[Phys-L] open-ended multi-stage inference

[John Denker <jsd@av8n.com>]

On 08/19/2012 07:10 AM, Chuck Britton wrote:
> JD's Twelve Coin puzzle reminds me of a box I built many years ago
> consisting of a Red LED, two D-cells and three rotary SPST on-off
> switches - all wired in series.

> The 'puzzle' is to 'flip' the switches enough to make the LED turn
> on.
>
> The REAL assignment is to figure out and write down an algorithm that
> will guarantee that the light will light in a minimal number of
> flips.

Very nice.

On 08/18/2012 11:29 PM, D.V.N. Sarma wrote:

> The candidates were blindfolded and three white caps were placed on 
> their heads. The blindfolds were removed. All the three raised hands. 
> Presently one of them said that his cap is white. How did he guess that?

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

These examples, and others, lead us to consider the idea of multiple
/stages/ of inference.

Once upon a time, for most of the last 2000 years, staging is something 
that students learned in geometry class.  The task of finding a proof 
for a high-school geometry proposition is important as an example of an
open-ended multi-stage question.
 -- It's like a chess game, where there are many possible first moves,
  many possible second moves, et cetera ... and it's not obvious until
  much later whether any given move was good or bad.
 -- It's like finding your way out of a maze, where there are many open
  doorways, and it's not obvious until much later which of them lead
  toward a solution.

This notion of staging and open-endedness stands in contrast to plug-and-chug 
problems, where the statement of the problem essentially telegraphs the method 
of solution.  What's even worse are the plug-without-even-chug trivia problems 
that can be answered immediately ... and which seem to make up the bulk of the 
current crop of standardized tests ... which in many cases seem to have become 
almost the sole focus of the K-12 educational system.

Nowadays, all too often, high-school geometry (except for "honors" geometry)
has been dumbed down to the point where no proofs are involved.  Instead
students learn the formula for the area of a circle and (separately!?!??) 
the formula for the area of an ellipse, et cetera.  No inference, just
equation-hunting.

Here's a "funny" story:  A while ago I was reviewing a state-mandated 
standardized test.  I ran across a question that involved the geometry of 
intersecting chords in a circle.
  http://www.ncsec.org/team21/images/mysterious2.gif
Given four points on a circle, A,C,B,D, the chords AB and CD intersect at point E.
They give you three lengths AE, BE, and CE, whereupon you are asked to find the
fourth length DE.  This is a multiple-guess, closed-book test.

When I first looked at it, I had no idea how to solve it.  I wasn't even 
convinced there was a unique solution.  I gave up and went on to other problems 
... until I discovered that several other problems on the test involved the 
same geometry.  That meant that figuring this out might make the difference 
between an F and an A+.

I have a terribly memory for theorems and formulas, but I am good at visualizing
geometrical relationships.  I convinced myself that three points determine a 
circle, so the fourth point was pretty well pinned down by the geometry, meaning 
the question was in principle answerable.  It also looked like DE was going to be 
roughly inversely proportional to CE, other things being equal.  That suggested 
that similar triangles might be involved.  Then with several additional /stages/ 
of reasoning I was able to prove that was the case.  That led immediately to several 
corollaries with served to answer a whole batch of questions.

This seems like a nifty problem ... the sort of problem that I might ask of a job
applicant, just to see how they handled it.  Some applicants would just know the
answer, which is OK but not interesting.  Most applicants would just give up.  The
interesting scenario is where they don't know but can figure it out.

However, I smelled a rat.  Figuring this out took me something like half an hour.
I had to rack my brain.  I had to test hundreds of hypotheses, almost all of which
turned out to be wrong.  Maybe if I had recently taken a good high-school geometry 
course I would have been more up-to-speed, but even so, the idea of multiple /stages/ 
of reasoning was completely out of character for this test.  All the other questions 
were of the plug-without-even-chug variety ... purely rote regurgitation.

Sure enough, sad to say, when I got to the end of the test I discovered there was
a "reference sheet" that had not been mentioned in the instructions.  It included
a formula for intersecting chords.  What I had treated as an open-ended inference 
problem was reduced to an equation-hunting problem.

Things like this make me want to tear my hair out.

Given the choice, we would be far better off throwing out *all* the multiple-guess 
trivia questions on the existing test and instead focusing (almost) all our attention 
on inference skills.  I am all in favor of accountability and assessment, but we should 
assess something we actually care about.  It would be better to assess the students' 
ability to handle one or two nontrivial questions rather than 50 or 100 trivial 
questions.

Remember the proverb:  Be careful what you test for, you might get it.