Research Statement

My research interest is undergraduate education in the STEM disciplines: science, technology, engineering, and mathematics with a heavy emphasis on how students make sense of mathematics. This is reflected in the research projects I have conducted at the University of New Hampshire and in my plans for future research.

Recent Research Projects

My recent research projects both focus on student understanding of mathematical concepts while taking different perspectives on their output. In the first project I created a model for student understanding of the limit concept whereas the second project looked at how instructors can impact student understanding through a well timed trigonometric intervention.

A Model for Student Understanding of the Limit Concept. This work relied on both quantitative data, student responses to an inventory of limit questions, and qualitative data, task-based interviews, to develop a model for the ways in which students make sense of the concept of limit through a variety of problems. From this data I created a model that interprets student understanding through the student's problem-solving modality and whether or not that modality is in line with the context underlying the posed problem. My model has been shared as a poster at TRUSE in Orono, Maine in June 2010, presented in a paper at the SIGMAA on RUME conference in Portland, OR in February 2011 and published in the peer-reviewed conference proceedings.

Impact of a Trigonometric Intervention on Algebra-based Physics Students' Understanding of Equations Describing Simple Harmonic Motion. This project was an interdisciplinary collaboration with Dr. Dawn Meredith, a physics educator at the University of New Hampshire. We designed and administered several versions of a trigonometric review worksheet to assess its impact on students making sense of the equations that describe the oscillatory behavior of objects. Each version of the worksheet featured a different motivation for interpreting the change in the trigonometric function's argument from θ to 2πft. Our findings that the motivation included in the first version of the worksheet, distance travelled around a circle, yielded no significant improvements on student understanding were presented at the SIGMAA on RUME conference in Portland, OR in February 2012. The feedback received from this presentation, along with suggestions from physics colleagues, led to several alternate motivations. Adding probing conceptual questions and a motivation based on angular velocity to the trigonometric review worksheet led to an improvement in student performance visible in their pre-test and post-test scores.

My Dissertation

During the interviews I conducted for my recent research projects I often asked questions such as "Are you feeling all set for your exam?" and "How have you been preparing yourself?" as ice breakers. Student responses to these questions revealed a diverse range of preparation behaviors, from preferring to work alone to getting together with a group of friends. These answers, in addition to the behavior I had observed in my own students through my role as a teaching assistant, prompted me to look deeper at student study behavior. In particular, I began to wonder how students engaged with mathematics in self-formed groups outside of the classroom.

What Do Students Do In Self-formed Mathematics Study Groups? My dissertation work has pioneered a method for observing students working together outside of the classroom. Its primary focus is on establishing a description of self-formed student study groups through the identification of what roles students assume and which resources students utilize. To date this project has led to an invited poster describing my methodology for observing students outside the classroom setting and a presentation describing some of the preliminary patterns observed in student behavior at the SIGMAA on RUME conference in Portland, OR in February 2012. Although preliminary, my findings suggest that students approach the task of reviewing for an exam differently than they approach the completion of their homework assignments.

Plans for Future Research

My dissertation lays the groundwork for a variety of future studies. First, I intend to ascertain the generalizability of my results. From there I plan on investigating what commonalities exist between student groups studying mathematics and student study groups for other STEM-discipline coursework. The results from both these studies will lend themselves to addressing the question of determining which traits characterize effective study groups.

Generalizing My Dissertation Findings. Additional observation is needed to ensure that I identify a complete set of the roles that students assume, create a comprehensive list of materials utilized, and improve the generalizability of my dissertation's findings. The majority of the students I observed for my dissertation were second year students enrolled in a rigorous honors course and had been enrolled in the same honors mathematics sequences during their first year of college. Thus there's strong reason to believe that these students are unusual, although given that my study is the first of its kind, they do form an initial basis. The possibly special nature of the class and collection of students makes it important to work with other types of classes and students.

I plan on re-using my dissertation methodology to study more than the two study groups I focused on during this round of research. This is not an attempt at replication, as I anticipate that non-honors students will compare homework problem solutions and prepare for exams differently than students I have already observed and I will specifically be looking at how they do that. To do so, I will canvas a larger variety of student self-formed study groups and rather than following them all for an entire semester, I will observe only one or two meetings per group. This will allow me to determine the generalizability of my earlier claims and, if needed, add student group roles and utilized resources to my previous findings. After establishing the robustness of my findings, I have two related research projects I want to pursue.

Does Subject Matter Alter the Structure of Student Self-formed Study Groups? As there are many subjects in the STEM disciplines that draw upon mathematics, it is reasonable to ask what characteristics of student mathematics study groups are also present when students study math-intensive subjects in groups. Thus I am looking for collaborators in related disciplines, such as physics, engineering, and statistics, in order to access students in their classes for observation. I believe the methodology and findings established by my dissertation work are applicable to this project. Several study groups across disciplines will be observed for one to two sessions each. Their dialogue and actions will be coded and compared to the roles and behaviors that have been identified in my previous work. Within one year of starting this project, I anticipate finding publishable results.

Determining Efficacy of Student Self-formed Study Groups. A robust description of how students engage with course material outside of the classroom lays the groundwork necessary to start investigating what it means for a self-formed study group to perform effectively. I believe this can be determined by identifying which characteristics (roles, resource utilization) are prominent in groups that seem unable to make progress toward their study goals and are absent in groups that achieve their study goals. One way to accomplish this would be to invite groups of mathematicians, or experts, that often work together to be observed while they work on novel tasks. The traits that arise in the group of experts can then be compared and contrasted with those observed in the student, or novice, study groups. This study would occupy 2 years in order to generate the in-depth level of data necessary for analysis and drawing conclusions to such research questions. A methodology would need to be designed for identifying a group's goal and measuring whether that group makes progress towards meeting that goal.

Statement of Teaching Philosophy

I used to believe that my role as an instructor was simply to present the facts and the tools my students would need in their later courses in the form of theorems and algorithms. Since then I have come to realize that the facts and tools I actually impart on my students are the fact that mathematics is a discursive activity and the tools of multiple ways to make sense of important concepts and problem-solving behavior. In my classes I use a number of different pedagogical practices and technologies to help my students develop rich mathematical understandings, a productive disposition, and a strong sense of community.

I show my students that mathematics is a discursive activity through classes that offer plenty of opportunities for interacting, in both my seminar-type classes and lectures. I stress equal participation from all students enrolled in my classes and I strive to make all of my students feel that they are welcome to contribute their thoughts, questions, and answers to discussions that occur in class. To set the mathematical tone for the class, I frequently have problems on the board that either require students to recall material from previous lessons, motivate the topic to be covered during the day's class, or both. Since the problems often review concepts from the previous lecture, they are a low stress opportunity for students to present their work on the board. Not only is sharing solution strategies is a great way for students to solidify their understanding of the material, but it provides students with a chance to practice communicating their ideas and receive feedback on their work.

My goals of fostering discussion and community extend to my office hours where I strike a balance between leading problem-solving activities from the board and encouraging the students to come up and take the reins. I will often answer a question of "Is this right?" with "What does everyone else think?" to encourage students to draw upon each other's knowledge. This fosters a community within the classroom where students can feel comfortable challenging each other's responses, testing their own knowledge of the material, and learning to see mathematics as a social activity. This helps me realize my goal of making students feel their contributions are valued so that they are more likely to speak up and offer answers or present their solutions on the board.

I tailor my teaching activities to my audience. This means I prepare myself with multiple examples of a concept and a multitude of ways to explain them. For topics such as determining the volume of a solid of revolution, if drawing 3-D shapes on the board has failed to resonate with the students I turn to the physical materials I have on hand to construct physical demonstrations of the concept. In one instance, I stacked coins of varying sizes to illustrate the disk method of computation. Similarly, I have used Maple and Matlab to help my students engage with mathematics through visualizations of a concept they are struggling with and demonstrating alternate ways to check their calculations.

Perhaps the most important aspect of my instruction is the modeling of thought processes and what it can look like to "do" mathematics. I believe that if students are uncertain how to approach a problem they may find their solution attempts ineffectual when they face problems on their own. Thus in class I open dialogues that analyze what the problem is asking for and generate possible solution strategies for it whether I'm solving the problem or I'm facilitating discussions of student work. Because of this focus on doing meaningful mathematics, there are times that students ask a question and I do not have an immediate answer. When that happens, I share my thought process with my students as I work through the derivation of a possible explanation or turn it back to the class as a group challenge. Mistakes provide additional opportunities for modeling the mathematical behaviors of verifying answers and reviewing solution strategies. I believe that many students need to be shown that there is value in reflecting on their work and that sometimes you need to restart a solution to a problem from scratch.

It seems to me that many students enter the classroom with the impression that mathematics is the silent, rote repetition of memorized procedures and that the answer obtained by the end of the application of the algorithm, regardless of reasonability, must be right. Thus I feel it is my role to break my students' silence with an open dialogue discussing problem-solving strategies and to develop techniques for assessing the validity of one's work.

Publications