Practicing Like an Expert (instead of a math student)

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

mathshampooThe 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:

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Posted on June 5, 2014, in Homework, Teaching. Bookmark the permalink. 7 Comments.

  1. Thanks for sharing these thoughts! I find myself wondering about this in writing instruction as well: teachers of writing can spend so much time writing comments, but how do we know if students are learning anything from all the time we spend on them? (Put more starkly, how do we know if our time spent grading their writing translates into meaningful learning for them?)

    The piano lesson analogy is a compelling one. It’s much easier to do what we’re good at, but it won’t actually help us improve. How can we help students learn to identify areas where they need to work…and then give them the opportunity to do so?

    Making this whole process transparent and explicit seems like the way to go as well. Thanks for such a rich post!

  2. To me this feels like it would be very tricky to get right. If they have any choice over which questions to answer, the less confident students would find it very hard to resist the temptation to stick to questions they already know. If they have access to solutions then they would find it hard to resist the temptation to go straight to them before trying on their own.

    I haven’t read the report yet, so maybe my concerns are discussed in there, but how will you try to avoid these problems?

    • Phil, you’re spot on. This is definitely a brainstorm of an idea where the concept is far easier than the execution. I deal with some of those issues in the full paper, but again only from a theoretical perspective. Not usually how I liken to blog, but theoretical is the grad school life! I hope to try it out in the fall.

      • And another quick thing: I think this would take work not only teaching them the math concepts but also HOW to do homework. Not going to work automatically for the reasons you mentioned.

  3. I think it can work. Have you read Daniel Pink’s Drive? His big assertion (which is pretty ambitious but I think is true) is that 85% of people in this world are looking to do their best on a given task, without the need to be supervised. In fact, Pink suggests that we’re far more likely to be motivated intrinsically if we are trusted to do so. If we go by that assumption and you provide clear tools for your students to help them diagnose/select their areas of weakness to continue working, it can work.

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