Abrahamson, D., Bryant, M. J., Howison, M. L., & Relaford-Doyle, J. J. (2008, March). Toward a phenomenology of mathematical artifacts: A circumspective deconstruction of a design for the binomial. 

Paper presented at the annual conference of the American Education Research Association, New York, March 24-28.

We demonstrate the potential of an innovative design-based research methodology—manipulation of normative features of didactic routines—to illuminate and support the integration of constructivist and socio-cultural theoretical models pertaining to the roles of epistemic and material resources at play in students’ mediated development of mathematical concepts. We present three cases of individual middle-school students who participated in a study of probabilistic cognition, where the design was for students to coordinate their intuition for the outcome of an experiment with a stochastic device with its sample space that they build through combinatorial analysis. The activity sequencing was non-normative in that only having completed the combinatorial analysis did students recognize its pertinence to their grounding intuition, and that recognition is the focus of our analysis. By uncovering students’ heuristics as well as the pragmatics of the student–interviewer dyad as they communicate over properties of the learning materials and negotiate activities embedded in the design, we implicate a major challenge of struggling students to be their underdeveloped epistemic disposition—they do not expect to be able to “solve the world” and do not skillfully use mathematical representations let alone trust them to bear on their inquiry. To grow into STEM content, these students require early nurturing into problem-solving socio–cognitive norms.