
Vi Hart: Every school should replace calculus with recreational math - RiderOfGiraffes
http://www.newscientist.com/blogs/culturelab/2011/03/mathematical-artist-why-hyperbolic-space-is-awesome.html#
======
xyzzyz
I'm a Math and CS student and still wonder how was it even possible that I did
not start to hate math in high school. Things done there are dull and boring,
exercises are repetitive and do not teach anything substantial other than
applying another algorithm shown by a teacher. I found out that almost
everyone did not really understand not only the problems that were told to
solve, but also the solutions they gave -- and it was hard to blame them, for
it was mainly teacher's and education system's fault not to give them thorough
understanding of what is math really about.

My idea? Dump all that algorithmic crap and teach real problem solving. Do not
punish for bad performance, because it is really hard to actually come up with
a solution to a problem, and this skill is not something that can be learned
by doing the same thing over and over again. Problem solving also require less
concepts to be introduced and internalized, so more time can be spent on
actual thinking and discussing, and less on preparations.

This kind of math course may seem not to prepare to good to "university level"
math, but in fact it is quite opposite -- universities have to start almost
from scratch anyway, and it is easier to teach new concepts a person who has a
good understanding what math really is, than a person who only know tens of
algorithms, used in high school to grind all kind of "problems" given there.

~~~
neutronicus
I'm probably one of the few, but I didn't have any problem with the _paradigm_
of high school math education, just the _pace_.

As far as my later career, it's been my experience that rote manipulation
without conceptual understanding _can_ solve your problem, and that the
converse _isn't_ true. Sometimes, if you do the math right, it will contradict
your intuition.

I'm hesitant to endorse any paradigm that aims to make math more intuitive,
since at the end of the day that's a _lie_. One of the take-home messages of
mathematics education should be that these mechanical processes are _better_
than your own "problem-solving skills".

~~~
_delirium
I guess I've rarely had situations where I had to _manually_ execute most of
that machinery. I agree the rote machinery is quite useful to apply, but the
question for me usually is what to apply and how. Applying it is a job for
computers.

For example, it's quite important for me to know when I might need to compute
an integral, and what I'd do with it. But when it comes to banging out the
symbolic manipulations, remembering tables of what integrates to what,
recalling integration strategies for common kinds of integrals, etc.,
Mathematica is more skilled than I am, so I defer to its expertise, and
haven't tried to keep any of that stuff in my head since high school.

~~~
neutronicus
One case I have trouble doing in Mathematica, etc. are variable
transformations to increase numerical stability. A lot of times if you're
clever you can do a variable change on an improper integral with (integrable)
singularities to get an integral of a continuous, bounded, function over (0,
1). Working out this change of variables is something I find easier to do by
hand.

I also generally have trouble getting Mathematica to do all of the work I need
to do to come up with error bounds on numerical schemes.

------
Swizec
If you aren't familiar with Vi Hart, let me suggest some of the more awesome
videos that seriously tickled my geeky innards.

<http://www.youtube.com/watch?v=DK5Z709J2eo> (infinite series through
doodling)

<http://www.youtube.com/watch?v=heKK95DAKms> (graph theory awesomeness,
through doodling)

<http://www.youtube.com/watch?v=e4MSN6IImpI> (some lovely properties of binary
trees ... through doodling)

<http://www.youtube.com/watch?v=Gx5D09s5X6U> (snakes)

Especially that last one, it poses a question as to whether you can use a
grammar to show that a snake lives (doesn't bite itself). Unfortunately she
doesn't provide a solution to the problem, but me and a few mates went ahead
and proved that it is impossible to prove whether the problem is solvable
without simulating the snake.

------
HSO
I'm really not convinced by this. Sure, math education is far from ideal. But
the solution cannot be to make it more "touchy-feely" and make everybody feel
good.

Courant, in his book "What is mathematics?", says the field historically seems
to go through alternating phases, from wild, creative, adventurous phases in
which new things are discovered -- often inspired by "real-world" problems in
physics or engineering -- to more sober, rigorous phases in which another
generation takes over and basically cleans up the findings of their
predecessors (making proper definitions, constructing and refining the proofs
etc.).

But this is far from what math _education_ can be about.

The biggest obstacle to math education is the meme that mathematics is "only
for intelligent people". Mathematics at the basic, cleaned up level has almost
nothing to do with "intelligence" and a lot to do with memory. I know this
first-hand because I used to be really bad at maths and then one day decided
to learn it anyway (and did). Most often, when someone doesn't get some result
or can't figure out the derivation, it's because he doesn't remember or has
never learned the relevant facts. [EDIT: Also, if you think you need to be
"intelligent" (however defined) to do math, you will give up sooner when you
don't understand sth because you think "uh, I'm too dumb" instead of asking
"what am I missing".]

Mathematics is a kind of language, and as in any language, most people will
actually need to sit down and memorize vocabulary. It's mundane, it's not
fancy-creative, it's a pain in your behind, but once you succeed in automating
certain things, the process takes on a momentum of its own.

This "dragon mom" stuff I occasionally read about in the NYT is silly IMHO but
it's just as silly for a 22 year-old, who by her own admission doesn't even
know the standard curriculum of a plain vanilla mathematics education, to want
to be "the ambassador of mathematics"
(<http://www.nytimes.com/2011/01/18/science/18prof.html>). First you do the
hard work, then you can fly.

~~~
ajscherer
_But the solution cannot be to make it more "touchy-feely" and make everybody
feel good._

Why? That certainly isn't the type of response that would cut it in
mathematics, since it is just an assertion without even a hand-wave to support
it.

Also, I'm not sure you really understand her argument. Her point isn't that we
should to make math classes easier so that people feel good about math; her
point is that students might be more interested in math class if it contained
some mathematics.

The primary aim of math class up through calculus is to teach kids to perform
the calculations needed to do physics, which is needed to do engineering,
which is generally perceived (correctly I think) to be greatly beneficial to
society. This is fine, but there is a huge chasm between this and the types of
things that actual mathematicians study. (My brother once commented that if
people had any idea what was actually going on in the math department, there
is no way in hell they'd fund it).

The best counter to Vi's point, I think, is "why would we want to make kids
appreciate pure math?" It's really fun and really interesting, and personally
I would love for more people to get to experience math, but I'm sure the same
could be said of classical music, or poetry, or knitting or pretty much
anything else that people find stimulating. There may be some benefit to it,
but I don't think that benefit has been quantified.

btw... sitting down and memorizing vocabulary is an awful way to go about
learning a language. Communicating in a langauge every day, reading a ton and
trying to think in that language are much more effective.

~~~
HSO
I like/appreciate your reply a lot (except the first line ;) and would comment
more in response if I didn't have to run just now.

But let me at least thank you for this: "The primary aim of math class up
through calculus is to teach kids to perform the calculations needed to do
physics, which is needed to do engineering, which is generally perceived
(correctly I think) to be greatly beneficial to society."

This is the best summary I have yet seen about why math is taught the way it
is (in high-school, not in university) -- and by extension why many people,
incl. myself, feel or felt uncomfortable with math up to high-school.

~~~
Shorel
Specially because it should be economics and budgets and financial stuff the
focus of math in those formative years. And statistics too!

Not everyone will be an engineer, but almost everyone will have to deal with
money in their life. And the world would be 1000 times better if only people
could understand a little more statistics.

------
RiderOfGiraffes
When people start thinking about math education - as I have at considerable
length - they always come up with the idea that since "real math" is about
problem solving (which it is) and not about rote algorithms and manipulations
(which it isn't) then what we should do is come up with problems to solve, not
formulas to manipulate. We should encourage creative thinking, novel
solutions, and ways of approaching problems.

This is, actually, all true. Students should indulge in problem solving,
analysis, investigation, early proof, ideas of proof, argument, explanation,
and all the things that math is really about, and which make math useful.

After all, who except for science and engineering students ever differentiate
a polynomial? Knowning qualitatively and intuitively what rate-of-change
really means is useful. Differentiating x.sin(x) isn't.

But here's the problem.

To within 20% or so life expectency in the UK is about 70, and there are about
70 million people, so roughly at every age point (up to about 50) there are
about a million people.

How do you assess and grade the mathematical standard (whatever that means) of
about a million children?

How do you persuade a government full of lawyers and classics graduates that
what you're doing is better?

How do you assess the teaching, and the school, if not by the grades they and
their students achieve.

It's all about assessment.

So that's the first stumbling block.

The second is the question of getting teachers who are up to the task of
teaching in an open and interactive manner, rather than trying to teach rote
algorithms and manipulations.

But you don't need to worry about that - schools will never teach proper
mathematics. They'll continue to teach arithmetic and manipluation.

~~~
wisty
If you don't get the fundamentals down, you can't handle the applications. Wax
on, wax off - boring but effective. But you also need to spend time putting
those fundamentals to work.

The real question is, why do you need assessment? To test that most kids are
getting the fundamentals? Easy. Have a few core tests. To check that class
time is also devoted to creative stuff, and not just drills? That's a
management problem. You _can't_ test whether or not kids have had (say) 50% of
their time doing fun non-core work, but you can regulate that through other
means.

~~~
RiderOfGiraffes

      > If you don't get the fundamentals down,
      > you can't handle the applications.
    

Absolutely, which is why I advocatedoing the algebra, doing the manipulation,
and doing "the sums" until the act of doing them is almost reflex.

    
    
      > The real question is, why do you need assessment?
    

I'm assuming you're serious. Society demands assessment. Government demands
assessment. Employers demand assessment (and then ignore it and apply their
own tests). Students themselves are conditioned to demand assessment - there
are many instances of teachers or lecturers playing with the idea of not
giving grades, and mostly the students hated it. (whether it worked better is
another question)

FWIW - I advocate a radical shift in what's done, and am working subversively
on a ten year plan to make changes. Just becuase I can identify some problems,
don't assume I believe they are real, or insurmountable.

~~~
ANH
> Society demands assessment. Government demands assessment. Employers demand
> assessment.

My college didn't hand out GPAs and all evaluations were narrative. Only on
rare occasions has the lack of numbers attached to my performance prevented me
from obtaining what I wanted, and even for those cases I remain dubious about
the benefits.

~~~
gamble
How did they apportion scholarships or determine who received a spot in
programs with a limited number of openings? Institutions assess for many
reasons, but primarily because educational opportunities are rationed and they
need some method for separating the worthy from the unworthy without relying
entirely on personal testimonials or politics.

------
mwill
Vi Harts youtube is pretty entertaining, Here's a video that shows pretty well
what her videos are like: <http://www.youtube.com/watch?v=Yhlv5Aeuo_k>

The whole topic vaguely reminds me of Khan Academy, both seem to recognize
that the way math is taught in schools at the moment is fundamentally broken,
and doesn't engage students the way it should.

------
ANH
Freeman Dyson wrote this of Richard Feynman: "The reason Dick's physics was so
hard for ordinary people to grasp was that he did not use equations. The usual
way theoretical physics was done since the time of Newton was to begin by
writing down some equations and then to work hard calculating solutions of the
equations ... Dick just wrote down the solutions out of his head without ever
writing down the equations. He had a physical picture of the way things
happen, and the picture gave him the solutions directly with a minimum of
calculation."

edit: That's from Dyson's autobiography 'Disturbing the Universe'.

~~~
HSO
Which is interesting and awesome but so not relevant for general public math
education... ;-)

------
crasshopper
Every school should replace calculus with spreadsheets and a basic programming
course. Or maybe with statistics. Each is more practical than calculus.

 _(I don't mean to put down calculus; it's spiritually rewarding and, when
paired with a few other subjects, yields crazy awesome higher math.)_

~~~
jokermatt999
What basic programming language would you suggest? I started with Visual
Basic, and I found that it really didn't prepare me for programming in
general. It seems like the best combination of useful and easy would be either
Python or bash, but I'm sure other people can come up with better suggestions.

I think the key is to find something that's teachable to people completely new
at programming, but something that will be useful to them. I was thinking of
Python due to it just feeling "natural" to me, at least after coming from VB
and then Java. Bash is somewhat of a weird choice (especially considering most
students aren't going to be using Linux), but I was thinking about that due to
it being useful for immediate computer tasks, but it may not be enough of a
"real programming language". I wish Windows had a good equivalent scripting
language...

~~~
crasshopper
I'd start with Python. Try to figure out how to do one thing that you think
would be cool w/ python. Like learning to play the guitar by learning a song
you like.

Really it depends on what you want to do...but Python is widely used in
science, finance, math, and the web (thru django). So that covers all of what
I'd want. And it's a high-level (read: easy), widely used (read: supported)
language.

If I were just starting out I'd use zed shaw's book. As it was I found the
django tutorial good; and Norman Matloff's tutorial on scientific python.

I'd learn programming in linux. Install Ubuntu if you don't already use linux.
You can dual boot and leave the windows side for video games.

HTH

------
amitagrawal
The only reason Calculus seems so dry is that the ones teaching it are not
very competent and don't understand the subject quite well.

During my school I'd ask teachers all about limits and why we need it and
their attitude was - Don't get too involved - here are the formulas, go learn
them and solve the exercises.

I later discovered that there were some amazing applications of limits
(calculus) but wasn't taught the way they were meant to. This is the reason
why I never found Calculus to be interesting.

------
rohanprabhu
If taught properly, and with the proper illustrative exercises, calculus is
recreational. Period.

~~~
psykotic
Here's the thing. Most of calculus's power comes from its use as a mechanical
calculational system. Hence the name. Archimedes's proof of the surface area
and volume of a sphere is far more insightful and explanatory than the
standard calculational proof using integration that any unthinking freshman
can crank through with ease. As a rule, calculation proceeds by reducing a big
problem into a sequence of small, easily dispatched problems, but human
understanding usually works the opposite way. So, it's not that calculus
cannot be taught in a way that highlights human understanding, but the
greatest strength of calculus is that it produces answers whether you
understand what you are doing or not.

------
alain94040
I know people here (the top 0.1% of the high school class) love to bash math
education in particular. I'd like to ask how you would apply your views on
teaching a topic such as History for instance.

Sure, rather than learning the dates of middle-age battles, it would be more
fun to interpret a play, in costume, or rebuild a battlefield with legos.

But can you spend your entire school year this way, or could this
"recreational" teaching need to be combined with more traditional teaching?

~~~
crasshopper
Use primary sources.

Have individuals or teams look at an original primary source, interpret it,
then discuss with the class, who all interpret something from a related
source. Students put their heads together, discuss/debate and post some
interesting original research to the Net.

~~~
shou4577
In mathematics?

Primary sources are way above the level of most college mathematics students,
much less high school or younger students. Almost anything written in the last
50 years requires very advanced mathematical ideas, while anything written
before that uses outdated notation which is very difficult to understand.
Moreover, the primary sources for common ideas are often spread out over huge
periods of refinements - going from a small idea to a general notion over the
course of years, perhaps involving dozens of papers by multiple authors. One
notable exception to this: Euler's papers. These might be readable by some
students, yet when I see them presented in college courses, they are often
presented by a secondary source for clarification.

The other thing I don't agree with is your use of the word "interpret." I
don't think that it is common to find a mathematician who believes that
mathematics is open to interpretation.

Lastly, original research in mathematics is hard. It's not the interpretation
of previous ideas (although it is occasionally relating concepts previously
thought to be unrelated - some consider this the most important type of
mathematical result, but it is much harder to do than you might think), but
the formulation of new ideas. Even to know if an idea is new requires a great
deal of mathematical training.

Perhaps I'm wrong about this, but I think that for young mathematics students
your ideas are unfeasible. Remember that the original sources for a lot of
basic mathematics are hundreds of years old. Even Euler's famous writings on
geometry (the exception that I said might be workable) were largely a
rewriting of previous ideas into a coherent whole - basically a textbook - and
could not really be considered an original source.

Edit: my apologies, I misunderstood. I did not realize that we were talking
about teaching history, in which case I defer to someone of greater
experience.

~~~
crasshopper
shou,

alain94040 asked what we would change about the way History is taught.

------
sp332
The head of the math department at my college actually taught one of these
classes. He would assign two students each class period to take notes. He said
that way, the rest of us could pay attention :) He would stand at the front
and ramble about whatever came to mind, and scribble on an overhead
transparency. He would start with a topic and take it wherever we asked:
multiple dimensions, knot-tying, origami, computational geometry, ray-tracing.
The course was called "3D Math" or something, so everything was relevant :)
There were no tests, and we got to choose our own final projects. The notes
and overheads were duplicated as study guides for the final, and the questions
were taken right out of them. It was the most informative math class, if only
because we were all doing our own thing and learning from each other.

I did a 3D extension of the Lyapunov fractal images that Mario Markus
published in Scientific American in 1992. Markus's 2D:
[http://charles.vassallo.pagesperso-
orange.fr/en/lyap_art/lya...](http://charles.vassallo.pagesperso-
orange.fr/en/lyap_art/lyapdoc.html) and my 3D
<http://www.youtube.com/watch?v=-omN5ZM3Jho> with a moving cross-section
<http://www.youtube.com/watch?v=ePRJfF3pwqg>

------
Tycho
Incidentally there is a textbook on calculus that Donald Knuth said he read in
his first year at university, and after doing all the problems in it he moved
from beginner to top of the class (and thereafter, renowned computing
scientist). I think the author was named Thompson, but I haven't seen the book
myself. Maybe some people here have?

~~~
akuchling
Maybe it was "Calculus Made Easy", by Silvanus P. Thompson? That's an old book
that explains the basic idea of integration and differentiation, but doesn't
try to be rigorous at all. [http://www.scribd.com/doc/8533492/Calculus-Made-
Easy-by-Silv...](http://www.scribd.com/doc/8533492/Calculus-Made-Easy-by-
Silvanus-P-Thompson)

~~~
hsmyers
Non scribe D version: <http://djm.cc/library/Calculus_Made_Easy_Thompson.pdf>

------
ihumanable
I had no interest in math until I had Calculus in High School. I then went on
to pursue dual degrees in Computer Science (because I <3 Programming) and Pure
Mathematics (because Calculus made me <3 Math).

Maybe it was just an exceptionally good teacher, but the more I learned about
Calculus the more I feel it was the subject itself. There was something
seductive in the idea that by changing the way you think about something, like
the area under a curve, you could calculate the previously impossible. The
core of Calculus (at least the part you would learn in High School),
derivatives and integrals, are easily enough explained.

I'm glad I had Calculus or I wouldn't have gone on to pursue mathematics so
doggedly, I wouldn't have gotten to learn the strange and beautiful world of
numbers, I may have missed out on one of my great loves.

That said, I'm sure in the hands of a less than stellar teacher, Calculus is a
painful thing.

------
mtalantikite
When I was in school majoring in math we always thought back to how poorly
motivated the subject was growing up. So we decided to explain some pretty
basic group theory (permutations, etc) to our friend's dad who taught
elementary school.

When he brought in the topic with some basic shapes to illustrate the concepts
to his kids (4th grade, I think), they all went crazy for it. We then gave him
other concepts from a geometry class, which they loved as well
(<http://www.gang.umass.edu/~kusner/class/462hw>).

I think catching them at an early age -- sparking their natural curiosity and
giving them confidence in the subject -- is most important, before they're
disheartened by all the years of computation and applying algorithms.

------
mcdaid
Well some of the videos look interesting but I don't agree with the title,
what is recreational math?

It would be great if Maths could be made more interesting and students make
their own unique discoveries etc. However as a former maths teacher I can tell
you that structure exists for a reason.

In a class of maybe 30 students for one hour, the teacher can afford to give
at best only 2 mins average time to each student. Some students may work very
well independently, but others would take advantage of the freedom to mess
about.

Even if the students were focused how is a teacher with so little time per
student supposed to pick up a random piece of work and then decide if the
child is doing something of worth.

The Maths curriculum needs to be rethought for the information age, but the
solutions need to be practical.

~~~
sid0
I think the Khan academy has a solution to this particular problem. Salman in
his TED talk proposed that students view the videos in their own time for the
basics, then engage in problem solving in class so that the teacher can devote
much more attention to each student.

~~~
RiderOfGiraffes
Experience with this shows that the kids simply don't watch the videos. It
sounds brilliant - let the kids watch videos together, they can eat pizza,
drink cola, laugh, text, watch the video again to get the bits they missed,
and then come to class to work the problems and get help.

The experiment has been done - the vast majority of the kids don't watch the
videos.

~~~
sid0
Is there a record of the experiment online somewhere?

~~~
RiderOfGiraffes
I'm pretty sure I read of it on a web site lunk to from here, but I don't have
a record of it. I have seen it reported from at least two different people who
tried it, but don't have a record of where I read it. Sorry.

EDIT:

I've found one reference to a teacher who's trying it but has some kids who
don't watch the videos:

[http://www.msteacher2.org/forum/topics/flipping-the-math-
cla...](http://www.msteacher2.org/forum/topics/flipping-the-math-
classroom?xg_source=activity)

Here's another in the comment stream for the Khan TED talk that suggests the
idea:

[http://www.joannejacobs.com/2011/03/khan-use-video-to-
flip-e...](http://www.joannejacobs.com/2011/03/khan-use-video-to-flip-
education/comment-page-1/#comment-159231)

There are more, but they require filtering/finding:

[http://www.google.co.uk/search?q=math+homework+flip+video+%2...](http://www.google.co.uk/search?q=math+homework+flip+video+%22don%27t+watch%22)

~~~
sid0
Thanks. That's somewhat disappointing, I guess.

------
duck
I have to ask... does the name Vi come from where I think it does? If so...
that is awesome, and then I would have to ask is there a improved version of
her named Vim?

------
upgrayedd
Everyone here on HN always seems to say the same thing: the way math is being
taught in most schools sucks, we need to change it. Math is glorious, it's the
kicks, etc.

I'm curious why everyone is so enamored with mathematics here though; I've
tried to understand what is so captivating about the subject but have failed
to do so for years.

I've never gotten past trigonometry and algebra myself, and this idea of
Calculus being more than the tedious integration work that textbooks I tried
to self teach from said it was is intriguing.

Enlighten me, how is mathematics not just a bunch of rules and methods to
solve problems within constraints, and that some professional practitioners
are forced to prove...

How many of you programmers and software engineers here even interact with the
Calculus you learned in high school/college on a day-to-day basis in your
apps/software projects?

------
anactofgod
"You don't actually need to know a lot of the groundwork and the basics to
appreciate the exciting bits of mathematics. The more technical things are
still awesome as tools and are necessary in some places. But luckily I am
usually with other people who do know that stuff."

How about this not being an either/or? Anything that exposes more people to
the beauty and power of the maths, and convinces some of them to more of the
"technical things" is goodness, IMHO.

------
daoudc
It's so true that the way that maths is taught at schools needs a complete
revamp. At secondary school level it should be taught as a creative subject
from the start: choose some rules and see what happens.

------
forkandwait
Just one question -- does the US have a shortage of engineers because math
education isn't touchy-feely enough now?

