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\contentsline {chapter}{\numberline {1}spacer}{1}
\contentsline {chapter}{\numberline {2}spacer}{2}
\contentsline {chapter}{\numberline {3}spacer}{3}
\contentsline {chapter}{\numberline {4}Phenomenographic Analysis}{4}
\contentsline {subsection}{\numberline {4.0.1}Application of Phenomenographic Analysis in this Study}{4}
\contentsline {subsubsection}{Phenomenographic Analysis of What Students Think Proofs Are}{5}
\contentsline {paragraph}{Codes}{11}
\contentsline {paragraph}{Preliminary Categories}{11}
\contentsline {paragraph}{Main theme and its relationships to minor themes}{12}
\contentsline {paragraph}{Categories and Relationships}{12}
\contentsline {paragraph}{Dimension of Variation and Critical Factors}{12}
\contentsline {subsubsection}{Phenomenographic Analysis of How Students Attempt to Understand Proofs}{12}
\contentsline {paragraph}{Codes}{15}
\contentsline {paragraph}{Preliminary Categories}{16}
\contentsline {paragraph}{Main theme and its relationships to minor themes}{16}
\contentsline {paragraph}{Categories and Relationships}{16}
\contentsline {paragraph}{Dimension of Variation and Critical Factors}{16}
\contentsline {subsubsection}{Phenomenographic Analysis of Reasons for Teaching Proof}{16}
\contentsline {paragraph}{Codes}{25}
\contentsline {paragraph}{Preliminary Categories}{26}
\contentsline {paragraph}{Main theme and its relationships to minor themes}{26}
\contentsline {paragraph}{Categories and Relationships}{26}
\contentsline {paragraph}{Dimension of Variation and Critical Factors}{26}
\contentsline {subsubsection}{Phenomenographic Analysis of How Students Attempt to Apply Proofs (When not assigned)}{26}
\contentsline {paragraph}{Codes}{27}
\contentsline {paragraph}{Preliminary Categories}{27}
\contentsline {paragraph}{Main theme and its relationships to minor themes}{28}
\contentsline {paragraph}{Categories and Relationships}{28}
\contentsline {paragraph}{Dimension of Variation and Critical Factors}{28}
\contentsline {subsubsection}{Phenomenographic Analysis of Whether students exhibit consequences of inability (such as avoiding recursion)}{28}
\contentsline {paragraph}{Codes}{29}
\contentsline {paragraph}{Preliminary Categories}{29}
\contentsline {paragraph}{Main theme and its relationships to minor themes}{29}
\contentsline {paragraph}{Categories and Relationships}{29}
\contentsline {paragraph}{Dimension of Variation and Critical Factors}{29}
\contentsline {subsubsection}{Phenomenographic Analysis of How familiar and/or comfortable are students with different (specific) proof techniques: induction, construction, contradiction?}{30}
\contentsline {paragraph}{Codes}{32}
\contentsline {paragraph}{Preliminary Categories}{32}
\contentsline {paragraph}{Main theme and its relationships to minor themes}{32}
\contentsline {paragraph}{Categories and Relationships}{32}
\contentsline {paragraph}{Dimension of Variation and Critical Factors}{33}
\contentsline {subsubsection}{Phenomenographic Analysis of Whether students notice structural elements in proofs}{33}
\contentsline {paragraph}{Codes}{34}
\contentsline {paragraph}{Preliminary Categories}{34}
\contentsline {paragraph}{Main theme and its relationships to minor themes}{34}
\contentsline {paragraph}{Categories and Relationships}{34}
\contentsline {paragraph}{Dimension of Variation and Critical Factors}{35}
\contentsline {subsubsection}{Phenomenographic Analysis of What do students think it takes to make an argument valid?}{35}
\contentsline {paragraph}{Codes}{36}
\contentsline {paragraph}{Preliminary Categories}{36}
\contentsline {paragraph}{Main theme and its relationships to minor themes}{36}
\contentsline {paragraph}{Categories and Relationships}{37}
\contentsline {paragraph}{Dimension of Variation and Critical Factors}{37}
\contentsline {subsubsection}{Phenomenographic Analysis of Whether students incorporate structural elements in proofs}{37}
\contentsline {subsubsection}{Phenomenographic Analysis of Combined Data}{38}
\contentsline {section}{\numberline {4.1}Does this go anywhere? Interview}{38}
\contentsline {paragraph}{Codes}{39}
\contentsline {paragraph}{Preliminary Categories}{39}
\contentsline {paragraph}{Main theme and its relationships to minor themes}{39}
\contentsline {paragraph}{Categories and Relationships}{39}
\contentsline {paragraph}{Dimension of Variation and Critical Factors}{40}
\contentsline {subsubsection}{Combined Themes / Categories}{40}
\contentsline {subsubsection}{Combined Relationships}{42}
\contentsline {subsection}{\numberline {4.1.1}Analysis of Homework and Tests}{42}
\contentsline {subsubsection}{Proofs}{42}
\contentsline {subsubsection}{Pumping Lemmas}{42}
\contentsline {chapter}{\numberline {5}Results}{44}
\contentsline {section}{\numberline {5.1}Criteria for Results}{44}
\contentsline {section}{\numberline {5.2}What do students think a proof is?}{46}
\contentsline {section}{\numberline {5.3}How do students approach understanding a proof?}{49}
\contentsline {section}{\numberline {5.4}What do students think a proof is for?}{52}
\contentsline {section}{\numberline {5.5}What do students use proof for, when not assigned?}{55}
\contentsline {section}{\numberline {5.6}Do students exhibit any consequence of inability in proof?}{57}
\contentsline {section}{\numberline {5.7}What kind of structure do students notice in proofs?}{59}
\contentsline {section}{\numberline {5.8}What do students think it takes to make an argument valid?}{61}
\contentsline {paragraph}{Definition based reasoning}{63}
\contentsline {paragraph}{Generalization and transformation based reasoning}{63}
\contentsline {section}{\numberline {5.9}Combined Description}{64}
\contentsline {section}{\numberline {5.10}Diagram of Conceptualizations}{65}
\contentsline {section}{\numberline {5.11}Outcome Space}{66}
\contentsline {section}{\numberline {5.12}Critical Factors}{67}
\contentsline {section}{\numberline {5.13}Earlier Paper Material}{67}
\contentsline {subsection}{\numberline {5.13.1}Categories of Experience of Entering Students}{67}
\contentsline {subsection}{\numberline {5.13.2}Representation/Symbolization in Pumping Lemmas}{68}
\contentsline {subsection}{\numberline {5.13.3}Abstract Model for Proof by Mathematical Induction and Recursion}{69}
\contentsline {subsection}{\numberline {5.13.4}Proofs by Induction}{71}
\contentsline {subsection}{\numberline {5.13.5}Pumping Lemmas}{71}
\contentsline {chapter}{\numberline {6}Interpretation/Discussion}{72}
\contentsline {section}{\numberline {6.1}Interpretation}{73}
\contentsline {subsubsection}{Interpretation of What Students Think Proofs Are}{73}
\contentsline {subsubsection}{Interpretation of How Students Attempt to Understand Proofs}{73}
\contentsline {subsubsection}{Interpretation of Reasons for Teaching Proof}{74}
\contentsline {subsubsection}{Interpretation of How Students Attempt to Apply Proofs (When not assigned)}{75}
\contentsline {subsubsection}{Interpretation of Whether students exhibit consequences of inability (such as avoiding recursion)}{76}
\contentsline {subsubsection}{Interpretation of How familiar and/or comfortable are students with different (specific) proof techniques: induction, construction, contradiction?}{76}
\contentsline {subsubsection}{Interpretation of Whether students notice structural elements in proofs}{76}
\contentsline {subsubsection}{Interpretation of What do students think it takes to make an argument valid?}{76}
\contentsline {subsubsection}{Interpretation of Whether students incorporate structural elements in proofs}{76}
\contentsline {subsection}{\numberline {6.1.1}Productive and Counterproductive Beliefs}{77}
\contentsline {subsection}{\numberline {6.1.2}Productive and Counterproductive Momentum}{77}
\contentsline {subsection}{\numberline {6.1.3}Published papers}{77}
\contentsline {paragraph}{Categories of Experience of Entering Students}{78}
\contentsline {paragraph}{Representation/Symbolization in Pumping Lemmas}{78}
\contentsline {paragraph}{Abstract Model for Proof by Mathematical Induction and Recursion}{81}
\contentsline {subsection}{\numberline {6.1.4}Helping Students Discern Abstraction}{84}
\contentsline {subsection}{\numberline {6.1.5}Algebra}{85}
\contentsline {subsection}{\numberline {6.1.6}Geometry}{86}
\contentsline {subsection}{\numberline {6.1.7}Seeing a Broader Context}{86}
\contentsline {subsection}{\numberline {6.1.8}Mathematics tests in high school that involve proving}{87}
\contentsline {section}{\numberline {6.2}Discussion}{87}
\contentsline {subsection}{\numberline {6.2.1}Importance}{88}
\contentsline {subsection}{\numberline {6.2.2}Interpretation of Results}{88}
\contentsline {subsection}{\numberline {6.2.3}More Discussion}{89}
\contentsline {paragraph}{Helping Students Discern Abstraction}{89}
\contentsline {paragraph}{Helping Students Discern Abstraction}{90}
\contentsline {paragraph}{Algebra}{91}
\contentsline {paragraph}{Geometry}{91}
\contentsline {paragraph}{Seeing a Broader Context}{91}
\contentsline {paragraph}{Mathematics tests in high school that involve proving}{92}
\contentsline {section}{\numberline {6.3}Previously Published Work}{92}
\contentsline {subsection}{\numberline {6.3.1}Categories of Experience of Entering Students}{93}
\contentsline {subsection}{\numberline {6.3.2}Categories of Experience of Entering Students}{93}
\contentsline {subsection}{\numberline {6.3.3}Representation/Symbolization in Pumping Lemmas}{94}
\contentsline {subsection}{\numberline {6.3.4}Representation/Symbolization in Pumping Lemmas}{96}
\contentsline {subsection}{\numberline {6.3.5}Abstract Model for Proof by Mathematical Induction and Recursion}{97}
\contentsline {subsection}{\numberline {6.3.6}Proofs by Induction}{99}
\contentsline {subsection}{\numberline {6.3.7}Abstract Model for Proof by Mathematical Induction and Recursion}{99}
\contentsline {subsection}{\numberline {6.3.8}Pumping Lemmas}{101}
\contentsline {section}{\numberline {6.4}Results of Combined Investigations}{101}
\contentsline {section}{\numberline {6.5}Categories}{101}
\contentsline {section}{\numberline {6.6}Critical Factors}{102}
\contentsline {subsection}{\numberline {6.6.1}Abstraction}{105}
\contentsline {subsection}{\numberline {6.6.2}Definitions}{107}
\contentsline {subsection}{\numberline {6.6.3}Symbolization}{107}
\contentsline {subsection}{\numberline {6.6.4}Structure}{108}
\contentsline {section}{\numberline {6.7}What do students think a proof is?}{109}
\contentsline {section}{\numberline {6.8}How do students approach understanding a proof?}{112}
\contentsline {section}{\numberline {6.9}What do students think a proof is for?}{113}
\contentsline {section}{\numberline {6.10}What do students use proof for, when not assigned?}{115}
\contentsline {section}{\numberline {6.11}Do students exhibit any consequence of inability in proof?}{116}
\contentsline {section}{\numberline {6.12}What kind of structure do students notice in proofs?}{117}
\contentsline {section}{\numberline {6.13}What do students think it takes to make an argument valid?}{118}
\contentsline {paragraph}{Definition based reasoning}{119}
\contentsline {paragraph}{Generalization and transformation based reasoning}{119}
\contentsline {section}{\numberline {6.14}Combined Description}{120}
\contentsline {subsection}{\numberline {6.14.1}Generalization meets preference for examples over definitions}{120}
\contentsline {subsection}{\numberline {6.14.2}Recognize problem solutions, comprehend problem solutions, create problem solutions}{122}
\contentsline {section}{\numberline {6.15}Diagram of Conceptualizations}{124}
\contentsline {section}{\numberline {6.16}Outcome Space}{125}
\contentsline {section}{\numberline {6.17}Critical Factors}{126}
\contentsline {section}{\numberline {6.18}More Discussion}{126}
\contentsline {paragraph}{Helping Students Discern Abstraction}{126}
\contentsline {paragraph}{Helping Students Discern Abstraction}{128}
\contentsline {paragraph}{Algebra}{128}
\contentsline {paragraph}{Geometry}{128}
\contentsline {paragraph}{Seeing a Broader Context}{129}
\contentsline {paragraph}{Mathematics tests in high school that involve proving}{130}
\contentsline {chapter}{Bibliography}{131}