Comparing Incorrect and Correct Examples in Algebra Classrooms

1. Comparing Incorrect and Correct Examples in Algebra Classrooms Kelley Durkin Bethany Rittle-Johnson Russell Ramsey Vanderbilt University

2. Research Questions • How do teachers make use of incorrect and correct examples in algebra classrooms? • Which recommended practices do they follow?

3. What are incorrect examples?

4. Why use incorrect examples? • Past behaviorist theory and people’s intuition don’t support using incorrect examples (Skinner, 1961). • Teachers in our study expressed the most concern about these incorrect examples. • Afraid they might overwhelm and confuse students.

5. Why use incorrect examples? • Math education researchers have long advocated for using incorrect examples in math classrooms (Bell, 1993). • Suggested that teachers directly confront students’ misconceptions and view errors positively (e.g., Bell, 1993; Borasi, 1994; McIntosh, Stacey, Tromp, & Lightfoot, 2000; Swan, 2007). • Strengthening use of correct procedures and improving knowledge of concepts (Durkin, 2009; Huang, Liu, & Shiu, 2008; Siegler, 2002).

6. Recommended Practices • Case studies and teaching design studies have introduced recommended practices for use with incorrect examples (Bell, 1993; Borasi, 1994; McIntosh et al., 2000; Swan, 2007) .

7. Recommended Practices: How to Introduce and Identify Errors • Incorrect examples can be helpful when introducing new material. • Teachers are encouraged to: • Provide time for students to decide whether an example is correct or incorrect. • Use small groups. • Reach a class consensus on whether an example is correct. • Deep level: Asking multiple students to explain why Alex or Morgan is incorrect • Surface level: Asking for a show of hands (Bell, 1993; Borasi, 1994; McIntosh et al., 2000; Swan, 2007)

8. Recommended Practices: How to Discuss Examples • Teachers are encouraged to: • Encourage multiple students to share and justify their ideas to the class. • Ask Jon who he thought was correct and why, then ask Megan if she agrees and why. • Support an effective discussion of errors and correct concepts. • Prompt for what was different between the two examples that made one correct and one incorrect. • Include a summary of the main ideas after the discussion. (Bell, 1993; Borasi, 1994; McIntosh et al., 2000; Swan, 2007)

9. Which is correct? Examples • Focused on teachers’ use of Which is correct? examples. • Two hypothetical students solved a problem—one solved the problem correctly, the other solved the problem incorrectly due to some common misconception. • A series of explanation prompts were included in an effort to promote student reflection.

11. Method • Participants • 8 out of 12 teachers used Which is correct? examples • On average, each teacher videotaped 3 or 4 examples, for 28 videos total • Coding Scheme • An experienced teacher used a strict coding scheme based on recommended practices for all videos. • Reliability coding is currently being conducted.

12. Results: How to Introduce and Identify Errors • Incorrect examples can be helpful when introducing new material. • Teachers are encouraged to: • Provide time for students to decide whether an example is correct or incorrect. • Use small groups. • Reach a class consensus on whether an example is correct.

13. Prior Knowledge of Students • Most often using incorrect examples to review previously learned information. • 20 out of the 28 videos were on lessons from the first two chapters of students’ textbooks.

14. Results: How to Introduce and Identify Errors • Incorrect examples can be helpful when introducing new material. • Teachers are encouraged to: • Provide time for students to decide whether an example is correct or incorrect. • Use small groups. • Reach a class consensus on whether an example is correct.

15. Results: How to Introduce and Identify Errors • Incorrect examples can be helpful when introducing new material. • Teachers are encouraged to: • Provide time for students to decide whether an example is correct or incorrect. • Use small groups. • Reach a class consensus on whether an example is correct.

16. Identifying and Introducing Errors • When Labeled as Correct/Incorrect

17. Results: How to Introduce and Identify Errors • Incorrect examples can be helpful when introducing new material. • Teachers are encouraged to: • Provide time for students to decide whether an example is correct or incorrect. • Use small groups. • Reach a class consensus on whether an example is correct.

18. Results: How to Introduce and Identify Errors • Incorrect examples can be helpful when introducing new material. • Teachers are encouraged to: • Provide time for students to decide whether an example is correct or incorrect. • Use small groups. • Reach a class consensus on whether an example is correct.

19. Identifying and Introducing Errors • Format of Studying Examples

20. Results: How to Introduce and Identify Errors • Incorrect examples can be helpful when introducing new material. • Teachers are encouraged to: • Provide time for students to decide whether an example is correct or incorrect. • Use small groups. • Reach a class consensus on whether an example is correct.

21. Results: How to Introduce and Identify Errors • Incorrect examples can be helpful when introducing new material. • Teachers are encouraged to: • Provide time for students to decide whether an example is correct or incorrect. • Use small groups. • Reach a class consensus on whether an example is correct.

22. Identifying and Introducing Errors • Student Participation in Identifying Correct

23. Results: How to Introduce and Identify Errors • Incorrect examples can be helpful when introducing new material. • Teachers are encouraged to: • Provide time for students to decide whether an example is correct or incorrect. • Use small groups. • Reach a class consensus on whether an example is correct.

24. Results: How to Discuss Examples • Teachers are encouraged to: • Encourage multiple students to share and justify their ideas to the class. • Support an effective discussion of errors and correct concepts. • Include a summary of the main ideas after the discussion.

25. Discussing Examples • Student Participation in Discussion

26. Results: How to Discuss Examples • Teachers are encouraged to: • Encourage multiple students to share and justify their ideas to the class. • Support an effective discussion of errors and correct concepts. • Include a summary of the main ideas after the discussion.

27. Results: How to Discuss Examples • Teachers are encouraged to: • Encourage multiple students to share and justify their ideas to the class. • Support an effective discussion of errors and correct concepts. • Include a summary of the main ideas after the discussion.

28. Discussing Examples • Content of Discussion

29. Results: How to Discuss Examples • Teachers are encouraged to: • Encourage multiple students to share and justify their ideas to the class. • Support an effective discussion of errors and correct concepts. • Include a summary of the main ideas after the discussion.

30. Results: How to Discuss Examples • Teachers are encouraged to: • Encourage multiple students to share and justify their ideas to the class. • Support an effective discussion of errors and correct concepts. • Include a summary of the main ideas after the discussion.

31. Discussing Examples • Content of Summary

32. Results: How to Discuss Examples • Teachers are encouraged to: • Encourage multiple students to share and justify their ideas to the class. • Support an effective discussion of errors and correct concepts. • Include a summary of the main ideas after the discussion.

33. Results: Summary • Provide time for students to decide whether an example is correct or incorrect. • Encourage multiple students to share and justify their ideas to the class. • Support an effective discussion of errors and correct concepts. • Use to introduce new material. • Use small groups. • Reach a class consensus on whether an example is correct. • Include a summary of the main ideas after the discussion.

34. Consistency of Practices • Rated how often each teacher followed recommended practices in each lesson

35. Conclusions • Wide variability in how teachers implemented instruction with incorrect examples and which recommended practices they followed. • Many of the recommended practices that teachers implemented were directly supported by our materials. • Future work should consider the best ways to support teachers using these recommended practices.

36. Conclusions • We need to emphasize the benefits of using incorrect examples to teachers. • Teachers may be hesitant to use incorrect examples when introducing new information. • Encouraging teachers to use incorrect examples with the recommended practices should improve student learning.

37. Additional Information from the Correct and Incorrect Examples presentation

38. General Results • Mean of 15 minutes on each example (SD = 9.30, range from 0.42 to 37.47 minutes) • 61% of the time examples used in the middle of a class period • 21% of the time used at the end of class • 18% of the time used at the beginning of class

39. General Results • After example, 39% of the time continued with a lesson on the same topic • 21% of the time completed practice problems • 21% of the time started a new lesson • 18% ended the class for the day

40. Discussing Examples • Comparisons Made During Discussion

41. High vs. Middle School Lessons