Unpacking Worked Examples: Teaching Associative Property

1. Learning to Teach Fundamental Mathematical Ideas in Elementary Classrooms: The Case of the Associative Property Meixia Ding & Kayla Copeland Temple University Meixia.Ding@temple.edu Kayla.Copeland@temple.edu AERA, 2015

2. Introduction We explored the process of PTs’ transformation of the learned specialized content knowledge (SCK) from PD settings to elementary classrooms. Focused on the associative property (AP) of multiplication

3. Theoretical Framework SCK is a key component of mathematical knowledge for teaching. The core of SCK is to unpack a mathematical concept or learning goal, making it visible and learnable for students(Ball et al., 2008). A focus on teachers’ choice and use of examples may serve as a window on teachers’ SCK (Chick, 2009).

4. Theoretical Framework To unpack a worked example, teachers’ sub-skills such as representation uses and questioning are critical (Ball et al., 2008). Representation Use • Make connections between concrete and abstract representations • Concrete  Abstract Questioning • Deep questions elicit students’ self explanations

5. Research Questions • How do PTs learn to unpack worked examples through representation uses and questioning in PD settings? • How do PTs unpack worked examples through representation uses and questioning in the elementary classrooms?

6. Methods Participants: Anna, Cindy, & Kate • Strong interests in learning to teach early algebra • Varied in content knowledge and beliefs

7. Methods Part of a 1-year long study

8. Task Analysis: Worked Example “Upright bass strings come in sets of 4. Suppose one box holds 2 sets of strings. If a musician orders 3 boxes, how many strings will there be?” Example of a concrete representation: = BIG IDEA: Associative Property! (3×2)×4 3×(2×4)

9. Task Analysis: Worked Example Textbook limitations: • 4×3×2 cannot be explained based on meaning of multiplication. • AP is directly presented.

10. Findings: PD setting • PTs embraced teaching for meaning in summer training. • PTs struggled with teaching for meaning in pre-lesson study. • Likely due to the conflict with prior knowledge and experience. • Instructor suggested using pictorial representations and insisted on questioning the meaning of each step.

11. Findings: Enacted Teaching • Worked examples

12. Findings: Enacted Teaching • Representation Use

13. Findings: Enacted Teaching • Teacher Responses Closer inspection: Asking for meaning of individual numbers Asking for number sentence with three numbers

14. Textbook Worked Example

15. Textbook Worked Example Anna’s lesson – Unpacking the string problem

16. Textbook Worked Example Cindy’s lesson - A student explaining how he got 6×4

17. Textbook Worked Example Kate’s lesson – An incorrect interpretation of the AP

18. Conclusions • Anna and Cindy made an effort in transforming the learned SCK, however, both lacked the ability to grasp students’ input to facilitate reasoning toward the expected direction. • Experienced frustration • Kate made the least amount of effort transforming what was discussed in the PD setting. • Aimed to stress only procedures and was satisfied with her enacted lesson. • Experienced the least amount of cognitive or emotional conflict.

19. Conclusions • Challenges found in unpacking worked examples: • Objective of AP not deemed as a goal • Misconception: The more examples the better • PTs demonstrated partial ability to transform subcomponents of SCK (representations & questioning). • Missing representation connections • Missing deep questioning • PTs own knowledge and beliefs mattered in transformation of SCK.