Pairs of Skills

A way to structure some quick, efficient practice

Efficient Practice

I've been interested recently in efficient practice. Practice is important, but it's also easy to be lazy and assign students questions 1-31 odd. All that practice often isn't any better than four or five problems because it becomes mindless. I want practice to be efficient. How can I spend five minutes of class time on practice and get the best bang for my buck?

Related to that idea, one research result that has always bugged me a bit is the benefit of interleaving practice. The idea is simple: rather than blocking practice into a bunch of problems on finding the volume of a cylinder, then a bunch of problems on finding the volume of a cone, then a bunch of problems on finding the volume of a sphere, it's better to mix those problems together. When students solve a bunch of the same type of problem in a row they aren't retrieving anything from memory. They're just copying what they did before, and 20 problems might not be any better than two or three.

Here's the issue. If your experience is anything like mine, throwing a bunch of interleaved problems at students tends to leave them frustrated, stuck, and not practicing at all.

Pairs of Skills

I want my students to practice. I want that practice to benefit from some interleaving, so students are thinking and not just mimicking. But I don't want to overwhelm students and leave them paralyzed because I'm interleaving tons of random skills all the time. My solution has been to do focused practice on a pair of complementary skills whenever I can. Here are a few examples:

• When working with scale drawings it's important to know how to multiply by 1/2, 1/3, 1/5, etc. To start our scale drawings unit we did some focused practice multiplying whole numbers by unit fractions, mixed in with dividing by unit fractions.

• One common context for proportional thinking is speed, and students often mix up how to solve simple speed problems. So we did some focused practice on "if you're going this fast for this long, how far did you go?" and "if you're going this fast and you want to go this far, how long will it take?"

• When working on circumference we did some quick practice switching back and forth between being given the diameter and finding circumference, and being given circumference and finding diameter.

• As part of fractions review work we went over finding GCFs and LCMs, and did some focused practice going back and forth between those two skills.

When observing this type of practice I often feel at a certain point like students are getting something, but then I change the question type and a student mistake shows me that actually we have more work to do. It's easy as a teacher to fool yourself into thinking practice is going well when you're repeating the same skill again and again.

Here's an example: on Tuesday I introduced my students to circumference problems. We did a few guided activities, talked about where pi comes from, a bit of math history, et cetera. Then I had them pull out mini whiteboards some practice. We did maybe three "here's the diameter, what's the circumference" problems. Students are a little shaky at first and then improve. Then I say "ok, now let's go the opposite direction, I'll give you circumference, what's the diameter?" The first one a bunch of students get wrong, we talk about inverse operations, and the second one most students get. Here's where I might think to myself, "ok students really get it! They can go from diameter to circumference and circumference to diameter!" Then I give them another diameter to circumference problem. Answers are all over the place. A few students are multiplying by two, mixing it up with radius to diameter, a bunch are dividing by 3.14, it's a mess. When students were answering before I wanted them to be thinking "ok to find the circumference I multiply diameter by 3.14." Really they were thinking "ok I just need to multiply by 3.14," with no connection to what they were actually solving. So I do a quick reminder, connecting what we're doing to the picture of a circle, the meaning of circumference, and the idea of a proportional relationship. Then we do a bit more practice switching randomly between diameter to circumference and circumference to diameter. It wasn't perfect, but it was helping students make the connection between finding circumference and multiplying by pi, rather than mindlessly copying.

It all seems obvious in retrospect. I want students thinking, not mimicking. But there's a pretty narrow window between giving students predictable practice where they aren't thinking, and giving students practice that's so unpredictable they get stuck and give up. Interleaving is a helpful practice strategy because it promotes thinking, and memory is the residue of thought. This pairs of skills strategy doesn't work everywhere, but I'm finding it really helpful at promoting thinking without increasing the difficulty so much that students get stuck. I’m starting to look more and more for places where I can put a pair of skills together for some quick, focused practice. I don't have tons and tons of time for practice in class, and the value of homework is limited because students can't get feedback the same way they can in class. This strategy is a great way to get some high-quality practice in just a few minutes.