1. Describing Teaching from a Constructivist Perspective
Duane Graysay
Kim Johnson
Shiv Karunakaran
The Pennsylvania State University

2. Theoretical Framework

3. The Context
The lesson chosen for the study involved statistical concepts taught as part of a course in teaching with technology for pre-service secondary math educators.
Learning goals outlined in the plan involved concepts related to statistical regression.
It seems relevant to describe the context, which is where we began to develop our plan. We started by thinking about the lesson and the kinds of study that might be reasonable given the lesson and the teaching situation. I do intend to be more specific about the learning goals that we chose to emphasize in our study.

4. What is mathematics teaching?
A system of actions through which a teacher attempts to create opportunities for change in the mathematical knowledge, skills, or understanding of an individual.
I think it is important to describe our attempts to create a definition of teaching that is consistent with a constructivist paradigm on learning but that does not necessarily define EFFECTIVE teaching, or QUALITY teaching. This definition then influenced our efforts to characterize this system in terms of how it might facilitate learning.

5. Theoretical Framework
Attempts to describe teaching actions in a way that reflects our constructivist paradigm.
Models mathematics learning as the outcome of two processes: mathematical activity and reflection on one’s mathematical activity.
Influenced by Piaget (1977).
I intend to expand on these points, of course. Specifically, there are multiple theoretical models for describing the mechanisms by which learning occurs. We have chosen to think of mathematical activity as providing fuel for reflective activity, and that these two are the context within which the learner assimilates mathematical experiences into existing conceptual schema and through which those schema are adapted to accommodate these experiences.

6. Theoretical Model of Teaching
Characterizes teaching actions as including, but not limited to:
Activity supporting actions
Reflection supporting actions
Model-constructing actions
Influenced by Cobb and Steffe (1983), Confrey (1990), Simon (1995)
I think that the “stage-setting” actions are ones which should emerge as we draw on the data to answer our first question. We might mention the need for additional types of activity as part of our preliminary results.
Talk about how the references influence the development of the Model-constructing actions.

7. The questions:
To what extent can teaching actions within a statistics lesson be characterized using this model?
How do teacher and learner actions link to form chains of activity within the statistics lesson?
How do these teaching actions appear to relate to changes in students’ learning relative to coefficient of determination and residuals?
I plan to describe the first two questions as preliminary (not subordinate) to the third, and asked in service of supporting the answer to the third question. I want to make it clear that we do not have three separate questions that we are trying to pursue, but a primary question with two supporting issues.

8. Data Collection

9. Instructional Setting
A methods course for pre-service mathematics teachers geared towards teaching mathematics using technology
Classes were taught by a guest instructor over two class sessions
Students were given preliminary instruction by their teacher of record prior to the implementation of the lesson on how to use FATHOM and basic regression calculation on a graphing calculator
The sessions were videotaped and select small group interactions were audio-taped
Note: Given this particular lesson to study, a guest instructor was invited to teach since one of the researchers was the teacher of record, preliminary instruction dealt with different types of regression. Students were asked to answer 10 questions without any outside resources to figure out their basic knowledge of regression. The small group interactions were audio-taped to gain knowledge on how these interactions took place.

10. Preliminary Data Sources
Students were asked to complete a pre-assessment to gauge their understanding of basic regression concepts
What does the correlation coefficient tell us?
What is a residual?
Why do we need least squares regression?
Reviewed and coded the potential teacher actions within the lesson plans
Videotaped both class sessions
Audio taped small group activities
Pre-assessment questions used to gauge student understandings through interviews and classroom observations. The lesson plan allowed us to gain insight into what might occur in the classroom.

11. Coded the teacher actions from the transcriptions
The first question asks, “To what extent can teaching actions within a statistics lesson be characterized using this model?” In order to address this question we:
Transcribed and annotated both class sessions and small group activities
Coded the teacher actions from the transcriptions
Analyze transcripts to see if patterns emerge within the coded actions in the transcriptions
Notes: the lesson plan gave understanding into potential teacher actions for the lesson and allowed us to understand how the lesson would flow. The video tapes allowed us to view the interactions that may not have been captured in the audio tapes nor through field notes. The codes will be discussed during the analysis of the data.

12. The second question is, “How do teacher and learner actions link to form chains of activity within the statistics lesson?” In order to address this question we:
Coded teacher actions and student actions from transcripts and lesson plans
Analyze transcripts to look for patterns of interaction within these actions to find chains of activities
Student actions in response to teacher actions and vice- versa
Notes: coded teacher actions and student actions from the transcripts, some of the teacher actions were done through various handouts from the teacher.

13. Decided to chose two students from the class to interview
The third question that our study investigates is, “How do the teaching actions appear to relate to changes in students' understandings relative to coefficient of determination and residuals?” In order to address this question we:
Decided to chose two students from the class to interview
Insufficient resources to interview all the students
Identified ten students based on their answers to the teacher’s pre- assessment and potential for change in understanding
Contacted all ten, and randomly selected from those that responded that they would be willing to be interviewed
Recorded the group interactions of the two students selected to be interviewed
Collected handouts completed by students in the group
Interviewed two students to find out their understanding of the learning objectives
It was difficult for us to come up with a way to make sure that student understanding was taking place, pre-post testing would seem like a burden for the students and may not give us the information that we really were looking for. Assessment results do not always indicate understanding. Since we knew that interviewing all students would be impractical, decided to chose two to focus on, identified ten students based on their answers on the pre-assessment, which showed a range of knowledge, some low and some high, happen to find two students with relatively strong background, focused on these two students group interactions since we would later gather understanding of the individual student understanding, believe it would be important to understand what took place during their group interactions
Copied worksheet results from group members and individual to gain some more background on what they understood on the topics discussed
Interview questions were based on pre-assessment (i.e. presented some of the questions and asked students if they would revise them, why and how) and classroom activities from that day, the questions were relatively open ended and allowed the students to discuss what they found to be useful during the class session.

14. Analysis of the Data
So now that we have collected the data what do we do with it?

15. Two large class interactions Small group interactions w/ documents
Interviews with two students, along with pre-assessment
Video and audio data was transcribed, annotated using fieldnotes, and then coded using three codes for teacher actions & two codes for student actions
Evidence of student understanding
First there is of course the data sources
Then there is what we did with it
So, how does this help us answer our questions?
Give session details
Length
What content was taught
1. Effectiveness of theoretical model
2. Interplay between student and teacher actions
3. Connections of interplay to learning

16. Coding Details Three codes for Teacher Actions
AS: Activity Supporting Actions
RS: Reflection Supporting Actions
MC: Model-Constructing Actions
Two codes for Student Actions
A: Actions involving Mathematical Activity
R: Actions involving Reflections on Mathematical Activity

17. Examples of Coding: Teacher Actions: MC

18. Examples of Coding: Teacher Actions: RS
Talk about how this question is not asking students to regurgitate the Barrett article or the handout. This question seems to ask the student to reflect on the FATHOM scatter-plots.

19. Examples of Coding: Teacher Actions: AS

20. Examples of Coding: Student Actions: A & R

21. Preliminary Results
Directions on handout from teacher acts as proxy for teacher’s actions
AS and RS are communicated through the document
Collection of responses may represent MC
Model-supporting instead of Model-constructing
Chains of teacher and student actions exist
Usually begins with MC
MC seems to give teacher access to information enabling decisions on further actions
Emergence of other types of teacher/student actions via analysis
Relation between actions and student understanding
To be continued…

22. References
Cobb, P. and Steffe, L. (1983). The constructivist researcher as teacher and model builder. Journal for Research in Mathematics Education, 14(2),
Confrey, J. (1990). What constructivism implies for teaching. In R. B. Davis, C. A. Maher, and N. Noddings (Eds.), Constructivist views on the teaching and learning of mathematics (pp ). Reston, VA: National Council of Teachers of Mathematics.
Piaget, J. (1977). The development of thought: Equilibration of cognitive structures. (A. Rosen, Tr.) New York: Viking Press.
Simon, M. A. (1995). Reconstructing mathematics pedagogy from a constructivist perspective. Journal for Research in Mathematics Education, 26(2),