Editorial: This is the penultimate article in the retrospective series we’ve been doing all week here at CO9s. This one takes us back to 2006 one more time.
One of the things that fascinates me most about teaching math is seeing how people acquire and use problem-solving skills. And one of the things I like to think and write about the most is how people can approach problems in different ways — especially when those ways are not the standard ways of doing so — and why students make various conceptual mistakes when they try.
This article was written after a calculus homework set involving a pretty standard intro problem about the velocity of an arrow shot straight upward on the moon. (Where the **** do we math people get these problem ideas?) I was reading James Gleick’s biography of Richard Feynman at the time and was very keen on how important visualization is in problem solving. I had also been thinking (and posting) about how the ephemeral idea of “critical thinking” really just boils down to having a good intellectual B.S. detector and having the will to question whether your thinking can possibly be right or not. Put all that together, and this article is what you get.
Side note: One of the biggest sources of traffic for this blog comes from people who enter in “velocity of an arrow shot upward on the moon” into a search engine and end up with this posting. For those of you who have found yourselves here by doing so: DO YOUR OWN HOMEWORK.
Critical thinking, visualization, and physical intuition
Originally posted: October 11, 2006
Yesterday I posted my belief that “critical thinking” has at least as much to do with intuition as it does with what we normally call “thinking”. Namely, critical thinking has to do with — is activated by — having a sense of when something can’t possibly be right. Question: Where does that sense come from? And importantly, can it be taught? I’ve been reading through Genius: The Life and Science of Richard Feynman by James Gleick and have been struck by Feynman’s reliance upon visualization to make his dramatic contributions to quantum theory and other areas, and it makes me believe there’s a strong connection between visualization and the ability to solve problems that I’ve not heard mentioned often.
Take a look at the following for a possible clue. A homework problem reads: If an arrow is shot upward on the moon with an initial velocity of 58 meters per second, its height is given by H(t) = [insert formula]. When will the arrow hit the moon? And with what velocity will the arrow hit the moon?
Here are three student responses that involved little or no algebra:
- “0 feet [sic] per second. If it has hit the moon, it has stopped moving, so the velocity is zero.”
- “The answer to when the arrow will hit the moon is never. The arrow is not launched with enough velocity to reach the moon.” [Student goes on to give data supporting this conclusion.]
- “The arrow will hit with a velocity equal to 58 meters per second, because that is the velocity it started with and it will be the same coming down as it was going up.”
What do these answers have in common? They all rely upon visualizing the problem and having a physical intuition for the forces at work in the problem. The first two responses have incorrect visualization/physical senses. The third one has a correct sense. And a correct visual/physical sense for a problem, especially a contextual word problem like this, is almost indispensable for what I am calling “critical thinking”.
If the first student above had had the correct physical intuition for the problem, s/he would immediately see as absurd the notion that falling objects are motionless when they land. On this student’s paper, I wrote, “Go and throw a baseball straight up in the air and let it land on your head. I think you’ll agree it’s not motionless when it hits!” All such responses aside, though, there is a fine line between the moment when the arrow hits and the moments immediately following impact. At the precise moment of impact, the arrow still has velocity. Then the velocity is stopped by its collision with the ground, and moments later it is indeed at rest. What is the problem asking for, though? The velocity WHEN IT HITS. It can’t be motionless when it hits; what would it do, just hover above the ground? What would be imparting the decelerating force? These are questions of critical thinking, and they all stem from that initial sense that something can’t possibly be right, namely the idea that a projectile loses its velocity prior to impact.
The second student is closer to being right, because s/he is certainly right that an arrow can’t ever reach the moon with a piddling 58 m/s initial velocity. But of course the student has read the problem wrong and made the wrong assumption; s/he thinks we’re standing on Earth, trying to shoot an arrow to the moon. That’s a comical picture. Why? Because we visualize it. The failure to visualize the situation — or at least, the failure to react appropriately to a visualization that can’t possibly be right — is really what allowed an initial misreading of the exercise to go on to become an incorrect result.
Finally, the third student shows that if you do correctly visualize and sense what’s going on in the problem, you can often get around doing any math at all. Many “math problems” aren’t math problems at all, but problems about thinking. So another aspect of critical “thinking” appears not to be related to what we call “thinking” at all, but rather the ability to project yourself into an imagined physical situation with clarity, and respond accordingly.
Casting Out Nines 
Whether you are solving for the answer to a math formula or determining the root cause of a complex business problem, eliminating what can’t be is far more effective than guessing what could be. Sir Conan Doyle hit it when he said, “When you eliminate the impossible, whatever remains, however improbable, must be the truth.”