Rachel Rupnow: Investigating how we teach abstract algebra—and the creativity it can teach us

The following story was writtenin April 2019 by Michelle Corinaldi in ​ENGL ​4824​: Science Writing ​as part of a collaboration between the English department and the Center for Communicating Science.

“I’m just not a math person.”

    We’ve heard it from classmates, friends, and family members—or may even have said it ourselves. It’s often accompanied by a small shrug of the shoulders or a quiet shake of the head about something “I’ve just never learned to understand.” Many of us usually shy away from anything that resembles mathematical equations or symbols.

    To pass the required math courses in middle school, high school, and college, students “who aren’t math people” make efforts to commit to memory the basic sets of rules and procedures. But after high school—or even earlier—solving the more complex operations requires more than following a sequence of steps and the rules that apply and going beyond the memorization of the basic fundamentals. 

    Rachel Rupnow, unlike many of us, is a math person through and through. Rupnow, a Ph.D. student in the Department of Mathematics at Virginia Tech, is researching how math is not only taught and learned, but how it is generally understood in the classroom.

    The curricular change from procedural to conceptual, she explains, significantly contributes to the “leaks” that exist in the STEM pipeline, a popular metaphor used to illustrate the general educational pathway of students who demonstrate an early interest in science, technology, engineering, and math to—decades later—having a career in these disciplines. Although there are other factors that can influence these “leaks,” the transition in the academic focus can be perceived as jarring, and students who don’t have a good grasp of the material (but do have an interest in the disciplines) can be “leaked” from the pipeline and filtered out from majoring in, and subsequently pursuing, a career in STEM.

    As a result of the gaps that exist between K-12 and college levels, much research is being conducted to make mathematics education a more coherent experience for students. Directly contributing to these critical scholarly efforts, Rupnow is specifically focusing on the relationships between instruction and conceptual understanding of the curriculum in abstract algebra courses at the college level.

    According to Rupnow, students in abstract algebra are typically pre-service math teachers. And because these students, who are almost exclusively math majors (or double majors), are graduating to become middle school and high school math teachers themselves, they are in a unique position “to understand their content knowledge and their belief[s] about math teaching and learning [as well as to] see how this is going to start in the cycle of lower grades,” Rupnow said.

    To understand specifically these relationships between instructor teaching and student learning, Rupnow is conducting research on the implications of different teaching approaches, particularly inquiry-oriented and inquiry-based instruction.

    “In my research, I was looking at one instructor who used inquiry-oriented materials, which is [when] students are guided through the process of doing some activities. . .that kind of [create] a need for reinventing the ideas. . .of the formal definition that a mathematician would recognize from a textbook,” Rupnow explained.

    “The other class I was looking at, the instructor would lecture two of the days, and then use what we called ‘labs’ on the third day, and there, it was essentially activities to build up their intuition," she said. "But the focus wasn’t necessarily on reinventing the terms, it was more of ‘here you played with [this concept] a little bit, and you’ve gotten an idea of what’s going on, and now I’m going to tell you the standard mathematics.’”

    Her research so far has found that there was student success in each of the classrooms.

    She explained, “The instructors came into this thing with different approaches, and yet to a large extent, students seemed to take away similar ideas from the class, in terms of ‘I was doing something different than just being lectured at, [and] that was interesting.’”

    These understandings, according to Rupnow, generally point to a common expectation for students to explain their reasoning behind a particular solution. And personally for Rupnow, this direction for reasoning was transformative for her early interest in math and, later, the path of her educational career.

    “In high school, the expectation was that we would go up and write a solution to one of the problems from the homework on the board, and it was a common idea that there should be some sort of rationale for what you are doing,” she said. “And because my teacher [encouraged my rationale], that certainly had an impact on my desire to be a teacher.”

    Now as a teacher herself, she makes it a point to emphasize the importance for her students to understand the process of the problem, and not just the answer. Especially with the conceptual focus in the college classroom, Rupnow stresses that it is necessary for students not simply to rely on memorizing and “obeying the procedure without question, but. . .actually understanding how these ideas connect together and why they connect together.”

    To have an understanding of the meanings behind the rules and procedures is crucial, Rupnow acknowledges, for students to see and learn math in varied ways. And it is through these different ways of reasoning and thinking that she hopes students can embrace math as a place of—and for—creativity.

    With an evident enthusiasm, Rupnow says that math is “kind of like how you approach an art project or approach a writing project, and you’re just trying to put some ideas together, and kind of see a world that you can create. You can do the same thing with math.”

    To Rupnow, this is what makes math fascinating for her to learn, study, and teach.

    “Math is fun, and. . .certainly I want that to be the case in my own classroom,” she said. “[I want to] see what also makes this interesting to students, so we can perhaps get more and more people thinking about ‘how do I make math applicable to my life?’”

    And for students to be asking these kinds of inquisitive questions and making these deeper connections with the math, she ultimately hopes that they, too, can become math people.