Date of Award

June 2018

Degree Type


Degree Name

Doctor of Philosophy (PhD)


Helen M. Doerr


examples, learner generated examples, learning study, structured exercises, teacher learning, variation theory

Subject Categories



Exemplification is significant for the teaching of mathematics and revealing mathematical structure to learners. Structured exercises and learner generated examples (LGEs) are pedagogical tools intended to provide opportunities for learners to discern and generalize mathematical structure. The aim of this study was to understand how middle school mathematics teachers develop their knowledge about designing and implementing sets of structured exercises and LGEs and the factors that influenced their use and implementation of structured exercises and LGEs. Four middle grades mathematics teachers participated in a series of four learning study cycles focused on the design and implementation of tasks that incorporated structured exercises and/or LGEs. The data was analyzed through a lens of variation theory and theory of example spaces to understand the changes in teachers’ use and views of examples over time and the purpose, design, and enactment of examples. The Principle of Explicit Contrast emerged as a principle of both design and enactment with a number of associated design and enactment strategies. The Principle of Attending to Generation and Response emerged for the design of LGEs. The teachers thought about and developed their knowledge of task design and enactment through deliberate practice that included careful consideration of their own thinking about a class of examples and the aspects of that class of examples they attended to, the collaborative exchange of ideas with colleagues, the collective and individual design of sets of examples using patterns of variation, including structured exercises, and LGEs, and the revision or potential revision of such tasks. Factors that influenced and shaped teachers’ conceptualization and operationalization of structured exercises and LGEs included teachers’ perceptions of control of the examples, or lack thereof, teachers’ notions of student success, and teachers’ prior opportunities and experience with task design.


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