The interplay between Researcher Carolyn Maher’s questions and fourth grade student reasoning while exploring fraction comparisons

This analytic is the fourth in a series of analytics that describes teacher questioning and student responses and reasoning so that researchers, teachers and teacher educators can study patterns of teacher questioning techniques and responses from students. Research reveals the importance of effective teacher questioning and highlights the role that such questioning plays in the development of students’ mathematical reasoning (Klinzing, Klinzing-Eurich & Tisher, 1985; Martino & Maher, 1999). Questioning can serve as a springboard for further discussion, elaboration of incomplete thought, and a greater conceptual understanding of the mathematical problems at hand. Teacher questioning can give the teacher a glimpse into the mathematical thinking of the students that may not be apparent otherwise. Not only may it help the teacher estimate students’ understanding of the concept being discussed, but it may also provide the teacher with a better understanding of the students’ thought processes, judgments?, concerns, or hesitations when coming up with solutions to open-ended mathematical problems. Modeling of good questions may also have a positive effect on the students. They may begin to imitate those forms of questions both when interacting with peers and when working on their own. By asking students for explanations of their work or for further clarification, or by asking students how they would convince their peers of their solutions, teachers may stimulate students to initiate such discussions on their own or prompt them to formulate cogent justifications and reasoning. Such probing by teachers may also result in students’ posing these questions to themselves, which in turn could help clarify the solutions in their minds before presenting their ideas to others. The events in this series of analytics are selected from sessions conducted as part of the longitudinal/cross sectional research study of the development of students’ mathematical thinking and reasoning conducted at Rutgers University that spanned over twenty-five years (Maher, 2010). They highlight events that occurred during sessions conducted in urban, working class, and suburban settings, across a large range of age levels, by a variety of researchers, and amongst a number of content domains. They showcase discourse that occurred during classroom settings as well as informal learning environments. The five video narratives show different types of questioning demonstrated by the researchers and the associated student reasoning and argumentation. During these representative sessions conducted by Researchers Carolyn Maher, Amy Martino, Robert Davis, and Arthur Powell, varied forms of questioning were used. The questions and discourse moves are presented along with the students’ responses in building proof-like justifications. Specifically, probing questions, eliciting questions, and questions that encouraged student engagement seemed to be associated with higher-order thinking and reasoning. These specific types of questioning techniques utilized by the researchers are presented along with the formulation of solutions by students. We observe, also, extensions of student reasoning, as well as the connections that were made, enhanced, and refined in the solutions. A complete discussion of the classification of teacher questioning during these and other sessions as well as a description of the associated student argumentation, justification, and reasoning can be found in the “Interplay between teacher questioning and student reasoning” (Gerstein, 2017). This analytic looks primarily at questions posed by the researchers associated with engagement or the public display of students’ ideas. The argumentation that evolved and the classroom discourse can be observed, as well as increasingly sophisticated justification and reasoning of students. Questions associated with engagement show patterns of attention to the task and listening attentively to peers. They also reveal whether students were engaged in the activity and whether the input of others was clear to all. Some questions are associated with confirmation and agreement. Others are associated with establishment of understandings of other students. Researchers elicited student ideas as they encouraged students to formulate their own ideas and strategies. Using these questions, researchers encouraged students to make their thinking public or explicit or to make their explanation understandable to other students. Eliciting questions encouraged students to voice their ideas and solutions, explain their thinking and share their thoughts or solutions with others. In this analytic, Researcher Maher uses many questions to elicit student ideas and encourage engagement during a whole class discussion at the Colts Neck public school and elicited argumentation, reasoning and justification as students discussed the task “Which is bigger ½ or 1/3?” During this session other forms of questions, such as probing questions, were posed as well by the researcher and noted in this analytic. Probing questions are questions that elicit student answers past their initial replies. Researchers also asked a probing sequence of questions (Franke et al., 2009) or a string of probing questions. These questions were often associated with students’ attending to errors in their claims or enhancements of their explanations. Probing questions often consisted of a series of more than two related questions about something specific that a student said and included multiple teacher questions and multiple student responses. Probing questions include elucidating probes that were used to clarify a student response. The teacher used these questions to query a student about what he or she meant and to invite the student to elaborate on what was said. It also includes redirecting questions that asked students to consider analyzing the question further and think about the implications of the response and how it would relate to other solutions or statements. Questions were also included in the classification of probing if they were used to encourage critical thinking or augment analytical cognizance. This type of probing question was used when the teacher attempted to increase the students’ critical awareness about what their responses had been and what their underlying assumptions may have been in making a particular statement or offering a particular answer and thereby make them cognitively aware of the reasons behind their thinking. Probing questions were also used to ascertain whether the students had completely answered the question at hand. These analytics are designed to provide valuable insight into the dynamics of teacher questioning and student reasoning. Franke, M. L., Webb, N. M., Chan, A. G., Ing, M., Freund, D., & Battey, D. (2009). Teacher questioning to elicit students’ mathematical thinking in elementary school classrooms. Journal of Teacher Education, 60(4), 380-392. Gerstein, M. (2017). The interplay between teacher questioning and student reasoning. (Unpublished doctoral dissertation). Rutgers University, New Jersey. Klinzing, G., Klinzing-Eurich, G., & Tisher, R. P. (1985). Higher cognitive behaviors in classroom discourse: Congruency between teachers’ questions and pupils’ responses. The Australian Journal of Education, 29(1), 63-75. Maher, C. A. (2010). The Longitudinal Study. In C. A. Maher, A. B. Powell, & E. B. Uptegrove (Eds.), Combinatorics and Reasoning (pp. 3–14). Springer. doi:10.1007/978-0-387-98132-1. Martino, A. M., & Maher, C. A. (1999). Teacher questioning to promote justification and generalization in mathematics: What research practice has taught us. The Journal of Mathematical Behavior, 18(1), 53-78