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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}
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