Re: [Phys-L] widget rate puzzle ... reasoning, scaling, et cetera

[John Denker <jsd@av8n.com>]

On 12/31/2014 06:54 PM, I wrote:

> consider the following transmogrification
> of the original question:
>
>      It takes 5 minutes for 5 machines to make 5 widgets.
>      So, how many machines does it take to make 100 
>      widgets in 100 minutes?

Here is one possible answer:  In a certain factory,
it takes only *one* such machine to turn out 100
widgets in 100 minutes.

Spoiler:  Here's how it works, in this particular
factory:  The machine takes four minutes and 2.5
seconds to warm up.  Thereafter it can bang out a
new widget every 57.5 seconds.  That means the first
widget appears at the 5 minute mark exactly, in
accordance with the statement of the problem.  It
can bang out 99 more in the remaining 95 minutes,
with a few seconds left over.

This is not a contrived or unphysical possibility.
Things like this happen in the real world all the
time.  In addition to "warmup" issues, there are
also "pipeline" issues.  A garden hose can put out
a ml of water in a few milliseconds.  However, that
does not mean that any particular ml made it all the
way from the well to the outlet in a few milliseconds,
or even a few seconds.  It spent a long time in the
pipeline.  Similarly, if you buy a desktop rated at
10,000 bogomips, that does not mean that it can carry
out one instruction -- from start to finish -- in 0.1
ns or even 1 ns.  There is a lot of pipelining.

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

More generally, and much more importantly:  Different
games are played by different rules.  If you show up
to play baseball wearing your football pads and helmet,
without a bat or glove, you're going to have problems.

There is a style of question that I call "the spherical
cow in the ivory tower" where it is assumed that all
the information provided in the question is relevant,
and all the information not provided is irrelevant.
If you want to play by those rules, be careful,
because the real world almost never works that way.

This issue has been recognized for hundreds of years, if
not longer:
  http://en.wikipedia.org/wiki/As_I_was_going_to_St_Ives

There are some jobs where all the questions are well
posed ("Would you like fries with that?") but they
tend to not pay very well.  Living in the real world
requires dealing with ill-posed questions.  Seriously,
when was the last time a student came up to you and
said "I'm confused.  Here are the four possible things
that could be causing the problem.  Pick one."

In the real world, if the customer comes to you and
says he needs 100 widgets not just 5, and people's
lives depend on it, it would be insane to think that
buying 20 times as many machines would be necessary
or sufficient to solve the problem.

I am a big fan of the building-block approach:  Given
a complicated problem, you break it down in to small
blocks, learn each block separately, and then put the
blocks back together again, two at a time, three at
a time, et cetera, until you have a finished edifice.

A pile of bricks and a pile of sticks is better than
nothing, but you wouldn't want to live there.  A pile 
of bricks and sticks is fine as some preliminary stage, 
but it is important to move beyond that stage as quickly
as possible, even in the most basic introductory course.

In physics as in masonry, virtually all the creativity
and all the utility is associated with how you put the 
building-blocks together.  As long as students see
physics as structureless pile, there are going to be 
seeerious motivation problems.

The counterargument is that real-world problems are
hard.  They take time and effort.  If we start doing
real-world problems, we will have to drastically cut
down on the number of problems.  All the bureaucratic
and political forces are pushing in the opposite
direction, towards being a mile wide and an inch
deep.

That's all true, but I'm saying it's time to push back.
I'm not arguing for the opposite extreme;  as usual, 
I think all the extremes are bad.

Some people say they don't know how to teach reasoning, 
or "can't" teach it because they can't "measure" it. 
Gaaack!  Well, here's one way of doing it:

First:  New rules:  Whenever you see a problem like
this:

>     It takes 5 minutes for 5 machines to make 5 widgets.
>     So, how many minutes does it take for 100 machines
>     to make 100 widgets? 

your first reaction should be to say wow, that is
seriously ill-posed, seriously underspecified.  If
somebody wants a hypothetical answer, predicated on
a bunch of simplifying assumptions, we can do that
... but if they want a non-hypothetical real-world
answer, if people's lives depend on the answer, then
we are going to need a whole lot more information.

Secondly, even if you are giving a hypothetical answer,
be upfront about the hypotheses:
 -- assuming the cow is spherical
 -- assuming everything is nice and linear
 -- assuming the rate of widgets per machine per unit
  time is uniform across all of spacetime
 -- assuming there are no other significant factors
 -- et cetera

     This is one of the reasons that multiple-guess
     tests are an instrument of the devil:  They do
     not encourage /or even permit/ surfacing the
     assumptions.

Thirdly, rather than asking students to find "some"
element of the solution set, expect them to characterize
the /entire/ solution set.  That is, consider /all/ of
the plausible hypotheses:
 -- (easy) Can you come up with a scenario where the
  correct time is 5 minutes?
 -- Can you come up with a scenario where the correct
  time is 100 minutes, for perfectly reasonable
  physical reasons?
 -- (harder) Can you come up with a scenario where
  the correct time is more than 100 minutes?
 -- Can you come up with a scenario where the correct
  time is less than 5 minutes?
 -- Can you come up with a scenario where the correct
  time is in the middle, approximately the geometric
  mean of 5 minutes and 100 minutes?
 -- Can you come up with nontrivial upper and/or lower
  bounds for the time, i.e. bounds applicable to all 
  plausible scenarios?

I realize this takes time and effort.  You can answer
the question wrongly in half a second.  You can figure
out the "conventional" spherical-cow right answer in a
minute or so.  Figuring out what's really going on could 
take a couple hours ... or longer if you're a student
with no experience in such matters.

That's fine with me.  I'd rather teach one useful thing
than 100 useless things.  Students are not going to 
remember the useless things anyway, so why bother?

===========

I've run into the widget puzzle three times in the last
week, and at least five times in the last five months,
in books, in scholarly articles, and once on NPR ...
posed by big-shot "experts" in the field of "reasoning"
... not one of whom noticed that the puzzle is ill-posed.
They noticed it was tricky on one level, but failed to
notice that it is tricky on a whole other level.

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

There is a school of thought -- as exemplified by the
book by Heller&Heller, and by numerous PER papers --
in which the teacher takes a trivial problem and makes
it artificially harder, perhaps by forbidding the easy
method of solution, perhaps by introducing disfluencies
in the statement of the problem or whatever.  This just 
horrifies me.  

I always thought the teacher's job is was to start with
nontrivial problems and make them /easier/ by teaching the 
right approach, and occasionally introducing pedagogical
simplifications.

There are plenty enough hard problems in the world.  Even
some seemingly-simple problems are challenging if you take 
them seriously.  There is no need to introduce artificial 
barriers and impediments.  I didn't make the widget mfg
process complicated;  the complexity was already there.
Please don't shoot the messenger.  The goal is to recognize 
the complexity, and to deal with it properly.

Again:
 -- (easy) Can you come up with a scenario where the
  correct time is 5 minutes?
 -- Can you come up with a scenario where the correct
  time is 100 minutes, for perfectly reasonable
  physical reasons?
 -- (harder) Can you come up with a scenario where
  the correct time is more than 100 minutes?
 -- Can you come up with a scenario where the correct
  time is less than 5 minutes?
 -- Can you come up with a scenario where the correct
  time is in the middle, approximately the geometric
  mean of 5 minutes and 100 minutes ... for perfectly
  reasonable physical reasons?
 -- Can you come up with nontrivial upper and/or lower
  bounds for the time, i.e. bounds applicable to all
  plausible scenarios?