Inquiry-Based Learning: What, Why, & How? Mathematics Instructional Colloquium University of Arizona Dana C. Ernst Northern Arizona University Email: dana@danaernst.com Web: http://danaernst.com Twitter: @danaernst & @IBLMath Thanks to Stan Yoshinobu for providing some of the content. 1

About me • Ph.D. from University of Colorado. • Areas of research: ‣ Combinatorics of Coxeter groups and diagram algebras, ‣ A little math education. • Currently an assistant professor at NAU (dream job!). • Spent 4 years at Plymouth State University prior to NAU. • Number of IBL classes I had as a student: 0 • Taught first IBL class in Fall of 2009. 2

What? 3

What is inquiry-based learning (IBL)? • According to the Academy of Inquiry-Based Learning: ‣ IBL is a teaching method that engages students in sense-making activities. ‣ Students are given tasks requiring them to solve problems, conjecture, experiment, explore, create, & communicate. ‣ Rather than showing facts and/or algorithms, the instructor guides students via well-crafted problems. • Students are responsible for guiding acquisition of knowledge. • Often involves very little lecturing, and typically involves student presentations. • Sometimes called Modified Moore Method, after R.L. Moore. 4

Continually ask yourself the following question: Guiding Principle of IBL Where do I draw the line between content I must impart to my students versus content they can produce independently? 5

Our main objective How do we get here? Students answering questions Students asking questions 6

Why? 7

One minute version of why IBL • Our system needs an upgrade. • Unintended negative outcomes via traditional methods. • Research suggests IBL outcomes are better. 8

• When I started teaching, I mimicked the experiences I had as a student (i.e., I lectured). • By most metrics, I was a successful teacher (e.g., high evaluations, several awards). Why change? • Inspired by a Project NExT Workshop run by Carol Schumacher (Kenyon College), I decided to give IBL a try. • For 3 consecutive semesters, I taught an intro to proof course at Plymouth State University. • 1st two iterations taught via lecture-based approach. • 3rd time taught using IBL with emphasis on collaboration. • When I taught an abstract algebra course containing students from both styles, anecdotal evidence suggested students taught via IBL were stronger proof-writers & more independent as learners. My first IBL class 9

Some data • 4-5 million freshmen in HS. • 75% HS graduation rate. • 1.2 million bachelors degrees annually. • Less than 1% of BA/BS are in math. • Only 400-500 U.S. citizens/residents earn a Ph.D. annually. • Education is a self-populating institution! Conclusion? You are peculiar!!! We need to renormalize. 10

What is happening in STEM education? • There exists a growing body of evidence suggesting students are dissatisfied with learning experiences in STEM. • Math Education Research suggests that college students have difficulty with: ‣ Solving non-routine problems, ‣ Packing/Unpacking mathematical statements, ‣ Proof. (Schoenfeld 1988, Muis 2004, Selden and Selden 1995, 1999, 2003, Dreyfus 2001, Sowder and Harel 2003, Weber 2001, 2003, Weber and Alcock 2004, Tall 1994) 11

• About half of STEM majors switch to non-STEM. • Top 4 reasons for switching are teaching related. • Good ones leave, too. • Loss of interest. • Non-STEM majors offer better education/more interest. • Curriculum overload. • Poor teaching. • Weed-out culture. Talking About Leaving 12

The good news Evidence from the math ed literature suggests that active, learner-centered instruction leads to improved conceptual understanding, problem solving, proof writing, retention, habits of mind, and attitudes about math. (Boaler 1998, Kwon et al 2005, Rassmussen et al 2006, Smith 2006, Chappell 2006, Larsen et al 2011, etc.) 13

The Colorado study • Sandra Laursen, CU Boulder. • Statistically significant advantages for students in IBL vs traditional courses. Interview SALG Pre-post tests Transcript Data Gender IBL Non-IBL Class Observation 14

Force Concept Inventory (Posttest -Pretest)/(Max Possible Gain) Hake, Am. Journal Physics, 1998 15

Twin Pillars 1. Deep engagement in rich mathematics 2. Opportunities to collaborate. 16

How? 17

Instructor Obstacles • “That’s how I learned, and it worked for me...” ‣ But you are peculiar! • “I like inspiring lectures.” ‣ Inspiration is necessary, but not sufficient. • “I’m afraid the students won’t like it.” ‣ My kids like Gummi Bears, but that doesn’t mean they are good for them. • Control! ‣ If I lecture, then I dictate pace. ‣ If I write something on the board, then there is a good chance that it will be done correctly. 18

Student Obstacles The main obstacle: Most students do not enjoy direct instruction, but it is what they are used to. They expect to be passive, & they have had 14+ years of experience to develop deep- rooted beliefs about how STEM classes should operate & what is expected of them. It is vital that student expectations & roles are clearly established. To help reset these expectations students need to have a clear understanding of what IBL is & why they should care. 19

Marketing! • Most students do not come equipped with the skills and interests that we have. • What are the secondary goals of the course? How are these skills acquired? • Students need to know what their role is. 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." These are all very different from normal expectations. 20

Marketing! (Continued) • Students need to know what the instructor’s role is. • Expectations & goals need to be reiterated throughout the course. • Students need to know that it is ok to be stuck and that you will support them in this endeavor. • Use analogies: learning to play an instrument, learning to ride a bike, etc. • Tip: Get the students to tell you what the best way is to acquire the skills necessary for effective thinking! 21

Content delivery & knowledge acquisition • For content that is to be imparted to students, how will it be delivered? ‣ Assigned readings? ‣ Lectures? Pre-planned vs. by request? ‣ Worksheets? ‣ Screencasts (Flipped Classroom)? • For content that students produce, how & when will it be acquired? ‣ Task/problem sequences? ‣ In-class vs. homework? ‣ Worksheets? ‣ Collaboration vs. independence? 22

Keys to success • Effective marketing. • Return to your guiding principle. • Adjusting problems/tasks appropriately. • Patience & trust! • Community. • Build on positive experiences. • Pick a style that you are comfortable with. • Adapt, overcome, & improvise. 23

Examples Let’s take a look at two examples. 24

Proof-based classes A Moore method approach with collaboration 25

• My experience: ‣ Intro to Proof, Number Theory, Abstract Algebra, & Real Analysis. ‣ 10-30 students. • Encourage collaboration. • Typical grade determination: The big picture Category Weight Notes Homework 25% Mix of Daily & Weekly Homework Presentations & Participation 30% Students present problems from Daily Homework 3 Exams 45% Typically take-home exams 26

Optimization problem! Useful feedback for students Data to support grades Time required 27

• 5-10 “tasks” (e.g., exercises, proofs of theorems) are assigned each class meeting (Daily Homework). Due at beginning of next class. • Students are responsible for digesting new material outside of class. • Nearly all class time devoted to students presenting proposed solutions/proofs to assigned exercises. • Students (usually) volunteer to present. • My job: ‣ Facilitate discussion ‣ Keep us on track ‣ Mr. Super Positive ‣ Cross my arms and say, “hmmm” • Students may request mini-lectures or screencasts. Day-to-day operation 28

More on student presentations • Must present at least 2x prior to each exam in order to receive a passing grade for Presentation category. (Is this working?) • I take notes during presentation & add to spreadsheet: ‣ Who & what problem ‣ Exercise or proof ‣ Miscellaneous notes ‣ Score 1-4 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. 29

Daily Homework • These problems form the backbone of the class. • Problems from task sequence are assigned based on where we ended previous class. • Felt tip pens!!! ‣ Credit: Clark Dollard (Metro State in Denver) ‣ 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? 30

Advantages of the felt tip pens • I know what happened before class versus during class. • Students mark up their work in ways they never did before. • Students have (mostly) correct work by the end of class (pedantic details & logical structure). • Students have a record of what happened in class together with their homework. • When students look back at their notes they see their comments about what they were thinking & they see corrected mistakes. • Students love the felt tip pen approach. Numerous positive comments about how useful this is. • Grading of the Daily Homework is fast! 31

Weekly Homework • On week n+1, students choose 2 *-problems (subset of proofs) from Daily Homework from week n. • Proofs must be typed (LaTeX preferred) & well-written. • Email PDF using my naming convention. • Files stored on Dropbox. Use iPad to annotate PDFs & then email back to student. 32

Weekly Homework (continued) • Submitted on a non-class day. • Students forced to reflect on previous week’s work by reviewing their notes from Daily Homework. • Incorporates multiple rounds of revision. • 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. 33

Calculus sequence An IBL-lite approach 34

• 20-50 students (NAU has larger class sizes than PSU). • Class meets 4 days per week. • 4 midterm exams and a cumulative final. • 3-4 Daily Homework assignments per week (WeBWorK). • Homework worth 15% of overall grade. • 1 Weekly Homework assignment per week. Covers main topics from the previous week. More challenging than Daily Homework. • 3 class meetings are devoted to introducing new material, either via lecture or IBL-style worksheets. • 1 class meeting devoted to students presenting problems from Weekly Homework. Students annotate with felt-tip pens. • Presenters are not graded, but 5% of grade is for participation. The big picture 35

• Academy of Inquiry-Based Learning (www.inquirybasedlearning.org) • Journal of Inquiry-Based Learning in Mathematics (www.jiblm.org) • Small grants from AIBL • IBL Workshop at JMM 2012 Resources Thank you! 36