Teaching Mathematics Through Purposeful Discourse

The idea of elementary school students collaborating on mathematical tasks is longstanding.
Recent focus on creating purposeful discourse in math classrooms leads to the need to revisit discourse-promoting strategies.
The Common Core’s Standards for Mathematical Practice (CCSSI 2010) promotes opportunities for conjecture-making, situation analysis, and discussion of solutions among students.
Purposeful discourse aids in developing students’ conceptual understanding (NRC 2001) around high cognitive-demand tasks (Smith and Stein 1998).
Challenges exist in facilitating discourse; the necessity arises on how teachers can implement lessons promoting this discourse effectively.

Focus on the launch phase of a lesson, which prepares students for high-demand tasks.

The think-aloud strategy is adapted from literacy instruction (Davey 1983), which helps students monitor their thinking and comprehension (Bereiter and Bird 1985) in reading.
Effective think-alouds lead to better text comprehension in developing readers (Baumann, Jones, and Seifert-Kessell 1993; Kucan and Beck 1997).
Think-alouds encourage students to articulate their thinking.

When used to launch a mathematics lesson on a high-demand task, teachers should model:
- The type of thinking that fosters conceptual understanding.
- How to articulate one's thoughts or thinking processes.

The method of task setup is crucial as it determines student engagement: Jackson et al. (2012) suggest how task organization directly impacts the achievements of students and teachers in the lesson.

In a forty-hour year-long professional development project (All Included in Mathematics - AIM), second-grade teachers were introduced to think-aloud strategies.
Teachers utilized think-alouds for launching lessons on addition and subtraction story problems.
Reports from teachers indicate positive engagement, with students eager to share thoughts during think-aloud-launched lessons.

Consider the story problem: "Jason put 4 large plates and 8 small plates on the table. How many plates are on the table in all?" (Fuson 2009).
Teachers distinguish between two types of think-alouds:
- Think-Aloud 1: Prescriptive model suggesting operations done via keywords.
- Think-Aloud 2: Reveals teacher’s thought process, exemplifying how to think aloud, which aligns better with desired student learning objectives.
Teachers practiced using the second type of think-aloud effectively.

Example of a Prescriptive Think-Aloud
- "If you read it carefully… ‘in all’ means addition… 4 plates and then 8 more… 4 + 8 = 12".
The above method focuses on modeling a strategy rather than encouraging dynamic student input.

Teachers noted successes:
- Think-alouds set students up for success in related problems.
- Encouraged discourse during group work and led to precise vocabulary use (e.g., more, less).
Challenges identified by teachers:
- The balance between modeling thought processes and solving problems explicitly, risking a shift from think-aloud to mere procedure demonstration.
- Visual models could lead students to repeat the visual rather than engage critically with the problem.

Importance of establishing norms for collaborative discourse during think-alouds:
- Teacher Norms: The teacher clarifies that it is their time to think aloud, setting expectations for active listening from the students.
Example dialogue setting norms:
- Teacher: "It's my turn to share my thoughts, you need to just observe and think."

Ensure students understand the think-aloud is a time for teacher modeling rather than student responses and problem-solving.
Decide the purpose of the think-aloud in terms of what students should focus on.
Connect mathematical ideas to prior knowledge, using phrases that aid comprehension (e.g., "This is similar to…").
Use additional support for diverse learners: drawings, gestures, vocabulary, and synonyms.
Prepare and plan, possibly scripting talk points in advance to maintain focus on modeling thought without directly providing solutions.

The structured think-aloud serves to promote discourse, allowing students to take responsibility for their learning while discussing their peers' strategies and solutions.
Skills developed through repeated modeling assist students in engaging in constructive discourse, which leads to collaborative learning.