Implementing IBL in an Introduction to Proof Course Legacy of R.L. Moore Conference June 14, 2013 Dana C. Ernst Northern Arizona University Email: [email protected] Web: http://danaernst.com Google+: http://sgplus.me/dcernst Twitter: @danaernst & @IBLMath 1 

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 

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 

• 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 

• 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 

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 

• 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 

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 

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 

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 

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 

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 

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 

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 

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: [email protected] Web: http://danaernst.com Google+: http://gplus.to/danaernst Twitter: @danaernst & @IBLMath 15