Break It Down

An example

Equations

I'm teaching two-step equations right now. Equations like 2x + 1 = 9, 17x + 31 = 354, -6x - 4 = 14, and more.

Teachers generally have a similar experience teaching this skill. Some students just "get it." You show them the basics of inverse operations and how to organize their work. All of a sudden they can solve anything. Other students don't get it. It won't click. I can walk through the steps with them as many times as I want but there's something missing. There are students in between, of course, but there’s a big gap between those who get it and those who don’t.

How do you respond when some students don't get it?

Some teachers will go deep into representations. Tape diagrams, balance hangers, algebra tiles, and more. Some teachers will put the steps on a poster or create a notes sheet and refer students to the steps. Some will blame the students. They aren't trying, or there's a lot going on at home, or they should've been held back.

My response is rooted in the idea that solving a two-step equation requires knowing lots and lots of other small pieces of math that build up to the larger skill. Some students arrive to class knowing all of these little pieces. Some don't. Some students are better at figuring out these little pieces on the fly, others aren't.

Here's a rough list of all the little things that go into solving equations:

• Knowing addition facts, like 8 + 4 = 12

• Knowing subtraction facts, like 12 - 4 = 8

• Understanding the connection between addition and subtraction facts, like knowing that since 3 + 4 = 7, it must be true that 7 - 4 = 3

• Understanding the meaning of the equals sign

• Understanding the meaning of a variable in a single-variable equation (which is different from variables in a two-variable equation like y = 4x, also a 7th grade standard)

• Solving an addition equation by connecting it to a known fact, like solving x + 4 = 12 by knowing that 8 + 4 = 12

• Doing the same for subtraction equations

• Solving an addition equation by using inverse operations, like solving the equation x + 4 = 12 by finding 12 - 4

• Doing the same for subtraction equations

• Knowing multiplication facts, like 5 x 3 = 15

• Knowing division facts, like 15 / 3 = 5

• Understanding that division can be written with either a division sign or a fraction bar, and the two are equivalent

• Understanding the connection between multiplication and division facts, for instance knowing that 4 x 6 = 24 implies that 24 / 6 = 4

• Knowing that the expression 5x refers to 5 multiplied by a variable x

• Solving a multiplication equation by connecting it to a known fact, for instance solving 5x = 15 by knowing that 5 x 3 = 15

• Solving a multiplication equation by using inverse operations, for instance solving 4x = 24 by finding 24 / 4

• Understanding how the order of operations applies to expressions, like knowing that in both 3x + 5 and 12 - 3x, the 3x is evaluated first

• Understanding that 8 + 2x is the same as 2x + 8 and being able to move fluently between the two

• Evaluating expressions like 8 + 2x and 3x + 5 for different values of x

• Knowing how to evaluate negatives in expressions, for instance 12 - 3x

• Knowing that expressions like x + 3 - 3 and 5x / 5 are equal to x

• Generating the inverse operation to get back to x, for instance knowing that if an expression is x - 4, adding 4 will get back to x

• Understanding that doing the same thing to both sides of an equation will keep both sides equal

• Connecting an equation like 2x + 1 = 9 to two simpler equations, y + 1 = 9 and 2x = 8

• Checking whether the solution to an equation is correct, for instance finding that if x = 4, then 2x + 1 = 9, solving the equation above

• And more. None of that gets into operations with negative numbers, fractions, or decimals, which are each their own rabbit holes. Another type of equation we solve in 7th grade involves parentheses, like 4(x + 2) = 20. There’s probably lots of other stuff I left out. You get the idea, so I'll stop here.

That Was a Long List. Where Is This Going?

It's a truism in education that some students learn faster than others. And yes, I think that's true. But it's less true than we often think.

The biggest variable in how fast a student learns is what they already know. A student who already knows everything on that list above will learn equations much faster than a student who's shaky with half of those skills. That's not an immutable property of the student. It's something we can change through teaching. Students will be more successful learning equations if I take the time to teach mini-lessons on those skills, practice them, and help fill in pieces before we start solving two-step equations. And there are lots of skills! That was the point of that long list above. This isn’t something I can do in five minutes and then move on. It takes time, patience, and planning.¹

A second variable in how fast a student learns is the extent to which they can figure these micro-skills out on the fly. Some students are better at picking up little skills along the way. If they forget how to evaluate stuff like 8 + 2x and try to do the 8 + 2 first, they can figure out their mistake quickly and learn from it while also learning key ideas of solving two-step equations. Other students struggle to pick up little pieces while also learning a larger skill. When I break things down into little pieces and focus on one piece at a time, those students are going to be a lot more successful. When I don't break things down, many students will still learn — but some students are stuck spinning in place. If students are missing eight of the skills on that list above, learning equations is going to be incredibly frustrating for them.²

I'm not arguing that all students learn at the same speed. I'm arguing that teachers can reduce differences in learning speed by breaking concepts into small pieces and explicitly teaching all the little pieces. The pitfall is that it's easy to justify not doing this. Some students don't need that type of teaching. They either already have a lot of that knowledge, or they're better at picking it up as they go along. If we focus on that group of students, we can tell ourselves that our teaching works and there’s no problem. But other students don’t, or can’t, learn like that.

The hardest part about breaking things down is developing the content knowledge to do it well. It's taken me a few years of teaching 7th grade to understand all the little pieces that go into solving equations. I'm sure I'll keep adding more micro-skills to my list. It’s easy to say, “break skills down into small pieces and teach the small pieces one by one.” It’s way harder to develop the knowledge needed to know what all those skills are, what order to teach them in, how they fit together, and how to build them up to a whole that is greater than the parts.

1. Teaching multiple strands at once is a good strategy for teaching all these micro-skills. Trying to squeeze them all in at once at the start of a unit won’t work very well, it will be overwhelming for students and they won’t have enough time to consolidate their learning. Instead, I do these in small chunks in the weeks or months leading up to a unit.

2. I wrote here about the general idea that some students are better at “soaking stuff up” or learning on the fly. My basic mental model is that some students have more working memory capacity and a faster mental processing speed. Those students learn more through less structured teaching, teaching that doesn’t break things down into small pieces. This is just a mental model. I’m not arguing that working memory capacity and processing speed are the only important differences between students. But I think that mental model is useful for teachers, and it’s simple enough to hold in my mind while I’m teaching or planning.