chapters34StandAlone.toc

\contentsline {chapter}{\numberline {1}spacer}{1}
\contentsline {chapter}{\numberline {2}spacer}{2}
\contentsline {chapter}{\numberline {3}Methodology}{3}
\contentsline {section}{\numberline {3.1}Design of the Study}{4}
\contentsline {section}{\numberline {3.2}Parts of the Study}{5}
\contentsline {section}{\numberline {3.3}Population Studied}{6}
\contentsline {subsection}{\numberline {3.3.1}Proofs by Mathematic Induction}{8}
\contentsline {subsection}{\numberline {3.3.2}Purpose of Proof}{8}
\contentsline {subsection}{\numberline {3.3.3}Proofs Using the Pumping Lemma for Regular Languages}{9}
\contentsline {subsection}{\numberline {3.3.4}Student Use of Proof for Applicability of Algorithms}{9}
\contentsline {section}{\numberline {3.4}Data Collection}{10}
\contentsline {subsection}{\numberline {3.4.1}Interviews}{10}
\contentsline {subsubsection}{Student Conceptions of What Proofs Is}{10}
\contentsline {subsection}{\numberline {3.4.2}Documents}{11}
\contentsline {subsubsection}{Proofs Using the Pumping Lemma for Regular Languages}{11}
\contentsline {subsection}{\numberline {3.4.3}Observations from Tutoring and Help Sessions}{12}
\contentsline {section}{\numberline {3.5}Method of Analysis}{12}
\contentsline {subsection}{\numberline {3.5.1}Analysis of Interviews}{23}
\contentsline {subsection}{\numberline {3.5.2}Analysis of Help Session and Tutoring}{24}
\contentsline {subsection}{\numberline {3.5.3}Application of Phenomenographic Analysis to Why We Teach Proof}{25}
\contentsline {section}{\numberline {3.6}Method of Presentation of Results}{26}
\contentsline {chapter}{\numberline {4}Phenomenographic Analysis}{28}
\contentsline {subsection}{\numberline {4.0.1}Application of Phenomenographic Analysis in this Study}{28}
\contentsline {subsubsection}{Phenomenographic Analysis of What Students Think Proofs Are}{29}
\contentsline {subsubsection}{Phenomenographic Analysis of How Students Attempt to Understand Proofs}{31}
\contentsline {subsubsection}{Phenomenographic Analysis of Reasons for Teaching Proof}{33}
\contentsline {subsubsection}{Phenomenographic Analysis of How Students Attempt to Apply Proofs (When not assigned)}{35}
\contentsline {subsubsection}{Phenomenographic Analysis of Whether students exhibit consequences of inability (such as avoiding recursion)}{35}
\contentsline {subsubsection}{Phenomenographic Analysis of How familiar and/or comfortable are students with different (specific) proof techniques: induction, construction, contradiction?}{36}
\contentsline {subsubsection}{Phenomenographic Analysis of Whether students notice structural elements in proofs}{36}
\contentsline {subsubsection}{Phenomenographic Analysis of What do students think it takes to make an argument valid?}{37}
\contentsline {subsubsection}{Phenomenographic Analysis of Whether students incorporate structural elements in proofs}{37}
\contentsline {subsubsection}{Themes / Categories}{38}
\contentsline {subsubsection}{Relationships}{40}
\contentsline {subsection}{\numberline {4.0.2}Analysis of Homework and Tests}{40}
\contentsline {subsubsection}{Proofs}{40}
\contentsline {subsubsection}{Pumping Lemmas}{40}
\contentsline {chapter}{Bibliography}{42}