The Interplay between Researcher Martino’s Questioning and Brandon’s Reasoning: Utilizing Probing Questions to Invite Students to Make Connections

This analytic is the second 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 can serve to elicit in students a greater conceptual understanding of the mathematical problems at hand. Responses of students from teacher questioning can offer to teachers a glimpse into the mathematical thinking of the students that may not be apparent otherwise. Not only may might 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, judgment, 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. One outcome is that students may come to imitate similar ways of questioning when interacting with peers and also 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. It also helps to establish the norms and expectations of the classroom community. The events in this series of analytics are culled from the 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. This series shows how different types of questioning illustrated in the videos were associated with the production of students’ arguments, forms of reasoning, and making connections to structurally equivalent problem tasks. During these representative sessions conducted by Researchers Carolyn Maher, Amy Martino, Robert Davis, and Arthur Powell, several varied forms of questioning are noted. The questioning techniques employed by researchers were associated with ways the students formulated their solutions, extended their reasoning, made connections, or otherwise enhanced or refined their solutions. These questions and discourse moves often invited students to build proof-like justifications. Specifically, probing questions, eliciting questions, and questions that encouraged student engagement seemed to be especially effective in providing opportunity for students to express higher-order thinking and reasoning. A complete discussion of the classification of teacher questioning during these and other sessions as well as a description of the association between researcher question and student argumentation, justification, and reasoning can be found in the “Interplay between teacher questioning and student reasoning” (Gerstein, 2017). This analytic looks primarily at probing questions that were posed by the researcher that invited the student to make connections. Probing questions are questions that elicit student answers past their initial replies. Often the researchers asked a probing sequence of questions (Franke et al. 2009) or a string of probing questions. These questions were often used to enable students to see the error in their statement or to help them enhance their explanations. They 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 questions 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 allow the student to elaborate on what he or she had said. It also includes redirecting questions where the researcher asked the students to consider analyzing the initial question further and consider the implications of the response and how it would relate to other solutions or statements. This type of questioning invites a student to “fold back,” an element of Pirie and Kieren’s theory for the dynamical growth of mathematical understanding and its associated model (Kieren, Pirie, & Gordon Calvert, 1999; Martin & Pirie, 2003; Pirie & Kieren, 1994). Folding back is considered by some to be the “central key construct” of the model (Martin, 2008). Pirie and Kieren (1991) define folding back as A person functioning at an outer level of understanding when challenged may invoke or fold back to inner, perhaps more specific local or intuitive understandings. This returned to inner level activity is not the same as the original activity at that level. It is now stimulated and guided by outer level knowing. The metaphor of folding back is intended to carry with it notions of superimposing ones current understanding on an earlier understanding, and the idea that understanding is somehow ‘thicker’ when inner levels are revisited. This folding back allows for the reconstruction and elaboration of inner level understanding to support and lead to new outer level understanding. (p. 172) 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 researcher attempted to increase the students’ critical awareness about what their response 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 ask the students whether they had completely answered the question at hand. In this analytic, probing questions are used by Researcher Amy Martino in an interview conducted with Brandon which invited him to make connections between his solutions to two tasks that on the surface appeared different but were structurally similar, and ultimately enabled him to discover an isomorphism. Researcher Martino conducted this skillful task-based interview with Brandon, a student at the Colts Neck Public School on April 5th 1993 (Maher and Martino, 1998). Brandon, a 10-year-old 4th grade boy, explained how he had solved two problems in earlier class sessions. In the first class session, Brandon worked with a partner on the following task: Your group has two colors of Unifix Cubes. Work together and make as many different towers four cubes high as is possible when selecting from two colors. The students were asked to solve the problem and convince others that their solution was complete and that they had not recorded duplicates. In the second session, students, again working with partners, were given the following problem: A local pizza shop has asked us to help design a form to keep track of certain pizza choices. They offer a cheese pizza with tomato sauce. A customer can then select from the following toppings: peppers, sausage, mushrooms and pepperoni. How many different choices for pizza does a customer have? Find a way to convince each other that you have accounted for all possibilities. In both session the students had been encouraged to convince themselves as well as their peer that their solution was indeed complete. During a session on March 11, 1993, Brandon created a code for representing the presence or absence of specific toppings for the pizza problem by using a 1 to indicate the presence of a particular topping or a 0 to mark the absence of a topping. Using this system, Brandon created a chart and organized his data to account for all of the possible pizza combinations by delineating all cases of zero toppings, one topping, two toppings, three toppings and four toppings. Researcher Martino then used probing questions that encouraged Brandon think more deeply about this problem together with the towers problem and Brandon discovered that the two problems were indeed isomorphic. He then showed Researcher Martino how each line in his pizza table could be matched to a corresponding Unifix tower and showed how he could be sure that he had found all possibilities of both pizzas and towers. Although mainly probing questions were used during the session highlighted in this analytic, other forms of questions were posed by the researcher and are noted as well in this analytic. Questions were posed by the researcher that encouraged engagement or elicited students’ ideas. These questions were also associated with sophisticated justification and reasoning. Questions encouraged engagement when they kept the student engaged and on task, confirmed agreement, or were posed to check for understanding. The researcher elicited student ideas when she encouraged the student to formulate his own ideas and strategies. 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. Kieren, T., Pirie, S. E., & Gordon Calvert, L. (1999). Growing minds, growing mathematical understanding: Mathematical understanding, abstraction and interaction. Learning mathematics, from hierarchies to networks, 209-231. 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. Maher, C. A. & Martino, A. (1998). Brandon’s proof and isomorphism. In C. A. Maher, Can teachers help children make convincing arguments? A glimpse into the process (pp. 77-101). Rio de Janeiro, Brazil: Universidade Santa Ursula (in Portuguese and English). Martin, L. C. (2008). Folding back and the dynamical growth of mathematical understanding: Elaborating the Pirie–Kieren Theory. The Journal of Mathematical Behavior, 27(1), 64-85. Martin, L., & Pirie, S. (2003). Making images and noticing properties: The role of graphing software in mathematical generalisation. Mathematics Education Research Journal, 15(2), 171-186. 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. Pirie, S. E. B., & Kieren, T. E. (1991). Folding back: Dynamics in the growth of mathematical understanding. In PME Conference (Vol. 3, pp. 169-176). The program committee of the 18th PME conference