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\chapter{Conclusion} | |

\begin{quote}Harel and Sowder \cite[p. 277?] {harel1998students}by their natures, teaching experiments and interview studies do not give definitive conclusions. They can, however, offer indications of the state of affairs and a framework in which to interpret other work.\end{quote} | |

Are CS students' conceptualizations more like Harel and Sowder, or more like Tall? | |

Are the several schemes (Pirie Kieren, etc. complementary? reconcilable? Is one more likely than another based on cognitive neuroscience of language? (proofs are in a language after all)) | |

This research suggests that suitable question for a larger study | |

\section{ Recognizing an Endpoint} | |

A qualitative study is thought to be finished when an internally consistent | |

narrative, compatible with the data, both situating the data and explaining | |

them, has been produced. | |

For our research questions, a model, accompanied by a narrative combining | |

the information obtained from inquiry about these topics will complete the | |

work. Data from our extended student body, that provide a persuasive model | |

containing categories of conceptualizations, and that are closely enough related | |

that some insight about concepts differentiating adjacent categories can | |

be inferred, are thought sufficient to generate this narrative. The proposed | |

differentiating concepts are thought to have the potential to become material | |

for a larger survey, thereby providing a starting point for new work. | |

I expect to find a model similar to that of Harel and Sowder 1998[?], but | |

modified because of the different emphases on material in computer science | |

compared to mathematics. First, students of computer science should be very | |

familiar with the idea of consciously constructing, examining and evaluating a | |

process, from their study of algorithms. Because of this, the category internalization | |

might be subsumed by the category interiorization. | |

From empirical data, we know that there are students of computer science | |

who think that proofs might be irrelevant to their career; it would be hard to | |

imagine a mathematics student who thought so. CS students who do not think | |

proof is part of their career might be relatively content with conceptualizations | |

corresponding to outside sources of conviction. We found computer science | |

students whose conception of proof includes that a single example is sufficient | |

for proving a universally quantified statement. We found computer science | |

students whose conception of proof is that definitions are barely interesting, | |

and who find demonstrations based on definitions unconvincing. Because our | |

findings were not quantitative, we could not compare the population of categories. | |

Nevertheless, the relationships between categories, and the resulting | |

critical factors, might be different, especially in the area of Harel and Sowder's | |

internalization and interiorization. | |

Because the scope is broader, involving proof for deciding whether or not an | |

algorithm is suitable for a problem, I expect we will find more categories, | |

related to algorithms and their applicability. | |

The product of a phenomenographical investigation is categories of conceptualization | |

and critical aspects that distinguish one category from the previous. | |

One hopes that by identifying critical aspects, suggestions about what to emphasize | |

when teaching, and what to seek in assessments are also clarified. This | |

investigation is intended to develop insight into students understandings of | |

proofs, that are the meanings they have fashioned for themselves, based on how | |

they have interpreted what they have heard or read. By examining some of | |

these understandings, we might find directions in which to improve our teaching. | |

Moreover, observations about the conceptualizations of students early in | |

the curriculum can forewarn instructors, helping them recognize the preparation | |

of incoming students. Perhaps we could use this to prepare remediation | |

materials. | |

For example, we can use UML diagrams and ``trie'' data structures to emphasize | |

definitions for families of concepts. We can choose groups of examples, | |

and non-examples of proofs whose correctness turns on the qualification that | |

distinguishes a subclass from its immediate superclass. | |

Beyond this, one may hope that qualitative research suggests worthwhile questions | |

for larger scale investigations. | |

\section{ Perspective on Future Directions} |