

Math is not linear, so why do we teach it like that? - sublemonic
http://prezi.com/aww2hjfyil0u/math-is-not-linear/

======
blintson
Most of my school's funding is and was not dependent on their student's
skills, schools get funding for attendance. When I went to school they checked
attendance 6 times every day (once for every class), and gave you 3 bathroom
breaks/year/class. Taking attendance took about 5 mins/ class, every time you
went to the bathroom a teacher had to sign a form verifying you had
permission. This comes out to about* 340,000 HOURS of wasted time in ONE YEAR
FOR ONE HIGHSCHOOL. This isn't even considering many, many fundamentally wrong
things with how grades and classes are structured and credit is awarded. For
every competent teacher teaching a useful subject there are 3-4 incompetents
wasting people's time with sophistry and selectively blind adherence to stated
rules.

Public high school education in this country is a net negative. High school
"education" has nothing to do with teaching students skills, its there first
to benefit the people running the school, and second to make people obedient
for factory jobs.

Math & science people tend to be humble & introverted because they're usually
wrong about the solution to whatever problem they're trying to solve, and they
spend all their time doing math/science and not talking to people. This is a
bad thing. Every hour spent on spiffy presentations is an hour not spent on
telling people public school's are doing it wrong. The author wasted his/her
time on this* * , it's not going to change anything. Math/Science people who
want to improve the state of math/science education should spend their time
politicking, not science'ing.

* (5mins/attendance * 6 classes * 10 mins bathroom break form filling/a day * 2000 students * 34 weeks/year / 60mins/hour = 340K)

* * It is pretty cool, though.

~~~
timthorn
It's worth sanity checking statistics - 340000 hours is equal to nearly 39
years, which seems a bit on the high side for a single school in a single
year...

~~~
michael_dorfman
Exactly. Attendence checking doesn't take 5 minutes _per student per class_ ,
yet that's what's being claimed by the math....

~~~
olefoo
Actually if taking attendance takes 5 minutes of class time (which it easily
can with a class size of 30), then all students are blocked from being taught
while the teacher is busy so it does make sense to count that as (n+1) * 5
minutes where n is the number of students.

Fingerprint scanners for attendance would both speed up and parallelize this
operation.

~~~
cma
What could possibly go wrong with implementing a complicated biometric
fingerprint scanner based system for our schools?

<http://news.ycombinator.com/item?id=1221068>

~~~
olefoo
Well card swipes are obviously unworkable. You could maybe link it to
student's mobile phones...

But the real solution is to have decent schools with classes small enough that
the teacher doesn't have to take roll call because they know everyone.

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prakash
An interesting quote I came across today from Edward Tufte:

 _One more example. If you are teaching math, hand out the proofs on paper at
the beginning of class to all the students; then work through the written-out
proofs aloud in class, following the proofs on paper.

That way your students aren't merely making notes and recording your words;
instead they are thinking. I believe that students should THINK in class, not
take notes. So give the students your lecture notes and go through them
carefully in class, trying to insure understanding of each part as you go.
Your voice in effect annotates and explains the material on paper.

(Of course, these ideas apply widely, not just to teaching math.)_

[http://www.edwardtufte.com/bboard/q-and-a-fetch-
msg?msg_id=0...](http://www.edwardtufte.com/bboard/q-and-a-fetch-
msg?msg_id=00001B&topic_id=1&topic=Ask+E.T). -- It's the last paragraph under
"E.T. ON TECHNOLOGIES FOR MAKING PRESENTATIONS".

------
lotharbot
We teach it like that because we learned it like that.

We also teach it like that because states have various standards with
pretentious names and acronyms ("Essential Academic Learning Requirements" =
EALRs [1]) that require students to have specific bits of knowledge at
specific ages. This means that, instead of students exploring various
mathematical subjects in parallel, they're stuck going through them in exactly
the order presented.

I remember being in a room full of math profs and TAs discussing how to get
more students interested in becoming math majors. I suggested a 100-level
number theory course, with the rationale that it's simple, accessible,
mathematically interesting, and makes it clear that there's more to
mathematics than the "progressively-harder calculus-based classes" (up to
DiffEq) most hard-science majors end up taking.

[1] 252 page, 8+ meg pdf:
[http://www.k12.wa.us/Mathematics/Standards/K-12MathematicsSt...](http://www.k12.wa.us/Mathematics/Standards/K-12MathematicsStandards-
July2008.pdf)

------
tokenadult
It's time for a favorite quotation about mathematics again:

"What should every aspiring mathematician know? The answer for most of the
20th century has been: calculus. . . . Mathematics today is . . . much more
than calculus; and the calculus now taught is, sadly, much less than it used
to be. Little by little, calculus has been deprived of the algebra, geometry,
and logic it needs to sustain it, until many institutions have had to put it
on high-tech life-support systems. A subject struggling to survive is hardly a
good introduction to the vigor of real mathematics.

". . . . In the current situation, we need to revive not only calculus, but
also algebra, geometry, and the whole idea that mathematics is a rigorous,
cumulative discipline in which each mathematician stands on the shoulders of
giants.

"The best way to teach real mathematics, I believe, is to start deeper down,
with the elementary ideas of number and space. Everyone concedes that these
are fundamental, but they have been scandalously neglected, perhaps in the
naive belief that anyone learning calculus has outgrown them. In fact,
arithmetic, algebra, and geometry can never be outgrown, and the most
rewarding path to higher mathematics sustains their development alongside the
'advanced' branches such as calculus. Also, by maintaining ties between these
disciplines, it is possible to present a more unified view of mathematics, yet
at the same time to include more spice and variety."

Stillwell demonstrates what he means about the interconnectedness and depth of
"elementary" topics in the rest of his book, which is a delight to read and
full of thought-provoking problems.

<http://www.amazon.com/gp/product/0387982892/>

~~~
spamizbad
Wasn't there an attempt to do this in the 1960s? They tried to give students a
better mathematical foundation for more advanced maths that ended up
backfiring politically? Namely they were teaching rudimentary set and number
theory to K-5 kids. This came at the expense of kids ability to multiply and
divide, and when the press caught wind of this, the program was quickly shut
down.

It seems like any attempt to restructure the math curriculum will be met with
massive resistance from parents who were eminently satisfied with their own
(likely poor quality) math education, and want their children to have the
same.

~~~
tokenadult
You are referring to "new math," which is the kind of math instruction I had
when I was a child.

I am reading just now a very interesting book The Mathematics Pre-Service
Teachers Need to Know

ftp://math.stanford.edu/pub/papers/milgram/FIE-book.pdf

(posted on

<http://hub.mspnet.org/index.cfm/13083>

which warns that it may be a slow download)

which describes improved university courses for students who plan to become
elementary teachers. That is a big emphasis in the United States now--
international comparisons have shown that mathematics education of elementary
pupils in the United States is lousy largely because the mathematical
education of elementary teachers (at all levels) is lousy,

<http://www.ams.org/notices/200502/fea-kenschaft.pdf>

so United States mathematicians are trying to do something about that that is
more effective than the 1960s attempt at "new math."

Yes, Stillwell's book, mostly aimed at mathematics students who will go on to
be mathematicians rather than schoolteachers, is also an outcome of thinking
about curriculum reform. He describes his motivation for writing his excellent
book as attempting to understanding concepts of mathematics he still didn't
understand after he earned his Ph.D. at MIT.

~~~
lotharbot
_"mathematics education of elementary pupils in the United States is lousy
largely because the mathematical education of elementary teachers (at all
levels) is lousy"_

This is something I've noticed in the time I spent working with teachers. It
amazes me how many 4th-6th grade teachers don't understand fractions, but are
trying to teach them to kids!

~~~
Radix
What do you mean when you say they don't understand fractions?

~~~
lotharbot
I mean, I have watched groups of elementary school teachers work on the same
sort of problems they assign (fractions being one example) and struggle
mightily. They understand the basic concept of what a fraction is, but many of
them get bogged down in the algorithms because they don't really understand
what the algorithms represent.

What is a "common denominator" beyond "the thing you put fractions over to be
able to add them"? Many of the teachers I've worked with would struggle to
explain this to students.

------
hugh3
Is the US high school mathematics curriculum really as linear as described
here? Do you really work through, say, "geometry" in one go, and never revisit
it?

My mathematical education was a bit more like advancing on several fronts at
once. A chapter on geometry, then a chapter on basic algebra, then some more
advanced geometry, then some more advanced algebra, then some basic
trigonometry, which enabled you to understand some more advanced geometry,
then...

~~~
spamizbad
Assuming you don't take any "advanced" math in college, yes, it is in fact
that linear.

It's complicated by the fact that often you spend 2 weeks learning a concept
and then have the teacher say "It's going to be on the test but you're not
going to use it until halfway through next year so try and remember it!" Most
everyone forgets it, so we waste another 2 weeks re-learning it next year.

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nnutter
Sorry, I couldn't finish the presentation because of that stupid animation.

~~~
est
The animation is awesome, it makes the logic connection much more clear and
easier.

Prezi is really revolutionize traditional reading habits. It transforms
chapter-by-chapter/slide-by-slide, liner reading, to a more hierarchical and
intuitive way. It takes less effort to make sense, so you can focus more on
digesting and re-thinking, not only comprehending.

~~~
snprbob86
Prezi is very nice and I've seen a couple of presentations which have
benefited from the hierarchy and demonstration of relationships. However, this
presentation seemed to be animation simply for the sake of animation. The
spacial relationships seem to have almost no bearing on the logical
relationships. I too was unable to finish the presentation.

------
axiom
The most useful part of this link was finding prezi, which is awesome, and I
hadn't heard of it before.

~~~
_delirium
For some reason, over the past year or so it's become almost standard in the
humanities (but not in science/engineering). I'm in CS but occasionally go to
interdisciplinary conferences, where every CS person will have a PowerPoint,
while every humanist who has a digital presentation at all will have a Prezi
(to a close approximation).

~~~
eru
How did LaTeX-presentations fare?

~~~
_delirium
Ah, those happen too, though not as much at the type of interdisciplinary
conferences where you'd also have a lot of humanities people--- seems to be
mainly a thing for people who need equations in their presentation, esp. in
cs-theory. I sort of prefer latex presentations myself, but just because I
like editing source-code-like stuff rather than editing a presentation in a
GUI tool.

~~~
eru
> [...] seems to be mainly a thing for people who need equations in their
> presentation, esp. in cs-theory.

Indeed. At the International Conference on Functional Programming last year
LaTeX was the norm. I enjoyed the hand-written slides somebody had produced on
their tablet pc.

------
petercooper
A delightful misuse of Prezi for zooming around with needless effects from
place to place rather than moving within content.

~~~
applicative
I thought it was a little disorienting too, maybe just not having experienced
it -- but after a bit I saw that it perfectly matched the actual content, and
spirit, of her argument.

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roundsquare
Alison Blank (the author) should write a math book instead of making this
presentation. I think that would be a great way to prove her point (which I
agree with).

~~~
ohwaitnvm
I also agree, but how would you write a nonlinear book? "Choose your own
adventure - Math" would be a tricky textbook to produce.

~~~
roundsquare
Well, I don't think any book/teaching method would entirely dismiss linearity,
it would just make things less linear (as compared to the current way of
teaching).

As a start, I suppose you could have a topic in algebra... then use that to
discuss probability. From there, you can view something as geometry and teach
some of that, etc...

Maybe that doesn't work. So maybe, you teach math by asking a question and
exploring to find an answer. As you explore, you inspect different approaches
to a problem. Later questions can use earlier points as analogies.

To be honest, I don't know quite enough math to even give a list of
questions/topics. But, I imagine a good math teacher could do something of
this sort.

Its probably best to start at the University level and move your way down as
you learn lessons about teaching these things.

As for your idea, CYA-Math, I like it in some ways. "To see how this connects
to this, go to page 34. To see an analogy to something else, go to page 54." I
agree though very tough to build.

~~~
nitrogen
_though very tough to build_

A robust cross-reference system could keep the page numbers automatically up
to date. I'd edit the textbook in a wiki style interface, then use some
automated software to commit it to a specific order with page numbers.

This is one area where a digital textbook would be incredibly useful.

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applicative
I hope my (little) daughter gets a math teacher with A. Blank's sense.

A similar point is true of all parts of knowledge, of course: I was convinced
to send my daughter to a local hippy dippy lab school when I saw the teachers
facing the "But what if my child is Gifted?" question from yuppie parents. The
teachers vehemently rejecting separate 'tracks' for the 'gifted' in favor of
the Blank plan: if some students master something early, you needn't move them
on to the 'next' thing, since this is in some respects an illusion. Rather you
get them working on something cool that is 'off to the side' from the point of
view of the curriculum sequence, whereby _inter alia_ they learn the unlimited
character of possible knowledge and might strike something that would really
get them absorbed.

\--This horrified the linearize-their-way-to-Yale yuppies in the audience, but
it seemed like genuine wisdom to me. (That they were really losing people with
this also attracted me.)

We'll see how it works in practice. I have a feeling Blank's utopia would be
too expensive given the significance attached to education in the present age.

~~~
ronnier
I wouldn't count on your daughter having a wonderful math teacher. I suggest
teaching her yourself in addition to what she learns in school. My daughter
isn't old enough yet (7 months), but I plan on teaching her daily.

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albertsun
Because high school curriculums are much easier to plan when they are linear.
At my high school the curriculums for English and Social Studies were also
linear. Only in science did we get to choose which order we took Chemistry,
Biology and Physics in.

~~~
KonaB
_Curricula_ , not _curriculums_...

~~~
hugh3
Apparently Latin wasn't on the curriculum at that school. :(

~~~
albertsun
Latin was part of the curriculum, an optional one that I did take. Luckily the
lingua franca of this site is English.

------
semuelf
We teach math linearly, because we follow the child's brain development.

numbers are quite an abstract things, and proportion is even more so. the next
step, the percentage, add relative point of views too. you just can't teach
that to a child that literally can't tell his right from his left.

Well, you can, but he won't really understand it and it is effort down the
drain.

So I know that here we all are CS/EE/whatever, and probably knew more math
then our elementary school teachers, but there is a reason to these steps,
even if the said teacher never know about it.

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chrischen
Doesn't just apply to math. Applies to education in general. It's like someone
one day decided that it's ludicrous someone might actually _want_ to learn
something to _do something cool_ and decided to take the safer systematic
approach to education. Only problem is that this systematic approach actually
disadvantages those who want to learn, especially the ones with ADHD ('cause
it's harder for them to shift focus to something they aren't immediately
interested in).

------
snth
Because time is linear?

~~~
splat
Or is it? <http://en.wikipedia.org/wiki/Imaginary_time>

~~~
Maro
Imaginary time is a trick in quantum field theory; roughly: it goes away at
the end of the calculation.

------
robertk
Her rectangle problem is wrong. Here are four rectangles with the same area
and perimeter:

x = 3, y = 6; x = 4, y = 4; x = 5, y = 10/3; x = 7, y = 14/5

The formula is fix x > 2 (one side), let y = 2x / (x-2) (other side). Derived
from xy = 2x + 2y.

~~~
gort
As I recall the problem was not "there aren't any" but "there aren't any more"

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cliveholloway
I got dizzy reading that and gave up half way through. Stick to math and let
someone else design your presentations :)

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nazgulnarsil
the root of the problems in the education system lie in its prussian derived
methodology.

<http://www.johntaylorgatto.com/underground/>

compulsory schooling is stupid and wasteful, just like compulsory anything.

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GrandMasterBirt
I love everything about it. The message, the presentation. Ugh if only I was
taught this way, makes me jealous. This is how my calculus teacher in college
taught. It was incredible compared to anything before that.

