One current project examines theories of mathematical knowledge, including the relation between math induction and ordinary induction (Rips & Asmuth, 2007) and theories of children's acquisition of natural numbers (Rips, Asmuth, & Bloomfield, 2006, 2008).
Some related experimental work looks at how college students learn new (to them) math systems, including group theory and non-Euclidean geometry. Here the issues center on how best to organize the new material to enhance people's understanding and how to prevent earlier math knowledge from interfering with the new content. We're finding that emphasizing deductive relations (as opposed to topical ones) helps people identify the more important aspects of the new system. Similarly, emphasizing more holistic aspects of the system (as opposed to more atomic aspects) helps learners filter out competing, irrelevant facts.
Another reasoning project centers on understanding causality. We're exploring how different concepts of causality inform our judgments of events and whether formal theories of causality (e.g., Bayes nets) can give a systematic account of counterfactual conditionals (such as, If Martha hadn't gone to college, she would have gotten a job as a construction worker). For a review of current theories of causal thinking, see Rips (2008) and for the experimental findings, Rips (2010) and Rips and Edwards (in press). An examination of evidence and claims about Michottean "causal perception" appears in Rips (2011). Also check the Concepts page of this web site for research on causal theories of individual concepts (individual people or cats or tables).
You can also find some earlier papers on deductive reasoning and argumentation in the Publications Page.