Calculus expands ‘flipped’ model

If students in Yale math professor Jim Rolf’s section of “Calculus of Functions of One Variable II” last semester were watching online videos as they did their homework, they were doing their job.

The 10 to 15 minute videos, which feature Rolf lecturing on introductory calculus concepts aided by animated graphics, form part of an approach to teaching called the flipped classroom: Students watch brief video lectures covering new concepts as part of their homework and spend class time applying and extending those concepts with guidance from their instructor. Rolf taught a flipped section for the first time this fall and hopes to make flipped classrooms the new model for Yale’s introductory calculus curriculum. He will be joined in teaching flipped sections this spring by math lecturer Yu-Wen Hsu GRD ’13 and math professor Sam Payne.

According to end-of-semester evaluations, 72 percent of students in the flipped classroom agreed that the videos were helpful to their learning.

The flipped classroom is not new, nor is it unique to Yale. Over the last decade, it has grown in popularity with teachers of all grade levels across the country. In fall 2011, University President Peter Salovey co-taught a flipped classroom seminar, “Great Big Ideas,” where students watched lectures at home in preparation for classroom discussion.

While some instructors supplement their lectures with videos already available online, like those at Khan Academy, Rolf developed original videos for his Math 115 classes last summer in collaboration with Hsu, graphic designer Matt Croasmun ’01 DIV ’06 GRD ’14, and staff from the Center for Scientific Teaching and Broadcast and Media Center. The project drew funding from multiple sources, including the Information Technology Group and the Provost’s Office.

Hsu said Rolf’s version of the flipped classroom would follow what Rolf dubs the “ICE” framework: inform, confirm and extend. The informational videos in the inform portion are accompanied by quiz questions in the confirm portion, which are factored into students’ final grades. In the extend portion, which takes place during class, instructors introduce harder problems and higher-level mathematical applications.

“The videos definitely help, because [Rolf] can do so much more when he doesn’t have to do the basics of everything,” said Jihad Womack ’17, who took Math 115 with Rolf last term. “I liked that we didn’t go into class blindly.”

Although the questions accompanying the videos could be much more difficult than the concepts the videos introduced, Rolf reinforced his students’ understanding by reviewing the more complex material in class, Womack added.

Yet Womack expressed concern about the assignments piling up as the semester drew closer to midterm exams. Students in Rolf’s flipped section were expected to watch the videos and complete their accompanying quizzes in addition to handing in regular problem sets, making for an occasionally “overwhelming” workload.

Flipped classrooms also require a greater time commitment from their instructors. Payne, Hsu and Rolf said they are up early each morning to review student feedback and the results from the previous night’s quizzes. From those results, they tailor each lecture to target difficult concepts and minimize time spent on well understood concepts.

“[The videos] take a lot of the guesswork out,” Payne said. “When you’re in lecture, sometimes you’re trying to read faces to figure out what they don’t understand. Now I show up at 9 a.m., and I have feedback.”

Students who feel less comfortable asking questions in lecture, Payne said, might especially benefit from the videos. Should they need a review, they can re-watch the video for clarification; Hsu mentioned that because students watch the videos outside of class, they can learn and digest material at their own pace. And, because responses to and questions about the video are received anonymously, shyer students might be more inclined to inform their instructors when they are having trouble.

Rolf acknowledged that not everyone feels comfortable with the flipped classrooms. With time, he hopes other instructors will come to feel that the videos help teach more effectively and save valuable class time —time, Rolf said, that can now be spent on other things.

Rolf plans to make increasedclassroom use of Learning Catalytics software, which allows him to quiz students online and receive immediate feedback on what they do and don’t understand. In the future, Rolf said he hopes to invite professors from other schools, like the School of Music or School of Medicine, to discuss how math applies to a variety of disciplines.

Croasmun, who worked with Rolf on the design and narrative of the videos, said that the videos would enhance interactions between students and their instructors, not replace them.

“People think the flipped classroom is all about the whizz-bangs you can do on screen,” Croasmun said. “But, at the end of the day, the reason to flip a classroom is to increase student-teacher contact, to offload the information dump that usually has to happen to make the best use of class time. When [Rolf] walks in, students are already way ahead of where they would have been if he then had to lecture for half an hour. It’s clearing the way for [class] to be more interactive, so that the instructor can be most useful to students.”

Prior to coming to Yale, Rolf taught math at the United States Air Force Academy.