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Timothy Fukawa-Connelly and Stephen Buck

A model in use at Prospect Hill Academy in Cambridge, Massachusetts, provides a skills-based portfolio assessment for mathematics classes.

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Valerie Klein, Timothy Fukawa-Connelly and Jason Silverman

“Noticing” and “wondering” can help teachers slow down to respond more thoughtfully to student work.

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Nicholas H. Wasserman, Keith Weber, Timothy Fukawa-Connelly and Juan Pablo Mejía-Ramos

A 2D version of Cavalieri's Principle is productive for the teaching of area. In this manuscript, we consider an area-preserving transformation, “segment-skewing,” which provides alternative justification methods for area formulas, conceptual insights into statements about area, and foreshadows transitions about area in calculus via the Riemann integral.

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Kristen Lew, Timothy Patrick Fukawa-Connelly, Juan Pablo Mejía-Ramos and Keith Weber

We describe a case study in which we investigate the effectiveness of a lecture in advanced mathematics. We first videorecorded a lecture delivered by an experienced professor who had a reputation for being an outstanding instructor. Using video recall, we then interviewed the professor to determine the ideas that he intended to convey and how he tried to convey these ideas in this lecture. We also interviewed 6 students to see what they understood from this lecture. The students did not comprehend the ideas that the professor cited as central to his lecture. Based on our analyses, we propose 2 factors to account for why students did not understand these ideas.

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Timothy Fukawa-Connelly, Keith Weber and Juan Pablo Mejía-Ramos

This study investigates 3 hypotheses about proof-based mathematics instruction: (a) that lectures include informal content (ways of thinking and reasoning about advanced mathematics that are not captured by formal symbolic statements), (b) that informal content is usually presented orally but not written on the board, and (c) that students do not record the informal content that is only stated orally but do if it is written on the board. The authors found that (a) informal content was common (with, on average, 32 instances per lecture), (b) most informal content was presented orally, and (c) typically students recorded written content while not recording oral content in their notes.