Re: [Phys-l] what shall we do about math?

[John Denker <jsd@av8n.com>]

On 07/16/2009 12:13 PM, Steve Highland wrote:
> I've been visiting a couple of summer algebra classes this past week and I'm
> depressed by how impossible a situation the students and teachers find
> themselves in.
>
> Several of these students have failed the course up to three times.  They
> are forced to repeat it over and over.  The summer class is supposed to
> cover the same amount of material the regular class did, so the teacher told
> me they have to cover about a chapter a day.  And the one group I met today
> has four hour sessions each day.  That would fry my brain.
>
> This just strikes me as torture.  Are there any better approaches out there?
> I would think there should be some other option for students who can't pass
> the course than just forcing them to fail it repeatedly.

Yes, that seems like torture to me.

Also it adheres closely to the classic definition of insanity:  
doing the same thing over and over, expecting a different outcome.

To say the same thing in slightly more constructive terms, if 
Plan A doesn't work, try Plan B.  If that doesn't work, try 
Plan C.  I'm not even suggesting that Plan C is "better" in any 
reproducible sense; the point is to do something different.

  If nothing else, you might hope for some sort of Hawthorne effect.

> From previous messages I gather that the classes are heterogeneous, 
which always makes things harder ... but the same principle applies:
Plan B might work for part of the class, and Plan C might work for
another part of the class.

a) Among other things, you might try some of the ideas suggested by
Paul Lockhart
  http://www.maa.org/devlin/LockhartsLament.pdf
i.e. some of the more creative aspects of mathematics.  

b) At the opposite extreme, you might try working out some of the
physics and engineering applications of algebra.

My point is that either way, (a) or (b), if the kids has some idea
of what the subject was good for, they might be more interested in
studying the foundations.  

This generally requires a spiral approach:  a rough introduction to
the basic techniques, then some applications of the techniques, which 
then motivates a more serious study of the basics.  Then repeat......

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

Here's a specific symptom to look for:  Watch out for any sign that
(some) students are doing better on the "hard" problems than the "easy"
problems.  That's a dead giveaway that they are dying of boredom.  It
means you *need* to move on, even if you think their grasp of the
fundamentals is deficient.  Move on now, and spiral back later if 
necessary.

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

Here's another example:  Once upon a time a student was doing OK but
not great with factoring, prime numbers, and all that.  He looked at
the end of the chapter and saw a "starred" problem that suggested
that 4 could be factored as (3 + √5)(3 - √5).  He said he suspected
that there was something more to the story, something they weren't
telling.

I said oh yeah, what you see there is just the tippy-tippy-tip of 
the iceberg.  You know about the integers, the rational number
system, and the real number system.  When we talk about factoring,
usually we talk about integers, where some things factor and some
things don't.  Everything factors over the rationals.  But those
are not the only number systems known to man.  What you're seeing
in that example is a _number field extension_ (aka number field,
aka field extension) which is bigger than the integers but smaller
than the rationals.  Some things factor in this system, and some
things don't.  The complex integers (Gaussian integers) are a
particularly common and useful number field extension.

The next day I started talking about integers modulo 7, to make
the point that there are yet more ideas about factoring.  We
discovered that every nonzero integer has an inverse mod 7.  The 
inverse of 2 is 4 (and vice versa), the inverse of 3 is 5 (and 
vice versa), while 1 is its own inverse and 6 is its own inverse.
I pointed out that 6 is equal to -1 mod 7, so it is no surprise
that +1 and -1 were the only square roots of unity.

At this point the student grabbed for the brass ring.  He wondered
whether it was _always_ true that all inverses came in pairs,
except for 1 and p-1, in any modular arithmetic system.  That's
equivalent to asking whether +1 and -1 are the only square roots
of unity.

We did some experiments.  For starters we checked to see whether
inverses came in pairs in the modulo-61 system.  This was easier
than you might think, because I have a program that can compute
(very efficiently!) the inverse of anything mod anything, using
the extended Euclidean algorithm.

But what about the general case?  The guy had no idea what a proof
was, and had apparently never even seen a proof ... so he had no 
chance of proving the general case on his own.  So I did the proof
and explained it to him.

  I really pounded on the idea that this illustrated the difference
  between arithmetic and mathematics.  Finding the inverse of 3
  mod 7 is just arithmetic.  In contrast, when you start wondering 
  whether *all* inverses come in pairs, except for +1 which is
  its own inverse and -1 which is its own inverse that's real 
  mathematics.

  At the opposite end of the spectrum, I mentioned that modern
  cryptography is all about modular arithmetic.  

Another good trick with integers is to evaluate directly 2^6 mod 7, 
3^6 mod 7, 4^6 mod 7, 5^6 mod 7, and 6^6 mod 7.  Are you getting 
suspicious about the general case?  Guess the value of 42^250 
mod 251, without actually computing it.  For extra credit, find
a reasonably efficient way to compute it.

The point here is *not* that we should stop teaching basic algebra
and start teaching young kids about Galois fields.  Really really
not.  The point here is that you can't show up every day wedded 
to Plan A only.  You need Plan B and Plan C and a spiral approach.
As Paul Lulai mentioned, linking in some physics examples might 
help, if done properly.

I am quite aware that some HS algebra teachers have long since 
forgotten anything they ever knew about number field extensions
and/or modular arithmetic and/or physics ... so the particular
examples mentioned here will often not work in practice ... but 
I still think the spiral idea is sound, and can be made to work, 
with details that differ from case to case.

Do *something* different from what you did yesterday.  Find *some*
way to make the subject interesting.