
thoughts. :)

~~~
nileshtrivedi
> when most mathematicians look at a set of abstract symbols they don't "see"
> the symbols per se, they see what those symbols are meant to represent.

Might we benefit from a different set of symbols that actually convey the
geometrical meaning behind them? If instead of π, we used a glyph that shows a
circle over a diameter, instead of x for a variable, we show an empty
rectangle that shows that it's a placeholder for a value?

> Students who said the answer was 44, how did the students who said 26 arrive
> at their answer?

I came across a great example of this approach in Chess: The Complete Self-
Tutor by Edward Lasker. Instead of just showing the right answer for a chess
puzzle, he tells you what you did right in your answer and what you missed in
getting an even better answer. This was a printed book that was completely
interactive.

------
BgSpnnrs
_" Michelangelo decorated a ceiling, but I’m sure he had loftier things on his
mind."_

That's almost Douglas Adams levels of dry wit.

This is a fascinating article, as someone who's never really contemplated the
playfulness of maths....I mean, for sure the wonder of maths or the power of
maths - I've just never really put the pieces together to link that back to my
grounding in maths, beyond the practical, functional stuff I incorporate into
my daily life, without considering it's mathiness. Very interesting.

------
rtfeldman
This is intensely thought-provoking and beautifully well-written. It's not
directly about hacking, but it's the type of treasure that hackers love to
stumble upon.

It's worth noting the times when HN really delivers. I doubt I'd have come
across this anywhere else.

~~~
gizmo686
>It's worth noting the times when HN really delivers. I doubt I'd have come
across this anywhere else.

At risk of getting to meta, this piece seems to pop up everywhere. I don't
mean that in a bad way, but this is at least the fifth time over the course of
many years that I have seen this piece pop up on completely unrelated sites.

~~~
ColinWright
Not to mention it's been here on HN at least 8 times, probably 10 or more.
I've stopped bothering to direct people to previous conversations - they just
say the same things over and over again anyway.

------
adultSwim
I thought I had been taking math my whole life until I finally took a math
class.

~~~
jackmaney
Back when I was in academia, one of my favorite student evaluations from an
abstract algebra class had the following comment (paraphrased from memory):

"This course was like climbing Mt. Everest. It's difficult, and sometimes slow
going, but at the end, it's breathtaking how much you've learned and
accomplished."

------
RogerL
The article resonated with me on some level, because it does take a long time
to learn how to actually do math. If you are at the point of just doing
algebraic manipulations on equations to try to figure something out, you've
lost the battle (as opposed to using algebraic manipulations to _encode_ your
thoughts, and work out the details).

On the other hand, I think everybody really did see the beauty in geometry.
Yes, the initial manipulation to prove something about symmetric angles is a
bit silly, but you aren't being taught symmetric angles here, you are being
taught how to do a geometric proof. Which is not easy to learn, so you start
with something super simple and obvious. I didn't observe anyone in my classes
(well this was back in 1982, so memory is a challenge here) confused about
that point. And, as soon as we learned it, the requirement for formalism was
dropped. It was the same in algebra. In the first weeks you weren't allowed to
go directly from x + 3 = 1 to x = -2. You had to do something like x + 3 - 3 =
1 - 3; x + 0 = 1 - 3; x = 1 - 3; x = -2; with all the rules that you are using
written out. Annoying yes, once you grasp it, but once you proved you grasped
it that was the end of that, and we never had to do it again.

But, I had good teachers that always tried to explain the 'why' of what we
were doing, and did not make us engage in pointless formalisms. But you need
to understand terminology like ABC in geometry; when you get to tough problems
you'll be using it. So, learn it with the easy problems.

~~~
gizmo686
My best math teachers never made us engage formalisations. What they would do
is manipulate us into having an arguement, or construct an apparent
contradiction. We then discussed the problem until we all agree (often without
the Teacher talking). The inevitable result of this is that everyone learned
formalisation: because it makes for really convinving arguements.

------
mathattack
I hated math until I got to calculus. I never knew why, but I might as well
quite this mathematician... They took an exciting topic that interested me as
a youth and killed it with rote repetition.

The question I struggle with is... How do you really get by without math as a
mandatory topic? How can you teach science? How can you teach someone to
balance a checkbook? The current system is awful, but do we push the subject
to electives as a result?

~~~
ef4
> The question I struggle with is... How do you really get by without math as
> a mandatory topic?

That's the wrong question, because "mandatory learning" is a contradiction in
terms. It very clearly doesn't work. Beyond the most primitive pavlovian
conditioning, you can't force people to learn if they don't want to. Intrinsic
motivation is the dominating factor in how well a person can master an
intellectually demanding task.

The right question is, how do you inspire people to want to learn?

~~~
mathattack
True enough. And that's a tough question too. I waited from 4th grade to
calculus to get inspired again. And then I think it was the material more than
the teaching method.

------
dominotw
The author of this article teaches a course on coursera that I can't recommend
enough, he put so much effort into that course.
[https://www.coursera.org/course/maththink](https://www.coursera.org/course/maththink)
As you might have guessed its not really about 'math' in traditional sense.

------
dwaltrip
This article just blew my mind. I feel little silly, especially since I
majored in my math. I have never seen these types of ideas expressed so
clearly and powerfully.

He is 100% right. We should not be teaching the formulaic rules of basic
arithmetic until high school, and only then as a part of life skills class. It
should be taught as the art form that it is.

------
dnautics
As an erstwhile math major (I couldn't hack the honors basic algebra class -
the difference between a euclidean domain and a principal ideal domain got too
confusing; but I rocked proofs in analysis) I have to say that the author is
confusing Mathematics (which is an art) and Arithmetic (which is a skill).
Part of what make the opening farce absurd is that musical skills and painting
are not terribly necessary as a matter of life and death, or even to a certain
degree, quality of life or death; whereas understanding sums and compounding
processes ARE.

While SALVIATI is completely correct in his analysis of the situation, he
offers no solution, and I identify with SIMPLICIO more.

Perhaps the problem is that in our schools we conflate arithmetic with
mathematics. Surely, they are related, but perhaps they need to be delineated
and the difference understood.

~~~
saraid216
> whereas understanding sums and compounding processes ARE [terribly necessary
> as a matter of life and death, or even to a certain degree, quality of life
> or death].

That's not actually true.

~~~
dnautics
in an ideal world, I would agree with you. We do however live in the real
world.

~~~
saraid216
Please give one example applicable to a reasonable majority of human beings on
the planet at this current time where "understanding sums and compounding
processes" are a matter of life and death.

~~~
jameshart
It seems probable that the subprime mortgage crisis was a contributing factor
in a number of deaths, via suicide, stress-induced illness, or, with the help
of alcohol, violent or vehicular incidents.

~~~
saraid216
But

(1) The subprime mortgage crisis could not have been averted by a larger
percentage of the population understanding sums and compounding processes.

(2) The advent of civilization was a contributing factor to everything that's
happened in the last several thousand years, including the paper cut I just
got.

~~~
jameshart
you understand that interest-bearing loans are a compound process, right - and
that an understanding thereof might be of relevance to someone entering into a
loan agreement which they probably, if they really understood it, were not
going to be able to repay...

But let's not only blame the borrowers of subprime loans, let's also ask
whether the banks didn't entirely understand the risk models of the
derivatives they were compiling out of subprime mortgages because many of
their senior managers also didn't understand compounding processes, or,
possibly, sums...

~~~
saraid216
> an understanding thereof might be of relevance to someone entering into a
> loan agreement which they probably, if they really understood it, were not
> going to be able to repay...

Oh, it's relevant. It's sort of like how being able to load and fire a gun is
relevant to the decision whether or not to commit suicide. Sure, it can modify
one of the many, many details, but some people just make a noose and hang
themselves.

Keep in mind, this is your argument: if more people in the world understood
sums and compounding processes, this would have prevented the subprime
mortgage crisis and thus the many deaths that were inspired by the resultant
fallout. There is no possibility that anything else caused the crisis, and no
possibility that anyone is at fault for these deaths other than parents and
teachers.

This claim, if true, actually _absolves_ the lenders that you talk about
below, because they can be held responsible only for _their own_ understanding
of sums and compounding processes.

Amusingly, if you accept your argument, you can also make an interesting
inverse version. The fact that lenders understood sums and compounding
processes led to their employment at unscrupulous institutions which then
mandated their sign-off on high-risk mortgages, which then caused the deaths
of all those people.

Math, apparently, kills.

------
lmartel
I've read this article several times at this point (it does tend to pop up
everywhere) and it resonates with me but I'm not sure what to do about it.

I really want to experience the kind of math the author writes about; can
anyone recommend a place to start as someone who has only ever done "fake"
high school math? I'm in college now and I'm halfway through a computer
science degree; I've tried a few times to break into theoretical math classes
but I've found the bar for entry pretty high (especially when I only have room
for one or two courses), with most classes and even peers asking for years of
experience and "mathematical maturity." Have any of you ever succeeded in
learning some math outside formal curricula?

~~~
westoncb
Hey Imartel -- I've been in a similar situation. I think my first starting
point was "Mathematics and the Imagination" or "Gödel's Proof." Mathematics
and the Imagination is a good high level overview, and would provide some
foundational notions that'll reappear repeatedly -- but, it won't given you
any practice in mathematical methods. For that I would recommend "What is
Mathematics?" by Courant and Robbins. Can be pretty challenging, but you'll
actually get somewhere if you put effort into it. If you haven't had much
experience with proofs, it's worth focusing briefly on the process of proving
explicitly as a preliminary. That's actually a good thing to do with another
person or in a class -- can be pretty difficult to get some of the subtleties
involved, and to know when you've done things correctly or not.

I mention "Gödel's Proof" because it's the first thing I came across that
informed me I am in fact interested in mathematical systems. I'm more
interested in architectures than problem solving though. If you suspect that
might be the case for yourself, might check it out -- it's about 100 pages.

~~~
abecedarius
Courant and Robbins is really great. I'd recommend checking it out after the
Lockhart book (or together with it), as it's more textbookish; there's more
danger of bouncing off.

------
gizmo686
During high school, at the end of the year everyone in my Calculus class had
to make a math presentation (to go along with a paper). During lunch one day
we stood with our posters to explain our project the the students that
choice/were encouraged to attend. I explained my topic (re-ordering
conditionally convergent series) to a group of students. They seemed to
understand what I was saying, showed awe at some of the key insights/tricks,
and even interupted me to excitadly finish the arguement. Then, when I was
done, they pointed to the equations and asked me to explain them. My response
was that those equations are exactly what I had just said, and I repeated the
arguement while showing them where it is in the equations.

------
ArbitraryLimits
Richard P. Feynman

“Physics is like sex: sure, it may give some practical results, but that's not
why we do it.”

------
detcader
We really need to teach people _how_ to teach induction, which is only done
right when you put quotes around your Boolean statements; the "implies" symbol
gets jumbled up with everything else otherwise, and not using it at all is
passing up on a great tool. One can do simple proofs-by-induction without a
single English word, completely symbolically, and have it be understood
easily, if one uses quotes and correct LaTeX formatting (or good handwriting)

Induction doesn't just involve numbers and equality signs, it involves
_statements_ with variables inside of them, and non-programmers need to be
made well-aware of this (and taught Boolean logic early, PLEASE)

~~~
hcarvalhoalves
There was a time people were taught logic and geometry and math and history
all at the same time. Also, philosophy.

The reductionist approach of our education is it's major failure.

------
kazagistar
A classic issue that mathematicians, philosophers, and even computer
"scientists" have is the idea that they can somehow reason their way to the
truth. Oh, he makes a good argument. But the greeks made great arguments about
how the sun goes around the earth.

I could say that this or that argument is flawed. But really, the only valid
arguments are data. His data is severely lacking, and many modern methods of
teaching are much more supported by experimentation. Logic is a tool for
finding logically consistent imaginary realities. Science is a tool for
finding out about this reality.

------
calebm
I had to overcome my school-age brainwashing in order to enjoy math. One book
that helped me was "Who is Fourier? A Mathematical Adventure"
([http://www.amazon.com/ho-Fourier-Mathematical-Adventure-
Edit...](http://www.amazon.com/ho-Fourier-Mathematical-Adventure-
Edition/dp/0964350432)). I highly recommend it for those looking to find
enjoyment in math.

------
dominotw
Attempts to present mathematics as relevant to daily life inevitably appear
forced and contrived: “You see kids, if you know algebra then you can figure
out how old Maria is if we know thatshe istwo years older than twice her age
seven years ago!” (Asif anyone would ever have access to that ridiculous kind
of information, and not her age.)

I wince every time I see these examples in my niece's textbook. So ridiculous.

------
kazagistar
For music, the formal curriculum does involve memorizing songs and scales and
whatnot. The problem is that the best way to start a subject is to fuck around
with it (playing random songs you like on your guitar, drawing cool shapes and
noticing patterns in them, etc), but our schools are ill equipped for such
things, and enabling this style is difficult and incredibly expensive.

------
hcarvalhoalves
Reflects my experience. During school I had interest and good grades in all
classes, including physics and geometry (those are separate from math in
Brazil), the only exception being math (which covers algebra, polynomials,
etc.). How a kid can get an A in physics and a D in math is beyond my
comprehension, but I did. I only started appreciating math much later in life.

------
Myrmornis
_" The saddest part of all this “reform” are the attempts to “make math
interesting” and “relevant to kids’ lives.”"_

+1

Also s/math/science/ +1

------
ctdonath
In contrast, my 5- and 3-year-olds are enjoying DragonBox on the iPad, not
knowing they're learning Algebra.

------
whiddershins
This is how I feel about doing most coding tutorials versus the MIT intro to
CS through python, or project euler.

When coding is presented as: \- here is a problem \- how would you solve this
problem? \- here are some hints to get you started

it is incredibly fun for me. when it is presented as:

\- follow along \- look what you did!

it can be a bit dry.

~~~
abecedarius
Hi whiddershins, here's a little present for you then:
[https://github.com/darius/regexercise](https://github.com/darius/regexercise)

(It's not quite finished and I'd love to get feedback.)

------
stiff
Please read the article with a critical eye, some of it is complete non-sense,
for example:

 _CALCULUS: This course will explore the mathematics of motion, and the best
ways to bury it under a mountain of unnecessary formalism. Despite being an
introduction to both the differential and integral calculus, the simple and
profound ideas of Newton and Leibniz will be discarded in favor of the more
sophisticated function-based approach developed as a response to various
analytic crises which do not really apply in this setting, and which will of
course not be mentioned._

"Mathematics of motion", which makes it sound so simple, has in fact perplexed
philosophers and mathematicians for centuries and continues to perplex a great
many people even today, consider for example the Zeno paradox:

[http://en.wikipedia.org/wiki/Zeno%27s_paradoxes](http://en.wikipedia.org/wiki/Zeno%27s_paradoxes)

The ideas of Newton and Leibniz were hardly simple, they had some valid
intuitions and managed to do formal manipulations that led to correct results,
but in their day it was impossible to at all logically understand why what
they are doing works, and not for some god knows how complicated things, but
even for most elementary ones. You don't even have to go back to writings of
Newton or Leibniz, just have a look at a 19th century textbook of calculus to
see how noticeably strange and illogical the exposition of the subject was
even then, with "infinitely small quantities" and an air of mysticism about
it:

[http://archive.org/stream/elementsofdiffer00woolrich#page/n1...](http://archive.org/stream/elementsofdiffer00woolrich#page/n17/mode/2up)

This kind of approach simply doesn't make sense, even though it happens to
apparently produce correct results sometimes. Now, "function-based approach"
is a weird phrase, but I guess he means the common modern exposition of
elementary calculus using limits. This however wasn't developed in response to
"various analytic crises". The only explanation of this statement I see is
that he knows history of mathematics poorly and confuses the latter
developments by Lebesgue, Jordan etc. that led to what we now call real
analysis (inspired by considerations of nowhere continuous functions,
continuous but nowhere differentiable functions etc.) with the earlier and
more general lack of any decent understanding of how calculus works at all
that was solved by Cauchy, Weierstrass and others. It is their introduction of
what the author considers "unnecessary formalism" that made us finally really
understand "mathematics of motion" and satisfactorily resolved things like the
before-mentioned Zeno's paradox.

If it is only motivated appropriately, the concept of a limit is actually very
interesting and powerful. There is a ladder of granularity with which you can
treat computational problems, with the most elementary approach being always
trying to get the exact answer. However, the class of problems that can be
solved this way is very narrow. You can jump over this severe restriction by
getting a bound, with inequalities for example, or you could try to get an
equality in the limit (when n approaches y, the sought thing x approaches
w*z). Unfortunately in school people almost exclusively learn to look for the
exact answer, while in mathematics proper and in real world it is much more
common to look for approximations and limiting behaviour. Furthermore, since
the limit concept so powerfully extends the range of problems for which we are
able to state anything interesting, there are lots of mathematical disciplines
that rely on it to a great extent, for example probability theory (laws of
large numbers, central limit theorem, ...). You won't understand almost any
higher mathematics without learning limits first!

One can get an excellent and well motivated introduction to reasonably
rigorous calculus using limits in Courant's "What is mathematics?" in less
than a 100 pages, up to the point of understanding basic differentiation and
integration, the exponential function, power series etc. The problem is not
the formalism, but the teachers who can't motivate the material well enough
both mathematically and physically and students who are not always mature
enough to put in the amount of work necessary to understand calculus, which
for most of them will be by far the most difficult thing they ever attempted
to learn.

~~~
vlasev
The point he's making is against unnecessary rigorization of introductory
calculus and I think you are getting a bit too hung up on the "function-based
approach". I repeat - introductory calculus. There's lot of time and space to
make things more rigorous in a class like Analysis.

When I help students with calculus most of them have no trouble with the ideas
but the implementation that they are required to perform. I spend a lot of
time getting them to understand the simple particulars. There are ways of
teaching calculus that would dispense with some of the more rigorous aspects
and make it a much more bearable experience for most.

~~~
stiff
Introductory calculus classes are hardly ever rigorous. The only formalism you
see is functions and limits, and that's a very useful one, far from
unnecessary, it might just not always be motivated appropriately by poor
teachers who themselves have little understanding of its usefulness. Your
interpretation is also very far from what he has actually written. I edited my
parent comment to make what I mean more clear.

~~~
utxaa
it is true that rigour is eventually needed, but the principia, laplace's
celestial mechanics and countless other works (heard of euler?) were all
published before cauchy and weirstrass. all of the wonderful work in elliptic
functions by gauss, abel and jacobi was done before rigour was en vogue.
euclidean and non-euclidean geometries both flourished wonderfully w/o modern
rigour (when was hilbert's book on geometry published?)

you're also wrong about your examples. it wasn't nowhere differentiable
functions, but fourier series that motivated lebesgue. that's what the author
is referring to regarding analytical traps ie, monotone convergence.

it is also quite a leap to assert the arithmetical definition of limits solves
the zeno paradox!!! i few of my colleague's might disagree with you.

don't led your initial fascination with rigour (it can be addicting) get in
the way of your intuition. rigour is necessary, but it comes after - sometimes
to the chagrin of some. look at the teaching of modern algebra. pure
abstraction and rigour, with complete detachment of all the wonderful ideas,
and experiences that gave it rise.

up with triangles i say :)

~~~
stiff
_it is true that rigour is eventually needed, but the principia, laplace 's
celestial mechanics and countless other works (heard of euler?) were all
published before cauchy and weirstrass. all of the wonderful work in elliptic
functions by gauss, abel and jacobi was done before rigour was en vogue.
euclidean and non-euclidean geometries both flourished wonderfully w/o modern
rigour (when was hilbert's book on geometry published?)_

Now any hard-working student of university calculus can without problems
understand and reproduce the results of those treatises. I think this would
not be possible without the modern systematic methods, including the epsilon-
delta approach.

 _you 're also wrong about your examples. it wasn't nowhere differentiable
functions, but fourier series that motivated lebesgue. that's what the author
is referring to regarding analytical traps ie, monotone convergence._

I mentioned fourier in another comments. Those weird functions were postulated
in the discussion that arised somewhere in the same time period. I don't know
what the author is referring to, because he is irritatingly vague, especially
for a mathematician.

 _it is also quite a leap to assert the arithmetical definition of limits
solves the zeno paradox!!! i few of my colleague 's might disagree with you._

If you are familiar with the concept of a limit, you can notice that Zeno
considers a limiting process of two related quantities, time and the
difference in the position of Achilles and the Tortoise. Since in this
limiting process time get arbitrarly close to some definite value (the meeting
time of Achilles and Tortoise), but never gets equal to it, it stops being so
surprising that the distance between them never reaches zero, though it gets
arbitrarly close. The formalism clarifies what seems parodoxical when
described in natural language. I find this quite convincing, and I never found
a better explanation, altough I know philosophers still dispute this.

I never said rigour takes precedence over intuition. I just think it's the
inherent difficulty of calculus that stops students from understanding it, and
not the epsilon-delta stuff.

