My Math Education Blog

"There is no one way"

Monday, August 12, 2013

Extending Exposure

Not all students learn new math concepts at the same rate. Everyone knows this. Can anything be done about it?

Most schools attempt to make the problem more manageable by tracking students into Honors, Regular, and Remedial tracks, or some variation on that. And yes, while on balance this may not be optimal, it does narrow the range in any one track. But even then, the problem is not solved: every class is heterogeneous.

My general answer is that one should strive simultaneously for constant forward motion, and eternal review. To achieve forward motion, it is necessary to start on a new topic before the previous one is grasped by everyone. Eternal review is of course impossible, but much review can be built in to a course without harming the forward motion. The key is to find ways to extend exposure to a given topic.

In previous posts, I described two techniques that support this goal. The first was lagging homework. The second was separating related topics. Today, I will discuss how the curriculum itself and assessment strategies play into this issue, and show how those different components can be orchestrated in designing an effective course.

The curriculum itself: teachers often ask me for "the best way" to teach a given topic. There is no good answer to this, because the question itself is flawed. Different ways will work better for different students, and a good teacher should know many ways to teach anything important. One part of this can be achieved by using a tool-rich pedagogy. A related approach is to use multiple representations of important concepts: numerical, graphical, manipulative, geometric, applied, symbolic, and so on. Each representation offers a different entry point, and making connections between them deepens understanding. Using multiple tools and representations makes it possible to extend student exposure to a topic without boring the quicker learners.

Assessment: the quiz or test should not be the end of the road on a given topic. It helps students know what they know and don't know, and they should get a chance to work on test corrections for credit, with the knowledge that any topic (particularly the hardest ones) is likely reappear in a later assessment. One way to manage this is to have short, focused quizzes, but to make sure that more substantial tests are at least somewhat cumulative. I tell students that anything which stumped many of them on a test is sure to appear on a future test. It is not sufficient (let alone effective or fair) to only offer a cumulative assessment at the very end of the semester or course.

Here is a schematic representation of one way to combine some of these ideas. ("Recycle" = test corrections, done as homework, with a higher standard of explanation than one would expect on the quiz itself. Students can get help on those, but must write everything in their own words.)

This scheme extends exposure and at the same time ensures forward motion. Combine that with the above curricular and sequencing suggestions, and you're on the way to a class that will work better for a wider range of students.

--Henri

PS: for more teaching suggestions, see my cheerfully titled article: "Nothing Works".