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theses

This study traces three primary school students’ longitudinal development of mathematical ideas and ways of reasoning while solving a strand of counting problems. The students worked on well defined, open-ended counting problems of variable difficulty in various settings: pairs, whole class settings, task-based interviews and small groups. Video-taped data, transcripts, and student work are analyzed for cognitive growth in reasoning, attentive to the social elements of collaboration in problem solving. Data include individual and group co-construction of justifications for solutions. Video narratives (VMCAnalytics) describe the students’ learning progressions. Student dialogue and co-constructions that fostered their development are identified and displayed in the 13 published video narratives linked to the analyses. For each student, how do their recognition of patterns, use of strategies and representations, display of justifications and forms of reasoning about solutions to counting tasks develop over time and how might each journey be displayed with a learning progression using video data?

Analyses revealed local and global recognition for enumeration of outcomes (by recursive strategies), invention of composite operations, connection between tasks, rule generalization, and direct reasoning by cases, induction, controlling for variables. Particular forms of reasoning are identified for each student. The following cognitive and social factors revealed that learning occurred collaboratively, in a variety of settings. Students were attentive to the counter examples/arguments posed by others and worked to convince others about their arguments that were “proof like” in structure.

The longitudinal study showed how earlier ideas became the foundation for building later ideas, represented in more sophisticated ways. The results have implications for effective mathematical practices, such as collaborative learning, and attention to providing justifications for solutions. These pedagogical approaches can be incorporated in curriculum design, can supplement approaches to teacher professional development. The learning progressions can offer teachers an approach to formative assessment of student reasoning on solving counting tasks.

ETD_10662

Ph.D.

Includes bibliographical references

doi:10.7282/t3-426x-1369

ETD doctoral

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