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*From*: John Denker <jsd@av8n.com>*Date*: Sat, 18 Oct 2014 02:05:42 -0700

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.

**Follow-Ups**:**Re: [Phys-L] problems with the teaching of algebra***From:*Philip Keller <pkeller@holmdelschools.org>

**Re: [Phys-L] problems with the teaching of algebra***From:*Chuck Britton <britton@ncssm.edu>

**References**:**Re: [Phys-L] problems with the teaching of algebra***From:*Bernard Cleyet <bernard@cleyet.org>

**Re: [Phys-L] problems with the teaching of algebra***From:*"Anthony Lapinski" <Anthony_Lapinski@pds.org>

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