### A guest post by Rachel Chou

I have been a classroom mathematics teacher for 20 years. I have heard students use the phrase “the distributive property” more times than I can count. Many of them misunderstand what “the distributive property” even is. But maybe I think that because I don’t really know what “the distributive property” means either.

When we teach a new idea or concept, one of our responsibilities is to teach kids why this new idea or concept is important,

interesting and special. Let me start with the case of teaching kids about SSS congruence in the context of triangles. Students are taught that if three sides of one triangle are congruent to three sides of a second triangle, that the triangles must be congruent.

This is true. But the reason that this is so interesting is that this is only true for 3-sided polygons! Put 3 straws (of varying lengths) on a string and tie a knot, and you have a rigid figure. You will not get to decide what the angles are in the triangle. The 3 sides (of pre-determined length) define the triangle. The person who tied the knot cannot make choices about the measurements of the angles! But now, put four straws (of varying length) on a string and tie a knot, and you have yourself a very floppy quadrilateral. Infinitely many such quadrilaterals exist. You, the owner of the 4 straws on the string, can rearrange the angles in the quadrilateral in a myriad of ways. In this sense, SSSS does not determine a unique quadrilateral. Similarly SSSSS does not determine a unique pentagon. To teach students SSS congruence without pointing out why this is so interesting is harmful for two reasons. First of all, this is an amazing result. It is the our job to point out amazing results! Triangles are rigid figures in a way that other polygons are not. Second, if they do not point out why this is special, teachers take the risk that their students will over-generalize and assume that SSSS, SSSSS and the like also guarantee congruence. Children and adolescents (and likely adults too!) are extreme over-generalizers. A math teacher’s job is to actively work against their willingness to over-generalize.

So, in my classroom, many students have told me that they are using “the distributive property.” The problem here is that much of the time that they relay this to me, they are misusing it, or misunderstanding it. Students are taught that a(b + c) = ab + ac. They need not be taught anything this abstract. From a young age, children can understand that if you’d like to find 17 groups of 23, you would be able to successfully do this by finding 17 groups of 20 and then adding on 17 groups of 3. To understand what multiplication of whole numbers means is to understand that this is reasonable.

Instead of focusing on this, students are often taught to memorize “the distributive property.” But the problem is similar to that of the SSS conundrum. Students need to be taught that it is rather amazing and awesome and special that multiplication distributes over addition (and subtraction) because not many things do distribute! Square-rooting does not distribute over addition (but most students tend to make this mistake with high frequency), nor do exponents over addition. Some operations distribute over others, but to call this “the distributive property” is problematic. It leaves out the key word in the previous sentence: some.

There is little reason to ever introduce this fancy property name before the 8th grade. Young students can be taught to break multiplications into easier parts without having a name to attach to it. The name of the property only gives students a license to stop thinking about the operation of multiplication and to start applying a black-box rule that is true “because the teacher said so.” Instead, if a 7th grader comes across 10(x + 8), why not ask them to think about what it means to have 10 groups of (x + 8)? They can likely figure out what to do without a name to the property. And if they work through the multiplication more slowly, because they are being thoughtful and methodical about what it means to multiply, that is a win, not a loss. Our job is not to instill an ability to move swiftly through low-level operations without even thinking. It is to encourage students to make meaning out of the mathematics they are studying and to understand every step that they take in a calculation.

There’s also an issue of accuracy. When my students tell me that 10(x + 8) = 10x + 80 is true because of “the distributive property,” they are actually factually incorrect. While this fact may demonstrate the validity of the fact that multiplication distributes over addition, 10(x + 8) equaling 10x + 80 is not true because multiplication distributes over addition. Naming an idea is not the same as explaining why it is true. I could invent a law which states that mothers are always older than their daughters. I could call this the MOTD law, but it would be a bit odd if I started telling people that the reason I know my students’ parents are older than they are is “because of MOTD.”

So what might we do with the instruction of the ideas of distribution? Here is one idea. When students are a bit older and have the facility to work with a variety of operations, begin the instruction with true-false tasks that students work through and discuss in groups. For example:

- (3 + 4)^2 = 3^2 + 4^2?
- 5(3 + 4) = 5(3) + 5(4)?
- 11(x – 7) = 11x – 77?
- a(bc) = ab * ac?
- sqrt(a + b) = sqrt(a) + sqrt(b)
- (3x)^4 = 3^4·x^4?

After students engage in meaningful discourse about tasks such as these, then, add the vocabulary to make the communication of this idea easier for your mathematical community. Relay something to the effect of, “We say that one operation distributes over the other if we can apply the outer operation to all the terms connected by an inner operation.”

For example, multiplication distributes over addition because it is true that: a(b + c) = ab + ac.

- Stop and consider what was written just now. Do you believe that a(b + c) = ab + ac? Be prepared to defend your point of view.
- Do you think that addition distributes over addition? For example, is a + (b + c) = (a + b) + (a + c)?
- Does squaring distribute over multiplication? Why or why not?
- Does square rooting distribute over addition? Why or why not?
- In your groups, come up with examples of operations that do distribute over other operations, and examples that don’t work. Two lists have been started for you:

**Operations that do distribute:**

- multiplication over addition

**Operations that do not distribute:**

- addition over addition

The point isn’t to give kids a list of operations that do distribute over others. This would probably lead to pointless weird memorization. The point is to instill the idea in students that the idea of operations distributing over each other is interesting, powerful, and special. We should yearn for an era when kids never simply say “the distributive property” again, but instead wonder aloud, “Wait! does sine distribute over addition?” “Does the log function distribute over multiplication?”

—Rachel Chou

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Note from Henri:

Thank you Rachel. I largely agree with your analysis. More generally, when learning a new idea, students should see *non-examples*, to clarify the limits within which the idea applies. I have done this in my own work as teacher and curriculum developer. To introduce SSS and the like, I start with a lesson on *non-congruent* triangles: can you make two non-congruent triangles that satisfy SS? SA? AA? and so on — see *Geometry Labs*, 6.1. Likewise, for the distributive rule, we addressed this issue in Lesson 5B of *Algebra: Themes, Tools, Concepts*, a book I co-authored in the 1990’s. (Both books are free downloads on my website.)

I’ll add that an array model in lower elementary school and later the area model can provide a visual way to understand the distributive property of multiplication over addition. This can be reinforced by modeling the operations with Base 10 blocks, and later with the Lab Gear or other algebra manipulatives. These hands-on approaches are a complement to the sort of discussion you encourage in your article.

— Henri

I agree with Henri that the idea of showing non-examples is a crucial thread underlying Rachel’s approach here.

I think Rachel says something further, though: something like “don’t make a generalization until you have more than one example.” If you only have multiplication and addition, there’s not so much of a need to name the property and see the abstraction, until you have more operations to work with and can see that there are multiple situations where it’s valid (and others where it’s not).

I think there are deep mysteries in all these simple things. Whole number addition, essentially repeated “counting on”, is commutative. Whole number multiplication, repeated addition, is commutative. So why is whole number exponentiation, repeated multiplication, suddenly not commutative any more? Same question for associative.

On the other hand, as we go up this hierarchy, each operation distributes over its immediate predecessor — at least so far! Does that continue if we have the “tower of powers” repeated exponentiation operator? … now I’m curious and have to go investigate that.

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