Structured and Systematic Teaching

The little things

I wrote a very long post last week. One of my claims is that some students are like sponges; they just soak up knowledge, learning even if the teaching isn't very effective. Other students need much more structured and systematic teaching. I wasn't very specific about what structured and systematic teaching means so I want to try and define that here.

The short version is that structured and systematic teaching breaks concepts down into small pieces and makes sure students learn those pieces, building gradually toward larger ideas. Here are three examples of what that looks like in practice.

Find gaps in prior knowledge

Last week I was teaching a mini-lesson on rounding. We're doing circles soon and rounding comes up a bit. Separate from circles it's a good skill to give students a refresher on. Lots of students forget rounding but it's helpful to know and reinforces some good place value concepts. One student was clearly lost during the lesson and gave answers that were pretty far off. While students got started on their next activity I pulled up a chair to work with her. I started asking a few questions and it became clear that she didn't really know how decimals worked. I drew a little number line, with a 3 and a 4 and ticks in between to represent tenths. Ok so here's 3.1, here's 3.2. Where would 3.4 be? What about 3.9? 3.7? "Ohhh that's what those mean!" That was a great moment.

Structured and systematic teaching means seeking out moments like that. What are the missing pieces that are holding students back? How can I make sure I find and remediate them, rather than leaving it to chance? Some students, even if they missed the lesson on rounding, will figure it out when rounding comes up in the future. Others need more targeted support. I want to figure out who those students are and provide that targeted support.

Make sure the foundation is solid

Inequalities are one of the tougher topics I teach in 7th grade. They're pretty abstract. A lot of the skills build on equation solving but it's easy to lose sight of the parts that make inequalities different. One of the keys is to make sure students are solid with a few key skills before we start. The first is simple: making sure students know what the different inequality symbols mean. Some students have figured these out and become pretty fluent with them on their own; many more haven't. I spend some time in the weeks leading up to our inequalities unit reviewing what the symbols mean and dropping questions in my daily Do Now. What symbol means "less than or equal to"? What does the > symbol mean? The second key skill is visualizing positive and negative numbers on the number line. This helps students to check their work and ensure it makes sense, in particular for some of the tricky inequalities with negatives. I drop in a lot of quick practice putting integers on number lines and evaluating whether statements like 4 < -6 are true or false.

Structured and systematic teaching means figuring out what the key skills are that students need to know before learning a concept, then making sure students learn them. There are two easy ways to screw this up. One is to think that students need to review every single topic from previous grades before doing grade-level work. That's not true. Students don't need to know long division or the triangle area formula to solve inequalities. I don't need to reteach all of math; I need to focus on a few key skills before each unit. Second, this is more than a few minutes of review right before we start solving inequalities. I need to reteach with time for multiple cycles of practice and feedback. I don't mean spending hours and hours on it, I mean spending a few minutes at a time, a few times a week, for 2-3 weeks, so students can get confident with these skills.

Break skills into pieces

As someone who has known how to solve two-step equations for a long time, it's easy for me to forget all the different pieces that fit together for students to solve equations. Here's an example. To solve many two-step equations, students need to understand inverse operations. If the equation is 3x + 11 = 53, they should start by subtracting 11 from both sides. My students kept getting stuck and not knowing what to do here. We spent a lot of time solving one-step equations and they got really good at those. But for some reason this step seemed like a huge leap from what they already knew. I realized it was because when I gave students one-step equations, too often I gave them one-step equations like 4x = 12. They weren't solving this by using inverse operations. They knew that 4 x 3 = 12, so the answer has to be 3. This even extends to simple two-step equations. If they saw 5x + 1 = 11, they could do it mentally and realize that 5 x 2 + 1 = 11, so the answer must be 2. I realized that these are actually two separate skills. The kindof "fill in the blank" method is an important one to understand. But it's different from the inverse operations method. Very few students can solve 3x + 11 = 53 mentally. If students get really really good at solving “fill in the blank” equations, then I suddenly change to inverse operations equations, I’m setting them up for failure. Some will figure it out, but others won’t. I need to be clear about the difference between the two methods, and give students lots of practice with simpler inverse operations questions like 3x = 72 to build that skill gradually, not throw kids into the deep end.

Structured and systematic teaching means being clear and precise about when something that seems like one skill is actually two separate skills, and making sure all students get practice with both skills. If I don't realize they're different, some students will figure it out. But others won't, and my goal is to make sure every student learns, not just if they're lucky enough to figure it out on their own, but because I make sure to teach it.

A final note

There’s more to teaching than being structured and systematic. If all we ever do is practice the meaning of the inequality symbols and rounding then students won’t be able to apply their knowledge in new contexts in the future. I think of all this as the beginning stage of teaching a concept. There’s an important part that comes next where students solve more and more ambitious problems in new contexts. I’m not trying to diminish that. In my experience, the best way to set students up for success with that next stage is to be really structured and systematic early on.

It’s also easy to fool yourself. Some students — depending on your student population, maybe many students — will learn just fine without this structured and systematic teaching. If I don’t find gaps in prior knowledge, make sure the foundation is solid, and break skills into pieces, the students who need that type of teaching are likely to tune out. There’s a type of teaching where I pat myself on the back for the students who are learning, and blame the students who aren’t learning for not paying enough attention, and keep it moving. Structured and systematic teaching is about teaching in a way that sets as many students as possible up for success.