Purposeful Pedagogy and Discourse Model

Purposeful Pedagogy and Discourse Instructional Model: Student Thinking Matters Most (by Linda Jaslow and Aimee L. Evans)

In studying the Common Core State Standards for Mathematics (CCSSM), and in particular the Standards for Mathematical Practice (SMP), it becomes clear that what we do in the classroom will change both from the perspective of the teacher and the student.  The teacher will need a deep and connected understanding of the mathematics content and, during instruction, will need to provide experiences that allow the students to construct meaning for themselves through carefully crafted tasks and conversations.   Students will need to reason, communicate, generalize and challenge the mathematical thinking of themselves and others.  Student thinking matters most.

The purposeful pedagogy and discourse instructional model that we are using in the Arkansas CCSS Mathematics Professional Development Project, is based on the research of four sets of researchers:

·      Jacobs, Lamb, and Philipp on professional noticing and professional responding;

·      Smith, Stein, Hughes, and Engle on orchestrating productive mathematical discussions;

·      Ball, Hill, and Thames on types of teacher mathematical knowledge;

·      Levi and Behrend (Teacher Development Group) on Purposeful Pedagogy Model for Cognitively Guided Instruction.

This model is intended to support teachers to deliver strong mathematical content using critical best classroom practices as well as to develop a learning environment where their students regularly use the 8 Standards for Mathematical Practice.

Assessing Students, Professional Noticing, and Teacher Mathematical Knowledge

At the core of our model is assessing students (TDG-CGI model), which refers to taking a close look at student understanding.  While assessing students, we apply the concept of professional noticing (Jacobs et al.). 

Professional noticing is comprised of 3 teacher skills:

·      Attending to children’s strategies,

·      Interpreting children’s understanding, and

·      Deciding on how to respond on the basis of children’s understanding.

In order to assess a students’ understanding, we must look at the details of their thinking (what did they do) and then mathematically interpret these details.  While this may seem trivial, students’ strategies are complex and many deep mathematical operations and properties are embedded implicitly in their work.  It takes time to identify the important details in students’ thinking and then mathematically interpret the relationships and properties of operations that are embedded.  The ability to notice will help the teacher identify the mathematics available for exploration during the lesson(s) to follow. Since student thinking matters most, in the Arkansas professional development courses  the beginning of most classes will involve just making sense of and deepening our understanding of the details of students’ strategies and the mathematical ideas embedded in their strategies. 

The deeper and more connected a teacher’s mathematical knowledge is, the easier it is to see and interpret the details of student thinking.  Teaching mathematics requires a variety of types of knowledge as shown in the figure below. 

One type of teacher mathematical knowledge is specialized content knowledge – the mathematics behind the mathematics.  For example, it is not enough to know we can divide fractions by inverting the second fraction and multiplying.  A teacher must understand the mathematics that allows that strategy to work.  Teachers must also understand how children will approach various problems, how their thinking develops, and how students’ thinking is different than adults’ thinking.  This knowledge is called knowledge of content and students.  All of this comes together to create the critical part of professional noticing, identifying the details of children’s thinking and mathematically interpreting the details, which allows us to assess students’ thinking, which of course matters above all else.

Exercising Professional Noticing

Click here for a list of vignettes for this section: Exercising Professional Noticing.

Professional Responding, Purposeful Pedagogy, and Orchestrating Classroom Discourse

Critical instructional decisions are based on the mathematical interpretation of students understanding.  With specialized content knowledge and knowledge of content and students in place, we are ready to focus on our mathematical practice. The Purposeful Pedagogy Model (TDG; Cognitively Guided Instruction) and Orchestrating Classroom Discourse (Stein et al.) come together to give us a vision of such practice centered around the all important student thinking. 

The Purposeful Pedagogy Model has three components: assess students, set a learning goal, and design instruction. 

Elements for the design of the instruction are defined by the Orchestrating Classroom Discourse research.

Orchestrating Classroom Discourse outlines 5 practices for doing so:

1.     Anticipating likely student responses to cognitively demanding mathematical tasks;

2.     Monitoring students’ responses to the tasks during the explore phase;

3.     Selecting particular students to present their mathematical responses during the discuss-and-summarize phase;

4.     Purposefully sequencing the student responses that will be displayed;

5.     Helping the class make mathematical connections between different students’ responses and between students’ responses and the key ideas.

We will use the details of student understanding to set learning goals for our students, design instruction, and orchestrate classroom discourse.  In doing so, we are engaging in the comprehensive practice of professional responding. 

This is best understood by taking a look at another classroom vignette.
Click here for a list of vignettes for this section: Professional Responding, Purposeful Pedagogy, and Orchestrating Classroom Discourse.

While this type of exchange requires the classroom teacher to think very purposefully about instructional decisions and to think deeply about the mathematics embedded in students’ solutions, the effort is worthwhile.  The evidence comes from Cognitively Guided Instruction, an instructional model that emphasizes these very practices. Visits to CGI classrooms in Arkansas will reveal that children are thinking more deeply and flexibly about mathematics.  They are not simply solving problems that have no meaning to them; they are becoming young mathematicians capable of explaining their thinking, which matters most, and grappling with and making sense of the complexity of the mathematics.

How do we now take the information we have about students’ thinking and professionally respond in a way that is based on students’ understanding and designed to facilitate children’s thinking along a learning trajectory?  We must select or design appropriate mathematical tasks or problems.

Mathematical tasks should be selected that will facilitate children’s development.  Once we have identified the task, we should consider the following questions:

·      What do we anticipate students will do with the task? 

·      Will this task provide the experiences needed to further students’ development? 

·      Which of the strategies we expect are likely to help the most in making sense of the mathematics in the goals we have set for them? 

The next stage is to pose the task or problem and allow the students to solve the problem in a way that makes the most sense to them.  Our job is to monitor students to identify what students are doing, guide them as they work, and decide which students’ papers should be shared.

Back to Teacher Mathematical Knowledge

Once we have identified the best student strategies to meet the learning goals, we need to decide in which order to share students’ strategies and what mathematical connections should be the focus of the classroom discussion.  Again, the teacher’s mathematical knowledge, specifically her knowledge of content and teaching (Hill & Ball, Figure 1), will be critical in making decisions by being able to envision how the mathematics available through the students’ strategies connect to one another and to the mathematics concepts that are desired. 

At this juncture, the teacher’s knowledge of the mathematics meets the need to design or plan the discourse to take students deeper into the mathematics.  This involves both the sequencing of the presentation and also the selection and phrasing of the questions posed during the discourse.  There are likely multiple productive paths, but there are certainly some unproductive or problematic paths as well, and the teacher will need to choose well.  Student thinking matters most.

Seeing It All Together

The research of these four sets of researchers come together to create the instructional model that we are using in the courses for the Arkansas CCSS Mathematics Professional Development Project.  While this model, being a blend of the work of so many different projects, may seem complex at first, it is perhaps more straight forward when viewed using the graphic organizer below.  The key ideas that hold the model together are the importance of noticing the details of student thinking, interpreting those details, and using that information to design instruction comprised of discourse around student strategies aimed at a specific mathematical goal. In other words, what details do we see in our children’s work, how do we interpret their thinking, and where mathematically do we go from there?  In maintaining this focus throughout the professional development courses, it is our hope to support teachers in their journey toward achieving mathematical proficiency for their students as described in the CCSSM.  And always remember, student thinking matters most.


Hill, H., Ball, D. L., & Schilling, S. (2008). Unpacking “pedagogical content knowledge”: Conceptualizing and measuring teachers’ topic-specific knowledge of students. Journal for Research in Mathematics Education, 39 (4), 372-400.

Jacobs, V. R., Lamb, L. L. C. & Philipp, R. A. (2010). Professional noticing of children’s mathematical thinking.  Journal for Research in Mathematics Education, 41 (2), 169-202.

Jacobs, V. R. & Philipp, R. A. (2010). Supporting Children’s Problem Solving. Teaching Children Mathematics, 17 (2), 99-105.

Smith, M. S., Hughes, E. K., Engle, R. A., & Stein, M. K. (2009). Orchestrating Discussions. Mathematics Teaching in the Middle School, 14 (9), 549-556.

Stein, M. K., Engle, R. A., Smith, M. S. & Hughes, E. K. (2008). Orchestrating productive mathematical discussions: Five practices for helping teachers move beyond show and tell. Mathematical Thinking and Learning, 10 (4), 313-340.

              Thames, M. H. & Ball, D. L. (2010). What math knowledge does teaching require? Teaching Children Mathematics, 17 (4), 221-229.