chapters34S5tandAlone.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 Addressing Validation}{26}
\contentsline {section}{\numberline {3.7}Method of Presentation of Results}{27}
\contentsline {chapter}{\numberline {4}Phenomenographic Analysis}{29}
\contentsline {subsection}{\numberline {4.0.1}Application of Phenomenographic Analysis in this Study}{29}
\contentsline {subsubsection}{Phenomenographic Analysis of What Students Think Proofs Are}{30}
\contentsline {subsubsection}{Phenomenographic Analysis of How Students Attempt to Understand Proofs}{32}
\contentsline {subsubsection}{Phenomenographic Analysis of Reasons for Teaching Proof}{34}
\contentsline {subsubsection}{Phenomenographic Analysis of How Students Attempt to Apply Proofs (When not assigned)}{36}
\contentsline {subsubsection}{Phenomenographic Analysis of Whether students exhibit consequences of inability (such as avoiding recursion)}{37}
\contentsline {subsubsection}{Phenomenographic Analysis of How familiar and/or comfortable are students with different (specific) proof techniques: induction, construction, contradiction?}{37}
\contentsline {subsubsection}{Phenomenographic Analysis of Whether students notice structural elements in proofs}{37}
\contentsline {subsubsection}{Phenomenographic Analysis of What do students think it takes to make an argument valid?}{38}
\contentsline {subsubsection}{Phenomenographic Analysis of Whether students incorporate structural elements in proofs}{38}
\contentsline {subsubsection}{Themes / Categories}{39}
\contentsline {subsubsection}{Relationships}{41}
\contentsline {subsection}{\numberline {4.0.2}Analysis of Homework and Tests}{41}
\contentsline {subsubsection}{Proofs}{41}
\contentsline {subsubsection}{Pumping Lemmas}{41}
\contentsline {chapter}{\numberline {5}Results}{43}
\contentsline {section}{\numberline {5.1}Criteria for Results}{43}
\contentsline {section}{\numberline {5.2}What do students think a proof is?}{45}
\contentsline {section}{\numberline {5.3}How do students approach understanding a proof?}{47}
\contentsline {section}{\numberline {5.4}What do students think a proof is for?}{48}
\contentsline {section}{\numberline {5.5}What do students use proof for, when not assigned?}{50}
\contentsline {section}{\numberline {5.6}Do students exhibit any consequence of inability in proof?}{51}
\contentsline {section}{\numberline {5.7}What kind of structure do students notice in proofs?}{52}
\contentsline {section}{\numberline {5.8}What do students think it takes to make an argument valid?}{53}
\contentsline {paragraph}{Definition based reasoning}{54}
\contentsline {paragraph}{Generalization and transformation based reasoning}{54}
\contentsline {section}{\numberline {5.9}Combined Description}{54}
\contentsline {section}{\numberline {5.10}Diagram of Conceptualizations}{56}
\contentsline {section}{\numberline {5.11}Outcome Space}{57}
\contentsline {section}{\numberline {5.12}Critical Factors}{58}
\contentsline {section}{\numberline {5.13}Earlier Paper Material}{58}
\contentsline {subsection}{\numberline {5.13.1}Categories of Experience of Entering Students}{58}
\contentsline {subsection}{\numberline {5.13.2}Representation/Symbolization in Pumping Lemmas}{59}
\contentsline {subsection}{\numberline {5.13.3}Abstract Model for Proof by Mathematical Induction and Recursion}{60}
\contentsline {subsection}{\numberline {5.13.4}Proofs by Induction}{62}
\contentsline {subsection}{\numberline {5.13.5}Pumping Lemmas}{62}
\contentsline {chapter}{Bibliography}{63}