The moment-to-moment dynamics of student discourse plays a large role in students' enacted mathematics identities. Discourse analysis was used to describe meaningful discursive patterns in the interactions of 2 students in a 7th-grade, technology-based, curricular unit (SimCalc MathWorlds®) and to show how mathematics identities are enacted at the microlevel. Frameworks were theoretically and empirically connected to identity to characterize the participants' relative positioning and the structural patterns in their discourse (e.g., who talks, who initiates sequences, whose ideas are taken up and publicly recognized). Data indicated that students' peer-to-peer discourse patterns explained the enactment of differing mathematics identities within the same local context. Thus, the ways people talk and interact are powerful influences on who they are, and can become, with respect to mathematics.

We identify and document 3 cognitive obstacles, 3 cognitive affordances, and 1 type of integer understanding that can function as either an obstacle or affordance for learners while they extend their numeric domains from whole numbers to include negative integers. In particular, we highlight 2 key subsets of integer reasoning: understanding or knowledge that may, initially, interfere with one's learning integers (which we call cognitive obstacles) and understanding or knowledge that may afford progress in understanding and operating with integers (which we call cognitive affordances). We analyzed historical mathematical writings related to integers as well as clinical interviews with children ages 6-10 to identify critical, persistent cognitive obstacles and powerful ways of thinking that may help learners to overcome obstacles.

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