Approximating Practice in Mathematics Teacher Education: Supporting and Revealing Teacher Candidate Learning

Matthew Campbell, Ph.D.

West Virginia University

Northern Arizona University

STEM Education Seminar Series

March 3, 2022

Overview of Talk

• “Notice and wonder”
• Background on practice-based pedagogies in teacher education
• Details on my work with coached rehearsals and scripting tasks in secondary math methods courses
• Three considerations of using approximations of practice:
• Representing student voice
• Considering “authenticity” and “complexity”
• “Revealing” teacher candidates’ resources

Acknowledgements

• Erin Baldinger, University of Minnesota
• Foster Graif, St. Cloud State University
• Josh Karr, West Virginia University
• Sean Freeland, West Virginia University
• Rebekah Elliott, Oregon State University
• Ron Gray, Northern Arizona University

What do you notice? What do you wonder?

A Scenario

What do you notice? What do you wonder?

Teacher: How did you use the axis labels in your answer?

Foster: I thought of the x-axis as horizontal distance and the y-axis as vertical distance.

Teacher: it looks like the x-axis is time. So zero seconds after leaving home the student is 0 meters away. 50 seconds after leaving home the student is 100 meters away. What type of distance do you think this is? Horizontal or vertical?

Foster: horizontal.

Teacher: so given time on x-axis and horizontal distance on y-axis, what is this graph telling you about the student’s journey?

Foster: That the student’s distance was decreasing at one point. So they must have turned around for a little bit.

Teacher: How did you get that answer?

Foster: You can see the student's journey on the graph.  As he's walking, you can see that he first goes up then down and then up again?

Teacher: What other answers did people get?

Jane: I thought that Foster walked away then towards and then away again.

Teacher: How do you see that?

Jane: Well, the x-axis tells you where Foster is at any one moment in time and the y-axis is the distance.  So as time passes, the student first walk farther, then closer, and then farther again.

Teacher: What other ways are people interpreting this graph?

*Wait time

Teacher: Okay, we have to possible interpretations of this graph.  Think quietly about which one you think is best and why for a minute or two before pairing up with your shoulder-to-shoulder buddy and discussing what you each think.

The Current “Moment” of Practice-Based Teacher Education

• Ball & Cohen (1999) - Situating teacher learning in the practice of teaching
• Grossman & McDonald (2008) - Calls for a “common language of teaching” and for “pedagogies of enactment”
• Shifting from an exclusive focus on “knowledge and beliefs”
• Lampert (2010) - “The practice of teaching” — What is it that teachers do, and how do they do it skillfully?
• Two interrelated strands of ideas: “Core practices and “Pedagogies of practice”
• Recent critiques of practice-based teacher education and a focus on “practice as core” (e.g., Philip et al., 2019)

Pedagogies of Practice in Teacher Education (Grossman et al., 2009)

• Representations of practice
• Decomposition of practice
• Approximations of practice - Opportunities for enacting core components of teaching in contexts and situations of reduced complexity

My Background with Practice-Based Pedagogies

• Early doc program: Use of tools such as video, student work, “MKT tasks”
• Mid-doc program: Initial efforts to understand use of coached rehearsals
• Dissertation: “Responsive pedagogies of practice” - Design research on use of rehearsals in secondary mathematics teacher education (Campbell, 2014; Campbell & Elliott, 2015)
• Since 2014, at WVU (in collaboration with Erin Baldinger) - Design and use of coached rehearsals and scripting tasks in secondary math methods courses
• Focus on the practice of “responding to student errors in whole-class discussions”

Coached Rehearsals

• Teacher candidates (TCs) lead a defined activity with their peers and/or teacher educators acting as students, with a focus on a particular practice(s)
• Rehearsal is not microteaching; rehearsal is more bounded and more purposefully focused
• Teacher educator provides in-the-moment feedback, responding to questions, opening up conversations around problems of practice
• Reduced complexity provides space for experimentation and learning
• Not an oversimplification — includes complex instantiations of practice
• Focused on interactive practices — students are an important part of the context, and the rehearsals require TCs to be responsive to student contributions

Images from Coached Rehearsals

Using Coached Rehearsals

• “Instructional activities” focus on leading whole-class discussion, drawing on common routines in secondary math classrooms
• e.g., Number Talks, Sorting Tasks, Representation Talks, Comparing Quantities, “Going Over a Problem”
• One TC is the rehearsing teacher; other TCs play students (though “as themselves”)
• Focus on responding to student errors – “planted errors”
• Possible follow-up activities - debrief discussion, video annotation, implementation in classroom

Scripting Tasks

• TCs produce dialogues of classroom interactions and rationales for these scripts in response to a classroom scenario
• Less authentic than coached rehearsals
• Not an active interaction with others (in fact, TC is responsible for creating all of the dialogue)
• More time to make decisions about how to act
• Different affordances for TCs and teacher educators
• Linking action with rationale
• Insight into TCs’ view of students and classroom interactions

An Example of a Scripting Task

An Example of a Scripting Task

Scripting Tasks

• TCs produce dialogues of classroom interactions and rationales for these scripts in response to a classroom scenario
• Less authentic than coached rehearsals
• Not an active interaction with others (in fact, TC is responsible for creating all of the dialogue)
• More time to make decisions about how to act
• Different affordances for TCs and teacher educators
• E.g., larger numbers of TCs can engage with the same scripting task

Using Scripting Tasks

From Earlier …

Teacher: How did you use the axis labels in your answer?

Foster: I thought of the x-axis as horizontal distance and the y-axis as vertical distance.

Teacher: it looks like the x-axis is time. So zero seconds after leaving home the student is 0 meters away. 50 seconds after leaving home the student is 100 meters away. What type of distance do you think this is? Horizontal or vertical?

Foster: horizontal.

Teacher: so given time on x-axis and horizontal distance on y-axis, what is this graph telling you about the student’s journey?

Foster: That the student’s distance was decreasing at one point. So they must have turned around for a little bit.

Teacher: How did you get that answer?

Foster: You can see the student's journey on the graph.  As he's walking, you can see that he first goes up then down and then up again?

Teacher: What other answers did people get?

Jane: I thought that Foster walked away then towards and then away again.

Teacher: How do you see that?

Jane: Well, the x-axis tells you where Foster is at any one moment in time and the y-axis is the distance.  So as time passes, the student first walk farther, then closer, and then farther again.

Teacher: What other ways are people interpreting this graph?

*Wait time

Teacher: Okay, we have to possible interpretations of this graph.  Think quietly about which one you think is best and why for a minute or two before pairing up with your shoulder-to-shoulder buddy and discussing what you each think.

Importance of a Framework of Teacher Learning

Three Considerations from Research and Practice

1. Representing student voice in approximations of practice
2. Considering “authenticity” and “complexity” of approximations of practice
3. “Revealing” teacher candidates’ resources through engagement with approximations of practice

Representing Student Voice in Approximations of Practice

• The actions of teachers are necessarily in interaction with the actions/contributions of students
• Important to make purposeful choices about the aspects of students you want to represent (e.g., mathematical reasoning, identities, social dynamics, etc.)

Representing Student Voice in Approximations of Practice

Representing Student Voice in Approximations of Practice

Designing “Planted Errors” to Use in Rehearsals

Using Planted Errors in Rehearsals

Representing Student Voice in Approximations of Practice

• In classrooms, teachers respond to student sense making
• Student reasoning in rehearsals must be similarly authentic — both mathematically and as something the individual actually means
• Important for TCs’ learning for how to respond to students in ways that value student reasoning and that fit within a vision for how these conversations can play out

Considering the Authenticity and Complexity of Approximations of Practice

• Typically thought of in terms of setting (e.g., in classrooms with “real” students vs. in a university classroom)
• Other ways to think about authenticity and complexity
• Is the student thinking (and the way it is presented) authentic?
• Is the TC held “accountable” in the moves they use to navigate a situation (e.g., not just going through the motions of using talk moves)?

Considering the Authenticity and Complexity of Approximations of Practice

• Moving toward “more authentic” does not equate to “better” for teacher learning
• Example: Affordances of scripting tasks relative to coached rehearsals, despite being “less authentic”
• Linking action with rationale
• More control over what is being approximated (e.g., certain situations, representing race, gender, and issues of inequity)
• Opportunity for teacher educators to weave together different forms of approximations

Revealing Teachers’ Resources (and How They Might Develop)

• Approximations of practice as formative assessment
• Different approximations offer different opportunities

Revealing Teachers’ Resources (and How They Might Develop)

• For scripting tasks, key is not just looking at the script

From Earlier …

Teacher: How did you use the axis labels in your answer?

Foster: I thought of the x-axis as horizontal distance and the y-axis as vertical distance.

Teacher: it looks like the x-axis is time. So zero seconds after leaving home the student is 0 meters away. 50 seconds after leaving home the student is 100 meters away. What type of distance do you think this is? Horizontal or vertical?

Foster: horizontal.

Teacher: so given time on x-axis and horizontal distance on y-axis, what is this graph telling you about the student’s journey?

Foster: That the student’s distance was decreasing at one point. So they must have turned around for a little bit.

Teacher: How did you get that answer?

Foster: You can see the student's journey on the graph.  As he's walking, you can see that he first goes up then down and then up again?

Teacher: What other answers did people get?

Jane: I thought that Foster walked away then towards and then away again.

Teacher: How do you see that?

Jane: Well, the x-axis tells you where Foster is at any one moment in time and the y-axis is the distance.  So as time passes, the student first walk farther, then closer, and then farther again.

Teacher: What other ways are people interpreting this graph?

*Wait time

Teacher: Okay, we have to possible interpretations of this graph.  Think quietly about which one you think is best and why for a minute or two before pairing up with your shoulder-to-shoulder buddy and discussing what you each think.

From Earlier …

Teacher: How did you use the axis labels in your answer?

Foster: I thought of the x-axis as horizontal distance and the y-axis as vertical distance.

Teacher: it looks like the x-axis is time. So zero seconds after leaving home the student is 0 meters away. 50 seconds after leaving home the student is 100 meters away. What type of distance do you think this is? Horizontal or vertical?

Foster: horizontal.

Teacher: so given time on x-axis and horizontal distance on y-axis, what is this graph telling you about the student’s journey?

Foster: That the student’s distance was decreasing at one point. So they must have turned around for a little bit.

Teacher: How did you get that answer?

Foster: You can see the student's journey on the graph.  As he's walking, you can see that he first goes up then down and then up again?

Teacher: What other answers did people get?

Jane: I thought that Foster walked away then towards and then away again.

Teacher: How do you see that?

Jane: Well, the x-axis tells you where Foster is at any one moment in time and the y-axis is the distance.  So as time passes, the student first walk farther, then closer, and then farther again.

Teacher: What other ways are people interpreting this graph?

*Wait time

Teacher: Okay, we have to possible interpretations of this graph.  Think quietly about which one you think is best and why for a minute or two before pairing up with your shoulder-to-shoulder buddy and discussing what you each think.

Future Considerations

• What stays “stable” in the learnings shared here as you change content, contexts, focal practices, etc.?
• Using approximations of practice with practicing teachers versus teacher candidates — Similarities and differences?
• How to better account for preparing educators to disrupt inequities in classrooms (and, specifically, math and science classrooms)?

Thank you

Matthew Campbell

West Virginia University

mpcampbell@mail.wvu.edu