More details to come!

• The relationship between historical conceptions in mathematics and student mathematical conceptions

What are key obstacles that arise in the acquisition of mathematical knowledge that are epistemological in nature, and thus can be expected to appear at both the individual and societal levels?

Student Obstacles and Historical Obstacles to Foundational Concepts of Calculus (Part 1 of my dissertation)

What is the relationship between the internal cognitive rules that govern the construction of (locally) consistent sets of conceptions and our external socio-mathematical rules that govern the construction of consistent mathematics?

Nonstandard Student Conceptions about Infinitesimal and Infinite Numbers (Part 2 of my dissertation)

In addition to the ones Piaget and Garcia (1983) posited, what prominent mechanisms govern the development of mathematical knowledge?  How do societal elements, such as language, affect, constrain, and guide these mechanisms?

• The relationships between mathematical conceptions and conceptions in other domains (for instance, in the sciences, humanities, and in non-school domains)

What is the role of argumentation and justification in math, science, and the humanities?  Are there any similarities to how students develop argumentation skills in these three realms?  In what ways can we expect mathematical argumentation and reasoning to transfer to science and the humanities (and back)?

How have the domain boundaries between the humanities, sciences, and mathematics changed over time, and are there analogous processes that occur in students and in the classroom?