I find it interesting that when you talk to a user of a given technology, there is a mental model for how the technology operates. I read an observation from somebody just the other day, about the old Polaroid cameras. People would take a picture, it would come out, and then people would wave it, and perhaps blow on it, until the picture became clear. In their mental model of how the technology worked, the waving of the picture was a natural step. Few people ever found out that without waving, and without blowing, the picture would appear just as quickly.
Mental models are important. It is not a matter of whether you use one, because you do. How you interact with a technology is a function of your mental model of that technology. Good technology allows for very simple mental models that are completely adequate for interacting with it. If you understand someone’s mental model for a piece of technology you can predict how they interact with it.
I’ve made lots of observations of children using calculators (adults, too!) and have noticed several common mental models for what’s happening. Some of these models are completely dysfunctional (the calculator tells me that 5 + 3 is 53, and therefore it must be correct!), but they aren’t as common as you might think. Many of the children’s mental models seem completely appropriate and are shared by engineers the world over. To characterize it roughly, it is a mental model in which numbers exist as decimal expansions, and any of the operations + – x /are equally quick.
It is not unusual for a math teacher in seventh grade to tell his students that 6 / 10 is more simply written as 3 / 5, and then not appreciate that for the children, this doesn’t match any part of their experience. With a calculator view of the world, both 6 / 10 and 3 / 5 immediately ‘reduce’ to .6 and that is pretty much the end of it. And for most kids, the .6 is more visible from 6 / 10 than it is from 3 / 5, so what in the world does the teacher mean when he says 3 /5 is simpler? Well, of course, the teacher is referring to another mental model, one that is typically not shared by the kids.