In my professional role as a mathematics education researcher, I write, I analyze, I discuss, I critique, I synthesize, I read, I ponder.
I seek to change the world with my ideas.
My work as a researcher is centered on my quest to better understand students’ meaningful learning of mathematics. This guiding motivation is at the core of what I do on a daily basis. Currently (the past three days) this work has centered on representational fluency in solving equations and conceptions of functions.
I research students’ meaningful learning of mathematics through investigations of students’ creation and interpretation of multiple representations in solving problems involving linear equations created using paper-and-pencil and computer algebra systems. The activity of working within and translating among graphs, tables, symbols, and words is central to meaning making and understanding of mathematical ideas. To characterize sophistication in students’ representational fluency, it is thus important to be able to model and visualize (1) the number and type of representations students use to solve problems, and (2) the nature of how students use these representations to make progress in solving a problem. Results of an interview study of ninth-grade students’ equation solving with multiple representations created with CAS and paper-and-pencil exemplify some of the rich variation possible in students’ sophistication in representational fluency. This study prompts a prudent reflection on theoretical assumptions about learning in relation to students’ representational processes. What do students “see” in representations? Is the relationship between students’ representational fluency and understanding of core concepts bi-directional and cyclical? If so, what does this look like? Perhaps the interplay of students’ perceptions and conceptions is an important lens to bring to bear.
Conceptions of Functions
Another way I understand students’ learning of mathematics is by analyzing students’ verbalizations and inscriptions in quantitatively rich task situations. These analyses focus on qualitative characterizations of students’ conceptions of functions in solving tasks in the context of a carefully sequenced teaching experiment. Over time, we are crafting a story of how students’ learning builds to robust mathematical conceptions in goal-directed activity of analyzing functions and change. For example, we are exploring how students’ symbolization of functional relationships (as symbolic rules, e.g., Area=2Height^2 or y=ax^2) may be supported through their coordination of quantities (in the form of numeric tables and pictoral visualizations of growing rectangles, lengths, heights, and areas).
Another aspect of my work that complements my focus on meaningful student learning is the design and testing of tasks and instruction involving mathematics technology. There is much more to come…