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Kindergarten - Grade 9 in Southborough, MA
Fay Magazine: Fall-Winter 2019

A Method to the Math

by Daintry Duffy Zaterka '88
Creative problem solving in Upper School math classes turns the traditional class format on its head.
Spend five minutes in Extended Topics in Algebra, and you might think that the students run the class. After all, they are the ones doing most of the talking!

Extended Topics in Algebra (ETA) follows the problem-based learning model of teaching math. Upper School math teacher Craig Ferraro doesn’t lecture to his ETA classes. On most days, his students are the ones up at the board explaining their work, sharing their thinking, and asking each other questions.

Problem-based learning is a student-centered, interactive approach to mathematics where students discover and learn new concepts together by working through a carefully structured series of problem sets. At home, ETA students spend time thinking and working on a problem set for homework. In class the next day, they work in pairs to present one of the problems to the class, explaining their mathematical thinking and how they tried to solve it—whether they were successful or not. The model follows the premise that students who engage 
with the material by collaborating, questioning, and making discoveries will develop a much deeper understanding than if the information was just fed to them by a teacher.

Since its introduction at Fay three years ago, problem-based learning (PBL) has become a strong thread running throughout the Upper School math program. Pre-Algebra Advanced and Algebra 1 both spend the winter term immersed in PBL, geometry incorporates PBL in a unit, and ETA spends the entire year engaged in PBL.

Craig first introduced PBL to Fay after attending a conference at Phillips Exeter Academy, a pioneer in the approach. One week spent working on problem sets and presenting them to fellow conference attendees convinced Craig that the model resonated strongly with his teaching philosophy. “I’ve always felt that the person doing the work is the one doing the learning,” he says. “I realized that my understanding of the math concepts was enhanced by the approach because more was being asked of me as a learner.”

For some students, the introduction to PBL can be daunting. Unlike traditional math, where students repeat a series of similar calculations, PBL 
problems are by nature more open-ended. Often, the first step is just trying to figure out what kind of problem it is. “Some students are uncomfortable with the uncertainty,” says Craig, “but that’s the point. I want them to get used to wrestling with something and to be excited to meet the challenge of the unknown.”

Students usually settle into the routine quite quickly, notes Mathematics Department Chair Cassandra Papalilo. “It usually only takes one or two problem sets for the students to get the hang of it, and it helps when they realize that everyone in the class is experiencing the same uncertainty.” One important element of PBL is that it’s okay for students to acknowledge that they don’t know how to do a problem, to ask their classmates for help, or to try different methods of solving a problem while presenting it to the class. “What happens over time is that students become more comfortable talking about their work, asking questions in the moment, and making sure they understand why someone did something,” says Craig. The grading for problem-based learning homework and classwork is not based on the accuracy of the work but on specific behaviors that the teachers want to reinforce. Each problem set has a possible value of four points. First, the students must have tried each problem. 
“I tell them to spend a limited amount of time thinking through and trying each problem,” says Craig, “because these are bigger problems that you might spend two days thinking about.” Students also earn points based on the presentation of their problem in class, their participation in the discussions, and
taking active notes on the classwork.

Craig and Cassandra have seen concrete benefits from the introduction of problem-based learning. Craig notes that the year-end test scores for his ETA students have been some of 
the strongest he has seen. “Their level of understanding is so much higher,” he says. Cassandra also sees significant changes in the way her students approach their work. “I notice that PBL improves students’ flexibility in thinking, their ability to think critically and solve problems creatively, their collaboration and communication skills, and their independence and follow-through on a procedure.”

Perhaps the starkest difference is in the students’ level of engagement with the material. “Students will often come into class and start a problem on their own or be working with one another long after the class has ended as they try to figure something out,” says Craig. Ultimately, seeing the students invested in the learning process instead of just focused on getting an answer is gratifying, he adds. “They’re looking for more, and they want to be able to compare and talk about their methods as opposed to just looking for an answer.”
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