The more I post on this site, the clearer it becomes to me that what I need is a “basics” series: a series in which I explain how I introduce the fundamental building blocks of elementary mathematics. Only in the context of these definitions will the work I do with my kids make sense.

What I focus on with my kids are the fundamental mental models that underpin much of elementary mathematics. These include place value, the four elementary arithmetic operations, and the concept of equality. I primarily focus on these concepts and the interactions between them, largely forgoing “extras” like telling time, using money, naming unusual shapes, and many other things that often gets stuffed into modern curricula. In my opinion, these topics are best encountered informally “in the wild.” As a child becomes more proficient with the basic models of arithmetic, they will be able to apply arithmetic to a wider range of questions. However, in my opinion, including these topics in the curriculum wastes valuable time best spend on the fundamentals.

Another early concept I spend a lot of time on is the concept of a variable. Algebra is a generalization of arithmetic: it is arithmetic done with an arbitrary number instead of a specific number. The right time to start generalizing one’s observations about numbers is when one is working with numbers in a hands-on way. And similarly, the right time to build a robust mental model of a variable is when one is doing arithmetic.

When taken together, these basics form a large part of our elementary mathematics program. I hope this series helps explain how we approach our journey!

This is the introduction to a series of posts about our basic approach. Here’s are my first and second posts about place value. Here’s a post about arithmetic operations. I will soon also add posts about equality, variables, and a couple of detailed examples of our daily lessons.

Time for another puzzle! This one is an equation grid and it’s relatively straightforward for a child who has a good understanding of place value, the arithmetic operations, and equality. Note that a child does NOT need to be able to actually subtract 27 from 25 to be able to check whether 2 = 25﹣27. They merely need to be able to think about what happens if you try to take away 27 from 25.

Here’ the puzzle:

Click on the image to get a copy of a pdf of the puzzle.

And here’s a solution to the puzzle:

Ready for a paradox? I present you with the following two true statements about place value.

Many kids find place value challenging and don’t learn it well if at all.

Place value is a concept with an extremely simple mental model. It is an idea that gets absorbed very well with continued exposure.

How can those two statements be true at the same time? The issue is that most adults are so familiar with place value that they cannot separate the mental model from its many consequences. Therefore, they teach kids the consequences, expecting that as a result kids will grasp the concept. Unfortunately, that is not an effective method of teaching: it will neither work for all kids nor will be efficient with the subset of kids who are able to learn like this.

So what is the straightforward mental model I’m referring to? Restricting our attention to integers for simplicity’s sake, here it is:

To figure out a quantity represented by a given integer, all we need to know is that each digit represents how many copies of a particular quantity our number contains, where each such quantity is ten times greater than the quantity represented by the digit to its right. The rightmost digit represents how many ones our number contains.

To put it in more familiar language, the rightmost digit of a number represents how many ones the number contains, the next digit to the left represents how many tens it contains, the next digit to the left represents how many hundreds (which is equal to ten copies of ten) it contains, and so on, so forth.

“That’s it?” you say. “Everyone knows that. Kids know that starting from Grade 1 or 2. They learn about ones, tens, and hundreds, and that’s all this is saying.”

Actually, as it turns out, kids are largely uncomfortable with place value, and so are many adults. They are comfortable with many of the consequences of place value, such that as the fact that when we start counting, the sequence begins with

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, …

and the fact that adding a 10 to a number between 10 and 89 simply increases the first digit by 1, and the fact that multiplying an integer by 10 tacks a 0 onto the end of the integer, but they are uncomfortable with the mental model itself. This becomes clear when you try to teach them a new application of this model. Suddenly, kids who seemed to be totally fluent with place value when using it to add can’t figure out how to use it to subtract. Or if they can subtract, they can’t figure out how to use the model to multiply. And don’t even get me started on dividing.

What’s going on? Unsurprisingly, it’s the usual issue. Instead of being presented with a single, clear, coherent model, kids are bombarded with a wide variety of tools, tips and techniques. If a child has a naturally good memory, they can use all these techniques to get right answers. If they aren’t as good at this, then they can’t. Either way, they are getting no practice with the model itself.

So what can we do? We can present them with tools that let them explore the immediate consequences of the mental model itself. Once the model is robust, we can build on top of this model. When approached in this way, all the shortcuts and algorithms are easy to remember because they make intuitive sense.

Of course, as usual, absorbing a mental model takes a surprising amount of time! In my next post, I’ll explain how I introduce place value, and in later posts, I will explain how to use it in a way that builds facility with the model.

This is a post in a series about our basic approach. Here’s the introduction to the series. Here’s my second post about place value. Here’s a post about arithmetic operations. I will soon also add posts about equality and variables, as well as a couple of detailed examples of our daily lessons.

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.

This is the second in a series of posts about mental models. Here are the first and third entries in the series.

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!

This is the first in a series of posts about mental models. Here are the second and third entries in the series.