In my last few posts, I’ve spent a lot of time talking about how to work in the context of a child’s natural mental model. But what is a mental model?
To put it simply, a mental model is an internal representation of an operation or a concept. A mental model allows us to attach meanings to symbols. It’s what helps us makes sense of abstractions. It’s our answer to the question “What exactly is it?”
And that’s the crux of the matter. To construct a mental model, our brain needs to spend time grappling with what something is. Unfortunately, when we usually teach math, we don’t spend much time worrying about what things actually are. Instead, we teach kids how to calculate them and when to use them. Do those sound the same to you? They aren’t. And yet the distinction is surprisingly tricky.
Let’s take as an example the operation of addition. This is the arithmetic operation that a solid majority of people have a robust mental model for. If I asked you what it meant to “add” two numbers, you would probably be able to explain it to me in a variety of ways. You might tell me that we’re putting two numbers together. You might show me a picture of combining two groups. If I’m confused, you might show what addition means to me with two piles of counters. You understand what addition is. It makes sense to you. In other words, you have a solid mental model for this operation.
So far, so good. But now let’s think about another operation: the operation of multiplication. What does an expression like 3 ⨉ 5 mean? What exactly is it?
Some adults I’ve asked this question can’t answer this question at all except to say “15.” They clearly have no mental model for multiplication. However, the most common answer is to say “It’s 5 3s or 3 5s, whichever you prefer.” Is there something wrong with that answer? Surprisingly, there often is.
This is a fine answer as long as the explanation of why 5 3s and 3 5s are the same is at the tip of your tongue, and as long as you could quickly give me the same explanation for why 34 167s is equal to 167 34s. (In other words, your explanation shouldn’t be predicated on calculating 5 3s and 3 5s and noticing that they are both 15: it should work for any pair of numbers.) But if you’re sure that 34 167s and 167 34s are equal and yet you aren’t quite sure why, then it’s possible that your mental model of multiplication is underdeveloped. Or as I like to say, your mental model is fuzzy.
Why is that? Well, because the answer “3 ⨉ 5 is 3 5s or 5 3s, whichever you prefer” is actually pretty unsatisfying! As you’ll soon see if you ask a young child about it, it’s not obvious that those are the same. If you tell a child to figure out what 3 5s make, they’ll add 5 and 5 and 5 together. If you tell a child to figure out what 5 3s make, they’ll add 3 and 3 and 3 and 3 and 3 together. Those aren’t the same procedure. It’s not at all clear why those should yield the same answer.
If you’re sure that those are always the same but aren’t quite sure why, then like many children, you’ve probably spent a lot of time in school calculating answers and not much time figuring out what you were actually calculating. You didn’t spend much time developing your mental models. You’ve never had to think that much about the question “But what exactly is it?”
A good mental model should be sharp and unambiguous. If stated as a definition, it should be clear why it really is a definition and not something that may give multiple answers. It shouldn’t make you frantically search your brain for which interpretation is the correct one. It should give you an unassailable sense of what something really is. As you can see, the above model of multiplication fails those tests.
What’s the problem with fuzzy mental models? After all, whether you take 5 3s or 3 5s, you’ll get 15. And that’s regardless of whether you know why the answer is the same. So what’s the issue?
As a short answer, building one’s mathematics on fuzzy mental models is like building a house on quicksand. If you don’t have a sharp internal representation of a concept, it is difficult to reason about it. Kids without well-developed mental models usually have to get by with applying rules that are true “because the teacher said so.” Fuzzy mental models lead to symbol shuffling and eventually to a lack of mathematical confidence.
A longer answer, however, will require another post. If you’re interested, come take a look!