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.

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