Prospective elementary teachers at three universities engaged in online modules called the Virtual Field Experience, created by the Math Forum. The prospective teachers learned about problem solving and mentoring elementary students in composing solutions and explanations to nonroutine challenge problems. Finally, through an asynchronous online environment, the prospective teachers mentored elementary students. The researchers assessed the prospective teachers’ solutions and explanations to problems at the beginning of the semester, at the middle of the semester after completing the training in mentoring, and again at the end of the semester after the mentoring was completed. The researchers observed improvements in the prospective teachers’ abilities to write explanations to problems. Specifically, growth was seen in prospective teachers’ communication of their explanations and their ability to construct viable arguments and critique the reasoning of others (Common Core State Standards Initiative, 2010, Standard for Mathematical Practice 3), and attend to precision (Standard for Mathematical Practice 6).
In a course emphasizing interactive technology, 19 students, including 18 mathematics education majors, mostly in their first year, reinvented the definition of limit of a sequence while working in small cooperative groups. The class spent four sessions of 75 minutes each on a cyclical process of guided reinvention of the definition of limit of a sequence for a particular value, L = 5. Tentative definitions were tested systematically against a well-chosen set of examples of sequences that converged, or not, to 5. Students shared their definitions and the problems they were having with their definitions with their peers through whole class presentations and public postings on a course electronic forum. Student presenters received feedback from their peers both in person and through the forum. The approximation, error, error bound framework was used to help structure students’ thinking. The use of interactive examples with epsilon bands and movable N values, in which students could zoom in to adjust the value of epsilon or zoom out to find a value of N, proved especially helpful in the process. The changes in their tentative definitions show the difficulties students had as well as the learning that occurred.
Teaching mathematics for understanding requires listening to each student’s mathematical thinking, best elicited in a one-on-one interview. Interviews are difficult to enact in a teacher’s busy schedule, however. In this study, the authors utilize smartphone technology to help mathematics teachers interview a student in a virtual one-on-one setting. Free from physical constraints and preconceived biases, teachers can concentrate on building questioning, listening, and responding skills when noticing student mathematical thinking. Teachers engaged in four communication types when working with students through this technology: clarification, verification, and either extension or redirection.
This paper explores student interactions from the Virtual Math Teams-With-GeoGebra Project, a computer-supported collaborative learning environment that allows individuals to interact, collaborate, and discuss user-created dynamic mathematics objects. Previous studies of virtual math teams have focused on the coconstruction of a joint problem space and the ways collaborative meaning making can be accomplished in the online environment. Instead, this study explored the development of the students’ argumentation practices. The researchers used Toulman’s (1969) model to analyze and explain the structure of the online interactions and the argumentative practices that become normative among students. In particular, the researchers found that the students made increasingly detailed and mathematical descriptions of the data, developed more abstract warrants, and increasingly acted as if giving reasons was normative in the discussion.