In my last post, I started talking about what it means to have a good mental model of something. As you may recall, a mental model is an internal representation of an abstract concept, and a good mental model must be clear and unambiguous. It must be easy to work with. It must be a satisfying answer to the question “But what is it?”
I call models not satisfying the description above “fuzzy.” So what happens when someone has a fuzzy mental model? Let’s take a peek:
Student: Professor, can I use this rule?
Professor: Well, what do you think? Does that make sense?
Student: Hmmm. I’m not sure. How would I figure that out?
As you can see, the student has no faith that they can figure out what’s true or false. They have no confidence in their ability to reason. They think that their professor holds the keys to the mathematical rules.
A mental model is what gives you the ability to interact with a concept. If your initial mental model of the concept is fuzzy, then you never gain the ability to reason about it, since you don’t fully understand it. And if you don’t fully understand it, then all the rules you learn feel “handed down from above.” They can’t possibly make sense, because you can never check them against a robust internal representation. In the long run, this leads to a loss of mathematical confidence and an inability to tell the difference between correct and fallacious arguments.
A mental model is the foundation that you build your mathematical reasoning on. Without a clear model, any structure we put on top is shaky.
Sadly, we often neglect to mindfully form these models. And the reason why is both very understandable and yet very wrongheaded.
The example of multiplication will make this reason clear. In my last post, I criticized the definition of 3 ⨉ 5 as “either 3 5s or 5 3s” because it didn’t produce a clear mental representation. It is not at all obvious to a beginning student that those two quantities are identical. An ambiguous mental model is by definition fuzzy.
So how would I define multiplication to make the mental model clear and not fuzzy? Simple. I would define 3 ⨉ 5 as 3 5s. Period. End of story. Nothing more to see here.
I can immediately hear the objections to this definition, because I’ve heard them many times before. “But how are they supposed to actually multiply then? What, are they expected to calculate 100 ⨉ 2 by adding up a hundred 2s? No, thank you. I’ll stick to my ambiguous definition! Look how much easier it is to use!”
And to that I say: I see your point. It is absolutely true that choosing a clear initial mental model conflicts with the goal of having kids achieve quick computational proficiency. However, it produces much better results in the long run. Remember, a clear mental model is merely the foundation of a student’s engagement with the concept. Once the foundation is laid, computational proficiency can be built on top of it. And unlike with fuzzy models, this proficiency will endure.
In my next post, I’ll explain how to build on top of a clear mental model and how rigorously obeying these principles has turned out with my 8-year-old.