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Klein's double discontinuity revisited: what use is university mathematics to high school calculus?

Much effort and research has been invested into understanding and bridging the gaps which many students experience in terms of contents and expectations as they begin university studies with a heavy component of mathematics, typically in the form of calculus courses. We have several studies of bridging measures, success rates and many other aspects of these entrance transition problems. In this paper, we consider the inverse transition, experienced by university students as they revisit core parts of high school mathematics (in particular, calculus) after completing the mandatory undergraduate mathematics courses. To what extent does the advanced experience enable them to approach the high school calculus in a deeper and more autonomous way? To what extent can capstone courses support such an approach? How could it be hindered by deficiencies in the students' advanced experience? In this paper, we present a theoretical framework for an analysis of these questions, as well as a number of critical observations and reflections on how they appear in our own institutional context.