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Re: physics/pedagogy of coffee-mixing



John Denker wrote:

Suppose you have a cup of coffee and a cup of tea. In step 1, you transfer
one spoonful of liquid from the coffee-cup to the tea-cup. In step 2, you
transfer one spoonful of liquid from the tea-cup back to the
coffee-cup. Question: Is there more tea in the coffee, or more coffee in
the tea?

I know of several ways of analyzing this problem, some of which lead to the
right answer, and some of which don't. I my experience, there are quite a
few kids at the high-school level who
-- are strongly attracted to the wrong analysis,
-- are not convinced by the correct qualitative argument, and
-- are not even convinced by a detailed mathematical analysis.

So the questions for today are
*) Why do people have a hard time with this riddle?

It's more subtle than it looks like at first glance, especially if
you have never thought of this sort of thing before. It's worded to
imply that the answer is going to be hard to find and require lots of
math. It is asked as a multiple-choice question with two possible
answers, neither of which turns out to be right, so the students are
being mislead by the question. If you really want them to get the
answer, this is a bit unfair. If you are trying to make a point, OK,
but make sure that they to get the point by the time you are finished.

*) What can be learned from this family of mistakes?

What do you mean by "this family of mistakes"?

*) In what other circumstances might people be tempted to
make similar mistakes? What are the warning signs?

I'm not sure what you're driving at here, either. People make these
kinds of mistakes when the problems are phrased in such a way as to
appear more complex than they really are. I think a bright student,
who had done problems like this might reason as follows (after Martin
Gardner): Since the problem, as stated, manifestly must have an
answer, and since the amount of coffee/tea as well as the amount
transferred is not specified, the answer must be independent of those
values, so what if I pick something really simple? Let the amount in
each cup be exactly one spoonful. If I put all the coffee in the tea,
then I have two spoonsful of "coffeetea" which is half coffee and
half tea. If I transfer one spoonful of "coffeetea" back to the
original cup then I have the same amount of liquid in each sup as I
had before, and the fraction of coffee in the tea cup is exactly the
same as the fraction of tea in the coffee cup. Therefore the answer
must be that regardless of the amount initially in the cups or the
amount you transfer each way, as long as you have the same amount in
each cup and transfer the same quantity each time, the fraction of
coffee in one cup will equal the fraction of tea in the other when
you are done. What if you now repeat the process? Will the converse
proportions still be the same? I dunno. I just thought of this and
haven't worked it out, but I suspect they may be.

Most students are not sophisticated enough to do this, and a general
mathematical analysis is also going to elude them (it took me three
or four tries and several numerical examples before I got it right).
Qualitative arguments will just be confusing, and since they couldn't
produce the mathematical analysis themselves, they probably won't
follow it when given to them. The way to convince them is with
marbles. Put 19 white marbles in one box and 18 black marbles in
another. Then transfer 9 of the white to the black box, so the white
box has 9 white marbles and the black box has 18 black marbles and 9
white marbles. since the ratio of black to white is 2:1, when you
transfer 9 marbles back to the white box, you must transfer 6 black
and 3 white. The result is 12 white and 6 black in the white box and
12 black and 6 white in the black box. In other words, the proportion
is just reversed. As long as you choose numbers for the total and the
amount transferred that allow you to use whole numbers of marbles (12
marbles in each box and transfer 4 also works, as does 72 and
transfer 9), it is easy to demonstrate and easy for the students to
see and understand. Once they understand what is going on, I think
the general mathematical analysis might make some sense to them.

Now, since I have done the analysis, and it looks (at least to me)
like I got the right one, I probably can't figure out the wrong one.
So how about letting us know what some of the incorrect analyses are.

Hugh



--

Hugh Haskell
<mailto://hhaskell@mindspring.com>

Let's face it. People use a Mac because they want to, Windows because they
have to..
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