Re: [Phys-L] understanding and reasoning ... or not

[John Denker <jsd@av8n.com>]

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Let me combine threads from two different lists:
On 01/21/2015 02:38 PM and on 01/21/2015 04:07 PM,
Robert Cohen wrote:

[some placement quiz questions]

Those are some nifty questions.  Very informative.

> My purpose was to show that some students tend to interpret any
> problem that has numbers into a "math" problem that does not involve
> any interpretation or reasoning.

That's a fine hypothesis.  It fits the observations as 
well as any, and better than most.

Let's take another step down that road.  Let's try to fill
in the next level of detail.  I reckon it's important to 
figure out where the students are coming from.

If they are not understanding, what are they doing instead?

Let's consider a slight modification of the problem:

<scenario>
  1. According to Oreo's second law, F=ma, where F is the
  frobozz, m is the merf, and a is the alcamex.  Suppose
  the merf is 1 when the alcamex is 1 and the frobozz is
  1.  What then is the merf when the frobozz is 10?

  We can proceed as follows:
   0) We do /not/ attribute any meaning to the words.  They
    are just empty symbols.

   1) Instead we invoke the rule that says:
    a) Any variable that is mentioned is important.
    b) Any that is not mentioned must be constant and/or 
     irrelevant.  That is to say, the words "Other Things
     Being Equal" (OTBE) are implicitly part of every
     question.

   2) In our example, the question mentions only merf and 
    frobozz, so the alcamex must be constant.

   3) Therefore the merf must vary in direct proportion to the
    frobozz.
</scenario>

Note the key distinction: Mass has physical significance, 
whereas merf does not.  The word "mass" symbolizes something
real, whereas "merf" is just an empty symbol.

Now, one of the most fundamental rules of reasoning says
"Never mistake the symbol for the thing symbolized."
  https://www.av8n.com/physics/thinking.htm#sec-symbolized

"Sometimes" you can get away with unthinking manipulation of 
empty symbols, but sometimes you can't.  A pocket calculator
doesn't think about /numbers/ at all;  it just manipulates
the /numerals/ without understanding what they mean.

At this point, we are left with the question, what makes
students think it is OK to manipulate empty symbols, rather
than thinking, rather than paying attention to what the
symbols mean?

The answer is that they have been trained to do that!

Here's an example:  In kindergarten, kids are required
to add 10 plus 2.  They are drilled incessantly.  The
goal is speed and accuracy.  Speed and accuracy.  Now
the "interesting" thing is the grading scheme:  The right
answer is worth two points.  If the student writes "2",
that's worth one point.  I'm not kidding!  The answer is
12, but 2 is considered half right, because one of the
two digits is right.  The problem is being graded based
on the numeral, not the number.  It is being graded on
the symbols, without regard to what the symbols mean!

You can't make this stuff up.

This reminds me of the joke at the top of this message.
Questions are being answered (and graded) in a way that
completely perverts the reason for asking the question.

At a slightly deeper level, in high-school algebra class
they make a virtue of plotting y as a function of x, 
without regard to what the symbols mean.

Furthermore, our own Philip Keller wrote a nifty book 
explaining how to answer the vast majority of SAT 
questions without using more than a tiny fraction of 
the skills you were "supposed" to learn in algebra 
class:
  http://www.amazon.com/New-Math-SAT-Game-Plan/dp/098158960X

The existence of such books proves that the SAT has
virtually no construct validity.  In an ideal world,
there would be no market for such books, because the
test would measure something resembling actual reasoning
and understanding.

Different games are played by different rules.  If you
show up to play baseball wearing your football helmet
and pads, without your bat and glove, things are not
going to go well.

Physics plays by different rules.  The real-world rule 
that says mass does not change outweighs the disgustingly
artificial OTBE rule.  This requires noticing that the
question says "mass" (not "merf") and then looking
beyond the symbol to see the thing symbolized.

If you want students to play by new rules, you will
have to explain the new rules ... and explain them
more than once.  Kids have been drilled -- literally
since kindergarten -- to emphasize speed, and /thinking/
about the meaning would take too much time.

I don't claim to know exactly what is going on.
Each student is different, and there are undoubtedly
multiple processes that contribute to the observed
results.  I offer this as  a hypothesis that can be 
checked.  It may help explain some fraction of the
observations.

Last but not least, I would point out that this is
only a first step, or fraction of a step.  It is
looking under the lamp-post.  The level of reasoning
that is being probed by these quiz questions is only 
the flea on the penguin on the tip of the iceberg.
Serious understanding and reasoning are vastly more 
complicated.