By Zack Hill, posted August 1, 2016 —

Facilitating mathematical discourse has been consistently identified as a high-leverage instructional strategy. This is the first in a series of four blog posts that will examine various aspects of mathematical discourse. During the past ten years, I have been intrigued by student discussions that revolve around mathematics. The book Five Practices for Orchestrating Productive Mathematical Discussions (Smith and Stein, 2011) has had a huge impact on my teaching. If you’re interested in mathematical discussions, I highly recommend reading it. Many of the ideas in this series of posts will come from this resource or have been shaped by it.

The first step toward high-quality mathematical discourse is to choose a task that’s worth talking about. Consider the following two questions:

• How many sides does a triangle have?
• How is a triangle different from other polygons?
  

The first question doesn’t give students much to talk about. The answer is three. The second question requires more thought and could generate a good discussion about some other important questions: What makes a triangle a triangle? What is a polygon? In what way (or ways) are they different? Are they alike in some ways? If so, how? Consider two big questions as you decide whether a task is worth talking about:

1.   Is the task a high-cognitive-demand task?

One way of determining whether a task is worth talking about is considering the cognitive demand of the task. Smith and Stein (2011) identify four levels of cognitive demand:

a.   Memorization

b.   Procedures

c.   Procedures with connections

d.   Doing mathematics

Tasks that fall into the last two categories are high-cognitive-demand tasks and are complex enough to give students something to talk about. A few general characteristics of these types of tasks are that—

• They are open ended in that they don’t outline an explicit solution pathway.
• They require students to make connections to prior knowledge and experiences.
• They allow students to explore big mathematical ideas.
• Students can learn from engaging with the task.
• Teachers can gain insight into student thinking by observing how students engage with the task.

2.   Does the task support the mathematical goal of the lesson?

NCTM identifies establishing mathematical goals to focus learning as another high-leverage instructional practice (2014). This practice is closely related to facilitating a purposeful mathematical discussion. Smith and Stein warn against losing the point of a lesson as various solution strategies are shared, potentially allowing it to become “scattered in the ‘mathematical landscape’”(2011, p. 6). One way to ensure that the point of the lesson is not lost is to identify a clear learning goal, then make certain that the task you choose aligns with the goal.

For instance, if addressing the Common Core State Standard 4.OA.C.5, “Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself,” you might choose a task that specifically provides students with an opportunity to “identify apparent features of a pattern that were not explicit in the rule.” The Illustrative Mathematics task below could be used to give students the opportunity to engage with and discuss this specific aspect of the standard. This type of alignment increases the odds that the discussion will support the specific standard or aspect of the standard that you’re teaching.

Here are some great places to look for standards-based tasks:

• Illustrative Mathematics—Most of the standards have tasks that could be used for instruction.
• Graham Fletcher has a nice collection of standards-aligned three-act tasks that are great for students to discuss.
• This is a collection of formative assessment tasks that could be used as instructional tasks. It’s a Florida resource, so the coding changes slightly; for example, 2.NBT.B.2 becomes 2.NBT.2.2.

Your turn!

What resources or tasks are you currently using to engage students in meaningful mathematical discussions? Please share in the comments section or reach out on Twitter (@zack_hill). In the next post we will discuss in further detail how to plan for a task.

References

National Council of Teachers of Mathematics (NCTM). 2014. Principles to Actions: Ensuring Mathematical Success for All. Reston, VA: NCTM.

Stein, Mary Kay, and Margaret Schwan Smith. 2011. 5 Practices for Orchestrating Productive Mathematical Discussions. Reston, VA: The 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.