Abrahamson, D., Bryant, M. J., Gutiérrez, J. F., Mookerjee, A. V., Souchkova, D., & Thacker, I. E. (2009). Figuring it out: mathematical learning as guided semiotic disambiguation of useful yet initially entangled intuitions. 

In S. L. Swars, D. W. Stinson, & S. Lemons-Smith (Eds.), Proceedings of the Thirty-First Annual Meeting of the North-American Chapter of the International Group for the Psychology of Mathematics Education (Vol. 5, pp. 662-670). Atlanta, GA: Georgia State University.

When participants in inquiry refer to an object, they may, unbeknown to them, construct the object differently. They thus tacitly attribute different idiosyncratic senses for their respective constructions and consequently draw different inferences regarding the phenomenon under investigation. A single person, too, may shift between alternative constructions of a mathematical object, assigning them different senses, thus arriving at apparently competing conclusions. Only upon acknowledging the different constructions can the person begin to explore whether and how the differing conclusions are in fact complementary. Building on empirical data of students engaged in interview-based tutorial activities targeting fundamental probability notions, we explicate breakdowns such false-contradiction introduces into learning processes yet suggest opportunities such ambiguity fosters.