Rachel Arnold

Excellence in Teaching Spotlight by the Center for Excellence in Teaching and Learning at Virginia Tech

The Excellence in Teaching Spotlight, presented by the center to approximately nine Virginia Tech faculty members each academic year, recognizes a faculty member's effective, engaged, and dynamic approaches and achievements as an educator.

Among the goals of the Center for Excellence in Teaching and Learning are advancing experiential learning at Virginia Tech and improving student learning through research-based instructional practices and student-centered design. According to Arnold, her approach to teaching focuses on inspiring personal autonomy and curiosity: “My job as a teacher is to inspire my students to take up agency and authority for the mathematical ideas that they develop within my classroom. Their ideas and questions form the basis of all class discussion. To promote a disposition of inquiry and curiosity in my students, I engage them with active learning tasks that bring forth and generate their mathematical thinking. I tell my students that their goal when working on these tasks is to explore their ideas with their peers, reflect on their ways of reasoning, and articulate their questions. This helps students prioritize their process of learning over simply giving a correct solution. By lowering the stakes through mathematical play and exploration, my students develop skills that enable them to tackle new problems flexibly and creatively. As their coach, I help them transform their existing ways of reasoning into even more powerful ways of reasoning that extend beyond the scope of my course.”

In line with her teaching approach, Arnold serves as the primary investigator for the NSF-funded “Proofs Project.” This project focuses on instructional tasks that "provide students with opportunities to realize their personal intellectual need for connecting their initial ways of reasoning to formal mathematical logic, language, and notation. Such an approach engages students in a gradual enculturation into practices that are considered normative within the greater mathematics community.”

Peter Haskell, Arnold’s former graduate advisor and current colleague, provided this example of Arnold’s teaching effectiveness: “Near the beginning of Rachel’s career, many teachers of lower-division classes began posting “skeleton notes” in which the structure of a class is revealed, with plenty of blank spaces in which students can fill in details as those details are discussed in class and the teacher writes them on a document camera. I observed many classes taught with such notes, and I saw both the intended benefits (such as clear organization, time for more examples, and well-focused student notetaking) and some drawbacks (such as static teacher presence and student passivity). To a greater extent than realized by any other teacher I observed, Rachel reaped the benefits of the method while avoiding the drawbacks. She realized that the notes allowed her to spend more time moving away from the front of the room and stimulating students’ interactions with her and with each other. The notes preserved attention to topical structure and were a place to which Rachel could return to record the salient points made in the class discussions.”

Griffin Paddock, a former student of Arnold’s, described how Arnold played a crucial role in inspiring their decision to pursue a graduate degree: “Dr. Arnold is by far the best professor I've had at Virginia Tech and was pivotal in my decision to pursue graduate studies in mathematics. Her lectures were extremely engaging, and I found myself thinking about the problems in new ways as we did them in class and relating it to past and future material. I found the entire structure of the course to be designed for success, with homeworks and tests that were difficult but reinforced my understanding of the material in a huge way. Outside of the class, Dr. Arnold was also incredibly supportive and encouraging of questions about mathematics in general and how different subjects connected to the introductory analysis content we were doing in class. I discovered how much I loved mathematics in her class, largely due to how exciting the course and content was and how she conveyed the beauty of the subject. Dr. Arnold was a major factor in me pursuing summer research in analysis specifically and has been foundational in how much I enjoy analysis in general (enough to pursue a Ph.D. in it).”

Ashlyn McDonald, another former student of Arnold’s, expressed her gratitude for the significant impact Arnold had on her during her undergraduate studies: "Rachel was integral to my experience in mathematics and college in general. She changed my whole perspective on office hours and what it means to really struggle with and master difficult concepts. She helped me learn how to approach challenging problems, and I modeled my own teaching in grad school off of her patient and honest approach to making each student feel capable and valued. Rachel spends innumerable hours making sure her students feel supported, and she goes out of her way to design creative and versatile lessons that cater towards many different kinds of learners. She was a big motivator behind my desire to attend grad school, and she inspires me to better myself and support other women in mathematics every day. I am so grateful that I signed up for her Intro to Proofs class many years ago and gained a lifelong mentor and friend along the way.”

Arnold provided the following advice for new college teachers: “I truly believe that students learn best when they find their own personal meaning for the concepts being taught in your classroom. When you’re designing your instructional tasks, ask yourself how those tasks are helping students make connections to their own experiences. Value your students' ideas and perspectives by positioning them at the center of your classroom. If you want students to feel comfortable sharing vulnerably what they’re actually thinking, not just what they think you want to hear, think carefully about the ways in which you solicit their input. Use open-ended prompts like, ‘what kinds of things were you noticing when you were thinking about this task?’ or ‘which aspect of this activity was challenging for you?’. Avoid asking things like, ‘what’s the answer to this question?’ as I believe this marginalizes those students whose ideas are still emerging-- emergent ideas that are not yet fully formed are so important for learning! Create a safe space in the initial discussion where students can share their ideas without evaluating them as right or wrong. Once everyone’s ideas are on the table, then you can help them revise and synthesize their understanding.”