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.

The research team developed the Student Argument and Reasoning Assessment (SARA) to measure students’ abilities to construct viable arguments and critique others’ arguments. The SARA was originally developed and validated in the LAMP pilot study (NSF Award Number: 1317034). Items were developed by reviewing prior research on proof/proving (e.g. Healy & Hoyles, 2000; Knuth, 2002b), state assessments, and feedback from the external advisory board. The pretest assessment has 5 items: 4 items measure the ability to construct viable arguments, and 1 item assesses the ability to critique others’ arguments. Specifically Item 1 was designed to elicit a direct argument. Item 2 was designed to elicit an indirect argument or a direct argument. Item 3 was designed to elicit a counterexample argument, and Item 4 was designed to elicit an exhaustive argument. Item 5 was designed to assess students’ ability to see the generalization in a specific example and recognize that the structure in the example applied to all cases. These items address mathematical content at the Grade 7 level to ensure the Grade 8 students have the mathematical knowledge necessary to adequately complete the assessment as a pretest at the beginning of their Grade 8 year (i.e., this ensures the assessment is measuring argumentation skills and not mathematical content knowledge). The posttest assessment includes the same 5 items as the pretest and 4 additional items that address mathematical content that is taught to Grade 8 students during the school year—at the onset of the school year the students would not have the content knowledge to respond to these items on a pretest.

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 is a technical report detailing the methods and findings for each of the research studies in the LLAMA project. 

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.

Several recent studies have focused on helping students understand the limitations of empirical arguments (e.g., Stylianides, G. J. & Stylianides, A. J., 2009, Brown, 2014). One view is that students use empirical argumentation because they hold empirical proof schemes—they are convinced a general claim is true by checking a few cases (Harel & Sowder, 1998). Some researchers have sought to unseat students’ empirical proof schemes by developing students’ skepticism, their uncertainty about the truth of a general claim in the face of confirming (but not exhaustive) evidence (e.g., Brown, 2014; Stylianides, G. J. & Stylianides, A. J., 2009). With sufficient skepticism, students would seek more secure, non-empirical arguments to convince themselves that a general claim is true. We take a different perspective, seeking to develop students’ awareness of domain appropriateness (DA), whether the argument type is appropriate to the domain of the claim. In particular, DA entails understanding that an empirical check of a proper subset of cases in a claim’s domain does not (i) guarantee the claim is true and does not (ii) provide an argument that is acceptable in the mathematical or classroom community, although checking all cases does both (i) and (ii). DA is distinct from skepticism; it is not concerned with students’ confidence about the truth of a general claim. We studied how ten 8th graders developed DA through classroom experiences that were part of a broader project focused on developing viable argumentation. 

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.

Students’ difficulties with contrapositive reasoning are well documented. Lack of intuition about contrapositive reasoning and lack of a meta-argument for the logical equivalence between a conditional claim and its contrapositive may contribute to students’ struggles. This case study investigated the effectiveness of the eliminating counterexamples intervention in improving students’ ability to construct, critique, and validate contrapositive arguments in a U.S. eighth-grade mathematics classroom. The intervention involved constructing descriptions of all possible counterexamples to a conditional claim and its contrapositive, comparing the two descriptions, noting that the descriptions are the same barring the order of phrases, and finding a counterexample to show the claim is false or viably arguing that no counterexample exists.

This presntation addreses 4 research cquestions

The SchoolWide Lab middle school curriculum units are anchored in phenomena, aligned to the Next Generation Science Standards(NGSS) and purposefully integrate Computational Thinking (CT). To date, four units have been developed by a team of researchers from the University of Colorado Boulder in collaboration with middle school teachers from Denver Public Schools.