Because explaining ideas in mathematics is seen as a fundamental part of the work of doing and teaching mathematics (Hill et al., 2008; NCTM, 2000), whole-class discussions in which PSTs explain their thinking to assigned tasks may help these future teachers develop a robust understanding of the mathematics they will teach.  These discussions, however, must do more than simply provide chances to share thinking.  They should provide PSTs with opportunities to respond to others’ ideas (e.g., Author, 2008; Walshaw and Anthony, 2008). Others’ ideas become “thinking tools” that PSTs can use to develop their own understanding (Wertsch & Toma, 1995).  As learners evaluate the ideas of others, they build connections between their conceptions and those of peers, and identify conflicting ideas (Carpenter & Lehrer, 1999).

Author (2011) describes talk moves that teachers can use to facilitate whole-class discussion.  The first moves help students clarify and deepen their own reasoning.  Additional moves prompt students to respond to each other’s ideas.  These moves, often posed in the form of a question, ask students to repeat or rephrase a classmate’s idea, apply their reasoning to it, and add on to its contents.  Though the potential of these moves seems high, their efficacy has not yet been measured.  This session reports on data from two iterations of a study which sought to investigate the relationship between various kinds of teacher moves and PSTs’ content learning. 

Both studies (n=23 and n=26) were conducted in two sections of a mathematics class for college students seeking certification in elementary education.  The instructor used the same instructional tasks in both sections.  One focus of instruction in both studies was interpreting mixed number quotients of repeated subtraction division equations. 

In both sections, the instructor used teacher moves designed to help PSTs clarify and deepen their own reasoning.  However, only in the experimental sections did the instructor also use talk moves that prompted PSTs to respond to others’ contributions.  

Pre-and post-test results were scored using a criterion-referenced rubric by two raters until inter-rater reliability reached 90% or better.  Analysis of the pre-test data showed that most students in both studies were unfamiliar with interpreting mixed number quotients of repeated subtraction division equations.   In the first iteration of the study, significant differences were found between groups (t=3.29, p=0.0035) on the post-test item.  By contrast, in the second iteration, an unpaired t-test showed no significant differences between groups (t=0.30, p=0.77) on the post-test item.

This difference in findings between the two iterations led the researcher to investigate the following questions: Were there any differences in instruction – including the implementation of the control and treatment conditions – that might account for this difference in findings between the pilot study and final study?  Did the student talk in one iteration differ from that in the second iteration in ways that might explain the conflicting post-test results?   

            The researcher used mixed methods to address the research questions, identifying all instructional segments that pertained to interpreting mixed number quotients of repeated subtraction division equations from each section of both studies.  All data from the whole-class discussions were recorded and transcribed.  A coding scheme was developed to identify all instances of teacher talk moves that were featured in both sections of each study (i.e., talk moves used to help students clarify and deepen their own reasoning) as well as all instances of teacher talk moves that were featured in the experimental groups only (i.e., talk moves that encouraged students to respond to a classmate’s reasoning).  Student turns were coded by mathematical content using the same criteria used in the scoring rubric of the post-test item. 

Analysis revealed differences in the control conditions between the two studies.  In the control group of the first iteration, teacher turns that encouraged students to clarify and deepen their own reasoning were almost entirely absent.  Analysis revealed that the teacher-student interactions during this segment were similar to those typically associated with an IRE instructional format (Cazden, 1988).  In the second iteration, however, students were given regular opportunities to clarify and deepen their own reasoning.  In fact, 27% of teacher turns were coded as talk moves that helped students clarify and deepen their own reasoning.  Analysis also revealed that these moves were enacted following student turns that focused on important mathematical ideas such as explaining that the quotient in a repeated subtraction division equation represents the number of groups.  Because giving explanations has been associated with higher achievement (Webb, 1991), these additional opportunities to clarify and deepen their reasoning may have bolstered the PSTs' scores on the post-test item – and contributed to the non-significant finding between groups.

Differences between studies were found in the mathematical content of student turns.  During the first iteration, in the control and experimental groups, 25% and 40%, respectively, of student turns were coded as explaining that the fraction in a mixed number quotient represents a partial group of the divisor.  This difference between groups, however, was not maintained in the final study.  During the final study, in the control and experimental groups, the frequencies with which students in the two groups discussed this concept were approximately equal.  Further, this rate was lower than either of the rates in the first iteration.

These findings suggest that giving PSTs the opportunity to clarify and deepen their own reasoning may be just as important as encouraging them to attend to the thinking of others.  It also suggests that the efficacy of talk moves may be inversely related to the span of the mathematical focus of the discussion.  As more topics are introduced, the talk moves are applied to a greater span of topics and may lose some of their efficacy.

Author. (2008). Classroom Discussions: Using math talk to help students learn (2^nd ed.). CA: Math Solutions.

Author. (2011). Classroom discussions: Seeing math discourse in action [DVD/Facilitator’s Guide]. Sausalito, CA: Math Solutions.

Carpenter, T.P. & Lehrer, R. (1999). Teaching and learning mathematics with understanding.  In E. Fennema & T. Romberg (Eds.), Mathematics classrooms that promote understanding.  Hillsdale, NJ: Erlbaum.

Cazden, C. (1988). Classroom discourse (1^st ed.). Portsmouth, NH: Heinemann.

Hill, H.C., Blank, M.L., Charalambous, C.Y., Lewis, J.M., Phelps, G. C., Sleep, L. & Ball, D.L. (2008). Mathematical knowledge for teaching and the mathematical quality of instruction: An exploratory study. Cognition and Instruction, 26, 430-511.

National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: NCTM.

Walshaw, M. & Anthony, G. (2008). The teacher’s role in classroom discourse: A review of recent research in mathematics classrooms.  Review of Educational Research, 78, 516-551.

Webb, N.W. (1991). Task-related verbal interaction and mathematics learning in small groups. Journal for Research in Mathematics Education, 22(5), 366—389.

Wertsch, J.V. & Toma, C. (1995). Discourse and learning in the classroom: A sociocultural approach. In Constructivism in education. Steffe, L.P. & Gale, J. (Eds.). Hillsdale, NJ: Lawrence Erlbaum Associates, Inc.