Today, I didn’t have time to revise the sample problem I’m supposed to do in physics. So I worked the sample problem as given. We were doing standing waves. In the particular problem, there was a string with both ends fixed and we were told it was vibrating in the second harmonic. In this case, the wavelength is equal to the length of the string.
After my sample problem, students were given a problem where the string was vibrating in its fifth harmonic. A fourth of the class did it correctly, drawing it out and concluding that the wavelength must be 2/5 the string length. Half the class did it wrong saying that the wavelength was equal to the string length. And a third of the class said the wavelength was 5/2 of the string length. As I was walking around, I asked a student why they thought so many people were making these mistakes.
She responded without hesitation, “The sample problem was poorly designed. You shouldn’t have given us an example in which the wavelength and the string length were equal. That makes it easy for everyone to think that’s what you are supposed to do every time. Plus dividing by 1 or multiplying 1 gives you the same result, so it’s easy for people to mix it up. Next year, you should use a different harmonic to set up the problem.”
I wish my future physics teachers knew had to unpack a sample problem like that, and see how it might lead to over-generalizations and misinterpretations.