Early algebra learning and the interplay between Researcher Robert B. Davis’ questioning and 6th graders’ reasoning

DescriptionThis analytic is the fifth 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 in relation to 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 videotapes of sessions conducted as part of a 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 invited 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 also observe 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 probing questions and questions intended to encourage engagement posed by a researcher to provide opportunities for students to express their developing algebraic ideas as formal equations. Probing questions are questions that elicit student responses that move beyond 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 an 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. Such questions include elucidating probes intended 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 had been said. This strategy of questioning also includes redirecting questions to encourage students to analyze the initial question further and consider the implications of the response as it might 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 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. This analytic also looks at questions posed by the researchers that were associated with student engagement and the presentation of students’ ideas.

In this analytic, Researcher Robert B. Davis uses probing questions and questions to afford opportunity for engagement during a session at Kenilworth where a group of 6th grade students worked on the Guess My Rule activity. He also uses recognition and praise while encouraging students to do more. Researcher Davis led a number of sessions at the Harding Public School in Kenilworth, where he introduced 12 sixth-graders to algebraic ideas prior to their formal study of algebra (Giordano, 2008).
The activity Guess My Rule works as follows. One or more student make up a secret rule such as "Whatever number you tell me, I will multiply by three and add four." Other students attempt to guess the rule by providing numbers to the rule makers and analyzing the results that they provide in turn. Alternatively, students are given a table with two columns. One column is labeled with a square and one with a triangle. Using the values in the table, the students attempt to determine what operations need to be done to the value in the square column to produce the value in the triangle column. This activity was used as a means of introducing the concept of function. Specifically in this analytic, students are working on the following Guess My Rule task:

Problem 6
□ ∆
0 1
1 2
2 5
3 10
4 17
5 26

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.

Giordano, Patricia (2008). Learning the concept of function: Guess my rule activities with Researcher Robert B. Davis. (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.
Created on2017-05-14T00:19:21-0400
Published on2017-09-25T13:44:28-0400
Persistent URLhttps://doi.org/doi:10.7282/T3VQ35KX