Developing proofs is an essential
mathematical process that students should learn in high school classes
in order to move on to higher levels of math, according to the
University of Delaware’s Michelle Cirillo.
But too often, she says, students don’t understand it — making their
transition to college-level math classes particularly difficult.
“Proof is a central mathematical process and really is the gateway to higher math,” said Cirillo, an assistant professor of mathematical sciences
who has been awarded a prestigious national grant supporting her
research to find better ways to teach the skill. “But many people have
found that proof is one of the most challenging concepts for students to
learn and for teachers to teach.”
The National Science Foundation
(NSF) recently selected Cirillo to receive a Faculty Early Career
Development Award as she works to help high school teachers and students
improve the process. The $874,000 grant, “Proof in Secondary
Classrooms: Decomposing a Central Mathematical Practice,” will support
her research and education program for the next five years.
The highly competitive NSF awards recognize junior faculty for their
role as teacher-scholars and are given to those scientists and engineers
considered most likely to become the academic leaders of the 21st
Secondary Teacher Education
program, said her research led her to develop a possible intervention
to improve teaching the proof process in the context of geometry. She
tried it out in a small, exploratory study with good results.
“I wanted to show that if you teach proof in a different way,
students can be successful,” she said. “When I worked with teachers who
tried it, their students did succeed. And the teachers told me that for
the first time in their careers, they heard students give positive
feedback about learning proof.”
The key, Cirillo said, is breaking down the process of developing a
mathematical proof into smaller components, easing students into the
idea that a formal proof is necessary as a way for them to explain their
thinking. She developed a series of specific sub-goals for students to
master as they learn the overall skill.
“One of the goals is for students to understand proof as a
mathematical process,” she said. “Instead of following a teacher’s
demonstration of proving and then copying that process — a kind of
show-and-tell method that’s often used now — students can learn proof as
a way to reason.”
Cirillo will work closely with classroom teachers over the next five
years. Her first steps will include recruiting teachers in the area and
developing a series of 16 lesson plans and student assessment
In the second year of the research, she will have teachers from the
exploratory study pilot the new lesson plans, and she will collect
baseline data on a new group of teachers. That summer, teachers will use
the new approach to teach small groups of students. During the next two
school years, teachers will use the lesson plans in their classrooms.
Throughout the process, Cirillo said, lessons will be video-recorded,
and teachers will be valuable partners in assessing how the lessons
work and what should be changed to fine-tune them. Eventually, the
lessons may be compiled into a professional development program.
“My research model is a professional development model, where I
consider the teachers to be my partners and colleagues as we work to
solve a problem of practice,” Cirillo said. “Connecting research to
practice — that’s always my goal.”