Setting for my project: MATH 4343–Topology in SP15 as pilot for new mid-level course that will help transition students from computational lower-division courses to abstract a proof-based upper-division courses.
The Experiment: Use Inquiry-Based Learning and a Modified Moore method, and move from a content-based course to a skills-based course.
Background: This project is also grant-funded (for a course release in the pilot semester, and to offset staffing for two subsequent semesters) by the Educational Advancement Foundation.
Many proponents of the Inquiry-Based Learning movement in mathematics were once taught by the mathematical descendents of R. L. Moore, a legendary mathematics professor who had a distinguished career at the University of Texas, in a unique classroom environment many call a “Modified Moore Method.” While modifications vary, what most hold in common is that students are given a very brief course packet of axioms, definitions, theorems, problems, & exercises. There are no lectures, and no outside resources allowed. Thus, all learning in the course entails the students’ 100% ownership of knowledge. Class time is dedicated to students presenting proofs and solutions at the board, with their peers reviewing, critiquing, and validating the work. Work that is handed in is given careful formative feedback from the instructor, and credit is only given once a proof is correct and complete.
For this course, I plan to use course notes prepared by Charles Coppin and available on the Journal of Inquiry-Based Learning in Mathematics. My personal modifications for the method were clarified and decided during the institute: when in doubt, I will look to the math research community model. In particular, all work must be typeset for submission. Students may collaborate, but only if they submit a coauthored result together; coauthored results must be balanced by solo work to convey expertise and mastery of course material. Selected results will be reviewed — initially by myself, then later blind peer-reviewed by select student referees who have demonstrated their ability to discern validity — for publication in an online class journal/blog. I will allow duplicate credit for work that is concurrent but independent. (Though this is rare in modern math through electronic communication, preprint servers, and vetting of early work at subdisciplinary conferences, there are some famous examples in the history of mathematics, notably Liebniz and Newton’s co-invention of calculus, and more recently the development of the HOMFLY-PT polynomial in knot theory, where Polish mathematicials P & T developed their work independent of the American group H, O, M, F, L, and Y.)
Through the institute, I was also able to refine my goals for the course in terms of student learning outcomes, which fall into three categories: proof/logic/comunication skills, transfer of knowledge, and self-efficacy. In particular, my proof skills goals are that students will:
- understand and use basic logical constructions (~, ∃, ∀, ∨, ∧, etc.)
- identify the hypotheses and conclusion of a statement to be proven,
- recognize and reconstruct the underlying logical framework of a proof,
- validate each claim with reference to a definition or prior result,
- communicate logical arguments clearly and coherently
Transfer will be demonstrated by grading proofs from prior and subsequent courses with a rubric, yet to be finalized. The self-efficacy goals I have set for my students are that they can respond affirmatively to the statements:
- I can “do proofs.”
- I know how to attack a tough problem.
- I recognize when a proof/solution is valid/correct and complete.
- I can use these skills in my other classes.
My biggest challenge in the course will be in shaping student expectations and participation. Students are the variable, and the course will depend on their Buy-in, Motivation, Hard Work, Perseverance, Supportive Classroom, and Academic Integrity. (Big Ideas deserve capitalization.) I need to inspire all this, and everything hinges on it.
My biggest source of inspiration this week, aside from the Innovation Fellows themselves, was the theme of #100percentdigital and the flurry of ideas surrounding the excitement of the new iPads. Already I have tried to identify ways that I can make this course paperless, and I find myself wondering: can we (myself and the students) have a 100% digital workflow? Would second iteration in Spring 2016 be a good pilot iPad course? Other flashes of inspiration from the Institute that I do not want to lose sight of are the idea of rewarding failure, of productive discomfort, and if students might be motivated by the ability to earn badges for meeting course goals.