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.