Making the Invisible Visible: Exploring Students\u27 Mathematical Understanding

This study, involving action research and in-depth interviews, explored students\u27 perceptions of their conceptual understanding in a secondary Advanced Placement Calculus course. My goal as a teacher-researcher was to surface and unpack ideas that would inform my instructional practice. The research focus arose from observations that students could accurately complete mathematical tasks, but often did not know why they did what they did. Skemp (1987) saw this as instrumental understanding and contrasted it with relational understanding, or the knowledge of, not only what to do, but why. During action research, I gave students opportunities to reflect on their learning, to articulate their observations, and to share these observations with their classmates and me in an online format. Based on the content of student reflections, I adjusted instruction to address specific needs. In order to further explore students\u27 perceptions of their mathematical understanding and their experiences developing these understandings, I conducted a series of in-depth interviews with four purposefully selected key informants a year after completing the action research. These interviews provided the opportunity to explore some of the ideas and conversations posted online in order to capture a more complete picture of students\u27 perceptions and experiences. Four inter-related themes emerged from students\u27 reflections. First, visual representations helped students develop relational understanding. Second, repetition, in terms of modeling, review, and practice, supported students\u27 problem-solving ability. Third, explicit math-to-math, math-to-world, and math-to-self connections were central to understanding. Fourth, classroom climate factors contributed to students\u27 development of understanding. The most significant lesson I learned as a teacher throughout the course of this study involved the power of asking a simple question: What do you understand? In order to respond, students reflected on their own thinking, formulated, and then communicated their responses. These reflections, shaped by students\u27 past experience, gave me a view of student thinking I never had before, and affected my intentional classroom behavior. Another aspect of the impact of that simple question was associated with the affective domain and learning. Students perceived the willingness to ask the question as an indicator of teacher interest important to their understanding. Skemp (1987) called understanding relational, emphasizing relationships between concepts. From a student\u27s perspective, understanding may be relational in other ways as well, emphasizing relationships between and among the teacher and the students. The student thinking made visible by the reflections gave me valuable insight into student learning and into the nature of their understanding--instrumental or relational. Reading student responses was much like looking into a mirror at my own instructional practices with sometimes positive, sometimes less than positive impressions. I was reminded of the need to explicitly guide students to make important conceptual connections, thus making what could be perceived as invisible relationships, visible for students