In conceptual physics, I define acceleration with one sentence that we repeat ad nauseum:
Acceleration tells how much an object’s speed changes in one second.
Then we talk separately about the direction of acceleration:
When an object speeds up, its acceleration is in the direction of motion.
When an object slows down, its acceleration is opposite the direction of motion.
In Regents-level and Honors physics, I define acceleration via the slope of a velocity-time graph and via the equation a = Δv/Δt. The conceptual class used neither of these, yet seems to understand acceleration better than my Regents-level folks ever did.
In terms of the magnitude* of acceleration, since all of our problems involve constant acceleration, I ask them to use their mathematical instincts:
* Though I never use the term "magnitude"... I say the "amount of acceleration."
An eastward-moving roller coaster slows from 25 m/s to 15 m/s in 5 s. What is the amount of the roller coaster's acceleration?
Using the fundamental definition, student can reason: "Acceleration tells how much an object's speed changes in one second. The roller coaster changed speed by 10 m/s in 5 s. So every second, the coaster lost 2 m/s of speed. The acceleration is 2 m/s per second."
It helps that I have insisted on everyone writing the full relevant fact of physics in answer to every problem. When someone struggles, I ask him to repeat the definition of acceleration, and I can guide him to the correct answer. After doing this a few times, he gets the idea. And the concept is sticking... since we're not plugging blindly into an equation, I'm having fewer mistakes of the form of "the acceleration is 10 m/s because that's how fast the roller coaster moves."
Note the unusual statement of units. Rather than use the mathematical notation of meters per second squared, I'm exclusively writing acceleration units as m/s per second. When students have to write that out on every problem set, they continue the process of internalizing the meaning of acceleration.
As for the direction of acceleration: that's pretty easy for the class, given that we've practiced all year justifying from facts of physics. "When an object slows down, its acceleration is opposite the direction of motion. This coaster is moving east and slowing down, so its acceleration is west."
The only tricky part here is that I've had to stamp out the phrase "acceleration is moving west." Acceleration doesn't "move." Acceleration simply "is." My students initially complain when they lose points for saying the acceleration moves west. Then I show them a classmate's reply in which he says "This coaster is moving east and slowing down, so the roller coaster must be moving in the opposite direction of motion, so is moving west."* I explain that the language in the explanation is as important as the answer. (And, they know from experience that those points ain't comin' back, no one in the class has any sympathy for their loss of points, so they might as well just do things my way and get the physics right.)
* Not making this up.
The last fact I teach regarding acceleration is that object in free-fall gain or lose 10 m/s of speed every second. The next post will discuss quantitative and qualitative demonstrations relating to acceleration and free-fall.