Mental models

We had a department meeting last week in which we briefly discussed one of my list-item goals: to explicitly teach mental models instead of just procedures.

Some teachers wanted clarification about how I thought of a mental model. The short answers was: a mental model is what a concept means, instead of a procedure to get a certain kind of answer. This can be tricky because often students are looking for "the steps to get the answer"--they're not attuned to what something means. It's also common to accidentally "proceduralize" a question--that is, take a question that required applying one's understanding of the meaning of a concept, and instead turn it into a question that students can solve by following a procedure they've memorized. Typically this happens by giving the same type of problem repeatedly, allowing the students to notice the steps you follow to solve it. Then they can follow those same steps without really understanding where they came from or why they make sense.

Here's an example of this that occurred to me while writing questions for my final.

The Topic: What does the "end behavior" of a function refer to? Specifically, what does it mean to say (for example): as x->infinity, y-> 0; as x-> -infinity, y->0 ?

The original question: The target sort of question is to give students an equation, and have them describe its end-behavior. For example: Describe the end-behavior of f(x) = 1/x

I discovered that I had inadvertantly proceduralized this kind of question for students. The procedure most of them know at this point is:

1). Plug in a very large x-value (or a few).
2). If it's large then write: "As x->infinity, y-> infinity". If it's close to 0 write: "As x->infinity, y->0", etc.
3). Plug in a large negative value for x, etc.

This process usually yields correct answers to questions like the target.

Unfortunately, many of those same students were stumped when I asked the following:

Say that the function f(x) has the following end-behavior: As x->infinity, y -> 0. For each point, explain whether or not you think it's likely to be on the graph of f(x). [if you don't have enough information to know for sure, clearly explain why]

a). (0, 2) b). (99999, 0.9999) c). (9999, 9999) d). (9999, -0.00001)

It wasn't that students gave wrong answers (although many did). Many students didn't even know where to start. This is a clear indication that they don't really understand the /idea/ of end-behavior.

A similar phenomenon occured when I asked them the following:

Say the function f(x) has a horizontal asymptote at y = 3. What is it's end-behavior?

Many students didn't know where to start, because they had memorized procedures to /find/ the horizontal asymptote, without ever really thinking about what it means to be a horizontal asymptote, and how that idea is related to the idea of end-behavior.

So, to get back to mental models, what would the mental model(s) be for end-behavior? I think it's hard to capture what someone's mental model is exactly; it's whatever they use to think about what a question means.

When I see: "as x -> infinity, y -> 0". I always say out loud in my head "as the x-values get bigger and bigger, the y values get closer and closer to 0"

I also always picture a graph in my head something like this...

As the x-values of my points go to the right, the y-values of my points will get closer to 0 (the x-axis). In other words, as I keep plotting points, their height (the red dotted lines) will get shorter and shorter. (Actually I imagine an animation with the dots appearing from left to right, as the x-values get bigger)

Since the idea of end-behavior is a slight formalization of a very natural question to ask about a graph, I think a good way to introduce it would be to have students do their best to answer the natural question as best they can.

It could be a "describe the graph over the telephone"-type activity. Students could have a series of graphs, and their goal is to write a short (i.e. 1-2 sentence) statement explaining what the the trend at the "ends" of the graph are, so that someone else could re-create it only from the description. After discussing student descriptions, you could unveil the sort of description mathematicians have agreed upon in a context that makes sense.

This would also be a good lesson for a party game I like. Take a vertical strip of paper and fold it into (say) 4 sections. The first person in the group draws a graph. The second person in the group looks at the graph and describes its end-behavior. The paper is folded so the 3rd group member can only see the 2nd member's writing. The 3rd member has to draw a graph that matches the end-behavior described by the 2nd member, and so on. It's like "telephone" using paper, and switching back and forth between representations. At the end, the group checks to see if they ended with the same kind of end-behavior they started with.

To emphasize to students that they should be learning what the concept means, it would be good to give them assignments that require them to express this understanding directly--not use it to get an answer to another sort of problem.

Since the concept of end-behavior has a natural interpretation in terms of a progression, I think a nice assignment would be to have students create a short multi-panel comic strip explaining or illustrating the idea of end-behavior for someone in a younger grade (like 9th).

In my experience, some students won't produce anything useful, but many students will produce clearer explanations than anything I was likely to give; so re-distributing some of the better comic strips as "notes" would be a good follow-up.