Tuesday, December 2, 2014

On Review and Remembering

Recently, our high school math department met up with the middle school math department to discuss all kinds of things. Part way through the discussion, someone mentioned the need for a magic pill to help students remember what they've learned.

We all agreed. We have all experienced the frustration of believing that our students have mastered a concept, only to discover they can't remember what they've "learned" just days, weeks, and certainly months later. By the next year, we're hearing that they have "never" seen whatever thing we are expecting them to remember.

In the middle of this conversation, it occurred to me that maybe we are teaching students to forget. A traditional math classroom (including mine until I started SBG) looks like this:

1.  Teach a unit
2.  Review the unit
3.  Test over the unit
4.  Move on to next unit
5.  Start forgetting previous unit

The end of each of my units always had a review day, where I would have students practice a bunch of problems that were really similar to the ones they'd see on the test the next day. Only the numbers were changed. The next day, my students would (generally) do pretty well on the test. I would pat myself on the back for my good teaching ability, and away we'd go to the next unit.

Now I'm thinking . . . Do those big-review-days-that-look-just-like-the-test-right-before-the-test-day just train students to stuff in the information, hold it in for 24 hours, and regurgitate it the next day? If our students do very well on such a test immediately following such a review day, does it give us a false representation of what they've actually learned? What if the "forgetting" we see is really just "never learning"?

What should review look like? When do you review? What do you review? WHY do you review?


  1. (I tried to publish a comment and I can't tell if the internet ate it. I'll repeat myself but be briefer.)

    1) Cumulative tests. Simple but revolutionary switch for me as a math teacher. Also, essential for AP classes (I teach Calc AB) where you are always needing more time for review: the review is built into the course. When you get to May, students have seen limits consistently since September.

    2) AP-style Free Response Questions. For three reasons, two I'll mention in this heading. First, the AP always finds a slightly new way to ask something, so students can't approach problems in a plug-in-numbers kind of way. Sometimes they get a table, sometimes they get a function. Totally different situations. Sometimes they provide a rate, sometimes one must be derived: students have to be sensitive to this and learn how to assess these things. Second, AP questions are rich conceptually. A well-formed questions can literally cover 80% of the course in one (multi-step) question. So students may be focusing on the step that is "this unit," but connecting it to other steps which tie in earlier units.

    3) Finally, FRQs demand verbalizing: students must occasionally justify or explain their reasoning with words. Coaching students through this made me realize how helpful SENTENCES are to students' understanding of math. Teaching them sensitivity to terminology, like the nuanced but critical difference between "change" and "rate of change," or how to justify using a derivative to find relative extrema, rather than simply launching into doing it, makes them more cognizant of the math as they apply it. For me, verbalizing happens in FRQ quizzes and in test corrections.

  2. The trade-off with cumulative tests is you can't test the new unit as robustly as before. But the robustness comes over several tests, and the trade-off for students not jettisoning previous units is huge. And with the right questions, you can test quite a bit in two or three questions.

    My tests are roughly 1/3 new stuff and 2/3 old stuff. Forces students to address their weaknesses, because if they ignore limits or whatnot forever, they'll be missing questions forever.

  3. An immediate thought is that this symptom indicates the students never really understood the concepts they were supposed to be learning. There could be a lot of reasons for this, including your idea that review days send a wrong message or, as Tim implies, dropping the ideas in one unit once they've been tested also signals time to forget.

  4. One thing I've tried this year is "Throwback Thursday." We review an "old" skill each Thursday (or I try to do it each Thursday...), whether it's related to our current material or not. It's hard to be consistent and to feel like your relinquishing time that could be used to march forward.