Using Video as a Stimulus to Reveal Elementary Teachers’ Mathematical Knowledge for Teaching

1. Using Video as a Stimulus to Reveal Elementary Teachers’ Mathematical Knowledge for Teaching Presenters: Angela T. Barlow, Wesley A. Baxter & Angeline K. Gaddy

2. Introduction “Teachers may need to know subject matter differently than their students or non-teachers” (Hill et al., 2007, p. 122).

3. Original Purpose To examine elementary teachers’ perceptions regarding the resolution of mathematical disagreements among students

4. Purpose Demonstration of the potential for using classroom video of students engaged in mathematical disagreements as a stimulus for revealing teachers’ mathematical knowledge for teaching

5. Image of Triangle after Rotation Pre-Image of Triangle

6. Introduction of the Idea of PCK • Effective teaching research in the 1980’s was focused on facets of the education process such as classroom management, cultural awareness, and recognition of individual differences. • Shulman began to discuss the “missing paradigm.”

7. Introduction of the Idea of PCK • Subject matter knowledge was not enough in the classroom. • Shulman called for the development of a framework that would allow for multiple categories of content knowledge.

8. Deborah Ball’s MKT Ball, Thames, & Phelps (2008)

9. Van Hiele Levels • Level 0: The student identifies, names, compares, and operates on geometric figures (e.g., triangles, angles, intersecting or parallel lines) according to their appearance. • Level 1: The student analyzes figures in terms of their components and relationships among components and discovers properties/rules of a class of shapes empirically (e.g., by folding, measuring, using a grid or diagram).

10. Van Hiele Levels • Level 2: The student logically interrelates previously discovered properties/rules by giving or following informal arguments. • Level 3: The student proves theorems deductively and establishes interrelationships among networks of theorems. • Level 4: The student establishes theorems in different postulational systems and analyzes/compares these systems.

11. Van Hiele Levels Connection to KCS • Knowledge of the levels • Choosing appropriate tasks for the levels Connection to KCT • Knowledge of appropriate instruction • Moving students to Level 1 (remove irrelevant features such as appearance and focus on the properties)

12. Methodology

13. Video-based Tool

14. Participants

15. Video-based Tool

16. Image of Triangle after Rotation Pre-Image of Triangle

17. Describe the mathematical disagreement the students are having. • What mathematical understandings or misunderstandings led to the disagreement? • If you were the teacher, how would you resolve the disagreement?

18. Video-based Tool

19. Describe the resolution of the mathematical disagreement. • Was this an appropriate way to handle the disagreement? • Is it important to allow disagreements to occur in the classroom? • When a mathematical disagreement occurs, is it okay to tell students who is right? • Is the instruction similar to your own classroom? • Are you familiar with the Van Hiele Levels?

20. Data Analysis • What did participants perceive as the mathematical misunderstanding(s) that formed the basis of this disagreement? • What instructional strategies would participants use to resolve the disagreement? • What were participants’ ideas related to the teacher’s resolution of the mathematical disagreement in the video?

21. Results & Discussion

22. What did participants perceive as the mathematical misunderstandings that formed the basis for this disagreement? Prerequisite knowledge/experiences Changing triangle’s orientation

23. Congruent figures Emma, later Ann Rotation Frank Prerequisite Knowledge/Experiences Atypical triangles Emma Vocabulary Ann

24. Changing Orientation Caused triangles to look different Ann Changed the attributes of the triangle Beth, Cathy, Delia

25. What instructional strategies would participants use to resolve the disagreement? different triangles/figures measuring sides and/or angles teaching a different lesson individual or small group exploration To move students from Level 0 to Level 1…

27. What were participants’ ideas related to the teacher’s resolution of the mathematical disagreement in the video? the teacher’s strategy was appropriate the participants were neutral on the strategy the resolution was unsatisfactory

29. Conclusion

30. Conclusion • Need for assessing knowledge that teachers possess via means other than tests • Previous researchers have utilized video, but none have used video featuring a mathematical disagreement such as ours. • Participants failed to provide evidence of appropriate KCS, KCT, and KCC. • Implications of lack of MKT (confusion and a lack of understanding) • Our video-based tool was helpful in revealing teachers’ knowledge.

31. Study Limitations • We only focused on pedagogical content knowledge based on the Ball et al. model. • Different results may have been found if the mathematical disagreement involved a different focus. • Future research should utilize a video-based tool involving a mathematical disagreement with dissimilar content. • Further examination is needed when teachers do possess the required MKT.