Mathematical Thinking and Learning, 14(1), 55-80.

ABSTRACT: What are the nature, forms, and roles of metaphors in mathematical learning and teaching? We present and closely analyze three examples of idiosyncratic metaphors produced during one-to-one tutorial clinical interviews with 11-year-old participants, as they attempted to use unfamiliar artifacts to reason about realistic probability problems. The first metaphor, invented by the student and elaborated by the tutor, supported combinatorial analysis in one context but did not readily transfer to a proximate context; the second metaphor was invented by the tutor in response to the student’s conflation of intuitive and formal conceptualizations of a sample space—the metaphor evoked a mechanical device simulating a non-intuitive epistemic agent; and the third metaphor functioned as a “lemma” by temporarily shifting the dyad’s joint attention away from the focal medium, a computer-based random-generator simulation, to a vacant multimodal image construction space. Metaphor was thus both spurred by, and transformative of, discursive events: metaphor served individuals as semiotic means of objectifying their own evolving view of the artifacts, yet its expressive manifestation immediately modified their interlocutor’s attention to these objects thus creating new shared referents that honed and mobilized the negotiation of meaning toward the instructor’s didactical objectives.We speculate on situation parameters affecting students’ utilization of idiosyncratic resources as well as how metaphor may contribute to effective instructional mathematical discourse.