Implementing IBL in an Introduction to Proof Course

In this talk, the speaker will relay his approach to inquiry-based learning (IBL) in an introduction to proof course. In particular, we will discuss various nuts and bolts aspects of the course including general structure, content, theorem sequence, marketing to students, grading/assessment, and student presentations. Despite the theme being centered around an introduction to proof course, this talk will be relevant to any proof-based course.

This talk was given on June 14, 2013 at the Legacy of R.L. Moore Conference in Austin, TX.

Transcript

1. Implementing IBL in an Introduction to Proof Course Legacy of R.L. Moore Conference June 14, 2013 Dana C. Ernst Northern Arizona University Email: dana.ernst@nau.edu Web: http://danaernst.com Google+: http://sgplus.me/dcernst Twitter: @danaernst & @IBLMath 1
2. About Me • Assistant professor at Northern Arizona University • PhD from University of Colorado (2008) • Special Projects Coordinator for Academy of Inquiry- Based Learning (AIBL) • New MAA blogger at Math Ed Matters with Angie Hodge • Spent 4 years at Plymouth State University prior to NAU • Number of IBL classes I had as a student: 0 • Taught first full-blown IBL class in Fall of 2009 2
3. The Big Picture • Collaborative Modified Moore Method approach for an Intro to Proof course; appropriate for any proof-based course. • Most recent version of course: http://teaching.danaernst.com/mat320s13/ My Approach to IBL • Students should as much as possible be responsible for: ‣ Guiding the acquisition of knowledge; ‣ Validating the ideas presented. (Instructor not sole authority.) • Get out of the way and see what they can do. 3
4. • Produce examples/counterexamples. • Validate arguments. • Make conjectures. • Produce valid proofs. • Learn to write. • Develop perseverence. • Develop independence. Course Goals Category Weight Notes Homework 25% Mix of Daily & Weekly Homework Presentations & Participation 30% Students present problems from Daily Homework 3 Exams 45% Mix of take-home and in-class exams Grade Determination 4
5. • A taste of number theory • Baby logic • Set theory • Topology of the real line • Induction • Infinitude of primes • Irrationality of square root of 2 • Relations, equivalence relations, & partitions • Functions • Order relations Course Content FUN! 5
6. Problem Sequence • Notes consist of definitions & very few examples mixed together with exercises, problems, & theorems to prove. • As opposed to typical course, students responsible for proving the key (& interesting!) theorems of the course. • Source of notes available on GitHub: http://github.com/dcernst/IBL-IntroToProof Rules of the Game • Students should not look to outside resources • Internet, other texts, other faculty, math major cousins, etc. are forbidden. • On the other hand, students are encouraged to collaborate on homework & even take-home exams. 6
7. • 5-10 “tasks” are assigned each class meeting (Daily Homework). Due at beginning of next class. • Students are responsible for digesting new material outside of class (readings & screencasts). • Nearly all class time devoted to students presenting proposed solutions/proofs to assigned tasks. • My job: ‣ Facilitate discussion ‣ Mr. Super Positive • Students may request mini-lectures or screencasts. • Students type up subset of problems from previous week (Weekly Homework), graded harshly. Day-to-Day Operation Hang on every word. 7
8. Student Presentations • One student at a time talking & writing. • Must present at least 2x prior to each exam in order to receive a passing grade for Presentation category. • I take notes during presentation & add to spreadsheet. Grade Criteria 4 Completely correct and clear proof or solution. Yay! 3 Solution/Proof has minor technical flaws or is lacking some details. 2 A partial explanation or proof is provided but a significant gap still exists. 1 Minimal progress has been made. 8
9. Daily Homework • Problems from task sequence are assigned based on where we ended previous class. • Felt tip pens!!! ‣ Each student grabs a felt tip pen on way into class. ‣ Students use pens to annotate homework in light of presentation & related discussion. ‣ No penalty for use of pen. • Graded on ✔-system. What did they have done before class? 9
10. Weekly Homework • On week n+1, students choose 2 *-problems (subset of proofs) from Daily Homework from week n. • Proofs typed (LaTeX preferred, check out writeLaTeX). • Submit PDF on non-class day to Canvas. • Students forced to reflect on previous week’s work by reviewing their notes from Daily Homework. • Graded harshly on 1-4 scale (credit: Ted Mahavier): Grade Criteria 4 This is correct and well-written mathematics! 3 This is a good piece of work, yet there are some mathematical errors or some writing errors that need addressing. 2 There is some good intuition here, but there is at least one serious flaw. 1 I don't understand this, but I see that you have worked on it. 10
11. Marketing! • Students are asked to solve problems they do not know the answers to, to take risks, to make mistakes, and to engage in "fruitful struggle." • Students need to know that it is ok to be stuck and that you will support them in this endeavor. • Tip: Get the students to tell you what the best way is to acquire the skills necessary for effective thinking! • Students need to know what their role is & what the instructor’s role is. • Expectations & goals need to be reiterated throughout the course. • Use analogies: martial arts, playing an instrument, riding a bike, etc. 11
12. Student Feedback • The objectives of the course were made clear to me. Strongly Disagree Disagree Undecided/Neutral Agree Strongly Agree 0 4 8 12 16 20 100% Agree/Strongly Agree • The instructor accomplished course objectives. Strongly Disagree Disagree Undecided/Neutral Agree Strongly Agree 0 4 8 12 16 20 100% Agree/Strongly Agree 12
13. Student Feedback (continued) • The course was intellectually challenging. Strongly Disagree Disagree Undecided/Neutral Agree Strongly Agree 0 4 8 12 16 20 100% Strongly Agree • My general estimate of this course. Poor Fair Satisfactory Good Excellent 0 4 8 12 16 20 100% Excellent 13
14. My Favorite Student Comments • “...my favorite course here at NAU. [...] he has found the perfect way to teach this course. [...] The way Professor Ernst had us struggle through homework and then come together as a group and discuss the topics was very beneficial. I personally struggled through most of the material and when I finally got to the right concept I felt like I fully understood it because I personally came to that conclusion. Also, when I didn’t fully understand a topic, coming together and discussing it connected all the gaps I was missing. [...] As a future educator, I would love to mimic his style of teaching so I can share with my students the same satisfaction that I got out of this style of teaching.” • “Try, fail, understand, win.” 14
15. Possible Improvements • Categorize presentations (exercise versus proof)? • Modify requirement for minimum number of presentations? • Off load easy exercises to something like WeBWorK? • Change some theorems to prove or disprove type of problems. Thank you! Please contact me if you have questions, comments, and/or suggestions for improvements. Email: dana.ernst@nau.edu Web: http://danaernst.com Google+: http://gplus.to/danaernst Twitter: @danaernst & @IBLMath 15