ch8.tex

\chapter{Conclusion}

Harel and Sowder \cite[p. 277?] {harel1998students} state:
\begin{quote} 
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}

This chapter offers conclusions about the individual research questions.
Then, some conclusions from the combined data are proposed.

\section{What Students Think Proofs Are}
 
Looks like a series of transformations
of mathematical formulation
that starts with one formulation
(is true, vs. taken to be true)
uses logically justified
(some students are somewhat unsure about logic)
transformation,
and ends at the desired goal
whose unequivocal truth value is now
more evident than it was.

Is supposed to convince, though some students are not convinced.
Some lack of appreciation of universal vs. existential

Using the component model of a learning objective described in Marton and Booth\cite{marton1997learning}, we conclude that there are conceptualizations in the student population that lack depth about certain aspects of proof.
One such conceptualization lacks the idea that the ability to ascertain or convince is based on logical argumentation. Instead, conviction might be adequately supported by experimentation or construction.

For conceptualizations that grant that logical argumentation is the appropriate means, there can still be lacking a clear understanding of negation, and/or of quantifiers.
The idea of a universal statement being proved, is sometimes only superficially held.
This manifests in multiple ways, including examples offered as proofs, ideas that code
implementations offer as much assurance as any proof, and the lack of appreciation for definitions in the warranting of a transformation step of a proof.

One conceptualization, present in our student population, lacks the idea that a proof about elements of an abstract domain, by virtue of the generality of that abstract domain, is also about specific problems that can be encoded into that abstract domain.
This probably impedes transfer.
This conceptualization might be related to a conceptualization that so emphasizes
constructivism that it places a great deal of emphasis on examples to the detriment of attention to definitions.
Students, without the structures of concepts of the subject matter already learned,
may wish to build these concepts by learning to categorize many examples and from the categories, infer definitions.
This satisfying process of redeveloping definitions might be relatively inefficient compared to learning definitions, recursively the terms those definitions use.

There is a conceptualization that does not notice, or make much use of the idea, that the representations of an idea are not equal to the idea.
We may choose to represent an idea in mathematical formulation for ease of proof.
That same idea might be expressed in diagrams or pseudocode.
It might be easier for students to discern the utility of proof if generalization away from a specific notation could be achieved.

\section{How Students Attempt to Understand Proofs}
 



Some students do not attempt to understand proofs. Of those who do attempt to understand, some students attempt pattern matching with patterns they have learned, such as proof by mathematic induction, or proof by contradiction.
Some students look for warrants from one line to the next.
None of the students we interviewed mentioned divide and conquer, or seeing the role of lemmas.


\section{What Students Think Proof is For}
 
So that in algorithms class the professor can calculate and state confidently the resource utilization of algorithms

communicating mathematical ideas

Some undergraduates say that, for them, proof has no purpose for them outside of assignment completion.

Graduate students: convince the audience of the properties, e.g., resource utilization of an algorithm. If necessary, tailor the algorithm so that it can be proved.

\begin{itemize}
\item demonstrating resource utilization
\item communicating mathematical ideas
\item convincing readers of new algorithms of their properties such as correctness and resource utilization
\item Not about acertaining whether a context suits an algorithm
\end{itemize}

The conceptualization about the purpose of proof populate a domain of variation including no purpose (``professors are mathematicians and think that we should learn some math'', ``do developers need to know these prime number facts?'', ``I never use proof for anything unless I'm assigned'', ``I don't use proofs, I can write the code''.), and including convincing others of universal (within specified domain) truth of their assertions about algorithms.

Even when prompted with ascertaining that a recursive node counting algorithm would terminate when a finite tree structure was input, in an interview in which proof was the subject, some students did not offer proof as a technique.

Moreover, when asked whether they ever used recursion, students said they did not know how to ascertain that recursion was suited to the context of a problem. They used recursion when they were instructed to for problems they were assigned.

\section{How  Students Attempt to Apply Proof, When Assigned, and When Not Assigned}
The students we interviewed attempted to apply proof only when assigned.
For undergraduates, these situations included homework and tests.
For one graduate student, this included a manuscript, in which a proof was expected.

This implies that work is needed on helping students transfer the knowledge they gain about proof to situations in which they might apply it.
It might be useful to ponder in which course this effort to support transfer might occur. If the discrete structures course, for reasons of efficiency, does not exercise application in the target domain in which students are expected to apply proof, then the burden of supporting transfer is on the course in which the proofs are applied. Which is better might be investigated in future work.

\section{Whether Students Exhibit Consequences of Inability With Proof}

We conclude from student interview testimony, that some students never use some algorithms, or algorithmic approaches, including recursion. Though the students are comfortable using the technique in problems on homework or on tests, they do not feel confident in their judgment about contexts in which recursion is applicable.

They obtain their conviction, not from proof, which they do not attempt in situations in which they are contemplating algorithmic approaches. Instead they rely on supposedly authoritative commentary: ``My uncle works at Microsoft and he says they never use recursion.''

\section{Student Familiarity with Specific Proof Techniques}

Some students are most familiar with proof techniques that they find are like processes. ``First I identify a base case and prove it, which is easy. Then I write the induction step and prove it, though it does not convince me.''

 \section{How Students Use Structure in Proof}
 

 
\begin{itemize}
\item Not sure what lemmas are
\item not sure about how to use contexts in which assumptions are made, as in proof by contradiction or categorizing a problem space into cases.
\end{itemize}

It may seem surprising that students who are accustomed to writing and employing methods in algorithms do not transfer this knowledge from the representation of pseudocode to the representation of mathematical formulation.
It may help in perceiving the students' conceptualization to remember that the abstraction from code to interface is also found, in some conceptualizations, to be of no value. Some relatively successful students question the value of time spent on UML\textsuperscript{\textregistered} diagrams.

It may be that, as with definitions, some students do not see any benefit in working with an abstraction and correspondingly pay little attention to it.

Thus when we discuss the use of structure, instead of seeing it as a powerful generalization, students may see it as of no particular use.

\section{What  Students Think Makes a Proof Valid}
\begin{itemize}
\item follow process steps, ``or'' use logic
\end{itemize}

Though some students are not noticing the benefits, through generality, of structure in proof, such as establishing lemmas, some students are noticing a similarity between process steps as in code, and components of the argument by mathematical induction.

Some students find they can use this process-steps idea to produce proofs by mathematic induction that earn full marks, even while the students do not find the argument convincing themselves.

That some students do not see a connection between a base case and an induction step is seen in the use by Sipser\cite{sipser} of a problem about horses being all of one color, itself resembling a problem posed by Polya\cite{polya1954mathematics}.

There is a conceptualization that proofs can match a pattern, such as proof by mathematical induction, or be created using logic.

This may usefully be considered in a context in which definitions are not appreciate, and the nature of syntactic rules are, rather than representing generalization, are viewed as a domain of inquiry, apart from any specialization to other problems.

In this austere light we can perhaps appreciate why some students, as software developers, do not find application for proof in their work.

\section{Conclusions from the Combined Data}

We proposed several conclusions:

\begin{enumerate}
\item 
C1: Outside of using known proofs to demonstrate known resource consumption properties of known algorithms, some students do not grasp the idea that within a defined context a proof provides universal, unequivocal assurance.
As such they do not see how proof could be used to establish that a context is suitable for the application of an algorithm.
Some students do not know why they might want to use proof.
%we do not model, in undergraduate curriculum, why students should employ proof to do what they might want to use proof for. 
Some students, given difficulties with transfer, may not recognize that proof is helpful in determining situations to which knowledge can be transferred.

Some students' difficulty with generalization could easily contribute to their difficulties with transfer, may not recognize that proof is helpful in determining situations to which knowledge can be transferred.

\item C2: When applied to proof, some students' difficulty with generalization could easily contribute to their difficulty with transfer. Mathematical formulation, much used in the application of proof to situations students may encounter outside of class, is a form of generalization.
The perception that proof techniques exercised on number theoretic ideas are about number theoretic ideas rather than about proof techniques might discourage transfer of proof techniques.

\item C3: Some students have a difficult time recognizing that the same thing is being expressed when this thing is expressed using multiple representations. Some widely recognized especially good students can do this, reporting it is challenging but fun to switch between representations. 
These representations are of algorithms, data structures and logical transformation steps.
Some students report not noticing similar information being taught in more than one class, due to a change in the representation forms being used from one class to the next.
Some students, lacking ability to translate between representations,  are hindered by the use of different representations they encounter in proofs. Example is seeing recursion in code, seeing data structures in graphs, seeing proof by mathematic induction in mathematical formulation.

\item C4: Some students do not see that we are teaching them argumentation because we believe they will need to be able to understand and sometimes use argumentation in their work as programmers.

\item C5: Some students do not have a sense of direction about the transformations they are trying to achieve, reporting that they spend a lot of time attempting to try all rules of inference.
Some students do not match the granularity of progress by applying a small number of transformation steps with the granularity of progress they are trying to achieve. 


For those students for whom there is room for improvement, it would help them to know
that they are constructing an argument, of the form,
we can assert x because y,
and that the overall argument is built from smaller arguments,
and the smallest arguments are rules of inference.
Moreover, the structure of that statement the proof is to show give valuable guidance to the structure to the proof.
Moreover, the definitions of the entities appearing in that statement the proof is to show are important for finding reasons, for justifying transformations.

\item C6: students'
 attention to an item depends upon their
 (possibly naive) assessment of the importance of that item
 (``when definitions are given, most students tune out.
 I try to listen, but I stop taking notes \ldots i really prefer examples.'')
 
\end{enumerate}













\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 the report (see section \ref{thickNrich}), a narrative combining
the information obtained from inquiry about these topics,  completes the
work. 

It is not unusual for qualitative studies in education to generate implications for pedagogy.
A product based on this model takes the form of a brochure (see section \ref{brochure}) in which the critical aspects inferred during the analysis are emphasized within the curriculum.

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.







\section{Application of Findings / Implications for Pedagogy}

Without having experimented, we may speculate.
The explicit treatment of a role for argumentation might be beneficial.

In textbooks \cite{epp2010discrete,rosen2003} we find single counterexamples for showing such a
statement false, and the method of exhaustion for showing a finite universal
statement to be true.


This is used in Huth and Ryan\cite{huth2004logic}.

Recognition of when argumentation is producing proof could be more widespread
among the students. Asking students when an argument is complete enough to furnish a proof, for a given domain might be beneficial.

Examining generalization, we could use the proofs ``about numbers'' being applied in future CS topics.

Practice with mathematical formulation, to the extent that students find it
helpful, might be beneficial.
Practice with negation, both in English and in mathematical formulation, might be beneficial.

Practice with the ideas of existential and universal, both in English and in mathematical formulation, might be beneficial. Devlin\cite{devlinintro} is one source for this.

Explicit discussion of the role of proof in transforming a statement from one
representation into another, that does not generate additional information, but
may have a different impact or perceived utility might be beneficial.
Carnap\cite{carnap1958introduction} treats this topic. 

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\textsuperscript{\textregistered} 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.

The topic of naturalistic reasoning steps, and their availability given
the form of the premise and the form of a desired outcome, could be treated more extensively.
This is done by Huth and Ryan\cite{huth2004logic} and Velleman\cite{velleman2006prove}.

The topic of warranting transformation steps can be more strongly emphasized.
Mathematics students also seem as if they would benefit from this according to Alcock and Weber\cite{alcock2005proof}.

Transformation among representations uses generalization and specializations.
Transfer, the application of material learned in one course to a suitable context in another course, might easily by limited by a lack of facility with transformation among representation. (``Now that you mention it, I see it is isomorphic.'' ``Why did you not notice it until now?'' ``Because in data structures we used diagrams, and in discrete structures, we used mathematical formulation.'')


%\section{ Perspective on Future Directions}