Mathematical Discourse, Part 2: Planning for the Task (Anticipating, Selecting, and Sequencing)
By Zack Hill, posted August 15, 2016 —
last post, we discussed the first step toward high-quality mathematical discourse—choosing a task worth talking about. In this post, we’ll discuss some of the specifics of planning a mathematical discussion. I’m a big baseball fan and feel that a sports analogy
could be helpful here. When a pitcher faces a hitter, he has information that can be useful. The pitcher knows specific locations of pitches (high, low, etc.) and what types of pitches the hitter likes and which pitches he struggles with. By the time the pitcher throws a pitch, he has already anticipated what
could happen and selected and sequenced his pitches accordingly to plan for the best outcome: getting the hitter out. So, how does this type of strategic thinking translate to the classroom?
Just as in sports, much of the act of teaching happens quickly, and making decisions in the moment can be challenging. One of the most important steps in planning a task that you want to use is working through the task yourself. Smith and Stein (2011) identify anticipating student responses (correct and incorrect) as a
critical practice in planning for a mathematical discussion. It’s important to not just work through the task how you would typically complete it but also to try imagining how your students would approach the task. This is an opportunity to consider misconceptions that students may have about the topic as well as strategies
and representations they may use to make sense of the task. Doing this work collaboratively can help gain different perspectives on the task and how students may approach it. This work is crucial and puts you in a much better position to make decisions during a mathematical discussion. See some other
great reasons for “doing the math” in
Champagne and Flynn’s TCM blog post (March 14, 2016).
Selecting and Sequencing
After you have anticipated what students will do with a task, the next step is choosing which solution strategies (and from which students) you’ll highlight during the discussion and in which order you’ll share them (Smith and Stein 2011). We know that on occasion, students don’t do exactly what we have predicted. However,
that shouldn’t discourage us from developing a plan. Also, the plan can change depending on what students actually do with the task that you have selected. During a first-grade lesson I taught this year, I thought I would highlight increasingly efficient strategies but instead shifted the focus to
representations. These types of decisions and, ultimately, this part of the planning process are going to be directly related to your learning goal. You may choose to highlight a few different representations of the same idea to deepen students’ understanding of a concept. Or you may choose to highlight
only a couple of strategies and encourage students to think about the efficiency of each strategy. This point in the planning process is also a great time to think about who you bring into the conversation. I would strongly encourage you to consider students whose voices aren’t typically heard in
classroom discussions and strategically plan to highlight their work.
I have included a
modified version of Smith and Stein’s monitoring chart below. It has been completed for a revised version of Smith and Stein’s (2011) caterpillar task to better align to a fifth-grade standard. (5.OA.3: Identify apparent relationships between terms.)
A fifth-grade class needs 4 leaves each day to feed its 2 caterpillars.
a) How many leaves would the students need each day for 12 caterpillars?
b) Describe how the caterpillar pattern is related to the leaf pattern.
Notice how the anticipating of multiple approaches to the task has been completed (correct and incorrect). Also, note how the strategies have been selected and a potential sequence for sharing has been mapped out.
How do you anticipate student responses to tasks? How do you decide which strategies you’ll share and how to sequence them? What other frameworks do you find helpful? Please share in the comments section below or reach out on Twitter (@zack_hill). In the next post, we’ll discuss monitoring students’ work on a task and how to
plan for students connecting to the important mathematical ideas of the lesson.
Champagne, Zachary, and Michael Flynn. Math Tasks to Talk About (blog). The National Council of Teachers of Mathematics.
Smith, Margaret S., and Mary Kay Stein. 2011. 5 Practices for Orchestrating
Productive Mathematical Discussions. Reston, VA: National Council of Teachers of Mathematics.
Zack Hill has worked in education for fourteen years and is currently an elementary school mathematics staff developer for Pinellas County Schools in Florida. He is currently serving on the board of the Florida Council of Teachers of Mathematics (FCTM) and is a member of NCTM. He
earned his Master’s in Education from the University of Florida and is interested in how mathematical discourse supports understanding of mathematical concepts.