Science has a reputation of being "difficult."  Nevertheless, elementary science is generally extremely simple.  Here's an elegant example of a "simple but difficult" science principle. 

For science to become a useful tool rather than a useless and  mystifying "alternative way of thinking" teachers and learners both need to master the mysteries of the "simple but difficult."  The bouncing ball is an excellent place to start because it's a widely "unseen" principle essential to understanding physics.

The Bouncing Ball

What is the direction, up or down, of the acceleration of a freely bouncing ball at the bottommost point of its bounce, that is, at the instant its velocity changes from down to up?
Warning: You enter a maze when you attempt to answer this question.  (Interesting maze, though.)

I wrote this exam question for my first elementary physics course.  It was a give-away question to put test takers at ease.  And it also tested one very elementary course objective: "The learner will be able to recognize acceleration in new, unfamiliar situations and be able to distinguish it from velocity."  A person who doesn't find this question trivial and obvious is probably going to be mystified by a large part of the physics that follows.  A person who "sees" often suspects that there's a hidden "catch"; it's simply too obvious.  The answer is given in the last phrase of the question!

It was no "give-away"!  That first class got about 5% correct, where pure guessing would have resulted in 50% correct.  Most had learned the answer for the top of the bounce.  They had learned that "answer" and had not understood it.  (And I headed off to the School of Education and the Psychology Department for help.)

I have since used the bouncing ball on exams in every elementary physics course I've taught, and I have asked it of many, many people who studied physics elsewhere.  Most classes have gotten about 5% correct, except one class which had considerably less talent for science: they got the pure-chance 50%.

If I were to teach the answer to this specific question, I feel confident most classes would give me about 90% correct answers, just as they do for the top of the bounce.  But I am equally confident, they would still not understand acceleration in any useful way.  Unless and until I find some other way to bring about that understanding.  Learning is relatively easy; "seeing" is something else.

December 1, 2001
I have thought much about this fundamental puzzle of science education, with several models in mind.  Recently I encountered a model which seems at once simpler and more encompassing than any other.  It fits my observations remarkably well, and it offers hope for effective solutions in ways that are consistent with the recent encouraging work of the physics teaching community (at the U of Wash, for example).  The model follows the suggestions of Keith Devlin.  SEE: PDX.

What a weird potpourri of reactions to a simple question

It's too easy:  I suggested to a colleague he might try the question on an exam for his class.  He looked at it, and said, "That isn't even a problem!"  He felt he needed the exam space for more important topics.

It's too hard: However, other physics teachers have suggested that the question is too difficult unless the answer to it has been specifically taught.  The question tests for insight that is more general than are memorized "answers."   That makes is at Devlin's "fourth level" of abstraction, the realm of mathematics and the source of the effectiveness of science.  Some physics teachers test their students for this kind of insight; some don't.

Are you kidding?:  Another colleague gave me a quizzical look and gave the problem a careful working over.  He created a couple of imaginative possibilities of what it might mean, and since I could see he thought it had a catch, I said, "Beethoven wrote Beethoven's Fifth."
        He paused a moment, then smiled,  "No catch?"  I think he took my word that students frequently miss it, but I'm not sure he accepted just how many miss it.

Huh?: An acquaintance who just finished a college physics course said she wasn't sure what the question means, but she supposed the ball didn't have any acceleration, since it isn't moving at the bottommost point in the bounce.  She also said she didn't really feel she got much out of her physics course.  (And this response is very similar to that of almost all non-physicist former physics students I've posed the problem to.)

You've got to teach the answer: A very good editor said I will lose my readers, a very highly educated group of readers, if I don't give the answer as I try to use the question as an example of a "simple but difficult" concept which will not be understood if the answer is merely taught.  And she gave a very convincing argument that she is right!  (My counter argument is here.)

Who wants to be able to use physics?

  Most concepts taught in an elementary physics course require the primitive level of understanding of acceleration required by this question.  I doubt that any other understanding has greater potential for increasing the understanding of physics in the general population.  But it must be understanding, not just learning.

(I have not yet done a thorough Web search for possible solutions to this problem.  I've seen beautiful solutions to more difficult problems, techniques leading to understanding of more difficult concepts.  Perhaps someone has focused on this "simpler" concept which so many physics teachers see as "not even a problem."  See "Seek Eurekas.")

Note that Newton invented the calculus to handle this concept.  (Note also, that vector analysis—which physics teachers regularly throw out in front of students who are known to have a crippling fear of calculus—is much more recent than calculus and, in my opinion, much more subtle.)

I offer this as a possible approach for the bouncing ball:

    On your airplane's instrument panel, install a map, a velocity gauge, and an acceleration gauge.
    The map is fairly familiar: it shows where you are (but includes elevation).  The velocity gauge is not familiar (those are speed gauges!):  it's an arrow in three dimensions which shows both magnitude and direction of motion (that is, it shows velocity).
    Now here's where you need to stretch your insight a bit: the acceleration gauge.  Just as velocity is the rate of change of position, acceleration is the rate of change of velocity.  It, too, is a three-dimensional arrow and looks a lot like the velocity gauge.  The direction of the acceleration is the direction of motion of the tip of the arrow in the velocity gauge.
    Imagine you are in a car driving at constant speed on a straight stretch of road.  The car has a velocity gauge and an acceleration gauge.  (Navigation is easier in two dimensions as all airplane pilots are aware.)  Ahead lies a right-angle turn to the right.  The turn is along a circle.  Then the road is again straight.  Imagine what the velocity and acceleration gauge will show as you approach the turn, as you go through the turn, and as you head off in the straight line after the turn.  When you have a good feeling that you have it just right, go to the gauges.

    When you have taken this little step of logic, you will have taken a giant leap toward understanding the physics that gets missed by so many, year after year, century after century.

(See also the HELP! HELP! comment on the computer questions page.  I'm fairly confident that computer simulations can work well for aiding understanding of acceleration.  A strong movement in cognitive psychology tends to reject such approaches because of a feeling that real-life experience is essential and computer simulation is not real-life.  I believe they misidentify relevant parameters.  The type of information input is important, but not whether or not it's a simulation.  We get about as much information from a photograph–a simulation of the photon pattern entering the eye–as we get from the real-life view.  Physics teachers often consider their physics laboratory courses to be essential for the same reasons.  However, I got so little from my undergraduate physics labs that, when I started teaching them, I was certain I had never taken them.  Until the day we did a spectroscope experiment.  It all came back to me.  When I was a student virtually all of our "experiments" had to me seemed ridiculous, pointless rituals.  What a waste of time!  But the spectroscope had shown me that I had perception of ultraviolet!  And that "impossible" perception was a sensation I realized I had never before experienced, and so were all those other "pure spectroscopic colors."  What a discovery!)

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