We employ cut and paste contact topological techniques to classify some tight contact structures on the closed, oriented genus-2 surface times the interval. A boundary condition is specified so that the Euler class of the of the contact structure vanishes when evaluated on each boundary component. We prove that there exists a unique, non-product tight contact structure in this case.

This study proposes a framework for examining ways in which prospective teachers integrate mathematical knowledge acquired in advanced topics courses into explanatory knowledge for school teaching. Participants, all of whom had recently completed coursework in abstract algebra, were asked to explain concepts connected to the school mathematics curriculum, such as division by zero and even numbers. The analysis shows that students used three distinct explanatory strategies: Abstract Mathematical Argument, Analogy, and Rules. Students experienced competition among and within strategies. Moreover, students often derived different conclusions using different strategies. When faced with this conflict, they felt compelled to choose a strategy and to draw a conclusion derived from applying that strategy, regardless of sense-making. Students' ability to integrate explanatory strategies appeared to depend on their possession of coherent mathematical meanings, suggesting that strategy integration is indicative of students' having a key developmental understanding of the underlying mathematical ideas.Keywords Abstract algebra . Secondary teachers . Advanced mathematics . Mathematical knowledge for teaching . Explanation . Coherence . Key developmental understanding Coursework in advanced mathematics topics such as abstract algebra is required in virtually all secondary mathematics certification programs for teachers. This requirement is justifiable on many grounds, including the existence of deep connections between concepts in abstract algebra and the school mathematics curriculum (Conference Board of the Mathematical Sciences 2001). However, recent trends in educational research suggest that we must study how prospective secondary mathematics teachers (henceforth: prospective teachers) establish and employ these Int. J. Res. Undergrad. Math. Ed. (2015) 1:63-90 DOI 10.1007 * Tanya Cofer tcofer@neiu.edu 1 Northeastern Illinois University, Chicago, IL, USA connections. Early studies such as those by Begle (1979) and Monk (1994) examined the effect of subject matter knowledge on teaching indirectly by comparing the number of mathematics content courses taken by teachers to measures of student mathematics achievement. Not surprisingly, the results were complicated and mixed. Though Monk found that the first few undergraduate mathematics courses taken by prospective secondary teachers modestly improved their students' subsequent performance, simply requiring prospective teachers to take additional advanced mathematics courses did not continue to improve student achievement. Looking more closely at teachers' subject matter knowledge, Shulman (1986) described an absence of focus on subject matter among the research paradigms for the study of teaching. He noted that research missed questions such as, BHow does the successful college student transform his or her expertise in the subject matter into a form that high school students can comprehend?^, BHow does he or she employ content expertise to generate new explanations...

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