"There is no one way"

Sunday, October 7, 2018

More on Extending Exposure

I have written quite a few posts in which I argued that extending student exposure to mathematical concepts is one key to reaching the whole range of students. This is based on the simple observation that students learn math at different rates, and that extending exposure by making simple changes to our routines can benefit all students: those who pick up new ideas quickly, and those who need more time. If schools heed NCTM’s recommendation to eliminate tracking, making these simple changes becomes even more important.
On this blog, the most popular post on extending exposure is Lagging Homework, and it links to other posts where I describe additional strategies (separating related topics, lagging assessments, and more.) If you haven’t read those posts, you should. The problem is, it does involve a bit of clicking around. Thus, I decided to combine all that information in a single longer article on my Web site.
However, before I do that, I need to share a few more related tidbits which hadn’t made it into those posts. Writing about those here will help when I’m ready to put it all together.
The article is written. Read it here: Reaching the Full Range.

Homework as Preparation

A number of people, over the years, have told me that they don’t agree with my lagging homework system because they like to assign homework that prepares the students for the next day’s lesson. That, my friends, is not a disagreement! I love that idea.
Lagging homework is not a rigid system that requires homework to be assigned exactly one week (or day, or month) after the corresponding class work. My main point is that on most days, you should not assign homework based on the day’s lesson. A week’s delay, more or less, provides many advantages which I described in my original post, the main one being extended exposure to each topic. This in no way precludes homework that sets up the next day’s lesson, as long as it is not usually based on the day’s lesson. I described the characteristics of such preparatory problems in this post, based on Scott Farrand’s approach to in-class warm-ups.
In general, such problems are actually examples of long-lagged homework, based on ideas that were introduced the previous semester, or the previous year, or whenever. Such long lags can also be used for review (better than take precious class time for that). On the other hand, if preparing for the next day’s lesson requires completing homework about today’s lesson, and this needs to happen frequently, then I strongly discourage that as the collateral damage on some of your students is substantial. (Unfortunately, it is the students who suffer from this policy will get the blame.) Work based on the day’s lesson should be done in class, perhaps as warm-ups the next day.

Two Units

I had the privilege of teaching in long periods for my whole career. One way I used what is sometimes known as a block schedule was to focus on two units at any one time. For example, here is the outline of semester 2 in a “Math 2" class I taught before my retirement:
(My point is not to recommend this exact sequence, which depends on many
department-specific assumptions, but to use it as an example of what is possible.)
Here are some of the advantages of that approach.
  • Any one day or week is more varied, which is helpful in keeping students interested and alert. Note in particular that we tried to match topics that are as unlike as possible.
  • It takes roughly twice as many days to complete a unit. This is good for students who need that extra time.
  • This takes nothing from students who pick up ideas quickly. In fact, they appreciate the variety.
  • It makes it easier to balance challenging and accessible work: if you hit a difficult patch in one topic, you can ease up on the other one. More generally, you gain a lot of flexibility in your lesson planning.
  • If your work on one topic hits a snag, you can emphasize the other topic while figuring out what to do.
  • Perhaps most importantly, it carries a message to the students: you still need to know this when we’re working on something else.
In the long period, it is possible to hit both units in every class period, for example by introducing new ideas on one topic during the longer part of the period, and applying already-introduced ideas on the other topic in the remaining time. (Homework is typically on one or the other topic.) Can this approach, or a version of it, be used in traditional 50-minute classes? I don’t know. I am guessing that the answer is yes, but I have not tried it. It might involve, for example, focusing on each topic on alternate days, while the homework is on the other topic .
But, you ask, is this not confusing to students? Don’t they prefer focusing on one single topic? If they do, that is only because that is what they’re used to. At my school, this was a department-wide policy, and once they’re used to it, it does not even occur to them to question it. In teaching, the biggest obstacles to making changes are the cultural ones. The only way to tackle these obstacles is departmental collaboration, and a step-by-step approach: don’t make all the changes at once!


  1. I appreciate your identification of extended exposure as a crucial aspect of instruction. It helps me to think about other aspects of instruction and how they relate. For example, one effect of making connections between two mathematical ideas is that it creates exposure to both, thus extending.

    I believe that one of the reasons that homework typically is not as effective as we'd like it to be is that while it is a way of extending exposure to the mathematical ideas and skills, it rarely serves to deepen the ideas. It sometimes deepens the skills, but usually all that happens is that the problems get messier as the homework assignment goes on. It is difficult to provide homework problems that a student can do in isolation, and that also deepen their understanding. Textbooks traditionally make little or no attempt to do this. Technology provides some ways of supporting homework that deepens the exposure to mathematical ideas, but I am unaware of much progress on this front, and I hope that this will get more attention.

    I'm really enjoying your blog posts Henri! Your thoughts resonate with me, and they prompt me to think. Thanks.
    --Scott Farrand

    1. Thanks for the compliments!

      My sense is that not much depth can be expected from homework, and that most significant learning happens at school. Thus for me homework was short, and its main purpose was to trigger good conversations in class the next day, as groups "went over" it. I wrote about that here:

  2. I appreciate the reminders about lagging homework and the importance of extending student exposure. I'm looking forward to your next article about over-spiraling as well as the compilation! Thank you for sharing your work and wisdom.