A fundamental goal of mathematics education is to prepare students to be flexible and resourceful learners and users of mathematics. As contended by Adding it Up (NRC, 2001), a core element of mathematical proficiency—the goal of mathematics instruction—is students’ adaptive reasoning, defined as the capacity to justify claims by logically identifying relationships linking concepts (NRC, 2001). Adaptive reasoning is the “glue that holds everything together, the lodestar that guides learning” (p. 129).  Current mathematics education research has recognized the importance of students’ providing complete and correct mathematical explanations (Franke et al., 2009; NCTM, 2009; Common Core State Standards Initiative, 2010). Given this emphasis on students’ reasoning, our presentation aims to further understanding about how mathematics instruction supports students in providing complete and correct mathematical explanations.

As part of a larger NSF supported research program on algebra teaching and learning, we measured changes in students' explanations and classroom practices that support these changes. Data led us to hypothesize that differences in curricula contribute to variation in classroom practices and student explanations. To capture these classroom practices, we developed and employed an observation protocol tracking both teachers eliciting student reasoning and students' responses.  To measure students’ reasoning, we adapted assessments (e.g., Shell Centre, 2003) that provided opportunities for students to explain and justify their reasoning.

We operationalize complete and correct explanations as follows.  First, students’ complete explanations must indicate the strategies used and reflect how the student thought about the task. The extent of insight these explanations provide is dependent on how students articulate their thinking in writing. Thus, the completeness of an explanation is assessed on the coherence and justification of reasoning, even if the reasoning resulted in an incorrect answer.

Second, correct explanations must support the accuracy of a procedure or solution in the context of the problem. With this, we are looking at how students draw on algebraic reasons for their answers, showing that both  their solution process and their answer make sense within the problem context.  This process may include drawing on the relationship between variables, some aspect of an algebraic representation, or using the context of the problem.

Our presentation will identify links between classroom practices and changes in students’ explanations over the course of a school year in two 8^th grade algebra classrooms using different curricula. This allows for a detailed exploration of how differences in curricula and instruction relate to variation in student explanations, helping us formulate hypotheses to be tested with a larger sample in future work.

Methods

Through a comparative case study, we investigated the following questions:

1. How do students' explanations and justifications in written algebraic tasks differ between two 8^th grade algebra classrooms? 
2. How do the teachers’ approaches to eliciting student explanations and justifications differ between these two classrooms?
3. How does variation in student performance on explanation and justification tasks relate to differences in instructional practices?

We analyzed data from two of ten 8^th grade algebra classes in our larger study. Teacher A used an NSF-funded curricula and Teacher B used a traditional, district-adopted text.  The instructional practices were recorded and analyzed using field notes, video, and our observation coding scheme. We observed that these two teachers differed in the explanations they elicited from students.

To analyze student work, we assigned a numerical score to students’ explanations and justifications using the criteria outlined above. By averaging these scores for each classroom, we were able to compare changes across classrooms. We also qualitatively characterized the types of explanations students provided individually and across the classrooms.

Initial findings

  Initial findings suggest important differences between the two classes. First, we observed that these two teachers differed in their instructional practices to elicit student explanations. Additionally, Class A, although a lower-tracked class, showed growth in the correctness and completeness of written explanations and justifications while Class B showed a decline. Further, the nature of students’ explanations and justifications differed across classrooms with class A drawing from a variety of methods to justify their answers while class B primarily recounted procedures used.

            Students in Class A were expected to provide explanations. Teacher A consistently pressed and supported student explanations through questioning, revoicing, drawing attention to representations and other visual cues, and building on previous knowledge. As norms around explanations were established, the frequency of pressing and supporting moves that Teacher A provided decreased. Concurrently, the nature of student explanations also changed.  Students offered justifications freely, building on each other’s explanations, basing their explanations on representations, previous knowledge, and contextual cues.

In contrast, students in Class B spent much of their time taking notes on specific procedures, working on exercises and showing steps in their work.  While Teacher B frequently elicited student participation, much of this participation consisted of suggesting the next step in a procedure or supplying the result of a calculation.  During independent or partner work time, students were primarily engaged in practicing procedures that had been demonstrated in class lecture.  During classroom observations, students rarely wrote explanations or justifications of their reasoning.  While the textual materials for Class B included contextual problems that might have provided more support for developing student explanations, these problems were not integral to the materials.

            The student assessments reflect differences between these two classrooms in students' explanations and justification of their reasoning. Students in Class B attempted 42% of explanation problems on their post-assessment and students in Class A attempted 64%. In terms of correctness, Class B lost 3% on their explanation score on the attempted problems from the pre- to the post-assessment, while Class A gained 3%.         

            From these initial findings, we see differences in classroom activities that may be related to differences in student explanations. This presentation will elaborate on these differences, and consider how later work can explore these conclusions in a larger sample of classrooms beyond this case study.