2-4-2 Homework

I’ve spent a lot of time this year thinking about homework, and I’ve come to some conclusions about my core values. They are:

• If homework isn’t useful for learning, I shouldn’t assign it.
• Homework shouldn’t take any student more than 15 minutes. The longer homework takes, the more inequitable it is likely to be.
• Homework should never depend on how well a lesson goes — if we get sidetracked in class or have to slow down unexpectedly, homework should still be useful for students.
• Homework is the best opportunity I have to integrate regular spaced and interleaved practice.
• If a student can’t complete homework one night, for whatever reason, that should not prevent them from learning in class the next day.

I attempted to shift halfway through last year to a model I learned about from Steve Leinwand, which he calls “2-4-2 homework”. In this model, homework assignments consist of eight problems. The first two address the topic we’re currently working on. The next four are mixed practice from other topics in the course. The final two involve some extended reasoning or explanation.

While I really like the elegance of this model, I’ve found it hard to follow through with. Too often I am writing homework at the end of the day when I want to go home, or the period before a class I’m teaching, and with three preps it’s easy to leave writing homework until the last minute. I end up just throwing random problems onto a handout and handing it out. I also wasn’t explicit enough with students about my goals for homework and the purpose of the structure. I do think it had some value in the mixed practice it provided, but definitely also some room for improvement.

One change I’m making is streamlining the way I write homework. I have a little three-section notebook I use for lesson planning, one section for each prep. In that notebook I’m going to start keeping a list of topics that should be the focus of mixed review. This will include prior topics that I know students are likely to get rusty with quickly. For instance, in my Algebra II class I might regularly ask them to solve a simple system of equations, or graph a quadratic function written in vertex form. It will also include skills students will need for upcoming units. If I’m going to start graphing polynomials in two weeks, students should probably be factoring every night, with some other polynomial arithmetic thrown in. Having a list of these topics instead of making problems up off the top of my head every day should make me more efficient in writing homework and also more focused in what I ask and what homework reveals about student thinking.

I usually enjoy writing the last two problems. I might ask students to explain why factored form reveals the zeros of a function during a unit on polynomials, or to find a function that doesn’t intersect the y-axis in a unit on rational functions. These aren’t meant to be the hardest problems they’ve ever seen, but they will hopefully elicit several perspectives. I’ve also found the Exeter problem sets to be useful places to find creative questions; they’re definitely worth flipping through for high school teachers.

I also want to be much more consistent with the way I review homework. I think that I could start each day by having students discuss one of the last two questions at their tables, and pick one or two students to share their perspectives. Then, if I notice any quick hits that I can clarify on the first six problems I can address those. Major issues get filed away for later. Then on to the rest of the lesson.

Another change I want to make is to tell students what the 2-4-2 structure is and why I think it is important — and to bolster that argument with cognitive science. With a clear purpose, when we go over homework I can frame my choices around the purpose of that problem. If there is confusion on one of the first two problems, I may table it and revisit it later in the lesson where that idea fits better. If there is confusion on one of the mixed review problems, I can tell students that we will revisit that topic in the future rather than trying to fix it all right there. I want reviewing homework to be a quick 2-3 minutes whenever possible, and that means not going down every rabbit hole a student is interested in exploring. It’s often frustrating for students when I move quickly through homework review, but making explicit the goals of homework can help to place that learning in the larger context of our goals in math class.

Hopefully these structures will help me be more consistent and more purposeful with homework the second half of this year. Here goes.