Michael Pershan is starting something awesome on the topic of feedback over at his blog, and he’s got me thinking.
I’m not thinking about rich problem solving tasks, or critiquing student argument or reasoning. That’s a pretty big challenge, and one I’m still wrapping my head around. But while problem solving is critical to learning math, so is practice. I assign homework every night — between 10 and 15 questions, most fairly simple, mixed review from the entire year. I polled students a few weeks ago, and they spend a median of 8 minutes on the homework each night. For these assignments, I’m thinking not about my feedback, which there often isn’t time for or is ignored as students are slow getting ready for class, but about the feedback the questions themselves give to students.
Case One: Problems that give their own feedback
Square roots are a favorite of mine to drop in homeworks. If you aren’t fluent in 2×2 multiplication, it’s good practice. If you are, automaticity with perfect squares from 169 – 625 will be a huge asset when we work with the Pythagorean Theorem. And it builds fluent number sense with the idea of square roots and perfect squares. Most importantly, I find that students rarely get this question wrong. Some forget how to find it; occasionally they divide by two, but for the most part, students guess and check until they get it — and if they write something else down, it’s laziness, not evidence of a misconception.
Case Two: Problems that are … meh
I like this problem. I really do. There’s no context, but the numbers are a bit unconventional. Students know that most equations are supposed to work out to whole numbers. If they don’t know how to do it, they usually know it. There are plenty of wrong answers — bad habits with laziness around negatives in equations, or misconceptions about inverse operations. These are definitely a problem if left unchecked — but my experience is that students know if they know how to do these, and know when they get them right.
Case Three: Problems that build bad habits
This is absolutely one of my least favorite questions in 8th grade math. With some practice, I can get every student to understand scientific notation. But no matter how many times we look at this question, or its various applications, students will, over and over again, tell me that the answer is . They constantly forget to rewrite their answer into scientific notation. There are plenty of reasons for this, one of which may be that this is a silly distinction, but I think a big part of the problem is that this question does not give it’s own feedback. A student could solve a hundred different problems and have no issue with writing the wrong answer every time — because they’re calculating correctly, but not attending to precision and answering everything the question is asking.
In summary: I’m only addressing a narrow aspect of feedback — students doing simple, straightforward practice, and the feedback that students get from that practice itself. But as I push students to do a little bit of math, every day, on their own, the habits and feedback they get from that practice becomes a significant part of their learning. Is this framework a useful way to maximize the benefits of that practice?