I’m still thinking about Emma Gargroetzi and Dan Meyer’s responses to the August New York Times op-ed on drill-based math teaching. The comments are fascinating. Here’s mine:

I notice that math teachers often draw a dichotomy between rich, open tasks and drill-oriented practice. I wonder if it would be helpful to try and articulate some of that middle ground, the rich tasks that also act as practice, practice that one can look back on and draw new connections, and any number of other places to bridge the gap and help teachers move more fluidly between open tasks and practice.

I still agree with my comment, but I’ve had trouble with what that articulation might look like. Here’s an attempt.

First, what is a rich task? I don’t think any one definition can capture the subtlety I find here, but a rich task has some (though rarely all) of the following qualities:

- Lends itself to multiple strategies
- Has a low floor for entry, whether through solving intermediate problems, making estimates, visualizing, or other places for students to recognize what they already know early in the problem
- Has a high ceiling, naturally leading to extensions or additional tasks
- Allows multiple representations, in particular visual representations
- Has an element of perplexity, provoking students’ curiosity
- Allows some experimentation or trial and error, and meaningful reflection on that work
- Lends itself to intuition
- Starts humble but leads to multiple useful mathematical ideas
- Values concepts and connections over procedures
- Gives students something to argue and collaborate about
- Involves ambiguity and requires making sense of mathematical ideas

Most of all, a rich task captures a slice of the richness of the discipline of mathematics. Rich tasks are hard for students; they involve new norms in math class, often require a positive disposition toward learning math, and can overwhelm students to the point where they aren’t learning. I think they should be used judiciously. But a large part of their value comes in exposing students to the beauty and complexity of mathematics.

Next, what is drill? I don’t like the word drill because of the connotations it brings in, but I do value practice. At a basic level, practice means retrieving ideas from long-term memory to strengthen connections, and often to make new connections as practice tasks increase in complexity.

I see these as two different purposes of math class, and purposes that aren’t necessarily in tension. While some folks might characterize one side as good and the other as bad, I think both rich tasks and practice have important places in math class, and useful opportunities for synergy.

A rich task can be used to introduce a topic by creating intellectual need for an idea, help students learn something new by taking what they already know and extending it a step further, or to give students an opportunity to apply what they know at the end of a unit. Those are very different purposes, and each purpose relies on choosing tasks thoughtfully, facilitating with clear goals, and supporting students to find success.

At the same time, a rich task can be practice. Ben Orlin’s Give Me and Open Middle are great examples. Practice can lead to a rich task, where students practice a skill they already know, then step back to look at patterns in their work and learn something new. Practice can incorporate elements of a rich task, and rich tasks can be interspersed with practice. Studying worked examples is a great bridge between rich tasks and practice that gets students thinking, while also focusing their thinking on specific ideas.

Rather than thinking of these ideas in opposition, I think of them on perpendicular axes. I start planning with a goal for a lesson, and based on that goal I think about what will help my students reach it. I want to offer richness, and I want to offer practice, and I want to find as many opportunities as I can to do both in ways that build off of each other.