Like the concept (and upvoted). Hate the “inverse” terminology—when I’m already confused by a math problem, I don’t want to add something that makes me flip a concept around every time I want to use it.
(Thanks for the upvote, first of all.) I’m not exactly sure what you mean here. We could easily give inverse speed its own name, like “slowness”. But would that help? Your wording makes it sound almost like you might be objecting to the very idea of adding “the inverse of speed” to your concept inventory, in which case I’m wondering what aspect of the post you actually liked!
To you, “speed” means thing X. To solve the problem, you need thing Y, which is the inverse of thing X. To me, speed means thing Y, so if you went around saying “inverse speed”, I would not only have to perform an “inverting” operation on “speed” every time the term came up, I would wind up with thing X after I did it, which is not useful to solving the problem.
Wait—really? Are you saying that you interpret “speed” to mean exactly what I mean by “inverse speed”? Hours per mile instead of miles per hour? So that when someone says they’re going 5 miles per hour, you don’t think “they’ll have gone 5 miles after one hour” but rather “1/5 of an hour will have passed after they’ve gone one mile”? And if I wanted you to think “they’ll have gone 5 miles after one hour”, I would have to say “they’re going 1⁄5 hours per mile”?
That’s not what I got from your original comment at all. When you said
I usually think of speed as a thing you use to get to a specific destination, so fiddling with the distance doesn’t make sense unless you’re fiddling with your route
-- well, I didn’t really understand what you meant, but I thought you were talking about thinking in terms of a two-dimensional map with different routes to the same place having different distances.
I suppose in retrospect I can make sense of it by interpreting the phrase “specific destination” as referring to keeping the distance fixed (and letting time vary).
But I still find it hard to believe that’s what you mean—for one thing, it would imply that you shouldn’t have been confused by the original problem at all, and instead should be more confused by something like this:
“If you want to average 40 mph on a two-hour trip, and went 20 mph for the first hour, what should your speed be for the second hour?”
(If you find this easier than the other problem, then that means your idea of speed is like mine rather than being inverse to it.)
So that when someone says they’re going 5 miles per hour, you don’t think “they’ll have gone 5 miles after one hour” but rather “1/5 of an hour will have passed after they’ve gone one mile”?
I don’t think any of those things. I wonder about how far they’re going at that speed, and if the answer is “up the block” I think “oh, they’ll be there soon” and if the answer is “to the moon” I think “that’s going to take forever”. I do not naturally think in numbers.
And if I wanted you to think “they’ll have gone 5 miles after one hour”, I would have to say “they’re going 1⁄5 hours per mile”?
No, if you want me to think that they will have gone five miles after an hour, you tell me they’re going someplace five miles away and it’ll take them an hour to get there.
I thought you were talking about thinking in terms of a two-dimensional map with different routes to the same place having different distances.
Well, I was, but this was incidental to my point.
I suppose in retrospect I can make sense of it by interpreting the phrase “specific destination” as referring to keeping the distance fixed (and letting time vary).
Yes.
you shouldn’t have been confused by the original problem at all
I was not confused in the way you were. I was confused in a different way, which has nothing to do with how I read the English word “speed” and everything to do with how my brain generates error messages when presented with math problems.
“If you want to average 40 mph on a two-hour trip, and went 20 mph for the first hour, what should your speed be for the second hour?”
Well, this is also confusing (my brain generates “sixty” automatically, but I don’t actually know if that’s an answer to this problem or just the result of seeing “40″, “20”, and “average” in that order, and I would have to do work to find out). It is not differently confusing than the first problem. I don’t get any farther or stop any earlier before I want to seek assistance. (I don’t even know if this is the same problem or not.)
I wonder about how far they’re going at that speed, and if the answer is “up the block” I think “oh, they’ll be there soon” and if the answer is “to the moon” I think “that’s going to take forever”. I do not naturally think in numbers.
Well, neither do I (I naturally think in terms of operations and transformations), so that’s not the relevant distinction. The relevant distinction is between “wow, they’re already far away” (speed) vs. “wow, they got there quickly” (inverse speed).
Let me see if I can generate, in your mind, something analogous to the confusion that existed in my mind. I probably won’t succeed, but the idea of attempting is too interesting to resist.
Here are two questions that are easy to answer:
(1) If I travel for an hour and spend a lot of time on the first part of my journey, how much time will I spend on the second part? (Answer: not much.)
(2) If I travel a mile and go a large distance during the first part of my journey, how far will I have to go during the second part? (Answer: not very far)
And now here are two questions that are confusing:
(3) If I travel for an hour and go a large distance during the first part of the journey, how much time will I have to spend on the second part? (Answer: Huh? That depends on how much time you spent going that large distance during the first part.)
(4) If I travel a mile and spend a long time on the first part of the journey, how much distance will I have to cover on the second part? (Answer: Huh? That depends on how much distance you covered during that long time you spent on the first part.)
(Thanks for the upvote, first of all.) I’m not exactly sure what you mean here. We could easily give inverse speed its own name, like “slowness”. But would that help? Your wording makes it sound almost like you might be objecting to the very idea of adding “the inverse of speed” to your concept inventory, in which case I’m wondering what aspect of the post you actually liked!
To you, “speed” means thing X. To solve the problem, you need thing Y, which is the inverse of thing X. To me, speed means thing Y, so if you went around saying “inverse speed”, I would not only have to perform an “inverting” operation on “speed” every time the term came up, I would wind up with thing X after I did it, which is not useful to solving the problem.
Wait—really? Are you saying that you interpret “speed” to mean exactly what I mean by “inverse speed”? Hours per mile instead of miles per hour? So that when someone says they’re going 5 miles per hour, you don’t think “they’ll have gone 5 miles after one hour” but rather “1/5 of an hour will have passed after they’ve gone one mile”? And if I wanted you to think “they’ll have gone 5 miles after one hour”, I would have to say “they’re going 1⁄5 hours per mile”?
That’s not what I got from your original comment at all. When you said
-- well, I didn’t really understand what you meant, but I thought you were talking about thinking in terms of a two-dimensional map with different routes to the same place having different distances.
I suppose in retrospect I can make sense of it by interpreting the phrase “specific destination” as referring to keeping the distance fixed (and letting time vary).
But I still find it hard to believe that’s what you mean—for one thing, it would imply that you shouldn’t have been confused by the original problem at all, and instead should be more confused by something like this:
“If you want to average 40 mph on a two-hour trip, and went 20 mph for the first hour, what should your speed be for the second hour?”
(If you find this easier than the other problem, then that means your idea of speed is like mine rather than being inverse to it.)
I don’t think any of those things. I wonder about how far they’re going at that speed, and if the answer is “up the block” I think “oh, they’ll be there soon” and if the answer is “to the moon” I think “that’s going to take forever”. I do not naturally think in numbers.
No, if you want me to think that they will have gone five miles after an hour, you tell me they’re going someplace five miles away and it’ll take them an hour to get there.
Well, I was, but this was incidental to my point.
Yes.
I was not confused in the way you were. I was confused in a different way, which has nothing to do with how I read the English word “speed” and everything to do with how my brain generates error messages when presented with math problems.
Well, this is also confusing (my brain generates “sixty” automatically, but I don’t actually know if that’s an answer to this problem or just the result of seeing “40″, “20”, and “average” in that order, and I would have to do work to find out). It is not differently confusing than the first problem. I don’t get any farther or stop any earlier before I want to seek assistance. (I don’t even know if this is the same problem or not.)
Well, neither do I (I naturally think in terms of operations and transformations), so that’s not the relevant distinction. The relevant distinction is between “wow, they’re already far away” (speed) vs. “wow, they got there quickly” (inverse speed).
Let me see if I can generate, in your mind, something analogous to the confusion that existed in my mind. I probably won’t succeed, but the idea of attempting is too interesting to resist.
Here are two questions that are easy to answer:
(1) If I travel for an hour and spend a lot of time on the first part of my journey, how much time will I spend on the second part? (Answer: not much.)
(2) If I travel a mile and go a large distance during the first part of my journey, how far will I have to go during the second part? (Answer: not very far)
And now here are two questions that are confusing:
(3) If I travel for an hour and go a large distance during the first part of the journey, how much time will I have to spend on the second part? (Answer: Huh? That depends on how much time you spent going that large distance during the first part.)
(4) If I travel a mile and spend a long time on the first part of the journey, how much distance will I have to cover on the second part? (Answer: Huh? That depends on how much distance you covered during that long time you spent on the first part.)