Re: [Phys-l] Seeking sources of good algebra materials

[John Denker <jsd@av8n.com>]

On 07/08/2009 03:48 PM, Steve Highland wrote:
> This is somewhat off topic for this list, but I'm hoping some folks will
> steer me in the right direction ...
>
> This morning I visited a high school Algebra I class here in Duluth and I
> found a batch of students who seem to dislike what they are doing and a
> teacher who told me she thinks she doesn't know enough to help them.
>
> I noticed she was giving them sets of problems to work on from a test manual
> that came with the text, but she said the students come from such a wide
> range of preparation backgrounds that she can't fit their needs.
>
> There has to be something better out there than batches of 100 problems all
> of the sort that ask you to simplify something like the square root of
> 50y^3.
>
> Where does one find problems that the students would find more educational
> than this stuff?


I don't pretend to understand the situation;  it would take me a great
deal of observation and experimentation to understand and/or solve the
problem.

However, to answer the question, and as a hypothesis that you can test,
I recommend the _Elementary Algebra_ book by Terry H. Wesner,  It has
the great advantage of being freely downloadable.
  http://www.totallyfreemath.com/

I like this book.  Like most first-rate works, there is not any one
thing that makes it first-rate.  The book presents the subject in a
more-or-less logical order, which is something readers often take for 
granted, but is in fact a tremendous achievement, given the complex
interconnections of the ideas.  The explanations seem about right to
me, not glossing over important points, while not belaboring minor
points.  The diagrams are practical, not flashy.  The exercises seem 
reasonable, i.e. not too few, not too many, and certainly not all alike.  
Many are story problems that implicitly make the point that the subject 
has real-world applications.  The book is like a good pair of work shoes: 
not trendy, not gaudy, not kooky ... just comfortable, solid, and practical.

> There has to be something better out there than batches of 100 problems all
> of the sort that ask you to simplify something like the square root of
> 50y^3.

Talking about "batches of 100 problems" seems like a strange way to phrase 
the overall question.  I don't have enough information to understand the
situation, but I'd be willing to bet that replacing one batch of 100
exercises with a "better" batch of 100 exercises isn't going to help much.
It sounds to me like the whole approach is wrong.  Any bug that is caught
using batches of 100 exercises should have been caught earlier.  Any
difficulty that can be solved using batches of 100 exercises can be better
solved by other means.

Here's a story I like to tell:  Once upon a time, I was taking an 
introductory electronics course.  The professor said 
  "Linear circuit.  Two points determine a line."  
He drew a graph.  
  "Open-circuit voltage, short-circuit current.  Again: linear circuit.  
  Point-slope also determines a line.  This point and slope gives you 
  the Thévenin equivalent.  The same slope with this point gives you 
  the Norton equivalent.  Useful.  Know it."  
And that was it.  The elapsed time was less than a minute.  There was 
no homework on the subject, except insofar as every homework for the 
rest of the year used Thévenin equivalents and Norton equivalents when 
needed.  It was part of the vocabulary for discussing things.  The 
students learned that Thévenin equivalents and Norton equivalents were 
simple and highly useful tools.

  *) Part of this can be described as brevity:  a word to the wise
   suffices.
  *) Part of this can be described as the spiral approach:  During
   the course, we saw maybe 100 Thévenin equivalents ... but that
   does not mean we needed to see 100 of them all at once on the
   first day.  An important idea will come up again and again, in
   due course, organically.

At another school I observed another approach to the same topic:  The
professor introduced the subject by lecturing for 90 minutes on Thévenin
equivalents, Norton equivalents, wye-delta transformations, blah de blah
de blah.  Even having seen it, I cannot believe he was able to fill up
all that time.  And then there was homework:  He assigned a horrible rat's 
nest of resistors and voltage sources, and required the students to find 
the open-circuit voltage and short-circuit current.  The circuit violated
a basic principle of circuit analysis, namely that you should assume that
the circuit makes sense.  The homework was very time-consuming.  And then
there was the lab work.  The students were required to build said rat's
nest circuit, measure the open-circuit voltage and short-circuit current,
and compare this to what was predicted by the homework calculation.  This 
didn't work out too well, because:
 a) The professor had stocked the lab with 10% resistors, thereby saving 
  half a cent per resistor.  In a circuit that combines a dozen or so 10% 
  resistors, the overall uncertainty is quite large.  
 b) Furthermore, the students were instructed to implement the voltage 
  sources using regulated power supplies.  Alas the circuit was such that 
  some of the voltage sources would have needed to source a /negative/
  current, which ordinary bench power supplies cannot do.  
So ... what did the students learn from this?  They learned that Thévenin 
equivalents and Norton equivalents were horribly complicated, and didn't 
work, i.e. didn't make accurate predictions!

  On top of all that, there is the opportunity cost:  think of all the
  better things they could have done with all that lecture time, 
  homework time, and lab time!


I've seen plenty of teachers who assume that if it's good to do one exercise,
it must be great to do 100 exercises (all alike, all at once).  I conjecture 
that the teacher mentioned above was trained this way, and doesn't know any
other way.  I've seen whole departments where that approach is part of the 
culture;  the Thévenin fiasco was, alas, not an isolated incident. 

Fortunately for me, I've not encountered such "fiasco" courses closer than 
arm's length, e.g. when tutoring or mentoring other people.  I wouldn't 
be able to pass such a course.  If I couldn't get out of the course, I'd go 
insane.  I've tutored students who were flunking the course (even though they 
could easily do all the work) just because they couldn't stay awake through 
all the busywork.

Again:  I don't have enough information to understand the situation, but still
I'd be willing to bet that replacing one batch of 100 exercises with a "better" 
batch of 100 exercises isn't going to help much.  Options include:
 -- Maybe some of the students need better pre-algebra background.
 -- Maybe some of the students need another discussion of the principles
  behind today's assignment.
 -- Maybe they need to see a few more worked exercises.
 -- Maybe some of them need to put a bookmark in today's topic, go on to
  something else for a while, and spiral back later.
 -- A heterogeneous class is always a challenge.  Maybe it's time to divide
  the class in half.  You work with one half, shoring up the foundations.
  Meanwhile, appoint some smart-alec kid to lead the other half in studying
  some tangentially-related "enrichment" topic.  (There's no shortage of
  such topics.  If anybody wants suggestions, please ask.)
 -- A few dozen things nobody has mentioned yet.
 -- Combinations of the above.