\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} There is an amount of attention that students give to their coursework. That level of attention is not necessarily even. That level of attention could be influenced in multiple ways. A student may pay attention in one way when having just heard that something will be on a quiz. A student may pay attention in another way, when having just heard that something will not be on a quiz. While a relatively relaxed, attentive, reflective mode might be deemed appropriate by an instructor, as a response to hearing that something will not be on a quiz, and while some students respond that way, it can also happen that a student's response is to think beyond the class, such as wonder about missed telephone calls. While it may seem obvious to an instructor that anything discussed in class deserves attention, interview results have shown that some students are not aware that anything to which class time is devoted should be regarded as worthy of attention. Their prior experience could easily have trained students to think that instructors attach points towards a grade to important learning objectives. There should be no surprise when students operate as if points imply importance. Students can easily err, supposing that no points implies unimportant. Instructors provide scaffolding to assist the development of student concepts. Some students take advantage of such scaffolding. That students sometimes take advantage of scaffolding implies that they can. It is probably true that students will use the ability to take advantage of scaffolding, even using material that might not have been intended for scaffolding. Some students learning about proof scaffold this new material with what they know about programming. The graceful transition from mathematical proof to programming language semantics illustrated by structural operational semantics (see, e.g., H\"uttel \cite{huttel2010transitions}) shows that scaffolding proof learning in computer science and engineering by programming can have good results. At the point in the curriculum at which proof is being taught, it could be that some students do not yet have the maturity of understanding of programming languages to afford this benefit on their own. It might be that explicit scaffolding of proof with programming language examples would improve the results. 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. Application of findings about students of mathematics to students of computer science is fraught by differences in the preparation and interests related to algorithms. One likely difference is motivation: students of mathematics know that proof is the principal means of discourse in their community, but students of computer science might not be aware of the importance of proof to their work. Not all differences favor students of mathematics. In particular, the categories internalization and interiorization of Harel and Sowder’s 1998 model\cite{harel1998students} are apt to be, in students interested in algorithms, more closely related, than in students of mathematics. There may be a difference regarding abstraction. Both mathematics and computer science deal in abstraction, and students in both disciplines struggle with it. \cite{mason1989mathematical,hazzan2003students}. In mathematics, following Vi\ete, \cite{viete2006analytic}, single letter variable names are used. These are thought to support the learning of abstraction, for example, Gray and Tall \cite[p. 121]{gray1994duality} observe we want to encompass the growing compressibility of knowledge characteristic of successful mathematicians. Here, not only is a single symbol viewed in a flexible way '' and in computer science abstraction, one way to exhibit abstraction is UML diagrams. Because the `trie'' structure and International Standards Organization ISO standard 11179 are computer science approaches to management of definitions, it could be that computer science students would be more accessible to noticing the desirability of concept definitions over concept images (see R\"osken and Rolka, \cite{rosken2007integrating} and Rasslan and Tall \cite{rasslan2002definitions}). It would be interesting to know whether any of several approaches reported by Weber [?] could be used, perhaps in modified form, for instruction of students of computer science. The Action Process Object Schema approach of Dubinsky \cite{dubinsky2002apos} sounds compatible with computer science students' interests. An approach due to Leron and Dubinsky uses computer programming \cite{leron1983structuring}, another \cite{leron1995abstract} is directed more to learning group theory than to learning proof construction. Also specific to students concerned with algorithms, we may wish to extend the notion of social constructivism from that of Piaget \cite{}, [?] and of Vygotsky, [?] where it was necessarily a person with whom the learner was communicating, and therefore with whom it was necessary to share a basis for communication, to include a compiler and runtime execution environment, as students of computing disciplines must also comply with rules (e.g., syntax) used in these systems. Recalling the work of Papert and Harel\cite{harel1991constructionism}, we might call this constructivism with constructionism. Constructionism is an approach to learning in which the person learns through design and programming. A cluster of related problems exists, which includes what students conceptualizations are, about some elements of proof they should understand: \begin{itemize} \item what internal representations do students use? \item Is there a gamut of internal representations, and does that help with abstraction? \item mathematization, which is the ability to represent problems in mathematical notation \item interiorization, which is the ability to examine and discuss the process of creating proof \item comprehension of simple proofs, which is the ability to see that, and why, an argument is convincing \item proof analysis, which includes the ability to analyze simple proofs to recognize structure \item problem recognition, which is the ability to see that a problem is one that matches a known solution technique \item transformational approach, which is considering the consequences of varying features of the problem \item axiomatic approach, which is the exploration of the consequences of definitions \item construction of valid arguments, which is to synthesize deductions with component parts, including warrants \end{itemize} \section{Application of Findings} \section{ Perspective on Future Directions}