All classes are heterogeneous, some more than others. In other words, some of our students don't pick up new ideas as quickly as their peers. Can something be done about it?
The answer depends on how you explain the discrepancies. Too often, I hear things like: "They should not have been allowed in this class. They don't work. They are not abstract thinkers. They can't remember the most basic things." And so on. If you think it's just that some students are incapable of doing math, well, then there's little you can do about it, and you're off the hook.
But what if what we do as teachers increases the gaps?
- If we rely on a lot of memorization with little or no understanding, we are favoring students who are more docile, and have better memory. The gap between them and those who are less eager to please or more forgetful will increase. Part of the answer to that is a tool-rich pedagogy, which can help replace or complement memorization with understanding.
- If our lessons are boring, once again, we favor obedience over genuine engagement, and the gaps will increase. Part of the answer to that is using "low threshold, high ceiling" problems and rich activities, which can be more motivating for a wider range of students.
But even making progress on those two fronts is insufficient: some students just need more time. I know: there's a lot to cover, and if you had to wait until everyone got it, you'd never get anywhere, and moreover, who wants to penalize the stronger students for the benefit of the others, who may not get it no matter what we do? We've got to keep up the forward motion!
Well, I contend that you can have more forward motion, while at the same time giving enough time for students who need it. Here is a two-prong approach:
1. Lag homework
Today's homework assignment should be about last week's class work. The main reason for doing this is to give students more time to absorb the ideas. Their initial engagement with the topic is entirely in your presence, which helps to make it sink in. An additional benefit of this system is that if something turns out to be difficult and to need more time, you won't need to change the homework assignment frantically during the last minutes of class. You may also see that thanks to better understanding, more students complete the homework, and you may get fewer questions the next day. (All these effects have been mentioned to me by teachers who started lagging homework.) I recommend a one-week lag (or so), but I'm told that even a one-day lag is helpful.
Frequently asked questions:
- Don't students need to practice new ideas? Yes, but they should do that in class at first, where they can get help from you and each other.
- How do students feel about it? A few of them notice this is different from what they're used to, and initially may feel some anxiety, but they quickly get used to it.
- What if your colleagues don't approve? If they're not ready to join you in this practice, tell them you're piloting this idea, and you'll get back to them with the results in a few months. I predict they will join you.
- How does it work at the beginning and end of the term? At the beginning of the term, you might give homework on a topic from last term, instead of spending precious class time on it. At the end, you might schedule a buffer week for review. But in any case, this is a general guideline, which obviously need not and cannot be implemented in a rigid manner.
Previous post on this: Lagging Homework (Do I repeat myself? Very well, then, I repeat myself.)
2. Lag assessments
Do not quiz on this material until the week following the homework week, and have students work on quiz corrections the week after that. That way you have dramatically stretched student exposure to this topic, perhaps from one week, to four weeks! Trust me: some of your students do need that much time. Why not give it to them? You are not taking anything away from kids who learn fast. Quite the opposite: this system allows you to move on to new topics at a good clip!
You may be worried that students will forget everything by the time the quiz comes around. My experience is the opposite: giving more time for students to develop their understanding makes for longer-term retention. When I announce a quiz, my students usually want to make sure it's not on something recent, that they haven't yet had time to fully grasp. (Instead of worrying about remembering, they're worrying about understanding!)
But really, if your students can't hang on to the ideas for an extra week, then why teach those ideas at all?
Previous post on this: Extending Exposure
My third excellent technique to extend student exposure to ideas is to separate related topics. This is an easy form of spiraling, but it does require re-mapping your course. It might wait until next year — but lagging homework and assessments? You can start on Monday. Or, at any rate, after winter break.
What kills me is that most people who hear about these ideas do not try them. Such is the force of habit and tradition. "There's no time" is a common objection to making changes to one's teaching routine, but it doesn't apply to this! Lagging homework and assessments does not take any more time: it's just rearranging the time you were already planning to take.
Of course, if all your students learn math at the same rate, you should ignore these suggestions!
One more link: Practical Implementation