Skip to content
Permalink
master
Switch branches/tags

Name already in use

A tag already exists with the provided branch name. Many Git commands accept both tag and branch names, so creating this branch may cause unexpected behavior. Are you sure you want to create this branch?
Go to file
 
 
Cannot retrieve contributors at this time
executable file 201 lines (201 sloc) 15.7 KB
\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}{13}
\contentsline {section}{\numberline {2.1}Phenomenography and Variation Theory}{16}
\contentsline {subsection}{\numberline {2.1.1}Variation Theory}{17}
\contentsline {subsection}{\numberline {2.1.2}Application to Proof in Computer Science}{19}
\contentsline {subsection}{\numberline {2.1.3}An example of applying variation theory}{20}
\contentsline {subsection}{\numberline {2.1.4}Variation Theory and Conjunctions}{21}
\contentsline {section}{\numberline {2.2}Constructivism}{22}
\contentsline {subsection}{\numberline {2.2.1}Piagetian Constructivism}{23}
\contentsline {subsection}{\numberline {2.2.2}Social Constructivism}{25}
\contentsline {section}{\numberline {2.3}Mathematics Education}{28}
\contentsline {chapter}{\numberline {3}Methodology}{31}
\contentsline {section}{\numberline {3.1}Design of the Study}{32}
\contentsline {section}{\numberline {3.2}Parts of the Study}{33}
\contentsline {section}{\numberline {3.3}Population Studied}{34}
\contentsline {subsection}{\numberline {3.3.1}Proofs by Mathematic Induction}{36}
\contentsline {subsection}{\numberline {3.3.2}Purpose of Proof}{37}
\contentsline {subsection}{\numberline {3.3.3}Proofs Using the Pumping Lemma for Regular Languages}{37}
\contentsline {subsection}{\numberline {3.3.4}Student Use of Proof for Applicability of Algorithms}{38}
\contentsline {section}{\numberline {3.4}Data Collection}{38}
\contentsline {subsection}{\numberline {3.4.1}Interviews}{38}
\contentsline {subsubsection}{Student Conceptions of What Proof Is}{39}
\contentsline {subsection}{\numberline {3.4.2}Documents}{40}
\contentsline {subsubsection}{Proofs Using the Pumping Lemma for Regular Languages}{40}
\contentsline {subsection}{\numberline {3.4.3}Observations from Tutoring and Help Sessions}{40}
\contentsline {subsection}{\numberline {3.4.4}Other Sources}{40}
\contentsline {section}{\numberline {3.5}Method of Analysis}{41}
\contentsline {subsection}{\numberline {3.5.1}Analysis of Interviews}{48}
\contentsline {subsubsection}{Traditional Phenomenographic Analysis}{50}
\contentsline {subsubsection}{Modified Thematic Analysis}{50}
\contentsline {subsection}{\numberline {3.5.2}Analysis of Help Session and Tutoring}{51}
\contentsline {subsection}{\numberline {3.5.3}Example: Application of Phenomenographic Analysis to What Students Think Proof is For}{53}
\contentsline {section}{\numberline {3.6}Method of Creating Summary Report (for Member Checking)}{54}
\contentsline {section}{\numberline {3.7}Method of Addressing Validation}{54}
\contentsline {section}{\numberline {3.8}Method of Presentation of Results}{54}
\contentsline {chapter}{\numberline {4}Phenomenographic Analysis}{56}
\contentsline {section}{\numberline {4.1}Phenomenographic Analysis of What Students Think Proof Is}{57}
\contentsline {subsection}{\numberline {4.1.1}Categories}{57}
\contentsline {subsection}{\numberline {4.1.2}Illustrative Quotations for Categories}{58}
\contentsline {subsection}{\numberline {4.1.3}Relations}{58}
\contentsline {subsection}{\numberline {4.1.4}Critical Factors}{60}
\contentsline {subsection}{\numberline {4.1.5}Dimensions of Variation}{62}
\contentsline {subsection}{\numberline {4.1.6}Validation}{67}
\contentsline {section}{\numberline {4.2}Phenomenographic Analysis of How Students Attempt to Understand Proofs}{68}
\contentsline {subsection}{\numberline {4.2.1}Categories}{68}
\contentsline {subsection}{\numberline {4.2.2}Illustrative Quotations for Categories}{69}
\contentsline {subsection}{\numberline {4.2.3}Relations}{78}
\contentsline {subsection}{\numberline {4.2.4}Critical Factors}{78}
\contentsline {subsection}{\numberline {4.2.5}Dimensions of Variation}{79}
\contentsline {subsection}{\numberline {4.2.6}Validation}{80}
\contentsline {section}{\numberline {4.3}Phenomenographic Analysis of What Students Think Proof is For}{81}
\contentsline {subsection}{\numberline {4.3.1}Categories}{81}
\contentsline {subsection}{\numberline {4.3.2}Illustrative Quotations for Categories}{84}
\contentsline {subsection}{\numberline {4.3.3}Relations}{85}
\contentsline {subsection}{\numberline {4.3.4}Critical Factors}{85}
\contentsline {subsection}{\numberline {4.3.5}Dimensions of Variation}{85}
\contentsline {subsection}{\numberline {4.3.6}Validation}{87}
\contentsline {section}{\numberline {4.4}Phenomenographic Analysis of How Students Attempt to Apply Proofs (When Assigned)}{88}
\contentsline {subsection}{\numberline {4.4.1}Categories}{88}
\contentsline {subsection}{\numberline {4.4.2}Illustrative Quotations for Categories}{88}
\contentsline {subsection}{\numberline {4.4.3}Relations}{89}
\contentsline {subsection}{\numberline {4.4.4}Critical Factors}{90}
\contentsline {subsection}{\numberline {4.4.5}Dimensions of Variation}{90}
\contentsline {subsection}{\numberline {4.4.6}Validation}{90}
\contentsline {section}{\numberline {4.5}Phenomenographic Analysis of How Students Attempt to Apply Proofs (When Not Assigned)}{91}
\contentsline {subsection}{\numberline {4.5.1}Categories}{91}
\contentsline {subsection}{\numberline {4.5.2}Illustrative Quotations for Categories}{91}
\contentsline {section}{\numberline {4.6}Phenomenographic Analysis of Whether Students Exhibit Consequences of Inability With Proof (such as avoiding recursion)}{92}
\contentsline {subsection}{\numberline {4.6.1}Categories}{92}
\contentsline {subsection}{\numberline {4.6.2}Illustrative Quotations for Categories}{92}
\contentsline {subsection}{\numberline {4.6.3}Relations}{92}
\contentsline {subsection}{\numberline {4.6.4}Critical Factors}{92}
\contentsline {subsection}{\numberline {4.6.5}Dimensions of Variation}{93}
\contentsline {subsection}{\numberline {4.6.6}Validation}{93}
\contentsline {section}{\numberline {4.7}Phenomenographic Analysis of Student Familiarity with Specific Proof Techniques }{94}
\contentsline {subsection}{\numberline {4.7.1}Categories}{94}
\contentsline {subsection}{\numberline {4.7.2}Illustrative Quotations for Categories}{94}
\contentsline {subsection}{\numberline {4.7.3}Proposed Next Step}{96}
\contentsline {section}{\numberline {4.8}Phenomenographic Analysis of How Students Use Structure in Proof}{97}
\contentsline {subsection}{\numberline {4.8.1}Categories}{97}
\contentsline {subsection}{\numberline {4.8.2}Illustrative Quotations for Categories}{97}
\contentsline {subsection}{\numberline {4.8.3}Relations}{99}
\contentsline {subsection}{\numberline {4.8.4}Critical Factors}{99}
\contentsline {subsection}{\numberline {4.8.5}Dimensions of Variation}{100}
\contentsline {subsection}{\numberline {4.8.6}Validation}{100}
\contentsline {section}{\numberline {4.9}Phenomenographic Analysis of What Students Think Makes a Proof Valid}{102}
\contentsline {subsection}{\numberline {4.9.1}Categories}{102}
\contentsline {subsection}{\numberline {4.9.2}Illustrative Quotations for Categories}{103}
\contentsline {subsection}{\numberline {4.9.3}Relations}{104}
\contentsline {subsection}{\numberline {4.9.4}Critical Factors}{105}
\contentsline {subsection}{\numberline {4.9.5}Dimensions of Variation}{105}
\contentsline {subsection}{\numberline {4.9.6}Validation}{106}
\contentsline {section}{\numberline {4.10}Phenomenographic Analysis of Combined Data}{107}
\contentsline {subsection}{\numberline {4.10.1}Categories}{107}
\contentsline {subsection}{\numberline {4.10.2}Illustrative Quotations for Categories}{109}
\contentsline {subsubsection}{Maybe it isn't argumentation}{110}
\contentsline {subsubsection}{Generalization}{111}
\contentsline {subsubsection}{Generalization vs. Argumentation}{111}
\contentsline {subsubsection}{Interiorization vs. Internalization}{111}
\contentsline {subsubsection}{Representation}{112}
\contentsline {subsubsection}{Structure}{112}
\contentsline {subsubsection}{Steps, Leaps and Direction}{112}
\contentsline {subsection}{\numberline {4.10.3}Relations}{113}
\contentsline {subsection}{\numberline {4.10.4}Critical Factors}{113}
\contentsline {subsection}{\numberline {4.10.5}Dimensions of Variation}{113}
\contentsline {subsection}{\numberline {4.10.6}Validation}{114}
\contentsline {chapter}{\numberline {5}Interpretation/Discussion}{115}
\contentsline {section}{\numberline {5.1}Interpretation of What Students Think Proof Is}{117}
\contentsline {section}{\numberline {5.2}Interpretation of How Students Attempt to Understand Proofs}{119}
\contentsline {section}{\numberline {5.3}Interpretation of What Students Think Proof is For}{121}
\contentsline {section}{\numberline {5.4}Interpretation of How Students Attempt to Apply Proofs (When Assigned)}{122}
\contentsline {section}{\numberline {5.5}Interpretation of How Students Attempt to Apply Proofs (When Not Assigned)}{123}
\contentsline {section}{\numberline {5.6}Interpretation of Whether Students Exhibit Consequences of Inability With Proof (such as avoiding recursion)}{124}
\contentsline {section}{\numberline {5.7}Interpretation of Student Familiarity with Specific Proof Techniques}{124}
\contentsline {section}{\numberline {5.8}Interpretation of How Students Use Structure in Proof}{125}
\contentsline {section}{\numberline {5.9}Interpretation of What Students Think Makes a Proof Valid}{128}
\contentsline {section}{\numberline {5.10}Interpretation of Combined Data}{128}
\contentsline {subsection}{\numberline {5.10.1}Argumentation}{129}
\contentsline {subsection}{\numberline {5.10.2}Generalization / Specialization}{131}
\contentsline {subsubsection}{On generalization}{135}
\contentsline {subsection}{\numberline {5.10.3}Generalization vs. Argumentation}{139}
\contentsline {subsection}{\numberline {5.10.4}Interiorization vs. Internalization}{140}
\contentsline {subsection}{\numberline {5.10.5}Representation}{141}
\contentsline {subsection}{\numberline {5.10.6}Structure}{141}
\contentsline {subsection}{\numberline {5.10.7}Steps, Leaps and Direction}{141}
\contentsline {section}{\numberline {5.11}Report: Navigating Through Space of Conceptualization of Proof}{141}
\contentsline {subsection}{\numberline {5.11.1}Context}{142}
\contentsline {subsection}{\numberline {5.11.2}Conjectures and Conceptualizations}{143}
\contentsline {subsubsection}{What if students are trying to learn ``what'', but not asking ``why''?}{144}
\contentsline {subsubsection}{What if students had a foundation in process/algorithms and did not have much experience thinking about justifications for specific reasoning steps?}{144}
\contentsline {subsubsection}{What if students were not sufficiently patient with themselves to notice careful distinctions?}{144}
\contentsline {subsubsection}{What if students do not listen with equal attentiveness to all that is said?}{145}
\contentsline {subsubsection}{What if students didn't realize that the work they see is a polished product?}{145}
\contentsline {subsubsection}{What if students hoped that process-step proofs would be sufficient?}{146}
\contentsline {subsubsection}{What if students were not finding a clarity benefit to mathematical formulation?}{146}
\contentsline {subsubsection}{What if students did not notice structure in proofs?}{146}
\contentsline {subsubsection}{What if students did not easily generalize from one representation and specialize an idea into another representation?}{147}
\contentsline {subsubsection}{What if students really did have trouble with negation?}{147}
\contentsline {subsection}{\numberline {5.11.3}Report}{148}
\contentsline {chapter}{\numberline {6}Validity and Reliability}{151}
\contentsline {section}{\numberline {6.1}Introduction}{151}
\contentsline {section}{\numberline {6.2}Validity and Reliability Goals in the Social Constructivist Paradigm}{152}
\contentsline {subsection}{\numberline {6.2.1}Trustworthiness}{153}
\contentsline {subsection}{\numberline {6.2.2}Credibility}{153}
\contentsline {subsection}{\numberline {6.2.3}Transferability}{154}
\contentsline {subsection}{\numberline {6.2.4}Confirmability}{154}
\contentsline {section}{\numberline {6.3}Approaches}{155}
\contentsline {subsection}{\numberline {6.3.1}Purposefully Seeking Diversity}{156}
\contentsline {subsection}{\numberline {6.3.2}Diversity in Participants}{156}
\contentsline {subsubsection}{Theoretical Sampling}{157}
\contentsline {subsubsection}{Saturation}{157}
\contentsline {subsection}{\numberline {6.3.3}Diversity in Sources}{158}
\contentsline {subsubsection}{Interviews}{158}
\contentsline {subsubsection}{Documents}{159}
\contentsline {subsubsection}{Other Sources}{159}
\contentsline {subsection}{\numberline {6.3.4}Alternative Explanations}{160}
\contentsline {subsubsection}{Looking for Supplementary Explanations and Supporting Data}{160}
\contentsline {subsection}{\numberline {6.3.5}Consistency Checking}{160}
\contentsline {subsubsection}{Peer Review}{162}
\contentsline {section}{\numberline {6.4}Specific Papers and Manuscripts}{163}
\contentsline {subsection}{\numberline {6.4.1}Validity and Reliability in Proofs Using the Pumping Lemma for Regular Languages}{163}
\contentsline {subsection}{\numberline {6.4.2}Proof by Induction}{165}
\contentsline {section}{\numberline {6.5}Statement of Researcher Bias}{165}
\contentsline {chapter}{\numberline {7}Related Work}{167}
\contentsline {section}{\numberline {7.1}Constructivism}{167}
\contentsline {subsection}{\numberline {7.1.1}Social Constructivism}{168}
\contentsline {subsection}{\numberline {7.1.2}Social Constructivism Applied to Mathematics Education}{169}
\contentsline {subsection}{\numberline {7.1.3}Radical Constructivism}{178}
\contentsline {section}{\numberline {7.2}Computer Science Education}{178}
\contentsline {subsection}{\numberline {7.2.1}Quantifiers}{178}
\contentsline {subsection}{\numberline {7.2.2}Symbols}{178}
\contentsline {subsection}{\numberline {7.2.3}Teaching Pumping Lemmas}{180}
\contentsline {section}{\numberline {7.3}Proof by Induction}{181}
\contentsline {section}{\numberline {7.4}Mathematics Education on Proof}{182}
\contentsline {section}{\numberline {7.5}Qualitative Research}{182}
\contentsline {subsection}{\numberline {7.5.1}Phenomenography and Variation Theory}{182}
\contentsline {subsection}{\numberline {7.5.2}Thematic Analysis}{182}
\contentsline {chapter}{\numberline {8}Conclusion}{183}
\contentsline {section}{\numberline {8.1}What Students Think Proofs Are}{183}
\contentsline {section}{\numberline {8.2}How Students Attempt to Understand Proofs}{185}
\contentsline {section}{\numberline {8.3}What Students Think Proof is For}{185}
\contentsline {section}{\numberline {8.4}How Students Attempt to Apply Proof, When Assigned, and When Not Assigned}{186}
\contentsline {section}{\numberline {8.5}Whether Students Exhibit Consequences of Inability With Proof}{187}
\contentsline {section}{\numberline {8.6}Student Familiarity with Specific Proof Techniques}{187}
\contentsline {section}{\numberline {8.7}How Students Use Structure in Proof}{188}
\contentsline {section}{\numberline {8.8}What Students Think Makes a Proof Valid}{188}
\contentsline {section}{\numberline {8.9}Conclusions from the Combined Data}{189}
\contentsline {section}{\numberline {8.10}Recognizing an Endpoint}{191}
\contentsline {section}{\numberline {8.11}Application of Findings / Implications for Pedagogy}{192}
\contentsline {chapter}{\numberline {9}Future Work}{195}
\contentsline {section}{\numberline {9.1}Pursuit of Conclusions}{195}
\contentsline {section}{\numberline {9.2}Helping Students Discern Derivation for Proof of Correctness}{197}
\contentsline {chapter}{Bibliography}{198}
\contentsline {chapter}{\numberline {A}II}{212}