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Real Analysis is a high-level mathematics class, described in the Centre College course catalogue as “a study of the theoretical foundations of single variable calculus.” The pace is swift, and the material is dense.
“Real Analysis explores all the underpinnings for math,” says Christine Shannon, PhD, who served as a full professor in the school’s math and computer science department from 1989 until her retirement in 2016. “What you’re doing is proving all the foundational ideas. It guarantees that the [concepts] that we use all the time and the assumptions that we make [are valid].”
While reading assignments might consist of only three or four pages, they cannot be completed quickly. “You would miss all the subtleties and nuances,” Shannon explains.
“You don’t process [the information] unless you actually sit down and try to write it yourself. Sometimes the first few things you try are just not going to work. And you just have to keep trying. You have to go back to the definition—to what you know and what you’re trying to show—and try to look at it some different way.”
Dr. Christine Shannon
In the early 2000s, Shannon noticed a change in students. “It became much harder to make [them] focus for a long time on a hard question,” says Shannon. “If they couldn’t get it pretty quickly, they would just give up.”
Interestingly, a new pedagogical approach—the flipped classroom—was emerging at the time. After studying the strategy, Shannon gave it her own spin, flipping her classroom halfway. “I think of the half-flipped class as providing scaffolding,” she says. “I give students things that will help them do better in class.”
Below, she explains what she means by this—and the strategies she uses before, during, and after each session to get and keep students engaged.
Shannon’s solution: Flip the class (almost)
The scaffolding in Shannon’s half-flipped approach includes elements that happen before, during, and after class. This, she says, gives students more time to absorb the material—and more ways in which to absorb it. “Students need some time with the concepts, and they need to see them again and again,” she says.
Shannon adds that she has not entirely eliminated traditional classroom methods from her approach. “I still present definitions and prove theorems at the board, and ask countless questions of the class as I develop those proofs,” she says. “As always, students present solutions to exercises that they have been assigned. But behind all this is an increased support, which encourages greater understanding of the definitions and the consequences of the theorems we are proving.”
Here, she breaks down what her approach entails and how it has helped her students.
For homework: Prevent rushed reading with tough focus questions
One flipped-classroom strategy that has been effective for Shannon is assigning reading in advance. By reviewing and studying material before class, students are already familiar with the concepts they are going to encounter and are better prepared to engage in discussions.
To ensure that they do not rush through the reading, Shannon adds a twist, giving them reading guides with specific questions to answer—ones that require some thought, not just looking for keywords. “They need a reason to go from one point to the other,” she says.
“So sometimes I’ll make a related assertion and ask if it is true or not. Other times I will ask for an example or counterexample. Often, I will require justification for a statement that is part of a proof in the text. I just want them to think about and internalize the concepts that may seem rather abstract.”
Dr. Christine Shannon
Before class: Assign quick quizzes to identify sticking points
A few hours before class, students must complete a pre-class quiz on the reading material. Shannon posts it on her school’s learning management system, so that students can have access at any point. This allows her time to read the responses before class and adjust the day’s lesson accordingly.
“If 30% of the class got a question wrong, they clearly didn’t understand it, so I could start class off by saying, ‘Oh, I noticed that a bunch of you didn’t understand this concept.’ And I could just try to explain it,” she says.
During class: Use group work to foster individual persistence
In earlier years, Shannon would work out a problem or a proof on the board and students would watch and learn vicariously. Today, though, she strives to get students as actively involved as possible. This is especially important as the theorems become more difficult.
“There are a few theorems that are just really hard,” she says. “Most students are probably not going to come up with the solution on their own. I want them to keep working on it, even if they get stuck.” To help them get unstuck, she enlists a common flipped-classroom approach of group work. She divides the class into groups of two or perhaps three students to discourage mere spectators. For the toughest theorems, she provides a worksheet to help students organize their steps and have a “roadmap” to the proof.
They collaborate, struggle with the problem together, and, at the end of class, share their work on the board. “Even if, ultimately, I have to present the correct solution,” Shannon says, “their engagement in this process has deepened their understanding of what can be very subtle ideas.”
After class: Incentivize students to reflect and redo what they missed
Concerned that her students were not paying sufficient attention to the comments she wrote on their homework papers, Shannon came up with a way to incentivize taking the corrections to heart. Each problem is worth 5 points, and for students who earn a 4 or 5, the grade is assigned and recorded. Otherwise, the students have until the next class period to submit a revision, for a maximum of 4 points. “A solution that was not at least worth a 4 was not much of a proof,” she says. “I point out all the issues and give them a second chance at it.” Those who do not take this opportunity earn a zero.
Shannon has seen excellent results because of this system. “The students who do this really improve,” she says.
“When students take the time to rewrite the things that they got wrong, the chances are much greater that the next time they come to something similar, they’re going to be able to do a better job of it.”
Dr. Christine Shannon
