I remember in one of my early calculus classes a student had to calculate the speed at which a boat was approaching a dock and came up with something around negative mach 3. Just last semester, I gave the standard problem of finding the maximum possible volume of a soup can cut from a small sheet of metal. One student’s can could hold a football stadium. Another had a negative radius.
We all have examples like these. I try to communicate that part of the problem-solving process is checking your final answer for common sense. I tell them that “-2 cm. I know this is wrong because a negative radius makes no sense, but I can’t find my error” gets more partial credit than “-2 cm.” Some things to watch for:
- In a geometry problem, does the scale make sense? For example, if you are finding the area of a region that fits inside a circle or rectangle, is the area actually smaller than the region that contains it?
- In an applied problem, does the answer seem reasonable in the context of the application? You probably made a mistake if your world population is growing at a rate of 3 people per decade or if your projected sales revenue exceeds the U.S. gross domestic product. See also the can of football stadiums above.
- Do the units make sense? If your volume is expressed in square meters or in candelas per newton, you made a mistake.
- Is a numerical check possible? Many problems ask you to find one function that is equal to another (e.g., trig identities). To test this, pick a random value and see if the functions agree on that value (not a nice value like 0 where they may agree by coincidence, but if the functions agree on 1.26, they are probably the same).
In proofs, I encourage my students to check the statement once before declaring victory. “Did you use all of the conditions in the premise?” “Did you actually prove the statement you say you proved?” Just those two checks would catch a lot of problems. “Can you point to something specific that justifies each step?” is a bit more of a check, but that one almost by definition corrects all errors. Another one that would be helpful is “Do you know what all of your notation means?” I don’t know many times I’ve had a student lay out a proof and mumble indecisively on one step. I point out that they did this, then ask where they think the mistake probably is.
It’s hard to say how much these tips really help, but I do get the occasional “I know this is wrong because” response, and I try to make good on my promise of additional leniency in scoring. But one thing I always point out is that these aren’t just tricks for getting through the class. I do many of the exact same checks in my own work as a research mathematician.
Bill Wood @MathProfBill is an assistant professor of mathematics at the University of Northern Iowa.
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