Logical Fun, Part I

Once a week, I teach a joint math class at my school which combines children of ages 6-10.  For these lessons, I generally try to pick an activity that I can present as a series of “problems” of increasing difficulty and depth.  Thus, the more advanced students (generally the older ones, but this really varies by topic) can go further and deeper whereas the ones who struggle with the topic can spend more time on the introductory problems.  Often, there is also a hands-on element to the activity.

This week, however, I taught a joint lesson in which there were no hands-on materials and no worksheets with problems or puzzles.  The whole class consisted of discussions, where students of all ages contributed greatly.

The topic was logic.  Most of the problems/puzzles we discussed came from Raymond Smullyan’s wonderful book What is the name of this book?

I loved how the students did not just look for “the one right answer” but looked for different interpretations, explored different definitions, and questioned “intended” answers.

Here is an example problem:

A train leaves from Boston to New York.  An hour later, a second train leaves from New York to Boston.  The trains are moving at equal speeds.  Which train will be closer to Boston when they meet?

The answer given in the book, and the one that I thought of right away with my experience with such problems, is that of course they are the same distance from Boston when they meet.

The students, however, started asking questions such as, What does it mean to meet?  Does it mean that the fronts are next to each other or the whole lengths of the trains are next to each other?  And how do we measure closeness?  Do we use the front of the train or the end of it?  The trains are not dots after all.

Different answers to these questions lead to different solutions.  We had truly wonderful and very mathematical discussions about how “it depends on many factors.”

And then one student asked whether it takes more or less than an hour to get from Boston to New York, so we discussed whether this would affect the answer to the problem.  Again, we decided that it depends on where the second train is before it starts off on its route.

I truly did not know what to expect from this lesson.  I was a little worried that the only response I’d get would be silence and confused looks.  However, I must say that this was one of the most enjoyable lessons that I ever taught.  And I was in absolute awe of the students, the freedom and depth of their thinking and their complete lack of fear in expressing their thoughts or of being wrong.

I wonder what would happen if I tried a similar lesson with much older students, or even adults.  Would they have the same freedom of thought or would they focus on looking for the one right answer?

Here’s another problem that we discussed with the students that perhaps requires a bit more brain twisting and that is a favorite of mine:

A man is looking at a portrait.  A passerby asks him whose picture is he looking at, and this is his response: “Brothers and sisters have I none, but this man’s father is my father’s son.” Who is in the portrait?

Stay tuned for a separate post about the students’ discussions of this one.

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