The Boogeyman
Things that are hard to talk about
A Lesson
My students always struggle with subtraction equations, equations like x - 3 = 5. I'll save my rant about this for a footnote.1 Anyway, I wanted to teach a mini-lesson on subtraction equations. I did a little choral response thing with a few subtraction problems. What's 8 - 3? (Signal, everyone shouts "5.") What's 10 - 1? (Signal, everyone shouts "9.") Then a few more. Then, "here's an equation: x - 3 = 5. This means, what number minus 3 equals 5? 8 - 3 = 5, so x = 8. Here's another equation: x - 1 = 9. What is x?" (Pause to think. Signal, and a mix this time. Some say 10, some say 8, some don't say anything at all.) I describe the idea again, and write out x - 1 = 9 with 10 - 1 = 9 beneath it. We do another one out loud. Then we do a few problems on mini whiteboards, first subtraction equations, then I mix in some addition equations. It goes ok, some kids pick it up quickly, some are still shaky. This is part one of a few mini-lessons on the topic, interspersed with rounds of practice and checks for understanding.
I find that this type of teaching is hard to describe for a lot of people. It can be a bit of a Rorschach test. To some people this would be one element of an inquiry lesson. To others it's textbook direct instruction.
Talking About Teaching
At the high school I attended the dominant pedagogy was lecture. It's the vast majority of what I remember. That’s the experience of lots of teachers. And that experience informs their teaching. Lecture is something they do but they’re kindof ashamed. Or lecture is something they avoid, and they avoid doing anything even similar to a lecture. Lecture is this big boogeyman that everything else is measured against. It becomes hard to talk about things that have stuff in common with a lecture (teacher-led, giving students explanations) but also have significant differences (class is broken into lots of small chunks, lots of questions and interactions with students).
That’s where much of my teaching lands these days, especially when we’re gaining traction with a new skill. It’s a good place to be. It’s good for middle school, it’s good for my students, I can see them learning math and gaining confidence. But it’s hard to describe this kind of teaching clearly because of the boogeyman of lecture.
Some people reading this will say “that’s literally direct instruction, it sounds like something straight out of a Direct Instruction program.” You’re right, but the vast majority of teachers don’t understand direct instruction in that way, which is the point of this post. Teachers who hear me talk about this type of teaching might think I’m lecturing at my students all day every day. Others think this is an innovative, progressive pedagogy. It’s neither.
Help me understand this. In the 6th grade standards, students are “solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.” Then in the 7th grade standards, students are solving “equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers.” It’s sneaky, but because negatives are included in the 7th grade standard, those equations include subtraction. But nowhere in the standards do students specifically learn to solve one-step subtraction equations. Many curricula follow that progression, and subtraction pops up out of nowhere in 7th grade. It’s a shame, because solving both addition and subtraction equations is a great exercise in understanding inverse operations. Subtraction equations are a big conceptual leap for some kids, especially the many kids who are much shakier with subtraction than addition, and they would learn a lot if subtraction was explicit in the standards.
Subtraction isn't generally an operation but rather a shorthand notation. Really, x -1 = 8 is just x + (-1) = 8, and then to solve, we use the additive inverse and zero properties.
But the issue, which you are calling out, is that subtraction is treated as an operation, and students are supposed to apply these concepts, but really don't deal with them in a meaningful way leading up to 7th grade.
I really enjoy your posts as they are remind me of how I taught and thought about teaching math (now in admin). If you don't use "direct instruction" at times, students will not progress. A teacher is a mentor for learning. Direct instruction is one technique to show students how mathematicians think. It also helps guide students to see patterns that they will never discover on their own (mainly due to disinterest). In fact, it actually creates more engaging classrooms when students have tools to discover rather than a pure constructivist classroom.
There is a note in the Progressions for the CCSS (EE 6-8, Revised 2023) on page 177 that states "Because students aren’t expected to do arithmetic with negative numbers until Grade 7, equations in standard 6_EE_7 are restricted to positive p and q. However, students in Grade 6 might solve equations of the form x – p = q where p and q are positive."
I found myself thinking about how many elementary students (and certainly 6th graders) think about subtraction as removal by default. So the problems you posed (8 – 3, 10 – 1) really don't activate anything useful for solving x – 1 = 9. I don't mean that as a critique, just as an observation. It's probably the students that have a more dense network of number relations under their belt that have more immediate success here. If I know all the partitions of 10, then I'd immediately recognize 1 and 9 one of those partitions. So for students with that "structuring" knowledge it doesn't depend on their knowledge of addition and subtraction.
I wonder if something like "list all the equations you can that use 3, 5, and 8" would be a helpful setup. I want to activate the idea that addition and subtraction are two sides of the same coin and I can use one to solve the other.
More to the point of your post though, it's easy to talk at students for long periods of time and it's also easy to provide students tasks to somehow "figure out". It's much harder to plan exactly what needs to be made explicit, visible, and directly explained and what aspects of a lesson benefit from inquiry and student exploration.