I’ve been thinking about the Mathematical Practice of looking for and making use of structure.
There’s a full description here: CCSS.Math.Practice.MP7 Look for and make use of structure.
Question: What’s the same and what’s different in these two expressions?
3(4+5) and 3(a+5)
A similarity we might notice is that there is a 3, a 5, parentheses, and an addition symbol in both.
One difference that’s pretty obvious is that 3(4+5) is comprised of all numbers, while 3(a+5) has two numbers and a variable.
Another difference we might notice is that the expression on the left has one particular value, namely 27, while the expression on the right has infinitely many values if we allow a to be any real number.
Your students might say this: In the first one we should follow the order of operations and in the second one we should do the distributive property.
Or, they might say that we should do the distributive property in both, and then immediately launch into it.
These latter two responses reflect what students think they’re being asked to “do” rather than focusing on what the expression is communicating.
Did you notice that both expressions contain a sum and a product? That’s structure.
Structure doesn’t tell us what to do. Structure tells us about, well, the structure of the expression. Here, the structure tells us we are dealing with a sum and a product.
I recently had the opportunity to talk to students enrolled in a course roughly equivalent to high school Algebra 1. When they saw an expression like 3(4+5), several of them said some stuff about the order of operations, but then changed the conversation to the distributive property, explaining that we should multiply 3 by 4 and 3 by 5, then add the result. It was all about what we should “do”.
The funny thing is, we had written the expression 3(4+5) to represent the thought process of a student, Danny, who had added 4 and 5, and then multiplied that sum by 3. When another student went to the board to re-create the thought process by explaining the expression, he talked about the distributive property, explaining “I think Danny used the distributive property and multiplied 3 by 4 and then 3 by 5 and then added 12 and 15.” Surprisingly, Danny agreed that yes, that is what he had done! This was absolutely not what Danny had done because we wrote the expression based on his description of adding first, then multiplying!
What do I think caused this confusion? My guess is that the students had been working on the typical stuff with algebraic expressions, like simplifying or combining like terms, and probably using the distributive property. They recognized the structure of a(b+c) and did what good little algebra students do.
The problem was, our task was not about writing an equivalent expression or simplifying or anything like that. We weren’t even evaluating the expressions to find a particular value. We were just discussing how the expression we wrote mapped onto Danny’s strategy, and whether the notation and the numbers represented how he thought about the problem situation.
We then extended the activity to think about “any number” in place of the 4.
Now the students really perked up about the distributive property. Funny thing is, the expression 3(a+5) represents the exact same thought process as 3(4+5): Find a sum, then the product of that sum and 3. We can’t know what the sum is unless we assign a value to a, but that doesn’t change the fact that this expression is about a sum, namely a+5, and the product of that sum and 3.
Many students think that the expression 3(a+5) is about the distributive property. It’s not. We can use the distributive property to rewrite the expression in another form that might provide more information or be more useful to us. But, the expression 3(a+5) is not about the distributive property, it’s about a sum and a product.
Let’s return to the earlier student comment about using the order of operations in 3(4+5) and the distributive property in 3(a+5). I have argued that both expressions call for finding a sum, and then a product. Students might reject that idea because we don’t know what a is. Their reasoning may be “If we don’t know a, we can’t find a+5, so we’re left with the only thing we can ‘do’ which is apply the distributive property. I mean, we have to ‘do’ something, right?”
I see this type of thinking as a huge hurdle for many algebra students. They’re looking for something to “do” rather than thinking about what the expression is meant to communicate to the reader. When we take time to think about what the expression is telling us, we can think about whether or not it would be useful to write the expression in some other, equivalent form. That’s the real power of algebra manipulations. We can transform expressions into different and more useful forms. Of course we have to learn what those allowable manipulations are, and probably practice them, but their purpose is to make the expressions more useful or informative.
This week’s tweak (whether you teach algebra or not): Point out the structure of an expression and talk with students about what the structure of the expression communicates to the reader. What is the author of the expression trying to convey about their thought process? Heck, you could even pose one or both of the expressions at the top of this post.
Please be sure to share how your experiences here!