Middle

The Classroom Argumentation Observation Protocol developed in LAMP measures teachers’ pedagogical practices in terms of teachers providing opportunities for students to engage in the mathematics learning experiences specified in the logic model. The protocol provides quantified scores for the types of claims a teacher uses, the explicitness of claims, the sophistication of the warranting, and the use of warrants and data. Open‑ended questions ask for the extent to which the observed lessons address LLAMA lesson objectives. LAMP established content validity of this protocol through an expert panel. At the onset of the LLAMA project, the protocol was revised. The primary revisions include (a) the inclusion of our recent understanding of generic example arguments (see Yopp and Ely, 2016 and Yopp, Ely, and Johnson‑Leung, 2016) and (b) asking about the percentage of students engaged in the classroom argumentation episode. The protocol was further revised during weekly Principal Investigator meetings and was piloted in Year 1. During Year 2, the research teams participated in an observation training to ensure interrater reliability among observers and maintain a codebook of decision rules pertaining to the coding of the observations. Minor modifications were made to the protocol during this training period. The team watched videos and scored the videos during multiple sessions and modified the rubric and wording accordingly, following those sessions.

Research suggests that if students use viable argumentation in their

middle school classes, then they will increase their complex

mathematical reasoning and mathematics achievement. This is a 2-page infographic detailing the results from a case study.

This book aims to advance ongoing debates in the field of mathematics and mathematics education regarding conceptions of argumentation, justification, and proof and the consequences for research and practice when applying particular conceptions of each construct. Through analyses of classroom practice across grade levels using different lenses - particular conceptions of argumentation, justification, and proof - researchers consider the implications of how each conception shapes empirical outcomes. In each section, organized by grade band, authors adopt particular conceptions of argumentation, justification, and proof, and they analyse one data set from each perspective. In addition, each section includes a synthesis chapter from an expert in the field to bring to the fore potential implications, as well as new questions, raised by the analyses. Finally, a culminating section considers the use of each conception across grade bands and data sets.

Students’ difficulties with argumentation, proving, and the role of counterexamples in proving

are well documented. Students in this study experienced an intervention for improving their

argumentation and proving practices. The intervention included the eliminating counterexamples

(ECE) framework as a means of constructing and critiquing viable arguments for a general claim.

This framework involves constructing descriptions of all possible counterexamples to a conditional

claim and determining whether or not a direct argument eliminates the possibility of

counterexamples. This case study investigates U.S. eighth-grade (age 13) mathematics students’

conceptions about the validity of a direct argument after the students received instruction on the

ECE framework. We describe student activities in response to the intervention, and we identify

students’ conceptions that are inconsistent with canonical notions of mathematical proving and

appear to be barriers to using the ECE framework.