(here are some excerpts from a paper I wrote for grad school about structuring math homework for better learning – the full paper is below)

The traditional structure of math homework (e.g. 1-67 odd) forces students to work hard, but not effectively, as all students blindly do the same assignment consisting of a similar number of each type of problem regardless of each student’s personal weaknesses. In “Practice Perfect,” Doug Lemov’s book on how to practice more effectively, the authors compare this type of practice to shampooing your hair, something we repeat daily but probably never improve on.

Math students ought to practice math the way that experts in other fields practice. When a musician learns a piece of music, they do not just play whole piece over and over. Instead, they workshop specific parts that they need to work on. When I was learning to play piano, my teacher would have me play difficult parts repeatedly – first each hand separately, then both hands together slowly, and finally at full speed. Students do the exact opposite on math homework – they do the problems with which they are comfortable and then leave blank those they do not know how to do. Thus, they are only practicing the very material that they do not need to.

If teachers would like them to engage with it differently, they need to make intentional changes in the structure. To make math homework more like expert practice, teachers should:

Force students to differentiate their homework experience.

Though the other suggestions below would be a helpful addition to traditional homework assignments, this first one would require a more radical shift. Instead of a linear assignment that encourages students to spend an equal amount of time on each part of the course, a math assignment should consist of minimal core problems for each learning objective that each student must complete, and a bank of other problems that the student could use to remedy any misconceptions. The set of core problems should be small enough that students can complete all and still have time to tackle their weak areas. Students should be instructed that a wrong answer should be a sign to reflect for a moment about what went wrong, perhaps even formally, and then immediately try more problems until they understand. To give space for this thoughtful type of work, a teacher might have to assign less work, but the quality of the work that is completed has the potential to be much higher.
(some thoughts on grading and accountability in the full paper)

Make homework objectives transparent.

With a differentiated homework assignment that required metacognition about weaknesses, students would need the tools to pick out their own weaknesses. When picking out problems from a math textbook to assign, I have an objective in mind for each group of problems. In retrospect, it seems obvious that I should simply share these objectives with students, paralleling Standards Based Grading for the wider structure of the class. Simply grouping math problems into learning objectives would help students focus their effort more effectively by allowing them to isolate skills and measure their success.

Ensure students can get immediate and actionable feedback

But even with clear learning objectives, students can’t make progress without feedback. Too often, students will power through their entire homework doing something wrong the entire time, encoding something in their brains the wrong way. Worried that students will copy answers out of the back of the book, teachers will assign problems that do not have attached solutions. We have to get over this fear – if students cannot check their work, or are not in the habit of doing so even when they are confident they got a problem correct, they risk not knowing that they are doing something wrong. Doing one problem wrong, fixing a misunderstanding and then doing a few more correctly will lead to far better results than doing five wrong and having to unlearn something incorrect a week later. For complex problems that only have a simple answer in the back of the book, teachers could post a solution guide that details not only the answer but the process that it takes.

No idea if this will work! I’m interested to try it out when I get back to the classroom. Here is the full paper:

One of my grad school professors taught me how to read.

Okay, so I knew HOW to read (hold your snarky math teacher comments, English folk), but I didn’t realize I had no idea how to read for an academic context. For one of our very first assignments, our professor set up a very sneaky experiment that taught me I wasn’t reading very well. He told the whole class that for one of our more dense and academic readings for the next week*, one person would be randomly selected to lead a class-wide discussion. This was early on in a program with a group of 22 all-star, experienced educators. Very scary.

I was terrified into competency.

Instead of just reading it straight through and perhaps highlighting, I wrote questions in the margins, connected various parts of the text, made a list of the main ideas, pulled out quotes that could generate discussion, and generally actively thought about the content of the article.

It turned out he was bluffing, which he revealed in class the next week. Phew, *changes underwear*.  But with this exercise, he made the point to us that the way we read that article was totally different from the way we probably read most of the other stuff. And more effective. I was thankful for this because for the rest of grad school, I read much more effectively. Even if I didn’t have time to read an entire article, I would spend a bit of time diagramming, writing questions in the margins, and actively engaging with the content. Instead of expending more effort, I used my effort more effectively.

How did I make it through so many years of education without knowing how to read? How much more could I have gained from both my high school and college education? How does this apply to our math students? How many of them are trying to do better by working MORE instead of by working MORE EFFECTIVELY? What can we do to show the how to do homework?

What’s a good meta-assignment that can show students how to do math homework effectively (without making them sh*t some bricks to learn the lesson)?