
What Silicon Valley gets wrong about math education again and again - occam98
http://blog.mrmeyer.com/?p=12782
======
Arun2009
The trouble with trying to arrive at any single definition of Mathematics is
that Mathematics is different things to different people. A research level
Mathematician might see it differently (finding patterns, abstraction, theory
- axioms and proofs) from an Engineer who has a purely practical interest in
it (cookie cutter methods and formulas). For everyday use Mathematics _is_ a
set of algorithms for doing stuff with percentages, fractions, basic
arithmetic etc.

Currently what's frustrating is that we subject everyone to the same level of
training. Those whose interest lie elsewhere are subjected to unnecessary and
meaningless torture, while those who are really interested in Mathematics as a
subject are left to their own devices to find additional training (which
thankfully is easier these days than when I was a kid, though it could be made
substantially easier still).

~~~
slowpoke
You beautifully sum up my experience with math at the university, as a CompSci
student in Germany. The absolutely most frustrating thing for me, in the
beginning, were the basic math lectures (which I, as a disclaimer, still
haven't finished). At first, of course, I did what most students do in their
first semester: bitch about the apparent uselessness of the subject.

That's about one and a half years back, now. I realized by myself that math is
actually pretty useful. The problem, as I still see it, is that _nobody
fucking tells you why_. They throw it at you and expect you to deal with it.
No further explanation - "It's math, we ain't gotta explain shit."

Personal eye openers for me were my internship (and continued assistant work)
at a department of the university that deals with image processing (which is
basically a boatload of applied math), articles like Wolfire's excellent
"Linear Algebra for Game Developers"[1], and plunging myself into a lecture
about Fuzzy Set Theory and Artificial Neural Networks (which again is applied
math).

I actually like abstractions, generalizations and logic reasoning (I wouldn't
study CS if I didn't). I still don't think that math is the only way to teach
those, but it's arguably the largest field of science dealing with these
things.

I don't really think the problem is math itself, it's the way it is taught to
most people - in the math way. Mathematicians teach math to do _more_ math,
which in turn is used to to _even more_ math. And that's fine - for a
mathematician. For literally every other field of science that needs math as a
tool, it's highly frustrating to be subjected to what you aptly describe as
"torture".

[1]: [http://blog.wolfire.com/2009/07/linear-algebra-for-game-
deve...](http://blog.wolfire.com/2009/07/linear-algebra-for-game-developers-
part-1/)

~~~
impendia
I am a math professor at a large state university in the US. A comment and a
question:

> Mathematicians teach math to do more math, which in turn is used to to even
> more math.

That is true of some mathematicians, and of some subjects. But I think most
mathematicians teach and learn math because they think it's cool, not because
it's useful -- either to learn more math, or for the "real world".

I might also add that I always appreciate it when teachers take a very long
view of things. For example I have trained in the martial arts, and most
teachers take the perspective that you want to master the art completely,
rather than just learn one or two things. Some people don't like this, but I
deeply appreciate it.

> The problem, as I still see it, is that nobody fucking tells you why. They
> throw it at you and expect you to deal with it. No further explanation -
> "It's math, we ain't gotta explain shit."

As a math instructor I am always looking for ways to improve, but I'm having
trouble extracting useful criticism from your remarks. "They throw it at you
and expect you to deal with it" seems to be true of challenging lessons in
anything.

I would cheerfully welcome general advice and suggestions from you or any
other HN readers. (But please keep in mind that both my interests and my
expertise skew heavily theoretical!)

~~~
luke_s
Slowpoke's comment really resonated with me, so perhaps I can provide some
concrete examples.

8 years ago, I graduated from my Software Engineering degree at Swinburne
University in Australia. As part of my degree I had to sit through 3 years of
engineering maths. In our first programming tutorial we were taught how to
write a program to display “Hello world”. It was immediately obvious to me why
we might want to write a program that displays messages on the screen and what
sort of problems this would be useful for solving.

In our first maths tutorial we were taught how to add, subtract and multiply
complex numbers. To this day, I still do not understand why I would need to do
that. Where do my complex numbers come from? What do they represent? When I
add them together, what does the answer mean? The only problems that I can
apply my maths education to, look exactly like those on the tests: what is
(-3.5 + 2i) + (12 + 5i) ?

It is possible that in my job I am besieged daily with problems that I could
use complex numbers to solve. But if so, I am totally incapable of recognizing
them!

Fourier series were a particularly egregious example – the subject started,
when the lecturer came in and wrote up 3 boards of dense maths, and said
something along the lines of “… and this is the formal derivation of a fourier
series!” It was as if somebody had tried to teach programming by explaining
the algorithm a complier uses for translating source code into machine code.
Then expecting the students to just figure out how to write actual useful
programs, all by themselves! I believe this is what Slowpoke was talking about
when he said: “The problem, as I still see it, is that nobody fucking tells
you why. They throw it at you and expect you to deal with it. No further
explanation - "It's math, we ain't gotta explain shit."

The way we were taught fourier series particularly hurt. Years later, I found
out by myself, that the things are actually incredibly useful. As it turns
out, that there are these things called fast fourier transforms, that programs
use all the damn time, to do fantastic stuff!

So, to give a very concrete example of how maths education in universities
could be improved: If only the first lecture on fourier series had instead
explained what they are used for, and why they were so important that we were
going to spend 5 weeks on them. Then perhaps I would have had a much better
understanding of what they are and how to use one. It helps so much, to be
able to think as the lecture is writing boards filled with formulas: “Shit!
Now I can use this to do X!”

Thanks for taking the time to listen. It’s such a shame that maths is being
taught this way, because mathamatics is both so very useful and very
important…

Edit: Actually I reflection I sat through 3 years of maths, not 4

~~~
jwhite
If you had enrolled in electrical engineering then by half way through second
semester of first year you probably would have been having lots and lots of
light bulb moments. Complex numbers and fourier series and transforms rapidly
become like oxygen for EEs. Essential but consumed almost without noticing.

You suffered from the university's lack of bandwidth -- only so many
lecturers, only so many lecture theatres, many other competing classes. They
try to find efficiencies where they can -- e.g. by giving all of the
engineering departments a common mathematics curriculum.

That curriculum is almost certainly driven by the needs of the EEs and
ME/CivE's much more than chem and software, and I'd guess software engineering
is looked at is if there's really no maths required. After all, software
engineers just program and write documents, don't they? The truth is of course
that there is just as much scope for maths, if not more, in a software
engineering or computer science course, it's just a whole other type of maths
that's needed.

~~~
luke_s
Hmmm ... It is entirely possible that the EE's went straight out of that maths
tutorial and into an electronics class that used complex numbers for a
practical application.

To be honest all us software engineers understood that we didn't make up the
majority of maths students. I would have been perfectly happy to hear about
how I could have used complex numbers to design circuits or build bridges. But
I guess the maths classes were designed to provide the pure theoretical
foundations upon which other, more subject specific classes would build. It
just so happens that for software engineering there were no subject specific
classes which made use of the maths.

------
troymc
Dan Meyer is the guy who gave this TED Talk:

[http://www.ted.com/talks/dan_meyer_math_curriculum_makeover....](http://www.ted.com/talks/dan_meyer_math_curriculum_makeover.html)

He taught high school math between 2004 and 2010, but became frustrated and is
now studying at Stanford on a doctoral fellowship, with interests in
curriculum design and teacher education.

I'm just glad that there's now a public debate about better ways to do math
education.

~~~
warfangle
I really wish I'd had a teacher like him back when I was in high school. I had
an algebra 2 teacher call me an idiot in front of the class ... and when I
went for after school help, he simply made fun of me the whole time.

Ugh.

Now I write lots of code. Yay!

------
nickolai
I agree that matching the variety of possible assignements by some generic
method is very difficult, though in my humble opinion, we already have a lot
of the tools required. There is a ton of improvements to do(lots and lots of
usability), but the base is there - and a project of such a scale would not be
unheard of.

> Its hard to communicate a fraction to a computer

> "Explain whether 4/3 or 3/4 is closer to 1, and how you know."

You just did that. Twice. Square root ? "(3/4)^(1/2)" or maybe "sqrt(3/4)".
There's no complexity in parsing that. I do agree it is not as natural as on
paper but maybe tablets will find a way to improve that. Thats what innovation
is here for after all.

>"Explain whether 4/3 or 3/4 is closer to 1, and how you know."

I am not familiar with the domain, but dont we have some automatic theorem-
proving tools? Validating the answer to such a question would look like a
perfect use case to me.[edit : clarified]

> [the description example]

Im not sure about this one. On the one hand i used to have a project in
college about reconstructing a picture from an incomplete description - and
its _hard_. On the other hand we are expecting a perfect description. Hence it
would be pretty much isomorphic to the code of a program used to draw the
picture. Matching the two images is also doable. Pseudocode would actually be
the best way to transmit this image .

~~~
cube13
This is the exact attitude that the blog post is saying is wrong. For
mathematics, computers are tools. Computers don't create the answers, they
assist the user in finding the answer. They're time-saving and error checking
devices, which are useful after the student learns the concepts inside and
out. They are not supposed to solve the problems directly.

>>"Explain whether 4/3 or 3/4 is closer to 1, and how you know."

>I am not familiar with the domain, but dont we have some automatic theorem-
proving tools? This would look like a perfect use case to me.

Theorem proving tools would work if the students wrote their answers in a
format that the tool would work in. In this case, it would be a natural
language proof instead of a formal proof, which simply isn't possible to parse
right now. Perhaps it will be in the future.

~~~
nickolai
> What does a student learn from this? They're learning the tool, not the
> process of solving the problem.

They will learn the process _if the tool is only used behind the scenes to
validate their answer_

~~~
cube13
I misread your post, and did a ninja edit. Sorry about that!

------
ddmeyer
I wrote this post. To summarize my argument in three lines:

There ARE different ways of defining mathematics, and some of them contradict
each other.

Silicon Valley companies wrongly assume their platforms are agnostic on those
definitions.

For better or worse, if you're trying to make money in math education, the
Common Core State Standards are the definition that trumps your or my
preference for recursion, computational algebra, etc, and those standards
include a lot of practices for which, at this point in history, computers
aren't just unhelpful, but also counterproductive.

~~~
pingswept
Elsewhere on this page, someone comments: "I would love to see one of these
'complaining about Khan Academy' articles with an attached example of the
author's Right Way to teach whatever subject."

I'd be interested in your response to that.

~~~
jfarmer
I agree the article was too strident, but if you watch his videos I think they
do a good job of illustrating his philosophy.

~~~
pingswept
I don't think that the article was too strident at all. (I was a high school
math teacher for years, and computers seem utterly useless to me when it comes
to teaching kids to think mathematically.)

I'll take a look at the videos, as you suggest.

~~~
jfarmer
Fair enough! Here's a good video: <http://mrmeyer.com/threeacts/speedoflight/>

I like the combination of the video and the info-animation. Really
interesting.

------
Jarred
I'm typing this on my phone as I sit in Geometry.

Math is presented as several pointless tasks that need to be memorized, in
order for the test in a week. For example, every exam tests if we "know" the
theorums from the tested sections. An example of a theorem we were tested on
is, "Each diagonal of a rhombus bisects two angles of a rhombus". Students can
see that that statement is true, but students don't know why (including me).

As a result, we don't see any applications of it besides the test, the final,
and future math classes.

I haven't seen any software that doesn't just make what teachers already do
faster. Everything I've seen for math is just incremental -- things like
putting a scantron online, or letting me find out what my homework is online.
At this point, technology isn't helping me understand math better.

~~~
Someone
_An example of a theorem we were tested on is, "Each diagonal of a rhombus
bisects two angles of a rhombus". Students can see that that statement is
true, but students don't know why (including me)._

The way to construct a rhombus is left-right and up-down symmetric. Because of
that, you can pick one up, flip it along a horizontal or vertical axis, and
put it back exactly where it came from. That operation keeps the diagonals
where they are, and puts those to-be-proven-equal angles on top of each other.

------
mc32
I'm I getting this right?

He's basically disappointed SV is attacking this problem with the methods they
understand best?

I don't see how one woud expect SV-type startups to adress maths education
without leveraging their strength --which is in computing. I would expect
experts in education to use other more pedagogical approaches.

It's like going on about a carpenter who wants to approach a problem with wood
in mind.

~~~
blake8086
I would love to see one of these "complaining about Khan Academy" articles
with an attached example of the author's Right Way to teach whatever subject.

I don't think there's really a route to improving education merely by
complaining about the approach of others.

~~~
fnoschese
I have routinely offered my own versions and contrast them with Khan's video
lessons. Watch this MSNBC clip about Khan Academy and my classroom at about
1:30 : <http://www.youtube.com/watch?v=GwE6iWEhtRk>

And this video about physics without lectures:
<http://www.youtube.com/watch?v=yKcjuIUxwo4>

Also read these: [http://fnoschese.wordpress.com/2011/10/28/newtons-3rd-law-
or...](http://fnoschese.wordpress.com/2011/10/28/newtons-3rd-law-or-how-to-
make-effective-use-of-video-for-instruction/)

[http://fnoschese.wordpress.com/2011/06/22/khan-vs-karplus-
el...](http://fnoschese.wordpress.com/2011/06/22/khan-vs-karplus-elevator-
edition/)

~~~
blake8086
That looks pretty great. I guess your approach only has a few downsides:

* you can only teach (I'm guessing) 300 or fewer students per year, so you can't really have the same reach as Khan

* your method costs orders of magnitude more (someone is paying your salary, and for facilities)

* you can only reach students who are physically near you

* students can only access this education on a very rigid schedule

I don't think your approach is bad, but I don't think you're really solving
the same problem. How could you scale your method to teach the entire world
for a few dollars per year per student?

~~~
fnoschese
I think you have to look at the trade-offs of scale. McDonald's serves 58
million people EACH DAY ([http://understandingbignumbers.com/how-many-people-
does-mcdo...](http://understandingbignumbers.com/how-many-people-does-
mcdonalds-serve-daily)) at extremely low cost. How does that affect the
quality of the food and the experience?

If Khan's method scaled (at lower cost) without sacrificing other features of
high quality face-to-face instruction, then I would expect people who pay
private school tuition (K-12) to push their schools to implement such an
approach in order to lower tuition.

But there's a reason why the elite pay $30K/year for K-12 private schools:
Small classes, strong student-faculty relationships, high-quality facilities,
and the exceptional educational experience that comes along with those
features that cannot be replicated on a large scale.

------
Duff
The problem here is that _everybody_ is wrong and everybody is right. There's
no magic method that will help everyone learn everything. The answer is that
we need to figure out the right mix of things, and empower students and
teachers.

My knowledge of Khan Academy-type methods mostly comes from some NPR features
about experiments in New York City. In the featured experiment, there was a
classroom of kids that were in a workshop setting. Some using computers,
others working at desks with access to a remote tutor, others talking to the
teacher.

The teacher was in a role more like a coach vs. a talking head at the front of
the room. Different students need help at different points, and using the
computerized system enabled the teacher to provide more individualized help.

Educational establishment types are afraid of change -- because their power is
derived from the cash flow of union duespayers whose jobs may be at risk.
Silicon Valley types see a way to tap into a ridiculously large stream of
money.

------
tylerneylon
Learning math from a human in a classroom works because it forms a local
culture that cares about math. Students have a chance to be motivated by what
the teacher cares about, and this is a very social phenomenon.

It is harder to express sqrt(3/4) on a computer in a meaningful way because
there is no ubiquitous and intuitive interface for doing this -- the article
has a good point here. But that doesn't mean it's a problem we can't address.

I like the idea of thinking of students in a classroom as young
mathematicians, because it respects the students and inclines us to teach in
alignment with real-world use cases -- it addresses the 'when will we ever use
this' question. From that perspective, it would be a mistake to exclude
technology based on its shortcomings. If it is genuinely useful for doing
math, then it is worth learning.

I see a lot of concern about math education -- I think most of it misses the
main problem, which is that there's zero motivation for people who are great
at math to become teachers. If I have a PhD in math, and I become a math
teacher, then (i) my math friends will think I will never publish a paper
again, (ii) they'll think I am no longer good enough for serious math
research, and (iii) I will take a huge pay cut vs typical math-PhD jobs. I
might as well add (iv) I would suddenly be working with many people who don't
respect or care about what I'm good at (students). I'm an optimist - I think
all of these are addressable in the long run. Questions about the virtues of
technology are secondary to the ecosystem and cultural perceptions around
teachers.

~~~
ghshephard
"It is harder to express sqrt(3/4) on a computer in a meaningful way"

I'd argue that you just did. Wolfram Alpha certainly has no problem with your
expression. Nor does Excel.

~~~
jfarmer
Fine. Express the homological proof of the Brouwer fixed-point theorem on a
computer.

Ideally in a way that is easily understood by both a computer and a
mathematician. Bonus points if it can be used as a teaching tool, too, so that
a (sufficiently mature) math student could read it and follow the proof.

Assuming you agree the above is Really Hard™, then we've established there's
an upper limit on how effective computerized math education can be (for now).

This leads to a bunch of interesting questions. Where is that limit? Can we
design a computerized version that satisfies the design requirements for that
system? Do we have the technology to make it happen? Is it even a technology
problem to begin with?

Yabba dabba doo.

------
px
We have arrived at a point where innovation in education is just beginning.
Khan Academy doesn't have all of the answers, but they have made a significant
contribution to the marketplace of ideas and tools, which will continue to
evolve and grow.

That type of innovation is the key to significant progress. And this sort of
discussion will drive it forward.

So many people seem eager to pick the winners and losers in this space right
now. It is a bit early for that.

------
backspaces
As lovely as symbolic mathematics is, it does have one major problem: it can
not be parsed.

I do not mean the symbols cannot be drawn etc, but even a simple expression
like "ab + c" can not be disambiguated. Is it "a*b + c"? or is there a
variable "ab" added to c.

This requires the idea of "closures" .. i.e. history/state in which the
expression appears. (Wolfram discussed it at a conference once, but I don't
have the reference alas.)

So the solution is either to simply introduce new symbols to facilitate
parsing (i.e. require multiplication symbols .. but there are more) or
introduce heuristics that use history as the human does.

One promising technology is touch screens and sophisticated trackpads (Magic
Pad for example) which lets us "draw" mathematics.

------
snowpolar
For me, traditional maths education in my country fails me (although my
country is ranked very highly in maths and the implications of failing maths
is huge as schools reject you)

I would say, what don't work for you doesn't mean it won't work for others.
Khan Academy, at the very least gave me hope that I can actually do some
maths, which is more important than giving up on maths totally.

------
jfarmer
Awesome! I was just talking about this with a friend the other day. My formal
education is in mathematics, but I've been writing software since high school.

I wish Dan made his argument in a less pointed way, though.

The underlying question is this: what does education look like on the web? Can
every subject be taught there?

In design, a skeuomorph is a derivative object that retains some feature of
the original object which is no longer necessary. For example, iCal in OS X
Lion looks like a physical calendar, even though there's no reason for a
digital calendar to look (or behave) like a physical calendar. The same goes
for the address book.

This is what I see happening in online education. I don't think it's a case of
"lol, Silicon Valley only trusts computers," but rather starting off by doing
the most literal thing.

Textbooks? Let's publish some PDFs online. Lectures? Let's publish videos
online. Homework and tests? Let's make a website that works like a multiple-
choice or fill-in-the-blank test.

These are skeumorphs. There's no reason for the online equivalent of a
textbook to be a PDF, it's just the most obvious thing.

For me it's 1000x more interesting to ask "On the web, what's the best way to
do what a lecture does offline?" than to say "Khan Academy videos are the
wrong way of doing it."

I think sites like Codecademy point the way when it comes to programming. The
textbook is the IDE.

What does that look like for math? It's much harder because, like Dan says,
computers aren't the natural medium for mathematicians, so there will always
be a translation step from math-ese to computer-ese.

Once you're past basic math and are working out of a higher-level textbook,
the exercises becomes very awkward to express on a computer in a way a
computer can understand.

And then grading -- oh boy!

Take this, for example. Here is the first exercise from my first-year college
calculus textbook (Michael Spivak's Calculus).

    
    
      Prove: If ax = a for some number a != 0, then x = 1
    

If you see that and think "That's easy enough to express in a way a computer
understands, and there are proof verification systems" you're missing the
point. No mathematician does his work that way. It would be like asking
someone to learn programming by reading through a book and instead of writing

    
    
      for(i = 0;i < n;i++) {
        printf("%d\n", i);
      }
    

you force students to write it down by hand in plain English instead of C. If
you're an engineer, think of all the ink spent on whether whiteboard
interviews are good or not. Asking mathematicians to do their work on a
computer will get the same kind of response.

(Empathy, brother!)

The above was just my stream of consciousness, but I've been thinking about
this for a bit now. Love this topic!

*: Yes, I've seen his videos. They're fantastic and do a great job illustrating his educational philosophy.

~~~
ezyang
There are areas of mathematics for which proof assistants do a much better
job; here is an interesting presentation using the theorem prover Coq as a
virtual TA for discrete math:
<http://www.cis.upenn.edu/~bcpierce/papers/LambdaTA-ITP.pdf>

I want to take objection to the presentation of your example. It should be
contextualized with the first chapter of Spivak's textbook, where he talks
about algebraic properties of addition and multiplication for "numbers" (which
aren't particularly well defined, but ostensibly since this is a Calculus
textbook they are actually real numbers.) You can do these proofs fairly
straightforwardly with rewrite rules, and proof assistants actually have
rather good support for this. (The real difficulty shows up when you get to
more complicated theorems, when we'd like to sweep rigor a little bit under
the rug.) Look at the end of this book chapter for an exercise which asks the
user to teach a proof assistant how to automatically show many simple
properties of groups <http://adam.chlipala.net/cpdt/html/LogicProg.html>

~~~
jfarmer
Thanks for the reply!

I'll read that presentation when I get back. First impression: the author
needs to learn about luminance as it applies to readability! Blue text on
black background == headache.

When you say "do a better job" what do you mean? I'd love to see papers about
it.

And yes, in that chapter you learn the field axioms and it's fairly
straightforward for a computer to verify the statement _given you present it
to the computer in the correct way._

That's my point, and I think Dan's point, too. Mathematicians don't do math
that way.

Like I said: "If you see that and think "That's easy enough to express in a
way a computer understands, and there are proof verification systems" you're
missing the point."

As for <http://adam.chlipala.net/cpdt/html/LogicProg.html>, it looks
relatively foreign to me. I'm going to read through it and figure it out, but
would it helpful for a first-year college student who went from AP Calculus
the previous year to Spivak's Calculus the next to be spending their time on
logic programming?

They've probably never even heard the word "lemma."++ It assumes you're at
least a little proficient with the language of mathematics.

Pointing to things like Coq as a solution is like telling an average computer
user, "Why use Dropbox when you can just use rsync and some shell scripts?"

It might be the basis for something better designed, but it's only part of a
larger, hitherto undiscovered, solution.

++: I'm describing my experience. Spivak was the first time I was introduced
to the proof-as-exercise paradigm that's the cornerstone of every higher-math
textbook.

~~~
ezyang
Proof assistants have close ties to logic and type theory, so it's not much
surprise that they're quite good for doing this small, subarea of mathematics.
(Indeed, I think in the not too distant future we will be teaching logic using
something like an interactive theorem prover. It makes a bit of the formalism
quite clearer.) There has been some push, especially in Europe, for using
these programs to prove other mathematics; it is certainly possible, although
few would say it is pleasant or resembles how ordinary mathematicians work.
But I think that, for specific areas of mathematics, these tools are useful
_today_. Especially when trying to teach students how to think formally;
you'll find that _formal_ corresponds closely to how the computer thinks about
things (and the whole problem is that mathematicians are not very formal at
all.)

CPDT is not a good first text; Software Foundations
<http://www.cis.upenn.edu/~bcpierce/sf/> does better for people with less math
experience.

------
arikrak
It's interesting that many of the points raised in the article demonstrate the
opposite point - that computers should have a greater role in mathematics, and
that the current system needs to be changed.

Currently, despite what they claim, schools basically just teach students to
follow set methods for solving problems with little thought, basically like an
algorithm. These processes can all be solved by a computer, so instead the
student can learn the actual thought processes involved and how to use
computers to solve real-life problems. Math education has not fully adjusted
to the existence of computers. Some of these skills themselves are difficult,
so teaching a large group of students in one class would probably not be so
effective. Instead, the best way to learn would probably be to use an
interactive tutorial combined with the option to ask someone for help.

------
SudarshanP
I think the author is fussing around the wrong problem. I guess the situation
is something like this.

Incompetent/bored math teacher < khan academy < better online learning
platform < Good math teacher.

The number of bad math teachers across the whole planet is staggering
considering that we are nearly 7 billion people on the planet especially more
so in the developing and underdeveloped countries. Khan academy sets a
reasonable good lowest common denominator education that our next generation
of students can receive both in quality and coverage. Competition will just
push this bar even higher.

To replace good teachers we may have to wait for deep AI even if wanted to do
that. Till then KA, Udacity etc. can produce more autonomous students who can
make better use of a great teacher's time.

~~~
SudarshanP
Did I say something inappropriate?

------
sbierwagen
That second problem is interesting. I'm gonna give it a try, though I know
essentially nothing about geometric proofs.

Inscribe 8 squares inside a circle such that point A intersects the circle,
point B intersects point C on the adjoining square, and point C intersects
point B on the proceeding square. For every square, draw a line from point D
to the center of the circle.

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maccylo
I might be missing the point, but math is just working with symbols - we don't
need to bend to fit the symbols, but we can make the symbols bend to fit us.
The way math was written 200 years ago won't be the way we write it in a few
decades, will it?

~~~
juiceandjuice
Yes, it will be. Your question is the equivalent of asking if English will
still be written the same in 30 years.

~~~
nkassis
would "u r wrong" have been understood as easily 30 years ago as today?
Language change, even 30 years is enough for some changes to happen. Now I'm
not saying that 30 years from now we will be talking in im speak but you can't
deny that english of today has changed in the last 30 years.

Mathematics has also changed over time. Trying to read mathematics documents
from Fermat's period would be rather hard today. In the 20th mathematics saw
some pretty drastic changes in the way it's expressed (someone can correct me
if I'm wrong on this). Check out this group who had some pretty big influence
on how math is expressed today.
<http://en.wikipedia.org/wiki/Nicolas_Bourbaki>

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tokenadult
Funny that I was just teaching that lesson about fraction comparison last
Saturday morning, but I was using fraction pairs that are a bit more
challenging (from the excellent textbook Algebra by the late I.M. Gelfand and
Alexander Shen).

<http://www.amazon.com/Algebra-Israel-M-Gelfand/dp/0817636773>

I just put problem 40 from the book, which I taught last week to children of
third-grade to fifth-grade age, into Wolfram Alpha's natural language
interface.

[http://www.wolframalpha.com/input/?i=Is+10001%2F10002+greate...](http://www.wolframalpha.com/input/?i=Is+10001%2F10002+greater+than+100001%2F100002%3F)

The Wolfram Alpha input and output is convenient for making the teaching
point, and could spark a discussion about problem 41, which is

41\. Which is greater, 12345/54321 or 12346/54322?

Of course a sensitive mathematics teacher is supposed to recognize at once
that what is really being asked for by the second problem is a way to
generalize when a/b is greater than (a+1)/(b+1) and when it is not. I will
wrap up that part of the lesson next week.

I have posted recently here on HN that "There will continue to be an important
role for in-person teachers,"

<http://news.ycombinator.com/item?id=3555728>

even after online teaching tools become much more fancy. I recommend some good
tools (I don't think Khan Academy is the best available online mathematics
teaching tool, but its price point is appealing) in that comment. I also
include links in that comment to thoughtful recent articles about improving
mathematics education. A skillful teacher will teach learners how to use
tools, when tools are suitable for getting the answer, and how to use the
unaided human brain and speech when that should be enough to get (and EXPLAIN)
the answer. A big part of mathematics learning is learning how to use
appropriate tools and methods in different circumstances. I don't decry online
mathematics learning tools; I model in the classroom using the good-old human
brain, sometimes with some help from pencil-and-paper or whiteboard-and-marker
calculations, to puzzle through challenging mathematical problems

<http://news.ycombinator.com/item?id=2760663>

and get reality checks on whether the procedure used to reach a solution is
correct or not.

AFTER EDIT: After posting this comment, I asked my Facebook friends (who
include a number of professional and amateur mathematics educators, including
homeschooling parents who have brought up International Mathematical Olympiad
gold medalists) about the blog post submitted here, and one of those friends
suggested that the blog post author look closely at the Art of Problem Solving

<http://www.artofproblemsolving.com/>

model of online mathematics education. "The medium is not the message, because
the medium is only stepping in to do (interactively) what you would do in
person if you could, and instead distributing the teaching resource more
widely, but basically in the same mode." I agree with that suggestion, and
with that comment on whether or not the medium is the message if online
mathematics education is well done.

~~~
codehotter
This problem was enough hard for me that I want to post my solution.

(a+1)/(b+1) = (a * (1 + 1/a))/(b * (1 + 1/a))

Which in turn, is the same as (a/b) * ((1+1/a)/(1+1/b))

This can only be greater than a/b if (1+1/a) / (1+1/b) > 1.

A fraction is greater than 1 if the numerator is greater than the denominator,
which means 1/a > 1/b. Dividing by a greater number leads to a smaller result,
so this happens if a < b.

12345 < 54321, so 12346/54322 is greater.

Did I overlook a much easier way to solve this, or is third grade much better
at math than I am now?

~~~
groaner
I suck at pure math, so I tend to think of this in some sort of analogy.

Suppose the fraction a/b is some statistic you are trying to measure, say, a
batting average or percentage of correct notes played in Guitar Hero.
(a+1)/(b+1) would be the new fraction after you got the next one right. By
getting the next one right, did you improve your score?

Of course, if you have a perfect record already, getting an additional 1-for-1
won't change anything. And if a>b, then you'd have to somehow score more than
1 point per attempt in order to maintain the same ratio, so (a+1)/(b+1) would
be lower.

~~~
cousin_it
I used to be a mathematician, and I think your comment is everything that math
education in school should aspire to be :-) Another example in the same vein
is Terry Tao's airport puzzle ([http://terrytao.wordpress.com/2008/12/09/an-
airport-inspired...](http://terrytao.wordpress.com/2008/12/09/an-airport-
inspired-puzzle/)), scroll down to Harald Hanche-Olsen's comment for the best
explanation.

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ChuckMcM
Dan really should check out the EPGY stuff from Stanford. Excellent
mathematics education, done online.

