Playing the believing game: Enhancing productive discourse and mathematical understanding

In mathematics classrooms, the practice of doubt pervades. However, Elbow (1986, 2006) contended that teachers must balance their practices of methodological doubt and methodological belief. The study reported here builds upon previous research which revealed the professor played the believing game (Elbow) and students were motivated to do mathematics. We addressed the question: How does a teacher (professor) play the believing game in a mathematics classroom? Videotapes, interviews, and field notes from an entire semester were collected and analyzed qualitatively. Although the professor was not consciously attempting to believe or doubt, we reveal when and under what circumstances they occurred. A temporal continuum of believing and doubting existed for the professor’s practice. Reserved believing and reserved doubting prevailed for the professor when she heard answers or comments she deemed incorrect and rich mathematical conversations transpired as she opened herself up to a deeper understanding of mathematics.     

Instances of mathematical thinking among low attaining students in an ordinary secondary classroom

This paper is a report of a classroom research project whose aim was to find out whether low attaining 14-year-old students of mathematics would be able to think mathematically at a level higher than recall and reproduction during their ordinary classroom mathematics activities. Analysis of classroom interactive episodes revealed many instances of mathematical thinking of a kind which was not normally exploited, required or expected in their classes. Five episodes are described, comparing the students’ thinking to that usually described as “advanced.” In particular, some episodes suggest the power of a type of prompt which can be generalized as “going across the grain.”     

Mathematics teaching practices with technology that support conceptual understanding for Latino/a students

We analyze how three seventh grade mathematics teachers from a majority Latino/a, linguistically diverse region of Texas taught the same lesson on interpreting graphs of motion as part of the Scaling Up SimCalc study (Roschelle et al., 2010). The students of two of the teachers made strong learning gains as measured by a curriculum-aligned assessment, while the students of the third teacher were less successful. To investigate these different outcomes, we compare the teaching practices in each classroom, focusing on the teachers’ use of class time and instructional format, their use of mathematical discourse practices in whole-class discussions, and their responses to student contributions. We show that the more successful teachers allowed time for students to use the curriculum and software and discuss it with peers, that they used formal mathematical discourse along with less formal language, and that they responded to student errors using higher-level moves. We conclude by discussing implications for teachers and mathematics educators, with special attention to issues related to the mathematics education of Latinos/as.     

Explanation,motivation and question posing routines in university mathematics teachers' pedagogical discourse: a commognitive analysis

This paper investigates the teaching practices used by university mathematics teachers when lecturing, a topic within university mathematics education research which is gaining an increasing interest. In the study, a view of mathematics teaching as a discursive practice is taken, and Sfard's commognitive framework is used to investigate the teaching practices of seven Swedish university mathematics teachers on the topic of functions. The present paper looks at the discourse of mathematics teaching, presenting a categorization of the didactical routines into three categories – explanation, motivation and question posing routines. All of these are present in the discourses of all seven teachers, but within these general categories, a number of different sub-categories of routines are found, used in different ways and to different extent by the various teachers. The explanation routines include known mathematical facts, summary and repetition, different representations, everyday language, and concretization and metaphor; the motivation routines include reference to utility, the nature of mathematics, humour and result focus; and the question posing routines include control questions, asking for facts, enquiries and rhetorical questions. This categorization of question posing routines, for instance, complements those already found in the literature. In addition to providing a valuable insight into the teaching of functions at the university level, the categorizations presented in the study can also be useful for investigating the teaching of other mathematical topics.