Analysis.tex

\chapter{Analysis}

\section{Analysis of Interviews}
\subsection{Themes / Categories}
\begin{itemize}
\item Definitions\\
Students divided into (1)those who found definitions boring, difficult to pay attention to, and undesirable compared to examples, from which they preferred to induce their own definitions, and (2) those who had caught on to the idea that definitions were the carefully crafted building blocks of reasoning.
\item Procedures
Students sometimes learned what was desired in a proof, but learned to produce it by procedure, and were not themselves convinced.
\item Context
Students asked whether the topics for examples and exercises, such as prime numbers, had relevance to programming, with which they had experience, but not unrelated to the topics.
Students did not know the context in which the proofs, or procedure version of proof, was applicable, so, for example, did not apply proof by mathematic induction to recursive algorithms, and did not know how to tell whether recursive algorithms would be applicable.
\item Concrete vs. Abstract
Some students felt quite comfortable with the application of rules of inference to concrete items, but had difficulty transferring application of those rules to mathematical symbols.
\item Symbolization
consistent with Harel and Sowder's 1998 categorization of concepts, we found students who would attempt to write in symbols, but not understand what was denoted, and consequently were uncertain about appropriate operations. Some of these students were glad to see a progression from pseudocode with long variable names to pseudocode with short variable names to mathematical symbolization (formula translation (FORTRAN) in reverse).
\item Applicability of single examples
Some students believed that a few examples constituted a proof. These examples were not generic particular, nor were they transformational, in the sense of Harel and Sowder's 1998 model.
\item Substructure
Students familiar with methods, in the sense of object-oriented programming, and with construction of programs involving multiple method calls, did not always recognize that proofs could be built from multiple lemmas, although they did understand that axioms could be applied.
\item What are proofs for?

\end{itemize}

\section{Analysis of Homework and Tests}
\subsection{Pumping Lemmas}
We wrote descriptions for each error. Some example descriptions
are in Table II.
A handful of students did exhibit their reasoning that for
all segmentations there would exist at least one value of 𝑖 that
would generate a string outside the language.
We categorized the errors as misunderstandings of one or
more of:
1) ∣𝑥𝑦∣ ≤ 𝑝 permits ∣𝑥𝑦∣ < 𝑝\\
2) 𝑥 is the part of the string prior to the cycle\\
3) 𝑦 is the part of the string which returns the state of
the automaton to a previously visited state\\
4) 𝑧 is the part of the string after the (last) cycle up to
acceptance\\
5) 𝑝 − 1 characters is the maximum size of a string
that need not contain a cycle, (strings of length 𝑝
or greater must reuse a state)\\
6) 𝑖 is the number of executions of 𝑦\\
7) There must be no segmentation for which pumping
is possible, if pumping cannot occur.\\
8) A language is a set of strings.\\
9) A language class is a set of languages.\\
Categories are shown in the chapter on results (labelled table iii).\\
 
 


\section{Help Session and Tutoring}
some students, who do know that any statement must and can, be
either true or false, thought implications must be true.