I’ve been reading a lot lately about the idea of a “modeling curriculum.” Not as in America’s Next Top Model and also not as in the teacher models the thinking and the student learns from watching and trying it themselves.  A modeling approach to teaching science and math means that the students work together to develop better and better conceptual models to explain situations. So in physics, you might roll two objects down a ramp and try to make a mathematical model to describe what was going on. At first you might include the weight of the balls in your model, but then you might observe that two objects with different weights behave the same, and so your model would change based on new data and new understanding.

Some of the studies of this kind of teaching show us that students come into situations with models already in their heads — they already have ideas about how balls fall, for example. Their models might not be the most accurate or easiest to use, and so as they encounter new situations and new demands, they change their models. While that’s happening, students might use lots of different competing ideas at once. One minute the same kid will go from making really accurate predictions about two balls of different weights rolling down an incline, but then say that gravity will make a bowling ball fall faster than a beach ball.

This week’s AlgPoW, Filling Glasses, asked students to match graphs of water level vs. time of glasses being filled at a steady rate, to pictures of the glasses. Students used many different models for thinking about the problem:

• Try to match the shape of the graph to the shape of the glass (e.g. count the wavy parts, look for straight graphs for straight glasses).
• Relate the skinniness of the glass to speed.
• Relate the skinniness of the glass to the steepness of the graph.
• Relate the height of the glass as a whole to the maximum height reached in the graph.
• Relate the skinniness of the graph to speed at which the glass fills and the speed at which the glass fills to the steepness of the graph.

What was most interesting, though, was the students who used different strategies at different moments. Students who are in the middle of learning often switch models based on small details or when a problem seems easier or harder for some reason.

Like this:

For this problem, you have to really visualize the glasses and their shape.

First, I looked at glass A. It starts out skinny for a tiny bit, then there is a huge bulge before it is a little skinnier. So the height would rise quickly for the shortest amount of time, then go slower, then finally go a little faster. I visualized the graph to be a slightly zigzaggy line that was not too tall. Graph 4 did not have any zigzags, and graphs 2 and 3 went too high. So, graph 1 matched with glass A.

Glass B is like a funnel, starting skinny and getting wider and wider as the top draws nearer. So the height would rise quickly at first and get slower and slower. Since there are no bulges in glass 2, the graph it matched up to would have to be zigzag-free. And the only graph without zigzags is graph 4.

Finally, glass C starts skinny, gets wider, gets skinnier, and then gets wider. The water will go fast at first, then slower, then faster, then slower. Graphs 2 and 3 are very similar, but only graph 3 starts out fast.

The student sometimes is looking for zig-zags, basically matching the shape of the glass to the shape of the graph. But in the case where there ar
e two possible zig-zag graphs that could match one glass, the student switches to a (more robust?) model of thinking about the width -> speed relationship (and maybe implying a speed -> steepness relationship?).
Or this:
Glass A= First of all glass A is the shortest so the line on the graph would be less steep. Also, since the glass is kind of round, at first the water would pour fast then gradually pour slower then after you get to the middle the water would gradually pour faster.
The thinking about how steepness relates to the shortness of the glass seems like a very different way of thinking about steepness than the speed idea that she uses after “Also,…”
Or finally:
i figured this out becauause if you look at the glasses and the graphs. the arches in the graphs are like the glasses when get bigger because you need to have more water and then it would fill it up.
There’s the shape kind of thinking there: “the arches in the graphs are like the glasses” but also some idea of the change in width of the glass affecting how it fills up.

Some “Filling Glasses” links in case you are interested: