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Re: [Phys-L] problems with the teaching of algebra

This is interesting and Lockhart's piece resonates. I have been working on
a project related to this that is very close to the finish...more on that
in a week or two (I hope). But for now...

Yes, most people never use algebra. But some do. And we start teaching
algebra mostly in grades 7, 8 and 9. If a student does not learn algebra,
there are a bunch of other interesting things that student will never do.
And those grades are still pretty early to be making curriculum decisions
that close doors.

On the other hand, by grades 10, 11, 12 students know a few things. Many
have a more reasonable assessment of their own talents and future
possibilities than we give them credit for. So when they assert in boredom
and frustration that they will never use this stuff, they are not wrong. I
visited a "lower track" algebra 2 class where the students were dutifully
trying to learn about "end behavior" and asymptotes of rational functions.
No context, no motivation...the despair was palpable. I couldn't help
wondering: we only have a finite number of hours left where these students
will be in a math classroom. Is THIS how we want to spend it?

I guess I am arguing for smarter curriculum decisions. But I do think that
the beginnings of algebra should be taught to all. The fundamental
abstraction behind algebra is useful. A student who does not learn at
least that much will also never write computer code or even be able to
handle a spreadsheet formula.

On Sat, Oct 18, 2014 at 5:05 AM, John Denker <jsd@av8n.com> wrote:

On 10/12/2014 12:37 PM, Anthony Lapinski wrote:

This is an interesting discussion!

Yes. It has been obvious for decades ... maybe centuries ...
that there was a problem. However, it's only in the last few
weeks that I've become aware of people actually doing something
about it, doing things I reckon have some chance of working.

Most of my friends -- in fact, most people in this country -- don't need
algebra. My
wife has a t-shirt that says, "Another day passed and I haven't used
algebra once."

Note that for 2000 of the last 2300 years, mathematicians
got by without algebra.

People find math useless and are turned off by it. What math skills do
most people need?
Arithmetic -- addition, subtraction, multiplication, division, fraction,
percents,
and decimals. Did I miss anything?

It's hard to say. It's hard to decide what sort of math
people "must" have or even "should" have.

I would say that algebra sits in a bad place, right on top
of a potential barrier. Algebra by itself is not good for
much, so if you take algebra and stop there, it's a poor
return on investment. You would be better off stopping
sooner ... or (!) stopping later, i.e. continuing on and
doing something interesting with algebra, e.g. physics
or computing or whatever.

Maybe the way math is taught needs to
be changed.

No doubt about it.

It's mostly numbers, rules, memorization, but no application or practical
use. It was like this when I was in school,

Note the contrast:

a) In the mind of the grade-school student, "math"
means "arithmetic" i.e. add, subtract, multiply,
divide, et cetera.

b) To a mathematician, math is something else entirely.
I know some pretty serious mathematicians who are not
very good at basic arithmetic.

I don't even know what to call this type of math. Maybe
"the art of mathematics". I think of it as "math with
panache", with style, with class.

Whatever you call it, it is woefully lacking in the schools.

Lockhart used music as a metaphor: Imaging forcing students
to learn the names of the notes (do, re, mi ...) and a bunch
of other formalism, without ever hearing any actual music,
much less performing any music, not to mention (gasp) composing
music ... or in any way enjoying music.

Highly recommended:
https://www.maa.org/external_archive/devlin/devlin_03_08.html
https://www.maa.org/external_archive/devlin/LockhartsLament.pdf


Math teaching really needs a revolution to make it more
practical,
relevant, and interesting.

Certainly it needs a revolution.

Sometimes -- but !not! always -- making it practical makes
it more interesting.

Sometimes it can be interesting even when it is completely
abstract and impractical. Some people pay money to buy
books of sudoku puzzles, presumably because they find them
interesting.

Here's another example that came across my desk recently:

Each student (or each 3-student team) gets a pair of
scissors plus a piece of paper with an arbitrary triangle
drawn on it. The mission, should you decide to accept
it, is to cut out the triangle using only a single
straight cut. Hint: you may fold the paper any way
you like before cutting.

This problem involves no arithmetic, yet it is intensely
mathematical. I cannot see any practical benefit to solving
this particular problem, but still it is interesting.
It has the advantage that you can pose it to people who
don't know any physics, don't know any algebra, and
couldn't multiply 44 by 5 without a calculator.

Here is a news story that features this puzzle, and offers
some hope that the much-needed revolution is starting:
Jessica Lahey
"Teaching Math to People Who Think They Hate It"

http://www.theatlantic.com/education/archive/2014/10/teaching-math-to-people-who-think-they-hate-it/381125/2/

The puzzle comes from chapter 6 of the following book:
Volker Ecke and Christine von Renesse
with Julian F. Fleron and Philip K. Hotchkiss
_Discovering the Art of Mathematics_
... _Games and Puzzles_

https://www.artofmathematics.org/sites/default/files/books/games-2013-06-06.pdf

The whole book is available online, free for all.

The book starts out with an uncompromising manifesto of
"art for art's sake" ... but as it goes along it mentions
a few bits of math that started out super-abstract but
found important applications.

Most people on this list probably have little direct
interest in puzzles of this sort, because we have a
lifetime supply of important physics puzzles to work
on. OTOH this is a pretty darn interesting puzzle.
Any solution the student comes up with will be
original and creative, because you can't solve it
by plugging into equation 14-3 in the textbook.
Students who have never been asked -- or permitted --
to do anything creative in class will be shocked.

In any case, we all have to live with the consequences
if the math department doesn't do a good job, so it
is nice to know that people are taking a good hard
look at ways of improving mathematical teaching.

FWIW I am not convinced that "games and puzzles" is
100% the right way to go. Puzzles have the advantage
that they don't require much specialized domain
knowledge. OTOH I reckon that with roughly the same
amount of work, one could mix in a few simple physics
ideas and get a lot of mileage out of that.

=============================

Here's another way of thinking about the issue.

Consider the paper
Gary A. Morris et al.
"An item response curves analysis of the Force Concept Inventory"
http://scitation.aip.org/content/aapt/journal/ajp/80/9/10.1119/1.4731618
http://dx.doi.org/10.1119/1.4731618

It contains 30 small graphs, one for each question on the
FCI. I find it really interesting. It tells us a lot
about the quality of the test questions, and even the
quality of the individual distractors.

To make sense of this you need some skill in interpreting
graphs. This is a skill that the ordinary grade-school
teacher can benefit from. Where is that skill supposed to
come from? It's not algebra per se, but it is one of the
things that gets covered in the algebra course.

As another example: Starting in the very early 1908s,
my mother had an Apple ][ in her classroom. She used
a primitive spreadsheet program to compute grades and
whatnot. Bear in mind that she was *not* a whiz at
basic arithmetic ... she couldn't multiply 7×8 without
stopping to think about it ... but she could do stuff
with the spreadsheet just fine. Again, that's not
algebra in the usual sense of the word, but it uses
the language of algebra, such as variables, functions,
et cetera.

Einstein said an education is what remains after you've
forgotten everything you learned in school. My version
of that would be: an education is what remains after
you've forgotten everything you /thought/ you were
learning in school. This is says something unflattering
about the way we teach, because it means that what we
overtly emphasize isn't really the important stuff.

I am cautiously optimistic. In some corners of the math
community, people are trying new things. There is
tremendous upside potential.

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