Abrahamson, D. (2008, March). Fostering the emergence of an embodied cognitive artifact: The case of the number line in a design for probability. In D. Abrahamson (Chair), D. Earnest (Org.), & H. Bass (Discussant), The many values of the number line–An interdisciplinary forum.

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

I trace the emergence of a mathematical instrument, the number line, in the context of student engagement with a situated-probability problem-solving interview task involving manipulatable objects. I argue that consequent semiotic–ontological ambiguity engendered struggle with generative conceptual confusion. Namely, as students conducted  combinatorial analysis to create the sample space of a random generator,  the objects they built to express the stochastic events served both as tick marks on an emergent number line and as outcomes, members of those  marked events. Negotiating the tickmark-vs.-member semiotic ambiguity  challenged, then facilitated the dyad’s discourse over the event-vs.-outcome learning axis, which I have implicated as key to deep  understanding of the binomial. Extrapolating from the data, I examine the phylogenesis-recapitulates-microgenesis conjecture (a reversal on Haeckel) by which the historical evolution of the number line may have proceeded from objects to inscription; I draw an explorative implication that mathematics instruction could follow suit, i.e., that students could reinvent the number line as an inscribed ordinal sequence of sorted objects.