The Basics

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.

Math Puzzle Monday!

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:

The Operations

In this post, I’m going to give a brief description of how I introduce each arithmetic operation. Due to my focus on mental models, I give a specific definition for each operation and then give the children a lot of time to explore them. This exploration is done in the context of place value.

I introduce the operations in the following order: addition, subtraction, multiplication, division. I introduce each new operation as soon as the definition and symbol of the previous operation have been internalized. I do not wait for math facts to be memorized or for secondary meanings to be learned. The following are my definitions:

Addition: I’ve talked about this operation briefly in my series about counting on. I introduce addition as “putting together.” In my experience, most children have some experience with this operation from daily life and can think about questions such as “What do we get if put 2 cookies and another 2 cookies together?”

Subtraction: I introduce subtraction purely “taking away.” This means that I do not treat it as obvious that subtraction also tells us how much bigger one number is than another. An expression like 5﹣3 means no more and no less than “what we get if we take 3 away from 5.”

The benefit of this definition is that this provides a robust mental model that makes it clear that subtraction is not commutative: that is, that 3﹣5 is not the same thing as 5﹣3. I’ve met many children without a clear mental model who do not distinguish between the two.

Multiplication: I introduce multiplication as “taking copies.” Specifically, I say that the first number tells us how many copies to take, and the second number tells us what number we’re taking copies of. For example, 3⨉5 is defined as 3 copies of 5 and 5⨉6 is defined as 5 copies of 6.

I do not treat it as given that multiplication is commutative from the beginning. Therefore, for my students 3⨉5 means 3 copies of 5 and does not mean 5 copies of 3, which would be written as 5⨉3.

Division: I introduce division as “splitting into a certain number of groups,” where the first number tells us how many things we are splitting, and the second number tells us how many groups we are splitting into. For example, 12 ÷ 3 is defined as the answer to “What do we get if we split 12 things between 3 people?”

As usual, I do not assume other meanings of the symbol from the beginning. For example, I do not treat it as given that 12 ÷ 3 also tells us the number of groups of 3 that we could split 12 into. In my experience, the fact that the answer to those two questions is the same is not at all obvious to most kids.

Some of these definitions require minor tweaking as we move into fractions, but overall, these mental models stay constant for my students throughout elementary math. This allows them to gain comfort with using them and results in excellent intuition about the operations and the relationships between them.

As I walk you through our lessons, you will see that we eventually learn all the standard properties of the operations that kids are often taught to take for granted. However, this does not happens early on as part of the definition! It happens organically as they gain fluency and as their mental models get more robust.

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

Place Value

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.

How to Use Clear Mental Models

In my last post, I discussed the pitfalls of fuzzy mental models﹣mental models that do not provide a clear, unambiguous picture of what operations and symbols mean. But getting a clear mental model is only the beginning of the story.

Let’s come back to our example from last time, multiplication. Say you decided to give a simple, unambiguous definition: 3 ⨉ 5 is equal to 3 5s, 6 ⨉ 7 is equal to 6 7s, and so on, so forth. What happens next? How do you move from a clear mental model to actually being able to multiply with fluidity?

The first thing you’ll notice when you start being mindful of mental models is that they take surprisingly long to absorb. It may take a child a few weeks to become comfortable with a new definition. Even after the definition can be fluidly recited, you’ll see that some “obvious” things may not actually seem obvious to your child. For example, anyone who’s used to multiplying knows that 1 ⨉ 5 is 5, and that in general, multiplying a number by 1 results in that same number. However, this may very well not be immediately obvious to your child! That’s because it can take a while to get comfortable with the idea of “putting together copies of a number.” At this stage, any kind of gentle practice will help the mental model become fully integrated and absorbed.

After the model is at your child’s fingertips, it’s time to start making observations using the model. Some observations turn out to be a lot closer to the surface than others, and they should be worked on first. However, it’s imperative to keep the mental model in mind when deciding what’s obvious and what’s not. For instance, the fact that

3 ⨉ 5 + 6 ⨉ 5 = 9 ⨉ 5

turns out to be much more obvious than the fact that 3 ⨉ 5 = 5 ⨉ 3! That’s because 3 copies of 5 and another 6 copies of 5 clearly make 9 copies of 5, whereas the fact that 3 copies of 5 is the same as 5 copies of 3 requires something else: the realization that an array shows a multiplication. The benefit of having an explicit mental model is that it becomes clearer what things should and what things should not be apparent to your child.

As you and your child make observations, it’s important to constantly refer back to the mental model. People already fluent with a model have a tendency to think that once a student has seen a definition a few times, the student understands it and is comfortable with it. Nothing could be further from the truth. Mental models take a long time to be internalized, and it takes even longer even for all the “obvious” observations to be made. And only after the “obvious” observations are made are you ready for the more sophisticated observations that make calculations easier.

You may well wonder whether this means that progress using mental models is very slow. Won’t a student take too long to learn to learn elementary mathematics if every single observation needs to be built on top of other observations, and if you constantly need to refer back to the original mental models?

The short answer is “no.” The longer answer is that the sequence of attained skills using these methods is very different. At the beginning, you sacrifice facility with calculation for conceptual fluency. However, in the long term, the student gains robust knowledge that is retained well and easily applied. While this starts out slowly, your student will eventually speed up and surpass learners who can calculate quickly immediately but whose mental models are fuzzy.

As a small case study, I have applied these methods with my 8-year-old since she was about 4.5. In the first few years, she was neither ahead nor behind: she had started on all four operations young and she had early exposure to algebraic concepts, but she had not spent time on the standard algorithms and she was slow on many of the standard 1st grade skills. We spent our math lessons making sure her mental models were well-integrated and making observations about how the various concepts we were working with related to each other.

The further we moved into mathematics, the more rewards we’ve reaped from her clear mental models. When we finally covered the standard algorithms, they each took about a day to learn, due to the clear models of the operations and place value. When we covered fractions, the time we had spent developing models of division made things much simpler. Despite our “slow” start, she’s currently working on high school topics such as algebra and proof-based geometry. Furthermore, she is very confident in her mathematics and she is able to write down clear explanations for everything she uses.

Of course, not every child will wind up accelerated through these approaches ﹣ acceleration in mathematics is a mix of both nature and nurture. However, every child will reap the benefits of the confidence and logical thinking that clear mental models bring. Clear mental models make math make sense.

The next question, of course, is a practical one. How does one use this idea to actually teach one’s child? In the coming weeks, I’ll provide lots of examples of the (deceptively simple) lessons I give to my children and other children I teach, and I will explain how they fit into this framework. This should make the ideas clearer. Stay tuned!

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

What’s Wrong With Fuzzy Mental Models?

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.

What Is a Mental Model, Anyway?

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.

Is “Counting On” Obvious?

I started teaching my older girl when she was 4.5. At this point, she could count, subitize, and show numbers on her fingers and that was about all. We started slowly and gently by introducing addition. And as soon as we got started, I got my first inkling of the fact that obvious things were not in fact so obvious. Since my goal was to make sure that my kids didn’t use any shortcuts that they didn’t thoroughly understand, this was a dilemma.

Let me illustrate. Do you know what 9+1 is? If you’re reading a math blog, you almost certainly know that it’s 10. Furthermore, you know that to add 1 to any number, you can simply take the next number in the counting sequence: 65+1 is 66, 118+1 is 119, and so on. Similarly, to add 3 to a number, we can count 1, 2, 3 starting from the number following our number. In fact, this works in general: to add something to a number, we can start at the number following our original number and count up to that something. This method of addition is called “counting on.”

Why does this work?

This is the kind of question that tends to stump people. For most of us, that fact is either completely intuitively obvious or memorized by rote. There is no “why.” It just is.

Except that if you’ve tried to teach this to young kids, you’ve probably noticed that they do not feel this way. If you ask little kids to put together 9 and 1, kids tend to start counting at 1. Assuming they don’t lose track, they’ll say the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 before they arrive at the conclusion that the answer is 10. They will not use the shortcut. It is in no way obvious to them.

That brings us to the next question. Since it’s not obvious to a little kid that 9 and 1 together make 10, what do we do? Most answers to this question fall into the following two categories:

(a) Teach a kid directly that this works. Instruct a kid that a number plus 1 is the next number and then practice, practice, practice.


(b) Let a kid discover this fact through experience. A typically developing child will eventually figure out that if you put a number and 1 together, you get the next number. After all, kids often have to figure out what happens if you take one more of a thing…

Out of those two options, I prefer (b). That’s the option that doesn’t rob kids of their innate numerical sense and instead replace it with “it’s true because the teacher told me so.” Unfortunately, option (b) isn’t very efficient, and it also tends to break down when a child is faced with concepts that aren’t often encountered in the wild. When that happens, even teachers who started out preferring option (b) tend to default to option (a). First, we’ll teach it, then we’ll cross our fingers that it’ll eventually make sense!

My preferred option is neither (a) or (b): it involves working with a child’s mental model of the objects and operations to derive each new step. Like option (a), it is directed and efficient. Like option (b), it makes deep sense to the child and leaves them with their mathematical confidence intact. It is, in fact, the best of both worlds.

This is the first in a series of posts about counting on. Here are the second, third, and fourth entries in the series. These lay out precisely how I teach counting on.

How It All Began

I’ve been teaching my kids math since before I became a homeschooler. I had never planned to pull my kids out of school (ah, the best laid plans of mice and men…), but I always figured I’d need to supplement their educations, and accordingly, I started my 8-year-old’s math lessons when she was about 4.5 and attending preschool.

As I had taught many products of our educational system over the years, I had deep knowledge of what could go wrong with their mathematical education. The main issue I’d seen over the years is something I like to call “symbol shuffling.” When they were doing math, they weren’t thinking logically: instead, they had a poorly memorized set of rules, and they tried to apply them. This would lead to equations like (﹣4)+(﹣5) = 9, because, after all, “two negatives make a positive.”

The problem, as I saw it, was that a lot of the mathematical symbols weren’t producing mental images of anything in particular. They were simply symbols on a page, and they were to be manipulated accordingly. The rules were rules not because they were logical inevitabilities: they were rules because the teacher had said so. In the context of that mindset, (﹣4)+(﹣5) = 9 makes just as much sense as (﹣4)+(﹣5) =﹣9. After all, how can something make sense when the symbols don’t actually refer to anything?

I was resolved to do better with my kids. So when my 8-year-old was 4.5, we started some very gentle math lessons, starting with simple addition and subtraction. My one overriding concern was that she needed to understand everything she used﹣I’d seen the outcome of the opposite approach far too many times. So I was determined to teach her to true conceptual mastery, and to let everything take as long as it needed to…

As you’ll see if you keep reading my blog, this worked far better than even I expected. Stay tuned!