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.
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.
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.
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
In this paper, we analyze a PD design, examining its activities and the sequencing of professional learning tasks. We use a theoretical framework typically used in pre-service teacher education to understand the design of one PD program. Our overarching goal is to theorize about how to design PD and sequence professional learning tasks for practicing teachers.
In this paper, authors analyze a PD design, examining its activities and the sequencing of professional learning tasks.
Beyond initial college preparation, secondary teachers in the United States have few professional opportunities to do and learn challenging mathematics, especially incollaboration with colleagues. The Mathematics Immersion for Secondary Teachers at Scale program engages sets of teachers in local school sites, connected synchronously and asynchronously to colleagues in other sites, in doing mathematics designed to promote experiences of mathematical immersion, community, and connection to the work of teaching.
The Mathematics Immersion for Secondary Teachers at Scale program engages sets of teachers in local school sites, connected synchronously and asynchronously to colleagues in other sites, in doing mathematics designed to promote experiences of mathematical immersion, community, and connection to the work of teaching. This study of two groups of sites over one year examines fidelity to the program as a model for systematically providing these opportunities, and the extent to which teacher participants experienced immersion, community, and connection in their collaborative work with the course facilitator and their local and distant colleagues.
This paper examines the use of Imaginative Education (IE) to create an NGSS-aligned middle school engineering curriculum that supports transfer and the development of STEM identity. In IE, cognitive tools—such as developmentally appropriate narratives, mysteries and fantasies—are used to design learning environments that both engage learners and help them organize knowledge productively. We have combined IE with transmedia storytelling to develop two multi-week engineering units and six shorter engineering lessons.
This paper examines the use of Imaginative Education (IE) to create an NGSS-aligned middle school engineering curriculum that supports transfer and the development of STEM identity.
This paper examines the use of Imaginative Education (IE) to create an NGSS-aligned middle school engineering curriculum that supports transfer and the development of STEM identity. In IE, cognitive tools—such as developmentally appropriate narratives, mysteries and fantasies—are used to design learning environments that both engage learners and help them organize knowledge productively. We have combined IE with transmedia storytelling to develop two multi-week engineering units and six shorter engineering lessons.
This paper examines the use of Imaginative Education (IE) to create an NGSS-aligned middle school engineering curriculum that supports transfer and the development of STEM identity.
