Acceleration as Simple but Vast Idea
So take the definition of average acceleration: a = Δv /Δt. Simple right? Well, yes, but not really. In order to really understand this definition, you’re going to have to explored a vast array of ideas, including ideas like
If an object is slowing down, its acceleration is in the opposite direction of its velocity
If an object is speeding up, its acceleration is in same direction as velocity
If an object is turning, there will be (a component of) acceleration in the direction of the turn.
It takes time to speed up or slow down. Some things speed up quickly and others speed up slowly and this has to do with an object’s acceleration.
Given a constant acceleration, speeds change linearly –an object gains (or loses) the same amount of speed in any equal interval of time.
A greater change in velocity in the same amount of time indicates a greater acceleration
The same change in velocity in less amount of time indicates a greater acceleration.
A greater acceleration will result in a greater change in speed in an equal amount of time.
A greater acceleration will result in an object taking less time to change its speed.
When an object is accelerating, the distances covered are not equal for equal times; objects cover more ground during times in which its moving faster and less when it’s moving slower.
Also, if you’re human, you’re going to have to become aware of and then wary of, a variety of other possible problematic ideas, like
Faster objects have more acceleration
A greater increase in speed means more acceleration
No velocity means no acceleration
Objects moving with same acceleration move in identical ways.
And we haven’t even begun to worry about having procedures for determining or estimating velocities at particular times or procedures for subtracting two velocities. We haven’t concerned ourselves with when it might be appropriate to model a situation with constant acceleration. We haven’t concerned ourselves with the difference between average and instantaneous velocity, or with strategies for selecting convenient intervals of time for carrying out one’s work. We haven’t talked about graphs and other representations. It turns out that acceleration is a high density idea.
Some Place Generative to Start:
Changing topics a little bit, one of the questions I have been thinking about is this: Given that there are so many ideas packed into definitions such as acceleration, which ideas are most generative? That is, which ideas serve as a good starting point for generating the entire set of ideas? Given such a good starting point, are there other ideas that come along for the ride? And I don’t mean logically generative–like you could derive certain ideas from others. I mean generative from a human learning perspective. What ideas serve as productive anchors or as productive leaping off points… So, now, I’m think, “Isn’t it odd to juxtapose the words anchor and leaping off point?” Like, one implies, “keeps you grounded somewhere.” The other implies “strong base from which to leave.” Those are totally different metaphors for generative starting place.
I also think about this a lot: Does the generative starting point need to be correct? or like a baby-version of correct? If I go with the anchor analogy, then yes, the generative starting point should be correct. It’s like “home base”–the place you are tethered too so you don’t get lost. It’s a trustworthy place to ground your thinking. But if I use the leap-pad analogy, then the most generative starting point can actually be a place you never return. It’s the place that launches you to the next place, which may be quite different, and possibly even wrong. I think we tend to operate under the tacit assumption that the starting points should be “anchors.” I think we have a hard time thinking about what a generative (but incorrect) launch pad would look like. I know I do. But still, I keep returning to the idea, because it has so many implications for how we might think about teaching, learning, and assessing progress.
Teach. Brian. Teach. 
Brian, you’ve really gotten me thinking about misconceptions over the last few months. When you have a class discuss how to approach something, often “mis”conceptions come out. Take, for example, your plan for teaching how the seasons work. Would you consider that a leap-off point? You’ve taught me that I make some dubious decisions in situations like that. Specifically, if I hear someone or lots of students saying wrong things, I try to “fix” their ideas too quickly without reaching down to the root of their understanding. If that’s a leap-off point, should you necessarily go back and point out how the initial ideas are wrong or just keep moving toward deeper understanding? -Andy
Yeah, I think the seasons is a good example of a leaping off point. You never intend students to return to some of their earlier ideas, at least as serious contenders for explanatory models. Now that I think of it, another good example is how the Physical Science and Everyday Thinking curriculum (PSET) builds a model of magnets. It starts in a place where a “charge separation” model of magnetism is constructed, only to later deconstruct that a model and build a new one.
I think often leaping off-points are almost always useful to talk about, even after you’ve left them, especially in retrospect, because they often give meaning to the newer models. Often the newer model only makes sense because of the limitations or faults of the older model. We do this in science all the time–we talk about earth-centered views of the solar system, we talk about Lemarckian evolution, we talk about the aether, etc.