Average velocity is another one of those basic calculus (really pre-calculus) topics that, like difference quotients, leave me at a loss for why students have such a hard time with them. There’s a very simple and common-sense definition, namely that the average velocity of an object with position s(t) from t = a to t = b is
(See? It’s just distance = rate * time solved for “rate”.) There are examples in the book and examples on the internet ad infinitum of how to calculate average velocities, and all of these are simple numerical calculations with absolutely no algebra involved. You have to know how to plug numbers into a function and then do basic arithmetic on your calculator. That’s all.
But students get so turned around. They calculate only the position at time t=b. They add up the positions at t=a and t=b and divide by 2 (“average”). They add in the numerator or denominator (or both). They get the fraction upside-down. And so on. Not all students of course, but many of them — a lot more of them than there should be. And in my calculus classes, it’s certainly not for lack of training data; we’ve done it in lecture, in group activities, in online videos, you name it.
With difference quotients, I can sort of understand where the difficulties might come from — it’s the algebra. But there’s no algebra at all in an average velocity calculation, and even if you struggle to get the concept, can’t you just memorize the formula for the time being? I try always to see student difficulties from the student’s point of view and remember that I was in their shoes once too, but honestly, I am finding it really hard to know where such a consistent mass misunderstanding of this particular idea comes from.
What’s with this topic? Anyone?
9 responses to “Average velocity”
We’ve been working on this topic for the past few days in my senior classes. We began by working from graphs of velocity vs. time (with constant acceleration).
We’ve done it by two different methods – finding distance traveled by the area under the “curve” and by averaging the endpoints.
A few of them still need to draw the graph every time when given the data – but they’re getting it. I think the graphical approach is helping.
My students too had difficulty with this last year. The thing that seemed like it messed them up the most is actually interpreting a s(t) graph, a v(t) graph, and an a(t) graph. They shut down once they saw those graphs; it didn’t matter that it was simple algebra, etc.
Once we finally got past that paralysis (not entirely successfully, mind you), average velocity became less difficult for them to comprehend. And the formula became easier for them to grasp and use.
But yeah, I feel ya.
The thing is, I’m not going for necessarily a deep grasp of the concept of average velocity here — all I’m asking is for people to use the formula correctly. Surely we don’t need to spend multiple days on that, right? Also, this is a college calculus class and we have exactly one day to think about average velocity, then it’s on to limits.
From my days as a student and as a grad TA, I seem to remember average velocity being instinctive when I was measuring a car travelling from point A to point B. (Or point B to point C, when I knew I had started at A…)
But when I met the same problem on a graph of a function, I had to work to remind myself that it was the same operation.
“and even if you struggle to get the concept, can’t you just memorize the formula for the time being?”
My bet is that to many students have done so with to many things prior to that you get them. They don’t really understand division and multiplication, they have memorized when and how to use what in hundreds of different cases but haven’t learned how to think about what it means. Some have learned to calculate v=(s2-s1)/(t2-t1) and then you come along and use a function to calculate the distance at different times. Even if they can plug number and calculate s(b) doesn’t meant that they understand that they just calculated a position, your formula is a totally new thing to memorize and I would guess that all the formulas they have in their head start to look very similar and hard to keep apart.
The hard part of being a good math teacher is to make the student think instead of memorize, and the more “pre memorization” my students come with that hard it is to change them since you need to go back at work at the level they really understand…and they hate you for it until they start to see the difference.
I think you mentioned one possible answer in your post: calculators. Today’s students are far too dependent on the damn things. That’s why I disallow calculator use on exams in my calculus courses.
I disagree. I don’t think that calculators are the problem with students not being able to think. It’s the questions we ask (or don’t ask) that has led them to “doing” instead of “thinking”.
I can definitely accept the hypothesis that in some cases, students become so dependent on calculators that their abilities to reason mathematically suffer. But in this instance calculators are definitely not the problem.
Calculating an average velocity — just calculating, not understanding — is a simple of matter of loading numbers into a formula and doing the arithmetic. An unhealthy dependency on technology doesn’t explain the kind of fundamental errors I’m describing here, which seem to be something neither computational nor conceptual in nature. I mean, what level of calculator dependency makes you think that you get average velocity by adding up two positions and then dividing by two? Even without calculators in the picture, it’s a simple matter of memorization of an exceedingly simple formula and then just plugging and chugging, and that’s the very thing that’s not happening.
To be clear, too, I do expect students to understand average velocity and not just compute it. But for many of them the wisest approach is to have them memorize first and let the understanding come with time (which we don’t have a lot of) and practice.
Just a comment. I learned about average and instantaneous velocity in physics. The concepts were not easy, and I am glad (in retrospect) that a science teacher was in front of the room.
It was also nice that we were doing that at about the same time we were playing with m(tan), and sliding the two points closer and closer and closer in math class.
I honestly believe that conflating the application and the abstraction is damned confusing.