Overt and Covert Participation in an Argumentative whole-class Discussion: Spread of Ideas about Quadratic Functions

Mathematical ideas are developed and spread during argumentative whole class discussions between a teacher and her students. The goal of the current study is to characterize how ideas about quadratic functions emerge and are spread during a whole-class discussion among ninth graders. To this end, we recorded both discussions between pairs of students and whole-class discussions led by the teacher. We used the approach of documenting collective activities as our methodological lens. The findings show that mathematical ideas about quadratic functions, like positive and negative range, increase and decrease range, minimum or maximum point, intersection with the axes, and more were spread in two parallel layers. Students participated in an overt layer in which they uttered their ideas in a public discussion. At the same time, they also uttered mathematical ideas privately with their peers in a covert layer. That is, whole-class discussions are not identical for all participants in that the covert layer turns these discussions into a unique experience.