1. DESIGNING AND FACILITATING WAYS TO ADVANCE IBL This study was supported by the National Science Foundation (NSF) under a grant (#0837443). Kyeong Hah Roh Arizona State University khroh@math.asu.edu

2. The Class Introductory Real Analysis Junior Level BA in Math or Secondary Math Education Students No Textbook Worksheets provided in class Class-note provided after topics are covered in class Definitions & Theorems without proofs Inquiry-Oriented Class Small groups (3~4 members per group) Students were asked to make and justify conjectures and to evaluate arguments.

3. Benefits of IBL Deep Understanding of Materials Promoting students’ mathematical thinking & reasoning Active Engagement in Learning Process Constructing & creating (or reinventing) knowledge by students themselves Collaboration & Communication

4. But … Designing and facilitating IBL is not a trivial task!

5. Challenges in Designing & Facilitating IBL Deep Understanding of Materials Promoting students’ mathematical thinking & reasoning Active Engagement in Learning Process Constructing & creating (or reinventing) knowledge by students themselves Collaboration & Communication

6. Questions might be… What & how to design tasks/curricular materials for IBL classrooms? How to facilitate student discourse in IBL classrooms? Issues with building up classroom norms Issues with dominating students Issues with shy students How to assess student work in IBL ?

7. This presentation is about What & how to design tasks/curricular materials for IBL classrooms? How to facilitate student discourse in IBL classrooms? Issues with building up classroom norms Issues with dominating students Issues with shy students How to assess student work in IBL?

8. DESIGNING TASKS/CURRICULAR MATERIALS Design Experiment Approach

9. Teaching, Student Thinking, & Design of Tasks Without the experiences afforded by teaching, there would be no basis for coming to understand what students understand or construct, or even for suspecting that students’ conceptions or construction may be different from those of instructors (Steffe &Thompson, 2000). What is students’ current understanding to given tasks? How do the tasks influence students’ learning? How does students’ understanding progress?

10. Iterative Nature of Designing Tasks Design Issue (Re)Design of Instructional Interventions P rediction of Student Learning Implementation of Interventions Analysis of Student Learning

11. Iterative Nature of Designing Tasks Design Issue (Re)Design of Instructional Interventions P rediction of Student Learning Implementation of Interventions Analysis of Student Learning

12. Iterative Nature of Designing Tasks Design Issue (Re)Design of Instructional Interventions P rediction of Student Learning Implementation of Interventions Analysis of Student Learning

13. Iterative Nature of Designing Tasks Design Issue (Re)Design of Instructional Interventions P rediction of Student Learning Implementation of Interventions Analysis of Student Learning

14. Iterative Nature of Designing Tasks Design Issue (Re)Design of Instructional Interventions P rediction of Student Learning Implementation of Interventions Analysis of Student Learning

15. Iterative Nature of Designing Tasks Design Issue (Re)Design of Instructional Interventions P rediction of Student Learning Implementation of Interventions Analysis of Student Learning

16. Iterative Nature of Designing Tasks Design Issue (Re)Design of Instructional Interventions P rediction of Student Learning Implementation of Interventions Analysis of Student Learning

17. CohortSemester# of Students # of Groups Sessions 1Fall 2006 62 (2)Three 50-minute classes for 15 weeks 2Spring 2007 22→205 (3)Two 75-minute classes & one 50-minute recitation for 15 weeks 3Fall 2009 21 (1)One 60-minute sessions for 10 weeks 4Spring 2010 113 (3)Two 75-minute classes & one 50-minute recitation for 15 weeks This study was supported by the National Science Foundation (NSF) under a grant (#0837443).

18. Cohort 4 (Spring 2010) Week 1Week 15 Chapter 1Chapter 2Chapter 3Chapter 4 Properties of Real Numbers Limits of Sequences Limits of Functions Continuous Functions Chapters 3 & 4 Exam Follow-up Interview 2 Chapters 1&2 Exam Follow-up Interview 1 Pretest

19. Cohort 4 (Spring 2010) The Participants (11 undergraduate students) Completed calculus and a transition-to-proof course Volunteered to participate and were willing to work in an inquiry oriented class Worked with the same group members for the entire semester Data Sources Written tests (photo-copied) Individual interviews (pen-casted & video-taped) Class Observation (field-note from Group 1 taken by a graduate student pen-casted in spring 2010, & video-taped from all three groups) Homework (photo-copied) Content logs from Group 2 taken by a graduate student (Michael McAllister) in summer 2010

20. Examples of Tasks from the Project Roh, K. (2008). Students’ understanding of the equivalent relationship to conditional statements. Paper presented at the annual meeting of American Educational Research Association, New York, NY. Knapp, J., & Roh, K. (2008). Students’ notion of convergence in advanced calculus courses. Proceedings of the 11th Conference of the SIGMAA on Research in Undergraduate Mathematics Education (SIGMAA on RUME), San Diego, CA. Roh, K. (2009). The nature of visualization and its impact on the teaching and learning of the notion of continuity of functions. Proceedings of the 12th Conference of the SIGMAA on Research in Undergraduate Mathematics Education, Raleigh, NC. Roh, K. (2010). How to help students conceptualize the rigorous definition of the limit of a sequence. Problems, Resources, and Issues in Mathematics Undergraduate Studies. Roh, K., & Lee, Y. (in press). The Mayan activity: A way of teaching multiple quantifications in logical contexts. To appear in Problems, Resources, and Issues in Mathematics Undergraduate Studies. Dawkins, P., & Roh, K. (in preparation). Mechanisms for scientific debates in real analysis classrooms.

21. CLASSROOM MANAGEMENT How to facilitate student discourse in IBL classrooms? Issues with building up classroom norms Issues with dominating students Issues with shy students

22. The Role of an Instructor Rephrasing: Taking student ideas and makes them concise or reinterpret them. Repeating: Repeating student utterances word-for-word. Clarifying: Attempts to make clearer what student utterances mean. Informing: Providing information by defining a term or procedure. Provoking: Creating students’ cognitive conflicts by directing or arranging interventions such as Debugging Steps: Processes of testing student conceptions or beliefs across instances Contrasting Prompts: A pair of statements which sound similar to, but are not logically equivalent to, each other Pivotal-Bridging Examples: Examples that contradict students’ conceptions or beliefs Devil’s Advocate: Incorrect or atypical arguments to evaluate Roh & Halani (in preparation)

23. The Role of Facilitators If you are the facilitator of your group, you are empowered by role and opportunity to enable all members of your group to share information freely with one another and help them attend fully to one another's perspectives. You must make an effort to promote participation of your group members in discussion as well as make sure that everyone is on the same page. You, as a facilitator, should promote participation, ensure equity, and build trust during group discussion.

24. The Role of Note Takers If you are the note-taker of your group, you are responsible for taking notes on the notebook that is provided to your group. You need to write down a copy of the final version of the proofs that your group come up with. However, the group notebook can also be used as a scratch paper. If someone needs a scratch paper when describing his/her idea, please pass the notebook with the pen & the notebook to the person. Please be sure that the notes written down contain enough information so that it is clear to the group members which exercises you are working on and include identifying information such as function definitions and equations.

25. eg. Cohort 4 Regular Sessions Tuesdays & Thursdays 75 min. each 15 weeks

26. Regular Sessions 1. Individual exploration a couple of minutes 2. All Purpose Go Around: Each student is given about one minute to share initial thoughts/ideas without interruption from others. 3. Group discussion (the duration was depending on tasks) 4. (Exchange Group: Regroup so that a couple of students from each group is in a new group as the “representatives" of their original group.) 5. Class discussion (the duration was depending on tasks) Some contents were provided by a traditional lecture style:

27. Protocol: All Purpose Go Around To the group Facilitator: If nothing else is mention in exercise in instructions, please initiate your group discussion by giving the group members a short time to think about the problem individually before sharing ideas. Then, you, as a facilitator, give your group members about 30 ~ 60 seconds each to present their ideas and thoughts. Please present your idea after everyone else in your group has chance to present their ideas. You should make sure that every group member has a chance to express their ideas and also ensure no one dominates the time given. You should distribute time properly as well as promote everyone's participation. Once everyone presents his/her idea, you can initiate group discussion by asking students to compare and contrast ideas suggested within your group.

28. Protocol: Exchanging Groups After discussion in your group, send two students in your group to two other groups. One student from each of the other groups join your group as well. You then ask those from other groups to present their ideas, thoughts, and opinions. Once they present their work, you, and perhaps one od your original group member, share your group's ideas, thoughts, and opinions with them as well. You may ask questions to them or receive question from them to clearly understand the ideas. This is an opportunity to share ideas, not to debate the validity of your argument. After sharing different ideas, send people from other groups back to their original groups. At this moment, your group members will return to your group as well. This is the time to compare and contrast your ideas and the other ideas presented from the other groups. This is also the time to debate the validity of all the ideas. Consider those from other groups as representing their groups. The idea of regrouping is to share different ideas. Therefore, you, as a facilitator, should give everyone a chance to present their groups' work to others when regrouping is made. Also, before regrouping, your group members should understand your groups' idea because each member in your group have to present your group's idea to those in other groups after switching members.

29. eg. Cohort 4 Recitation Fridays 50 minutes for 15 weeks Collaborative proof writing

30. Example: Collaborative Proof Writing during a Recitation Ex. Prove or disprove: {(-1) n +1/n} converges to 1. Final Proof Writing

31. The classroom environment that empowers students Time for individual thinking followed by All Purpose Go Around enabled the students to share various answers to the task in the group and recognize the incompatibility among students’ conceptions. Group discussion after All Purpose Go Around helped students adjust their understanding of the given tasks. The group exchange allowed students to view the task in terms of other group’s explanation and adjust their conception to fit that expressed by their original group mates previously. The final exchange with the original group affirmed the sense of mutual fit in their understanding and thinking of the given task. Dawkins & Roh (in preparation)

32. The students achieved acceptable compatibility in their conceptions relative to the task. The students heard a variety of explanations of the same task. Thus they reached a conclusion without instructor intervention beyond setting up the parameters and responsibilities of group discussion. Dawkins & Roh (in preparation)

33. ASSESSMENT OF STUDENT WORK Peer Evaluation

34. Midterm 30% Final 30% Assignment 20% Classroom Participation 10% Recitation Participation 10%

35. Evaluation of Participation Attendance 35% Instructor Evaluation 35% Peer Evaluation 30%

36. Peer Evaluation List ALL names of your group members including your own name. Give a score (0 as lowest to 10 as highest) for each member based on his/her contribution to your group during the group discussion of the day. Explain your evaluation. If you think nobody else contributed to the group more than you did during the group discussions, explain what difficulties you experienced while working in your group. Otherwise, please nominate one of your group members as one who contributed the most of the day. You should NOT nominate yourself. Explain why you nominate this student.

37. Questions? Kyeong Hah Roh Arizona State University khroh@math.asu.edu