By Zack Hill, posted August 1, 2016 —
Facilitating
mathematical discourse has been consistently identified as a highleverage
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 highquality 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 highcognitivedemand 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 highcognitivedemand 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 highleverage
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 standardsbased tasks:
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.