Flipping Differential Equations: An Initial AnalysisEric EagerMathematics Department - University of Wisconsin - La Crosse

The Flipped Classroom

• Flipped learning is an approach that allows educators to implement a methodology, orvarious methodologies, in their classroom by delivering course content through nontra-ditional means (Bermann and Sams 2012).

• Many educators are experimenting with the idea of a flipped classroom model. According tothe Flipped Learning Network (2012), membership on its social media site rose from 2,500teachers in 2011 to 9,000 teachers in 2012.

• The flipped classroom can come in many forms, depending on the instructor, his or herinstitution, the course being taught, and the student population being taught.

• Bloom’s Taxonomy, named after Benjamin Bloom, refers to a classification of the differentobjectives that educators set for students (learning objectives). The process of accomplishingthese learning objectives has heavily influenced the popularization of the flipped class-room.

• Nontraditional means of instruction create an environment where “school work is done athome” by using media (textbooks, class notes, videos) outside of class to accomplish the“transmission” phase of the learning process.

• Thoughtful in-class activities create an environment where “homework done at school” bygiving students multi-layered looks at the course material to accomplish the “assimilation”phase of the learning process.

• Students have access to video and print materials for learning both now and later (espe-cially important given textbook rental policies).

• Students practice learning new concepts independently (how to take and refine notes).

• Students work on hardest parts of the subject when the instructor is most physicallypresent, allowing for complex and multifaceted assignments and projects (instructor becomesthe consultant, the student the client).

• Students get more time to work on “sense-making” activities with an instructor present to“course-correct”.

My Motivation

• In my previous two times teaching MTH 353 (Differential Equations) at two different univer-sities, I found myself wanting more time to engage with students.

• This disappointment stems from the fact that even many of the trivial problems/demonstrationsdone in a differential equations course, as well as reviewing material from previous courses,can take the large majority of class time, leaving little time to engage in constructive dia-logue with both individual students and the class as a whole.

• The flipped-classroom environment has the potential to better expose students not only to thebasic material of a differential equations course, but also deep, time-consuming problemsby pushing the former into out-of-class time via video lectures and homework, leavingclass time for the latter.

Solution to the heat equation using separation of variables and Fourier series.

Course Overview

Course Description: Fundamental existence and uniqueness theory, linear independence andthe Wronskian, series solutions near regular singular points, Laplace transforms and systemsof first order linear equations. Fourier series and the method of separation of variables will beapplied to the heat equation, wave equation, and Laplace’s equation. Prerequisite: MTH 309(Linear Algebra) and MTH 310 (Calculus III).

Learning Objectives:

• Cultivate an understanding of the process of solving a problem by reviewing the importantinformation, developing a plausible mathematical model, obtaining solutions, and evaluatingthe results.

• Understand the basic theory, numerical techniques, and solution methods to elementary differ-ential equations.

• Explain and apply the basic concepts of calculus including the various forms of derivativesand integrals of continuous functions, their interconnections, and their uses in analyzing andsolving problems from other disciplines of science.

• Most importantly, be able to articulate mathematical ideas verbally and in writing, using ap-propriate terminology.

Course Materials and Methods:

• 104 video lectures

– PowerPoint-type lectures– Blackboard-type written examples– Pre-class Google questionnaire

• In-class exercises

– Mini-lectures guided by pre-class questions– Small quizzes twice a week– Practice problems and well-crafted application questions– Two class projects (Golden Ratio derivation and Fourier Transform exploration)

• Evaluation

– Two mid-term exams– Final exam

Results

• Students came into the class with weak recall from previous courses. For example, morethan half of the class was uncomfortable with many of the basic concepts from Calculus II(separation of variables, partial fractions, improper integrals, infinite series and power series).

• Students taking advantage of the pre-class questions excelled in the course (an averageof 0.5495 increase in final grade percentage per pre-class question asked), but many studentsdid not take the questions seriously (e.g. asking fantasy football questions), and subsequentlystruggled.

• Many students began skipping the class period, or left after the quizzes, possibly because Idid not grade in-class activities.

• Exam scores were similar to my unflipped Fall 2014 section (median scores 84.5, 88.5 and81.67 (Final Exam) in Spring 2015 vs. 86.5, 75 and 79.33 in Fall 2014).

• Many students had positive feelings about the course in general (seeupper right panel).

Daily responses topre-class questionnaireover time

ReferencesBergmann, J., Sams, A. Flip Your Classroom: Reach Every Student in Every Class Every Day. iste, Eugene, OR and ASCD, Alexandria, VA (2012)

Eager, E.A., J. Peirce, P. Barlow. Math Bio or BioMath? Flipping the Mathematical Biology Classroom. Letters in Biomathematics, 1(2) (2014) 139-155

Selected Student Reactions

• “Continue to use the video lectures. They were very well organized as far as content goes andallowed me to learn it during more convenient times.”

• “I really liked the setup of this course. It was easier to teach yourself when you could pausethe video and rewatch things that you didn’t understand.”

• “Frequent quizzes force people to keep on top of things. They also are important because theymake people come to class, which is important in a class with this setup.”

• “Keep doing the video lectures and the process of the inverted classroom.”

• “Towards the end you started doing quick summaries in class of what the online lectures cov-ered and I liked that since hearing the content a second time helped reinforce the material.”

• “The video lectures: They allowed me to create thorough and useful notes rather than desperatescribbles.”

• “I really liked the relaxed atmosphere of the classroom and how easy it was to throw questionsand ideas out and get responses not only from you, but other classmates.”

• “I think the inverted classroom style made me work harder and helped me get more out of theclass.”

• “This was like having the opportunity for office hours 3 times a week, without a change in myschedule and in a way that is arguably more efficient for the instructor. Good stuff.”

• “I think there should be less in class lectures. I was often a couple videos ahead of what wasbeing talked about in class.”

• “I think it’s possible that the inverse setup allows students to focus only on memorizing deriva-tions and how to solve problems, while glossing over the theorems and proofs that justify theseprocedures. This is because students can pick and choose which parts of the video to watch,which isn’t possible during a lecture.”

• “The inverted classroom was a hassle to deal with. Video lectures on all of the topics discussedwere/are already available online from Khan Academy, MIT Courseware, etc., so the personallecturing from a professor is largely why I pay for college. I do not think the inverted modelneeds to be entirely scraped, but more time lecturing in class would have been significantlymore beneficial than having the majority of class time as a glorified office hour.”

• “I am not a huge fan of how the class structure is set up because it is easier to fall behind inmy opinion. If there is a question I have when watching a lecture on D2L, I cannot ask it rightaway and expect an answer right away, possibly not until the next class. I like the traditionalclassroom setting with doing lecture in class and homework and projects outside of class witha few work days mixed in there.”

• “I did not like having pre-class questions be a part of my course grade.”

• “I think the whole class being inverted is not necessarily great because students are not use tothis format so they do not take advantage of the opportunity to ask questions in class. I thinkif the videos existed that would be great but for maybe 2 classes being lectured based and oneclass period being inverted.”

Conclusions and Future Directions• Not all students, even within the same major, are created equal, and the best way to

account for this is a flipped classroom environment.• Insufficient recall of prerequisite material was the biggest obstacle my students faced.

• A small fraction of students in the course never bought into the flipped environment, possiblydue to their success in the lecture-based format. These students were often defiant when offeredhelp, and struggled to succeed in the course.

• Flipping the classroom is very time-consuming. as I spent roughly 60 hours just develop-ing the videos for this course. Plan ahead.

Future directions for this course, and this work in general, include

• the continued documentation and modification of course materials and learning outcomes.• the dissemination of course materials in the form of publications, webpages, blogs, etc.