## Tuesday, April 19, 2011

### What Does That Graph Tell You? (Part I)

It's a bit of a dirty secret: students can plot points, find different trendlines on their calculators, label axes (if you harangue them) and produce graphs.

Wait, that's not the secret part.  This is the secret part:

Many students don't know what their graphs mean!

This isn't just a problem of graph interpretation (a post for another day?), but it's a serious problem for graph construction.  If you don't know what you're saying, how can you say it?
The more fundamental issue, IMHO, is that many students don't even think that graphs and diagrams and equations actually say anything!

What other explanation can there be for the number of silly graphs, diagrams, and equations that all of us teacher types see?  ...did you really mean that the car is speeding up (from your acceleration and velocity vectors) and slowing down (from your values of velocity)?  ...did you really mean that the object accelerates (because you drew a diagram showing unbalanced forces) and that it doesn't accelerate (because your Fnet equation = 0)?  ...did you really mean that the gravitational force increases (like your graph does, uniformly) as you move away from the Earth's surface?
In most cases, I argue that the students don't really mean most of these things, but instead that they're not attributing meaning to the graphs, diagrams, and equations that we get them to draw.  Why do we harp on them about these?

Graphs, diagrams, and equations are the most important and useful tools for constructing understanding, not just for communicating it.

Teachers and scientists know this, because we know how to use them, but it has to be part of the process to not only make the students draw these things, but to try to get them in situations where they have to use these in conceptually deep ways.  They don't see the value until they... see the value!  It's easy to give this short shrift, but it's vitally important not to.  Until then, they're checking off boxes - going through the motions.

We're working with optics in physics now, so we warmed up with a couple of easy graphs: magnification (M) vs. object distance (d_o) and image distance (d_i) vs. object distance (d_o) for planar mirrors.  We've ray traced and shortcut, looked at images and fields of view, figured out the necessary length for a full-length mirror, and this is a bit of a review (we've been doing convex mirrors for a couple of days) and a wrap-up in disguise.

Together, on the board (not always the best approach, but we were a little pressed for time this day), we figured out how to translate these conceptual sentences into graphs:

"The image is always the same size as the object."

There are actually a few places to trip up here.  There's hidden reasoning that you can't gloss over, if you want to really have the tools to tackle harder situations.

The image is always the same size as the object.
The magnification (the ratio of image height to object height) is always 1.
For every value on the horizontal axis (object distance), the vertical axis (magnification) has a value of 1.
The M vs. d_o graph is a horizontal line.

Let's try a bit harder one:

"The image is always the same distance behind the mirror as the object is in front of it."

There's a bit more reasoning to do here:

The image is always the same distance behind the mirror as the object is behind it.
The image distance is the same "number" as the object distance.
The image distance is the opposite (times -1) of the object distance, since the images are behind the mirror.
For every horizontal (d_o) value, the vertical value (d_i) is the same value, but negative.
If the object's right up against the mirror, so is the image, so the point (0,0) is part of the graph.
The d_i vs. d_o graph is a downward sloping line (slope -1) with an intercept at the origin.

Are these really "simple" graphs? Well, if you think about all of the (mostly unstated) reasoning that goes into them, you start to see why they can be difficult for students.  Remember that most students don't have a real understanding of what the slope, vertical or horizontal intercept, or vertical or horizontal asymptote really mean!  They can draw them and point to them on the graph, but they walk into physics without much of an idea at all of what they actually tell you, much less how to translate some physical conceptual understanding into a graph by using those features of the graph - we (and they) have to put some time and effort into figuring out how these are just different representations of the same information.  Once they've done that, it opens up huge new worlds and gives them tremendous reasoning power that they don't even know exist.