Here is a question that I have been fortunate enough to observe many students pondering over:

"A stone is dropped from a cliff. One second later, an identical stone is dropped from the same height. Describe the distance between the stones as they fall."

The question is a little bit interesting because it has this compelling wrong answer: The two ball's keep the same distance. I have seen this answer arise as simply an intuitive reaction, but I have also seen this intuitive reaction argued for in terms of school knowledge that students can call upon.

"Gravity acts the same on all objects.""Gravity acts the same and the initial position and velocity are the same"

"Both balls falls with the same speed of 9.8 m/s"

While each of these students has given the same answers, their thinking about the problem is different. The first student I might characterize as over-generalizing from a rule they've learned*. The second student I might characterize as conceptually equation hunting (e.g, "If I plug in all the same stuff, the number that comes out can't be different"). The third student I might characterize as not distinguishing acceleration from velocity.

While students' answers to the question don't tell me much, listening to them explain tells me quite a bit about how they are approaching the problem and what difficulties they might be having.

In general, the diversity of thinking is something that intrigues me. Partially because it is just plain interesting much in same the way that the diversity of life is interesting to biologists. But it also intrigues me because it places serious constraints on the possibility of "carefully sequenced instruction."

The diversity of thinking, of course, is not restricted to wrong thinking. Consider the three following arguments for the correct answer. I've seen each of these arguments in person.

Pulling Away Argument

After one second, the first stone will have picked up 10 m/s of speed. The second stone, however, hasn’t even started moving at all, at least not yet. So it’s speed is still 0 m/s. Since the first stone is moving and the second stone is only about to start moving (but not moving yet), the first stone will pull away from the first stone, making the distance between them greater. This trend of pulling away (and making the distance greater) will continue because the first stone will always be faster than the second stone by 10 m/s.

Shadowing Argument

We can think of the second stone as “shadowing” the first stone—it will be where the first stone was, exactly one second later. Since both the original stone and the shadow stone are falling faster and faster as they fall, the distance that each travels in that one second must be greater and greater as well. Thus the distance that the second stone covers (in order to “shadow” the first stone) becomes greater and greater, showing that the distance between them increases over time.

Parabola ArgumentThe position vs. time graph for free fall is a parabola. If you pick two points on a parabola that are one second apart near the origin, you will see that there will not be much separation on the y-axis. However, if you pick two points that are one second apart that are also far from the origin, there will be a big separation. This shows that distance increases as time increases.

Just as the wrong answer doesn't tell you much, knowing that a student gets the right answer doesn't tell you much. Each of the above arguments tells you something quite different about the student. The first student is thinking about acceleration as the accumulation of speed and about the consequences of relative velocities. The second student has invented a novel way of thinking about the problem in terms of speed-distance-time comparisons. The third student is bringing graphical tools to this problem in a successful way.

Like I said, I think the issue of diversity in thinking is interesting, but it is also important. So, what does the diversity of thinking imply? I'm going to be exploring this issue over the next week or so in blog posts. I hope to discuss its implications for assessment. It's implications for curriculum design. And it's implications for teacher preparation.

* Footnote: I've heard students use this argument to support all kinds of non-nonsensical answers. Students will say that a ball thrown straight down and a ball thrown straight out will hit the ground at the same time because gravity is the same. Students will also say that a ball thrown a mile into the air will take the same amount of time to reach the peak of its motion as a ball thrown an inch into the air because gravity acts the same. "Gravity acts the same" functions as a blanket statement to cover any problem where a student is asked to compare times. In fact, the only time these students get the question right is when they are asked, "If a bullet is shot from a rifle and another bullet is dropped at the same time from the same height, how will time compare for the two to hit the ground?"

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