We’ve been thinking about explanations in our research (see, e.g., https://arxiv.org/abs/2205.07938) and your example of explaining the wrong answer well is lovely.
I dislike these kinds of questions, because they’re usually posed at a point well before the wave equations are presented. At this point, you are largely working with verbal explanations and, as you point out, verbal explanations are much harder to pin down.
Mathematically, if A implies B, and you are working to the best of your ability, you can’t derive ~B (you may not be able to derive B, of course!) Verbally, this is not so clear; a lot of philosophy is people arguing about whether A implies B or ~B.
If the underlying logic is (as it is in physics) mathematical, then a verbal account of mathematical fact A can be loose enough that you can derive ~B, because the verbal account is also consistent with A’, which implies ~B.
In these explanations, there’s another factor: a lot of the talk is relying on intuitions for “ordinary solids”. One explanation I encountered when I googled referred to the “stiffness” of a hotter gas. While you might be able to cash this out in more formal terms, the temptation is to think about stiffness in one’s normal experience. (It might have been possible to get the right answer by imagining heating up a bicycle tire that’s already inflated; intuitively, the casing will be stiffer, “ring” at a higher frequency, etc.)
If you continue on in physics to relativity, quantum mechanics, etc, you end up dropping this kind of talk very quickly. This is why I don’t like these questions; it’s a bit useless to get good at them, because the more advanced you get the more you learn to rely on mathematical intuitions to get the right answer (and then perhaps informal folk talk afterwards, if you communicate to a popular audience.)
Personally i think the verbal kinds of explanations still have an important role in more advanced physics, as something you can present after the calculation or simulation. Things like ‘it kind of makes sense that we see this because...’ or ‘this may seem surprising but it actually makes sense if you look at it as...’.
I dislike these kinds of questions, because they’re usually posed at a point well before the wave equations are presented. At this point, you are largely working with verbal explanations and, as you point out, verbal explanations are much harder to pin down.
I agree that they’re not great test questions, but they can be excellent class discussion or homework problem set questions (as long as you encourage working together on homework, which can work in college but not usually in high school). If anything, using them well puts a much higher burden of understanding on the teacher to not only know the answer but also all the ways students are likely to go wrong in trying to reason about the answer and how to steer the discussion without just giving the answer.
In this case, yeah, I’m sure this question was posed at a point where the student doesn’t really know what T and P mean at a fundamental level, what makes a gas “ideal,” what the Maxwell-Boltzmann velocity distribution is and why, and a whole bunch of other relevant things. Given that, you should still be able to reason it out using dimensional analysis, the definition of kinetic energy, the idea that T is proportional to kinetic energy, and looking at some limiting cases and boundary conditions, but it isn’t easy.
I dislike these kinds of questions, because they’re usually posed at a point well before the wave equations are presented. At this point, you are largely working with verbal explanations and, as you point out, verbal explanations are much harder to pin down
What would the same question look like when presented in mathematical form?
At a quick glance I can’t see a concise way to express it.
That’s the point—to actually be precise you need both wave mechanics and statistical mechanics as background to even define the “speed” of a “sound” “wave,” the “temperature” and “pressure” of a gas, and how a gas gets “heated.”
At what appears to be this level of coursework, it should be something they’ve already discussed in class. Otherwise, it should be either asked with a lot more context, or as part of a discussion instead of on a test.
I’m assuming this is something like a high school level physics class, and that this specific problem hasn’t been discussed before the test. In that case I think it’s a sufficiently hard problem that if you’re going to ask it on a test, then it should be a multi-part series of questions that leads to towards the right mode of thinking. Maybe something like, “1) What happens to the average speed of molecules in an ideal gas when it gets heated up? 2) How, if at all, does this affect the speed of sound? 3) How, if at all, do changes to V and P affect the speed of sound in an ideal gas?”
We’ve been thinking about explanations in our research (see, e.g., https://arxiv.org/abs/2205.07938) and your example of explaining the wrong answer well is lovely.
I dislike these kinds of questions, because they’re usually posed at a point well before the wave equations are presented. At this point, you are largely working with verbal explanations and, as you point out, verbal explanations are much harder to pin down.
Mathematically, if A implies B, and you are working to the best of your ability, you can’t derive ~B (you may not be able to derive B, of course!) Verbally, this is not so clear; a lot of philosophy is people arguing about whether A implies B or ~B.
If the underlying logic is (as it is in physics) mathematical, then a verbal account of mathematical fact A can be loose enough that you can derive ~B, because the verbal account is also consistent with A’, which implies ~B.
In these explanations, there’s another factor: a lot of the talk is relying on intuitions for “ordinary solids”. One explanation I encountered when I googled referred to the “stiffness” of a hotter gas. While you might be able to cash this out in more formal terms, the temptation is to think about stiffness in one’s normal experience. (It might have been possible to get the right answer by imagining heating up a bicycle tire that’s already inflated; intuitively, the casing will be stiffer, “ring” at a higher frequency, etc.)
If you continue on in physics to relativity, quantum mechanics, etc, you end up dropping this kind of talk very quickly. This is why I don’t like these questions; it’s a bit useless to get good at them, because the more advanced you get the more you learn to rely on mathematical intuitions to get the right answer (and then perhaps informal folk talk afterwards, if you communicate to a popular audience.)
Personally i think the verbal kinds of explanations still have an important role in more advanced physics, as something you can present after the calculation or simulation. Things like ‘it kind of makes sense that we see this because...’ or ‘this may seem surprising but it actually makes sense if you look at it as...’.
I agree that they’re not great test questions, but they can be excellent class discussion or homework problem set questions (as long as you encourage working together on homework, which can work in college but not usually in high school). If anything, using them well puts a much higher burden of understanding on the teacher to not only know the answer but also all the ways students are likely to go wrong in trying to reason about the answer and how to steer the discussion without just giving the answer.
In this case, yeah, I’m sure this question was posed at a point where the student doesn’t really know what T and P mean at a fundamental level, what makes a gas “ideal,” what the Maxwell-Boltzmann velocity distribution is and why, and a whole bunch of other relevant things. Given that, you should still be able to reason it out using dimensional analysis, the definition of kinetic energy, the idea that T is proportional to kinetic energy, and looking at some limiting cases and boundary conditions, but it isn’t easy.
What would the same question look like when presented in mathematical form?
At a quick glance I can’t see a concise way to express it.
That’s the point—to actually be precise you need both wave mechanics and statistical mechanics as background to even define the “speed” of a “sound” “wave,” the “temperature” and “pressure” of a gas, and how a gas gets “heated.”
So how instead should this category of questions be ideally presented?
At what appears to be this level of coursework, it should be something they’ve already discussed in class. Otherwise, it should be either asked with a lot more context, or as part of a discussion instead of on a test.
I’m not sure how to imagine ‘a lot more context’ for this question, can you provide an example?
I’m assuming this is something like a high school level physics class, and that this specific problem hasn’t been discussed before the test. In that case I think it’s a sufficiently hard problem that if you’re going to ask it on a test, then it should be a multi-part series of questions that leads to towards the right mode of thinking. Maybe something like, “1) What happens to the average speed of molecules in an ideal gas when it gets heated up? 2) How, if at all, does this affect the speed of sound? 3) How, if at all, do changes to V and P affect the speed of sound in an ideal gas?”