Inquiry-Learning Strategies for a Hybrid Introduction to Proofs Course: A Preliminary Report

1. Inquiry-Learning Strategies for a Hybrid Introduction to Proofs Course: A Preliminary Report 14th Annual Legacy of R. L. Moore Conference Washington D.C., June, 2011 Elena Anne Corie Marchisotto California State University, Northridge

2. Mathematics 320: Foundations of Mathematics • Course website: http://moodle.csun.edu/Professor: E. A. Marchisotto Office Hours: on-line and by appointmentText: Notes adapted from those of Professor Ronald Taylor of Berry College, partitioned into themes and posted on moodle. • Course format: Hybrid. The course is structured to combine in-class instruction with on-line instruction and student collaborations.  The course meets one day a week in the classroom for discussion, student collaborations and presentations, and the remaining instruction is facilitated through the moodle website. • Course methodology: Modified Moore Method. Discovery/Inquiry Based Learning. Shared Responsibility for Learning among Students and Professor.  • Support: The construction of this course was supported by a Beck grant from CSUN, and incorporated materials provided by The Journal of Inquiry-Based Learning in Mathematics (JIBLM)  

3. Course Objectives • Mathematics 320 is the traditional first course in proofs that is designed to help Mathematics majors make the transition from computational to conceptual mathematics.   • Proof in mathematics serves many purposes simultaneously... Proof is respectability.  Proof is the seal of authority.  Proof, in its best instances increases understanding by revealing the heart of the matter.  Proof suggests new mathematics...Proof is mathematical power, the electric voltage of the subject which vitalizes the static assertions of the theorems.  Finally, proof is ritual, and a celebration of the power of pure reason (The Mathematical Experience, Study Edition, 1995, p.167). • To discuss what proof is, how it operates, and what it is used for, we need to have concrete examples of proofs.  To understand the power of proof, what role it plays in mathematics, and what it can and cannot do, we need to spend time doing proofs.  In this class we achieve these objectives using a modified Moore Method and combining the power of the internet with student-centered class discussion. 

4. Why Inquiry-Based Learning ? The modified Moore Method implemented for this class is intended to provide students with the motivation to acquire new knowledge, a perspective for incorporating new knowledge into their existing knowledge, and an opportunity to apply their knowledge. Inquiry is active learning, in contrast to learning driven by instructor lectures, which often involves only passive reception of knowledge. In particular this learning strategy provides opportunities to: • 1) develop general inquiry abilities that include posing and refining research questions, developing conjectures, and analyzing and communicating results with rigorous proofs.. • 2) develop an improved understanding of mathematical concepts. Inquiry activities can contribute to this knowledge acquisition process by providing a meaningful context for learning. • 3) discover proof techniques and proofs of theorems rather than duplicate them.

5. Function of the On-Line Component • Posted on moodle are class notes and all assignments: Exercises, Problems, Projects. Out-of-class activities (individual and collaborative) are designed to promote discovery learning, as well as shared responsibility for learning between students among themselves and with the professor. • Prior to each class meeting, students are required to do assigned readings of the notes, and then answer a set of exercises. Class meetings then focus on student presentations of exercises and collaborative discussions of the homework problem assignments (individual and group) that are due after each class session. 

6. Moodle Site: Class Information and Sample 2nd and 3rd Weeks • News forum • Course Information and Guidelines: Welcome Message; Syllabus; Requirements; Grading Guidelines; Appendices • January 31 - February 6: Class Meeting: Symbolic Logic/The Language of Proof • SubmitIndividual Answer to Problem 1.3.1 • Group Forum for Editing and Approving the Draft of the Group Submission • Submit Group Answer to Problem 1.5.1 Theme 2 Class Notes: Proof Methods Submit Individual Answers to Exercises 2.2 and 2.3 • Group Forum for Exercises • February 7 - February 13 Class Meeting : Proof Methods • Submit Individual Answer to Problem 2.1 • Group Forum for Editing and Approving the Draft of the Group Submission • Submit Group Answers to Problems 2.2-2.5 Theme 3 Class Notes: Mathematical Induction Submit Individual Answers to Exercises 3.1 and 3.2 • Group Forum for Exercises

7. Why a hybrid? Research has shown that on-line instruction provides a wide range of opportunities to address differences in learning styles and preparedness among students, and problems of time constraints.  Students benefit from greater flexibility in scheduling their learning and from working on assignments at their own pace, within a certain time frame. They are exposed to on-line resources that inform the teaching and learning of mathematics, and experience the power of collaborative work as they interact in on-line discussion forums, sharing the results of their work.    

8. Why Shared Responsibility for Learning? Activities to promote shared responsibility are based on two strategies with the following goals: • Students engage in work prior to professor-driven discussion of a topic. The goal is to help students become self-starters and independent learners. • Students collaborate with colleagues. The goal is that such discussion will promote increased understanding of mathematical arguments.

9. Grading Student grades in the class are computed on the basis of the following 850 points: • Individual Solutions to Exercises. Total Possible: 60 • Individual Solutions to Problem Sets for Themes 1 to 5. Total Possible: 120 • Group Solutions to Problems Sets for Theme 1 to 5. Total Possible: 120 • Student Presentations. Total Possible:100 • Midterm and Final Examination. Total Possible: 400 • Final Project (Paper, Surveys, Questionnaires): 50 points

10. Grades and Assessment • The grade distribution was 2A, 2A-, 3B+, 5B,1C+, 5C,1C-, 2D, 1F. • Student participation was better than in my previous taught Math 320 classes: there were virtually no absences. • Students work ethic appeared to be good: nearly all submitted all assignments. • Student responses to the discovery learning strategy was mixed. There was a decent amount of overall variability that I believe corresponded to the differences in ability – i.e., the best students were the most positive about discovery learning, while the weakest ones gave the more negative responses. • About half of the students were unfaithful to the modified Moore method in the sense that they consulted texts and websites for homework problems. • A majority of students reported positive experiences with respect to the weekly exercises and group collaborations. • A majority of students reported negative reactions to student presentations. • Nearly all students reported positively about professor comments on their work.

11. Results of Anonymous Assessment • Question 1. How would you describe your own success in discovery learning? • a. I found the process to be empowering almost from the beginning of class. 4 • b. I was initially challenged by the process but then empowered by gradually being able to construct proofs on my own and in collaboration with my group.6 • c. I found the process too challenging, and ultimately began consulting textbooks or the internet to find proofs.12 • d. OTHER: 3

12. Results of Anonymous Assessment • Question 2. How would you, on average, describe your understanding of the exercises on topics you had completed before class sessions involving student presentations of the topics? • a. Since I had worked on the exercises, the student presentations helped me come to a general understanding of most of the issues of the discussion 16 • b. Even though I had worked on the exercises, the student presentations made me more confused about my answers.5 • c. I often didn’t do the exercises, but the student presentations helped me understand the answers to them0 • d. I often didn’t do the exercises, and the student presentations didn’t help me understand the answers to them0 • e. OTHER:4

13. Results of Anonymous Assessment • Question 3. How would you, on average, describe your feelings about your work on individual problems that you completed after class discussion on the topic? • a. The internet notes and class discussion helped me understand how to approach the problems and I felt empowered by the fact I could do them. 5 • b. The internet notes class discussion did not provide sufficient information to know how to approach the problems and I felt frustrated because there were few examples to study. 18 • c. OTHER: 2

14. Results of Anonymous Assessment • Question 4. How would you, on average, describe your feelings about the results of group collaborations when you discussed the problems and formulated the group responses? • a. Our group collaborations helped me understand what I may have done wrong and how to improve my answers.15 • b. Our group collaborations made me more confused about my answers. 1 • c. OTHER:9

15. Results of Anonymous Assessment • Question 5. How would you describe, on average, the value of receiving detailed comments on your graded individual and group problems? • a. The comments were an important factor in my understanding of the correctness of answers and how to improve them.22 • b. The comments made me more confused about my answers.1 • c. I didn’t have the time to read the comments.0 • d. OTHER:2

16. Results of Anonymous Assessment • Question 6. Rank the following components from 1 to 5 (with 5 being most helpful) with respect to how much you believe they contributed to your learning • Internet Notes 3.5 • Exercises 3.8 • Individual Problems 3.96 • Group Problems 4.04 • Professor Comments on Graded problems 4.64 • Professor responses to questions (internet or in-class) 4.04 • Student Presentations 3.2 • Doing My Presentation 3.36 • Studying for Midterm 4.04 • Reviewing the Midterm in class 4.16

17. Reflections • My concerns are the following: • 1) that there were students who relied on the internet and other texts for their work and they may not be fully prepared for the next mathematics classes. I say this even though they passed the class (due to the structure of the grading system) and the fact that they reported on the post-class survey that they felt confident of their abilities to create and trouble shoot proofs. • 2) that CSUN may not be a place where discovery learning can succeed. One student in the class commented: Perhaps, classmates didn't devote sufficient time to the discovery learning process--they have other upper division classes, jobs as well. A colleague of mine observed: I've felt for quite a while that while discovery learning is a great approach in many cases, it often doesn't work for the type of students we have at CSUN, unfortunately. At least without a LOT of hand holding.