thesisThrough6.toc

\contentsline {chapter}{\numberline {1}Introduction}{1}
\contentsline {section}{\numberline {1.1}Research Questions}{3}
\contentsline {section}{\numberline {1.2}Qualitative Research}{5}
\contentsline {section}{\numberline {1.3}Phenomenography with Variation Theory}{6}
\contentsline {subsection}{\numberline {1.3.1}Conceptualizations}{8}
\contentsline {subsection}{\numberline {1.3.2}Relevance Structure}{9}
\contentsline {section}{\numberline {1.4}Phenomenography / Variation Theory for these Research Questions}{12}
\contentsline {section}{\numberline {1.5}Overview}{12}
\contentsline {chapter}{\numberline {2}Research Perspective and Epistemological Framework}{14}
\contentsline {section}{\numberline {2.1}Phenomenography and Variation Theory}{17}
\contentsline {subsection}{\numberline {2.1.1}An example of applying variation theory}{21}
\contentsline {subsection}{\numberline {2.1.2}Variation Theory and Conjunctions}{23}
\contentsline {section}{\numberline {2.2}Constructivism}{24}
\contentsline {subsection}{\numberline {2.2.1}Piagetian Constructivism}{24}
\contentsline {subsection}{\numberline {2.2.2}Social Constructivism}{26}
\contentsline {section}{\numberline {2.3}Mathematics Education}{30}
\contentsline {subsection}{\numberline {2.3.1}Phenomenology Applied to Mathematical Proofs}{31}
\contentsline {subsection}{\numberline {2.3.2}Phenomenology Applied to Problem Solving}{32}
\contentsline {chapter}{\numberline {3}Methodology}{33}
\contentsline {section}{\numberline {3.1}Design of the Study}{34}
\contentsline {section}{\numberline {3.2}Parts of the Study}{35}
\contentsline {section}{\numberline {3.3}Population Studied}{36}
\contentsline {subsection}{\numberline {3.3.1}Proofs by Mathematic Induction}{38}
\contentsline {subsection}{\numberline {3.3.2}Purpose of Proof}{38}
\contentsline {subsection}{\numberline {3.3.3}Proofs Using the Pumping Lemma for Regular Languages}{39}
\contentsline {subsection}{\numberline {3.3.4}Student Use of Proof for Applicability of Algorithms}{39}
\contentsline {section}{\numberline {3.4}Data Collection}{40}
\contentsline {subsection}{\numberline {3.4.1}Interviews}{40}
\contentsline {subsubsection}{Student Conceptions of What Proofs Is}{40}
\contentsline {subsection}{\numberline {3.4.2}Documents}{41}
\contentsline {subsubsection}{Proofs Using the Pumping Lemma for Regular Languages}{41}
\contentsline {subsection}{\numberline {3.4.3}Observations from Tutoring and Help Sessions}{42}
\contentsline {section}{\numberline {3.5}Method of Analysis}{42}
\contentsline {subsection}{\numberline {3.5.1}Analysis of Interviews}{53}
\contentsline {subsection}{\numberline {3.5.2}Analysis of Help Session and Tutoring}{54}
\contentsline {subsection}{\numberline {3.5.3}Application of Phenomenographic Analysis to Why We Teach Proof}{55}
\contentsline {section}{\numberline {3.6}Method of Addressing Validation}{56}
\contentsline {section}{\numberline {3.7}Method of Presentation of Results}{57}
\contentsline {chapter}{\numberline {4}Phenomenographic Analysis}{59}
\contentsline {subsection}{\numberline {4.0.1}Application of Phenomenographic Analysis in this Study}{59}
\contentsline {subsubsection}{Phenomenographic Analysis of What Students Think Proofs Are}{60}
\contentsline {paragraph}{Codes}{65}
\contentsline {paragraph}{Preliminary Categories}{65}
\contentsline {paragraph}{Main theme and its relationships to minor themes}{65}
\contentsline {paragraph}{Categories and Relationships}{65}
\contentsline {paragraph}{Dimension of Variation and Critical Factors}{66}
\contentsline {subsubsection}{Phenomenographic Analysis of How Students Attempt to Understand Proofs}{66}
\contentsline {paragraph}{Codes}{68}
\contentsline {paragraph}{Preliminary Categories}{68}
\contentsline {paragraph}{Main theme and its relationships to minor themes}{68}
\contentsline {paragraph}{Categories and Relationships}{69}
\contentsline {paragraph}{Dimension of Variation and Critical Factors}{69}
\contentsline {subsubsection}{Phenomenographic Analysis of Reasons for Teaching Proof}{69}
\contentsline {paragraph}{Codes}{72}
\contentsline {paragraph}{Preliminary Categories}{72}
\contentsline {paragraph}{Main theme and its relationships to minor themes}{72}
\contentsline {paragraph}{Categories and Relationships}{72}
\contentsline {paragraph}{Dimension of Variation and Critical Factors}{73}
\contentsline {subsubsection}{Phenomenographic Analysis of How Students Attempt to Apply Proofs (When not assigned)}{73}
\contentsline {paragraph}{Codes}{73}
\contentsline {paragraph}{Preliminary Categories}{73}
\contentsline {paragraph}{Main theme and its relationships to minor themes}{73}
\contentsline {paragraph}{Categories and Relationships}{74}
\contentsline {paragraph}{Dimension of Variation and Critical Factors}{74}
\contentsline {subsubsection}{Phenomenographic Analysis of Whether students exhibit consequences of inability (such as avoiding recursion)}{74}
\contentsline {paragraph}{Codes}{74}
\contentsline {paragraph}{Preliminary Categories}{75}
\contentsline {paragraph}{Main theme and its relationships to minor themes}{75}
\contentsline {paragraph}{Categories and Relationships}{75}
\contentsline {paragraph}{Dimension of Variation and Critical Factors}{75}
\contentsline {subsubsection}{Phenomenographic Analysis of How familiar and/or comfortable are students with different (specific) proof techniques: induction, construction, contradiction?}{76}
\contentsline {paragraph}{Codes}{76}
\contentsline {paragraph}{Preliminary Categories}{77}
\contentsline {paragraph}{Main theme and its relationships to minor themes}{77}
\contentsline {paragraph}{Categories and Relationships}{77}
\contentsline {paragraph}{Dimension of Variation and Critical Factors}{77}
\contentsline {subsubsection}{Phenomenographic Analysis of Whether students notice structural elements in proofs}{77}
\contentsline {paragraph}{Codes}{78}
\contentsline {paragraph}{Preliminary Categories}{78}
\contentsline {paragraph}{Main theme and its relationships to minor themes}{78}
\contentsline {paragraph}{Categories and Relationships}{79}
\contentsline {paragraph}{Dimension of Variation and Critical Factors}{79}
\contentsline {subsubsection}{Phenomenographic Analysis of What do students think it takes to make an argument valid?}{79}
\contentsline {paragraph}{Codes}{79}
\contentsline {paragraph}{Preliminary Categories}{79}
\contentsline {paragraph}{Main theme and its relationships to minor themes}{80}
\contentsline {paragraph}{Categories and Relationships}{80}
\contentsline {paragraph}{Dimension of Variation and Critical Factors}{80}
\contentsline {subsubsection}{Phenomenographic Analysis of Whether students incorporate structural elements in proofs}{80}
\contentsline {paragraph}{Codes}{81}
\contentsline {paragraph}{Preliminary Categories}{81}
\contentsline {paragraph}{Main theme and its relationships to minor themes}{82}
\contentsline {paragraph}{Categories and Relationships}{82}
\contentsline {paragraph}{Dimension of Variation and Critical Factors}{82}
\contentsline {subsubsection}{Combined Themes / Categories}{82}
\contentsline {subsubsection}{Combined Relationships}{84}
\contentsline {subsection}{\numberline {4.0.2}Analysis of Homework and Tests}{84}
\contentsline {subsubsection}{Proofs}{84}
\contentsline {subsubsection}{Pumping Lemmas}{84}
\contentsline {chapter}{\numberline {5}Results}{86}
\contentsline {section}{\numberline {5.1}Criteria for Results}{86}
\contentsline {section}{\numberline {5.2}What do students think a proof is?}{88}
\contentsline {section}{\numberline {5.3}How do students approach understanding a proof?}{90}
\contentsline {section}{\numberline {5.4}What do students think a proof is for?}{91}
\contentsline {section}{\numberline {5.5}What do students use proof for, when not assigned?}{93}
\contentsline {section}{\numberline {5.6}Do students exhibit any consequence of inability in proof?}{94}
\contentsline {section}{\numberline {5.7}What kind of structure do students notice in proofs?}{95}
\contentsline {section}{\numberline {5.8}What do students think it takes to make an argument valid?}{96}
\contentsline {paragraph}{Definition based reasoning}{97}
\contentsline {paragraph}{Generalization and transformation based reasoning}{97}
\contentsline {section}{\numberline {5.9}Combined Description}{97}
\contentsline {section}{\numberline {5.10}Diagram of Conceptualizations}{99}
\contentsline {section}{\numberline {5.11}Outcome Space}{100}
\contentsline {section}{\numberline {5.12}Critical Factors}{101}
\contentsline {section}{\numberline {5.13}Earlier Paper Material}{101}
\contentsline {subsection}{\numberline {5.13.1}Categories of Experience of Entering Students}{101}
\contentsline {subsection}{\numberline {5.13.2}Representation/Symbolization in Pumping Lemmas}{102}
\contentsline {subsection}{\numberline {5.13.3}Abstract Model for Proof by Mathematical Induction and Recursion}{103}
\contentsline {subsection}{\numberline {5.13.4}Proofs by Induction}{105}
\contentsline {subsection}{\numberline {5.13.5}Pumping Lemmas}{105}
\contentsline {chapter}{\numberline {6}Interpretation/Discussion}{106}
\contentsline {section}{\numberline {6.1}Interpretation}{106}
\contentsline {subsubsection}{Interpretation of What Students Think Proofs Are}{106}
\contentsline {subsubsection}{Interpretation of How Students Attempt to Understand Proofs}{107}
\contentsline {subsubsection}{Interpretation of Reasons for Teaching Proof}{107}
\contentsline {subsubsection}{Interpretation of How Students Attempt to Apply Proofs (When not assigned)}{107}
\contentsline {subsubsection}{Interpretation of Whether students exhibit consequences of inability (such as avoiding recursion)}{108}
\contentsline {subsubsection}{Interpretation of How familiar and/or comfortable are students with different (specific) proof techniques: induction, construction, contradiction?}{108}
\contentsline {subsubsection}{Interpretation of Whether students notice structural elements in proofs}{109}
\contentsline {subsubsection}{Interpretation of What do students think it takes to make an argument valid?}{109}
\contentsline {subsubsection}{Interpretation of Whether students incorporate structural elements in proofs}{109}
\contentsline {subsection}{\numberline {6.1.1}Productive and Counterproductive Beliefs}{109}
\contentsline {subsection}{\numberline {6.1.2}Productive and Counterproductive Momentum}{110}
\contentsline {subsection}{\numberline {6.1.3}Published papers}{110}
\contentsline {paragraph}{Categories of Experience of Entering Students}{110}
\contentsline {paragraph}{Representation/Symbolization in Pumping Lemmas}{111}
\contentsline {paragraph}{Abstract Model for Proof by Mathematical Induction and Recursion}{113}
\contentsline {subsection}{\numberline {6.1.4}Helping Students Discern Abstraction}{115}
\contentsline {subsection}{\numberline {6.1.5}Algebra}{118}
\contentsline {subsection}{\numberline {6.1.6}Geometry}{118}
\contentsline {subsection}{\numberline {6.1.7}Seeing a Broader Context}{118}
\contentsline {subsection}{\numberline {6.1.8}Mathematics tests in high school that involve proving}{119}
\contentsline {section}{\numberline {6.2}Discussion}{120}
\contentsline {subsection}{\numberline {6.2.1}Importance}{120}
\contentsline {subsection}{\numberline {6.2.2}Interpretation of Results}{120}
\contentsline {subsection}{\numberline {6.2.3}More Discussion}{121}
\contentsline {paragraph}{Helping Students Discern Abstraction}{121}
\contentsline {paragraph}{Helping Students Discern Abstraction}{122}
\contentsline {paragraph}{Algebra}{123}
\contentsline {paragraph}{Geometry}{123}
\contentsline {paragraph}{Seeing a Broader Context}{123}
\contentsline {paragraph}{Mathematics tests in high school that involve proving}{124}
\contentsline {section}{\numberline {6.3}Previously Published Work}{124}
\contentsline {subsection}{\numberline {6.3.1}Categories of Experience of Entering Students}{125}
\contentsline {subsection}{\numberline {6.3.2}Categories of Experience of Entering Students}{125}
\contentsline {subsection}{\numberline {6.3.3}Representation/Symbolization in Pumping Lemmas}{126}
\contentsline {subsection}{\numberline {6.3.4}Representation/Symbolization in Pumping Lemmas}{128}
\contentsline {subsection}{\numberline {6.3.5}Abstract Model for Proof by Mathematical Induction and Recursion}{129}
\contentsline {subsection}{\numberline {6.3.6}Proofs by Induction}{131}
\contentsline {subsection}{\numberline {6.3.7}Abstract Model for Proof by Mathematical Induction and Recursion}{131}
\contentsline {subsection}{\numberline {6.3.8}Pumping Lemmas}{133}
\contentsline {section}{\numberline {6.4}Results of Combined Investigations}{133}
\contentsline {section}{\numberline {6.5}Categories}{133}
\contentsline {section}{\numberline {6.6}Critical Factors}{135}
\contentsline {subsection}{\numberline {6.6.1}Abstraction}{137}
\contentsline {subsection}{\numberline {6.6.2}Definitions}{138}
\contentsline {subsection}{\numberline {6.6.3}Symbolization}{138}
\contentsline {subsection}{\numberline {6.6.4}Structure}{139}
\contentsline {section}{\numberline {6.7}What do students think a proof is?}{141}
\contentsline {section}{\numberline {6.8}How do students approach understanding a proof?}{143}
\contentsline {section}{\numberline {6.9}What do students think a proof is for?}{144}
\contentsline {section}{\numberline {6.10}What do students use proof for, when not assigned?}{146}
\contentsline {section}{\numberline {6.11}Do students exhibit any consequence of inability in proof?}{147}
\contentsline {section}{\numberline {6.12}What kind of structure do students notice in proofs?}{148}
\contentsline {section}{\numberline {6.13}What do students think it takes to make an argument valid?}{149}
\contentsline {paragraph}{Definition based reasoning}{150}
\contentsline {paragraph}{Generalization and transformation based reasoning}{150}
\contentsline {section}{\numberline {6.14}Combined Description}{150}
\contentsline {section}{\numberline {6.15}Diagram of Conceptualizations}{152}
\contentsline {section}{\numberline {6.16}Outcome Space}{153}
\contentsline {section}{\numberline {6.17}Critical Factors}{154}
\contentsline {section}{\numberline {6.18}More Discussion}{154}
\contentsline {paragraph}{Helping Students Discern Abstraction}{154}
\contentsline {paragraph}{Helping Students Discern Abstraction}{156}
\contentsline {paragraph}{Algebra}{156}
\contentsline {paragraph}{Geometry}{156}
\contentsline {paragraph}{Seeing a Broader Context}{157}
\contentsline {paragraph}{Mathematics tests in high school that involve proving}{158}
\contentsline {chapter}{Bibliography}{159}