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

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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

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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)  

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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. 

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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.

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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. 

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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
• Submit Individual 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

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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.    

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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.

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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 Possible100
• Midterm and Final Examination. Total Possible
  400
• Final Project (Paper, Surveys, Questionnaires)
  50 points

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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.

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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

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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 didnt do the exercises, but the
  student presentations helped me understand the
  answers to them0
• d. I often didnt do the exercises, and the
  student presentations didnt help me understand
  the answers to them0
• e. OTHER 4

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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

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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

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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 didnt have the time to read the comments.0
• d. OTHER 2

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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

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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.