Supercharging Three Common Practices in Math Classrooms 

By Kristin Klingensmith, Beatriz Font Strawhun, Brenda Robles

A Bit About WODB Activities

Which One Doesn’t Belong activities are designed to spur critical thinking. In a WODB, participants are presented with a set of items. They are asked to identify which one of the items doesn’t belong and explain their reasoning. The goal is to encourage participants to think critically, make connections, and justify their choices based on specific criteria. The key aspect of a WODB activity is that there can be multiple valid reasons for selecting any of the items as the one that doesn’t belong, making it a flexible and open-ended learning tool.

Now give it a go! Which would you say doesn’t belong and why?

Scenario B – Sowing Seeds for a Discussion

During a WODB warm up activity, the teacher has the students seated in groups. Students have a variety of tools, writing utensils, and their own copy of the WODB. Students are asked to circle which one does not belong and stop and jot why they would eliminate it. After having some individual think time, the teacher asks students to turn and talk about which one they eliminated and why they kept the other ones. Meanwhile, the teacher circulates the room listening to the turn and talk between shoulder partners. The teacher purposefully selects a shoulder-partner pair of students to share what they heard from each other. After they share, the teacher asks the class about the connections they can make to their own thinking. The teacher continues to ask a series of questions that invite other students into the conversation by linking their thinking to that of their peers and asking students to revoice others’ thinking. The teacher uses questions like

• Thank you, Sam. Justin, you also said you thought B did not belong? Was your reasoning similar to or different from Sam’s?
• How is your thinking related to Jing’s?
• Who can say back what Jing’s reasoning was?
• Marco, how was your argument different?
• Who agreed with Marco? Elena, say it for us one more time. What was the reason both you and Marco said C did not belong?

The teacher also encourages students to pose questions to each other as clarification is needed. They also mark the key observations and critical ideas about mathematics that students share by charting the contributions publicly to be used later. The warmup ends with the teacher saying that they will return to the claims and ideas that have been charted during the lesson.

Benefits of Turn and Talk in Scenario B:

Students receive all the same benefits expressed in Scenario A plus the following.

• The teacher has a better understanding of the students’ understanding of the concept by listening to the turn and talk.
• The teacher sets an entry point into the discussion by intentionally selecting which student pair will share first. This decision may be based on the most common thinking to surface in the group, the earliest access point into the conversation, a misconception that surfaced, or a generalized claim, to name a few.
• Shifting the focus of the turn and talk from just what did you eliminate to how are the remaining items connected increases the likelihood that students will engage with the mathematics.

Some tips for maximizing the use of a Turn and Talk:

• Layer learner-centered routines like asking students to engage in a stop and jot before having them do a turn and talk. This gives students a bit more time to get their own thinking in place before sharing with a partner. Including a stop and jot tethers oral communication with written communication which can help to deepen a student’s understanding.
• Vary the questions posed to include reasoning for both examples and non-examples, like asking students to reason about why a particular item was eliminated and why were the other kept.

Mathematical Representations and the Connections Among Them

National Council of Teachers of Mathematics (NCTM). (2014). Principles to Actions: Ensuring mathematical success for all. Reston, VA: NCTM.

Adapted from [Lesh, Post, Behr (1987) Representation and Translations Among Representations in Mathematics Learning and Problem Solving] and modified from [Huinker, D. (2015) Teaching Matters: Actions for Attaining High-Leverage Teaching in Every Mathematics Classroom]

Scenario D – Leveraging and Connecting Multiple Representations

A class is part-way through a unit on operating with negative integers. The teacher asks students to solve the negative integers problem using whatever representation and strategy they like. The teacher walks around to see how the students are working. The teacher then groups the students into triads in which each student has a different representation and/or strategy for arriving at the solution. For some triads, this includes student work that contains an error or misconception. The teacher asks the students in the triads to look at each other’s work to identify how each of the parts of the problem are represented even though the representations and approaches are different. Additionally, students are expected to construct an explanation in writing and be ready to share their collective thinking about how all three pieces of work are related to each other and the mathematics of the task.

Benefits of creating and discussing Mathematical Representations in Scenario D:

Students receive all the same benefits expressed in Scenario C plus the following.

• Students must make connections between and among various mathematical representations.
• Students reason about the mathematics more deeply as they translate it across different representations and/or strategic approaches.
• Students critique the reasoning of others; in some cases, this may include revision of an error or misconception.
• Students work together to construction an explanation in writing (which serves as another representation) and requires synthesis across the ideas shared by the three students.

Some tips for maximizing students use and connection of representations:

• Expect students to create and explain various mathematical representations and orchestrate opportunities that require students to make connections among the representations to reason about how the math is translated in each of them.
• Select or create problems that allow for a variety of representations and strategic approaches AND allow students to choose what works for them so that there will be a range of solution paths that can be shared and discussed.

Facilitating Whole Group Math Discussions

Scenario E – Having a Show and Tell

Students have been working on a high-level task. They have had a chance to work on their own and in a small group. While they worked, the teacher walked around to learn about their current thinking and pose questions to advance their thinking.

Following the small group time, the teacher pulls the students together for a whole group discussion. The groups of students take turns sharing their thinking by holding up their papers from their desks as their peers listen.  When one group finishes the next group starts. Each group gets to share, and by the end of the discussion, all six groups have shared their work.

Benefits of facilitating whole group discussion in Scenario E:

• The teacher positions students as holders of knowledge by creating space for them to share with a peer.
• Students have to be able to explain the work that they did with peers.
• Students get to listen to how other groups thought about and solved the task which may expose them to different ways of thinking.
• Students may acquire more academically precise language from hearing other groups discuss their work.

Scenario F – Sharing, Connecting, and Growing Collective Reasoning

Students have been working on a high-level task. They have had a chance to work on their own and in a small group. While they worked, the teacher walked around to learn about their current thinking and pose questions to advance their thinking. As the teacher visits and revisits each group, the teacher identifies the portion of the group’s work that the students will be invited to share during the whole group discussion. The teacher shares this with each group before moving onto another group.

Following the small group time, the teacher pulls the students together for a whole group discussion. The teacher strategically calls on group members to show and share the starred portions of their work, making a communal poster of work associated with the task. As the groups take turns sharing their thinking and building onto the communal work, their peers listen and are asked to make connections between the existing information and the new information being shared. Many of the groups share their thinking and by the end of the discussion, members from all six groups have actively engaged in the discussion by sharing and/or making connections between and among their work and what has been shared by their peers.

Benefits of facilitating whole group discussion in Scenario F:

Students receive all the same benefits expressed in Scenario E plus the following.

• Students are positioned as the authors of the work from which the collective group develops a common and shared understanding of the mathematics.
• The teacher sets the tone of the discussion and can work to ensure that the thinking of all groups surfaces in the communal work.
• Student thinking from multiple entry points can be honored since the teacher is selecting the portions of the work each group will share.
• Students have access to a shared visual record of their collective thinking which can serve as anchor for this (and future) discussions.

Some tips for maximizing the facilitation of a whole group mathematics discussion:

• Monitor and identify specific elements of work that can be shared by each of several groups to ensure that a variety of thinking is reflected, being sure to look for a range of entry points and opportunities to discuss common misconceptions or errors to get at the “why” and “how” of mathematical concepts, rather than just the “what.”
• Encourage students throughout the discussion to respond to and build upon each other’s ideas with the goal of prioritizing deep and shared conceptual understanding over mere reporting out of procedures used to arrive at a calculation.

In this article, we hoped to highlight the impact that different ways of implementing common practices have on students’ opportunities to engage with the mathematics and each other during a lesson.  Though all the illustrated scenarios offer some benefits to the learners, we can see that minor tweaks supercharge these common practices and significantly impact the opportunities afforded students.