[Phys-L] problems with the teaching of algebra

[John Denker <jsd@av8n.com>]

Hi --

Once upon a time, competence in basic algebra was a college
admission requirement.  Nowadays, in some places, it is not 
even a college /graduation/ requirement.

> From the point of view of the physics teacher, at the HS or
college level, this is a problem, because it means there is
a cohort of students moving through the system that has no
clue about algebra.  Some fraction of these will try to sign
up for physics, with disastrous results.  Sometimes I think
the world is one big conspiracy to make life harder for 
physics teachers.

This is trickier than it seems.  Let's be grownups and look
at it from the students' point of view.  There are a lot of
careers in this world that require a post-secondary education 
but do not actually involve doing algebra.  Sure, the folks
on this list use algebra all the time, not just professionally
but in everyday life, but the fact remains that most people
can get by without it, mostly.

Keep in mind that for 2000 of the last 2300 years, mathematicians
got by without algebra.  There is no reason to think Galileo 
ever wrote an equals-sign in his entire life.  Even Newton's
_Principia_ is mostly devoid of algebra.  It is safe to assume
Newton derived his results using algebra, but he explained them 
to others in non-algebraic terms, presumably on the grounds 
that most of his readers didn't know -- or didn't trust -- 
algebraic methods.

Example:  Today, your typical second-grade teacher could not 
solve an algebra problem if their life depended on it.  College
algebra is required for a teaching certificate, so you have to
wonder how these folks got where they are, but that's a question
for another day.

Another example:  There exist decent-paying high-tech jobs
for avionics technicians.  Training is typically a two-year
community college program, and does /not/ require algebra.
A lot of the training would be quicker and better if the
students did know algebra, but they don't, so various crutches,
euphemisms, and work-arounds are employed.

Another example:  Once I was sitting in on a meeting of the AT&T 
corporate board of directors, back when AT&T was one of the 10
largest companies in the S&P 500.  They spent a minute or two 
wondering whether it was possible to solve a certain problem.
I was under orders not to speak unless spoken to, but I couldn't 
stand it any longer, so I blurted out:  "It's one linear equation 
in one unknown.  I reckon the Bell Labs guys can solve it."  Note 
that the board members are considered "masters of the universe" 
types and get paid a couple hundred thousand dollars per year 
to serve on a board that meets only few times per year.

Don't get me wrong, I like math.  However, I will not project
my feelings onto others in defiance of the evidence.

People are seriously asking "Is Algebra Necessary" ... which 
cannot possibly be the right question.  I reckon some /subset/
of what is covered in the typical algebra sequence is useful.

Perhaps more importantly, there is a problem with the sequencing.
The typical algebra book is nearly devoid of real-world examples.
There is a certain subset of students who will tolerate that,
because they are patient and trusting ... but there is another
subset of students who are impatient and skeptical, and IMHO
have every right to be.  They want to see some relevance.  If
we could get some real-world applications on the table, they
would be more motivated to learn the tools to handle those
applications.  Fixing this sequencing problem is easier said
than done, but people are trying.  There are some preliminary 
claims of success:

   http://www.npr.org/blogs/ed/2014/10/09/354645977/who-needs-algebra

See also
   http://www.nytimes.com/2012/07/29/opinion/sunday/is-algebra-necessary.html?_r=0
   http://freakonomics.com/2012/07/31/dump-algebra/

I suspect that this is some of what was driving the "physics
first" movement:  If you can get some physics examples on
the table, it helps motivate the math.  This is good for the
students, but burdensome for the physics teachers.  Why is it
that people routinely ask the physics department to teach math, 
but never ask the math department to teach physics?  Life is 
seriously not fair.

Partially-baked suggestion:  Robotics and programming  There is 
a large-ish subset of students who are fascinated with robots.
This motivates them to learn physics and programming.  That 
in turn motivates them to learn algebraic ideas.

The Khan Academy is super-helpful for some students, but 
not for others.  At each level, they insist on everybody 
having a strong background (in math and everything else) 
before moving on to the next level.  This tends to push
real-world applications into the far distant future.

For most of the last 2300 years, all too many teachers and
textbook authors have acted as if the following things
were equivalent:
 -- The way in which mathematical results are published,
  in their polished form, as in Euclid's _Elements_
 -- The way in which mathematical results are invented.
 -- The way in which facts and methods are learned (and
  taught).

In contrast, it is obvious IMHO that the elegant logical
structure of Euclid's _Elements_ is just terrible pedagogy,
for all but a few students.  

According to legend, Euclid mocked a beginning student who
dared ask about applications.  IMHO that's just terrible
pedagogy.  We've got to do better than that.

Right now algebra is a mess.  The physics guys did not create
this mess, but I reckon we are going to be called upon to 
help clean it up, so we need to be paying attention.