The first two examples of naïve perspective have been directed at younger students.  Here’s a couple for older students.

Pendulums: The heart of the naïve perspective is to re-discover the magic in the science we teach.  And it’s not only basic matters in elementary school.  Take pendulums.  I often played with pendulums across my schooling, learning such key bits as the period of a pendulum does not depend on the weight of the pendulum bob, and more intriguingly that it also doesn’t depend on the vigor of its swing.  Let it sweep out a long arc and it takes no more or less time than just a little swing back and forth.  That’s why you had pendulum clocks; they didn’t slow down as they wound down.

The whole thing gets delightful when we look at the expression for the period of a pendulum:  T = 2π√l/g .  All intro physics book use this expression.  It’s a standard experiment for students.  You have them time the swing for different lengths and plug the results into the expression, and if you did it right your numbers work.

But it is a most curious expression and it is a lot more valuable to take a slightly different tack.  Don’t give the students the right answer first.  Have them measure the times for different lengths of string.  Then ask them to derive an expression for the relationship.  This a great exercise for calculators these days, because if you stick too close to the data you get very arcane cubic expressions.  This, in itself, is a good discussion, but it gets better.

Whatever expression they get, it will lack the trappings of the standard expression.  They are playing with the relationship between l and T, and there is no reason to invoke or introduce the acceleration due to gravity, “g”.  How did it get in there?  Clearly gravity has a lot to do with pendulums going back and forth, but still…where did “g” come from –all you have are numbers on “l” and “T”?  But if you look carefully there’s another more pressing issue; there’s an irrational number right in the middle there – π.  It is not unreasonable for there to be constants in such cases, but how could the constant be an irrational number? That means no one has ever measured the period of a pendulum to be what we say the period of a pendulum is.  You can’t.  Nor can you get it out of some average of results.  The simple fact is: what we know to be the right answer never actually happens.

This is a really critical matter to talk about with students who are serious about what they are learning in school.  Oftentimes, students take the simple bromides teachers and textbooks offer them as the truth with a capital “T”.  Science is about careful, objective observation and just a matter of facts.  But the sciences are much richer and more complicated than that.  Pascal said that the heart has reasons that reason knows not… which is lovely.  We might alter that a bit and suggest that science has reasons that facts know not.  In this case, something is understood as true, even though it never happens.