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.

Lesson Plan: Introducing Place Value

Required Skills: the ability to fluently count on, preferably the ability to read numbers up to 100 by rote (although this can be taught concurrently.)

Required Objects: any counters that can be used to distinguish between ones, tens, and eventually hundreds. Inspired by this excellent post, I use differently-colored poker chips, but you could also use labeled place value disks, or cardboard squares, or whatever appeals to you and is easy for your child to manipulate and remember. However, it’s best for these to neither be groupings of objects nor to be proportional to the quantities they represent.

In my last post, I discussed the fact that while the mental model for place value is simple and easily absorbed with exposure, most kids struggle to use the model fluently. This is an immediate outcome of the fact that we spend far more time teaching the consequences of the model than we do teaching the model itself.

So how can we teach place value effectively? The important thing is to present children with the abstract model in a shape that allows them to calculate effectively. The key feature of place value is the fact that the different digits in a number represent copies of different quantities, and that the quantity represented by a specific digit is ten times greater than the quantity represented by the digit to its right, with the rightmost digit representing ones. This is, in fact, the entirety of the place value model. At its heart, place value is a trading model and it should be presented as such. The consequences can wait.

Here’s how I introduce place value. I first introduce the child to the manipulatives we will be using. In my classes, ones are represented by blue poker chips and tens are represented by green poker chips. I explain that the green poker chips are worth ten times as much as the blue ones: that is, a green poker chip can be traded for ten blue poker chips. (Place value requires facility with the idea of a number as a unit, which is why I prefer to teach place value after counting on is mastered﹣counting on requires this facility as well.) We practice that for a bit, and then I explain how these new tools are related to the way we write numbers. I write an example of a two-digit number: for example, we may use a 53. I tell the child that the rightmost digit tells us how many blues to take, while the digit left of it tells us how many greens (in other words, tens) to take. And I have the child actually demonstrate this with the chips:

We then practice representing a few more two-digit numbers in poker chips. And that’s it for the introduction!

“Wait a minute. That’s IT?” you say. “This sounds exactly like what everyone else does, except for the fact that you’re using poker chips. And you said the poker chips aren’t essential, anyway. How’s this different from what other people do?”

Actually, you’re right. The above introduction is very similar to what most conceptual programs do to introduce place value. The difference is not in the introduction but in what I do next. Remember, my goal is to get kids to internalize mental models, and my expectation is that mental models take a long time to get integrated. To absorb the place value model, kids have to use it over and over and over again, until trading the quantities represented by the digits becomes second nature. When we introduce the poker chips, we introduce a way for children to use a highly abstract trading model in a hands-on way. The poker chips are not the destination but a tool. They allow children to actively engage with an abstraction.

Once I introduce the poker chips, they become our main model of number. We use them for all calculations and all operations. And in the process of using them, the mental model slowly and surely gets developed.

This is a post in a series about our basic approach. Here’s the introduction to the series. Here’s my first 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.