thesis2.bbl

\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
\providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR }
% \MRhref is called by the amsart/book/proc definition of \MR.
\providecommand{\MRhref}[2]{%
  \href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2}
}
\providecommand{\href}[2]{#2}
\begin{thebibliography}{100}

\bibitem{czeisler}
\url{projects.csail.mit.edu/video/talks/other/czeisler_sleep_talk.mp4} viewed
  January 18, 2016.

\bibitem{aakerlind2012variation}
Gerlese~S {\AA}kerlind, \emph{Variation and commonality in phenomenographic
  research methods}, Higher Education Research \& Development \textbf{31}
  (2012), no.~1, 115--127.

\bibitem{alcock2002definitions}
Lara Alcock and AP~Simpson, \emph{Definitions: dealing with categories
  mathematically.}, For the learning of mathematics. \textbf{22} (2002), no.~2,
  28--34.

\bibitem{alcock2005proof}
Lara Alcock and Keith Weber, \emph{Proof validation in real analysis: Inferring
  and checking warrants}, The Journal of Mathematical Behavior \textbf{24}
  (2005), no.~2, 125--134.

\bibitem{almstrum1996investigating}
Vicki~L Almstrum, \emph{Investigating student difficulties with mathematical
  logic}, Teaching and Learning Formal Methods (1996), 131--160.

\bibitem{arcavi1998teaching}
Abraham Arcavi, Cathy Kessel, Luciano Meira, and John~P Smith, \emph{Teaching
  mathematical problem solving: An analysis of an emergent classroom
  community}, Research in collegiate mathematics education, III \textbf{7}
  (1998), 1--70.

\bibitem{aspinall2000proof}
David Aspinall, \emph{Proof general: A generic tool for proof development},
  Tools and Algorithms for the Construction and Analysis of Systems, Springer,
  2000, pp.~38--43.

\bibitem{baranchik1998supplementary}
ALVIN Baranchik and BARRY Cherkas, \emph{Supplementary methods for assessing
  student performance on a standardized test in elementary algebra}, Research
  in Collegiate Mathematics Education \textbf{3} (1998), 216--233.

\bibitem{bausell1972factorial}
R~Barker Bausell, William~B Moody, and F~Neil Walzl, \emph{A factorial study of
  tutoring versus classroom instruction}, American Educational Research Journal
  (1972), 591--597.

\bibitem{ben1998constructivism}
Mordechai Ben-Ari, \emph{Constructivism in computer science education}, Acm
  sigcse bulletin, vol.~30, ACM, 1998, pp.~257--261.

\bibitem{beth1966mathematical}
EW~Beth and J~Piaget, \emph{Mathematical epistemology and
  psychology,(translated by w. mays), d}, Reidel. Dordrecht, The Netherlands
  (originally published 1965) (1966).

\bibitem{black1998inside}
Paul Black and Dylan Wiliam, \emph{Inside the black box: Raising standards
  through classroom assessment}, Granada Learning, 1998.

\bibitem{BBloom1984}
Benjamin~S. Bloom, \emph{The 2 sigma problem: The search for methods of group
  instruction as effective as one-to-one tutoring}, Educational Researcher
  \textbf{13} (1984), no.~6, pp. 4--16 (English).

\bibitem{bloom1956taxonomy}
Benjamin~S Bloom et~al., \emph{Taxonomy of educational objectives. vol. 1:
  Cognitive domain}, New York: McKay (1956).

\bibitem{booth1997phenomenography}
Shirley Booth, \emph{On phenomenography, learning and teaching}, Higher
  education research \& development \textbf{16} (1997), no.~2, 135--158.

\bibitem{booth2001learning}
\bysame, \emph{Learning computer science and engineering in context}, Computer
  Science Education \textbf{11} (2001), no.~3, 169--188.

\bibitem{bowden2004university}
John Bowden and Ference Marton, \emph{The university of learning}, Psychology
  Press, 2004.

\bibitem{boyatzis1998transforming}
Richard~E Boyatzis, \emph{Transforming qualitative information: Thematic
  analysis and code development}, Sage, 1998.

\bibitem{bransford2000designs}
John Bransford, Linda Zech, D~Schwarz, Bngid Barron, and Nancy Vye,
  \emph{Designs for environments that invite and sustain mathematical
  thinking}, Symbolizing and Communicating in Mathematics Classrooms (2000),
  275--324.

\bibitem{braun2006using}
Virginia Braun and Victoria Clarke, \emph{Using thematic analysis in
  psychology}, Qualitative research in psychology \textbf{3} (2006), no.~2,
  77--101.

\bibitem{brooks1999search}
Jacqueline~Grennon Brooks and Martin~G Brooks, \emph{In search of
  understanding: The case for constructivist classrooms}, ASCD, 1999.

\bibitem{carnap1958introduction}
Rudolf Carnap, \emph{Introduction to symbolic logic and its applications},
  Dover Books, 1958.

\bibitem{dahlin2007enriching}
Bo~Dahlin, \emph{Enriching the theoretical horizons of phenomenography,
  variation theory and learning studies}, Scandinavian Journal of Educational
  Research \textbf{51} (2007), no.~4, 327--346.

\bibitem{denzin1973research}
Norman~K Denzin, \emph{The research act: A theoretical introduction to
  sociological methods}, Transaction publishers, 1973.

\bibitem{norman2005sage}
Norman~K. Denzin and Yvonna~S. Lincoln, \emph{The sage handbook of qualitative
  research}, Sage, 2005.

\bibitem{devlin2012mathematical}
Keith Devlin, \emph{Introduction to mathematical thinking}, Keith Devlin, 2012.

\bibitem{devlinintro}
Keith Devlin, \emph{Introduction to mathematical thinking}, Keith Devlin, 2012.

\bibitem{dorfler2000means}
Willi D{\"o}rfler, \emph{Means for meaning}, Symbolizing and communicating in
  mathematics classrooms (2000), 99--131.

\bibitem{dubinsky2002apos}
Ed~Dubinsky and Michael~A McDonald, \emph{Apos: A constructivist theory of
  learning in undergraduate mathematics education research}, The teaching and
  learning of mathematics at university level, Springer, 2002, pp.~275--282.

\bibitem{ellenberg2014not}
Jordan Ellenberg, \emph{How not to be wrong: The power of mathematical
  thinking}, Penguin, 2014.

\bibitem{epp2010discrete}
Susanna Epp, \emph{Discrete mathematics with applications}, Cengage Learning,
  2010.

\bibitem{epp2011discrete}
Susanna~S Epp, \emph{Discrete mathematics with applications}, Cengage Learning,
  2011.

\bibitem{ernest1994constructing}
Paul Ernest, \emph{Constructing mathematical knowledge: Epistemology and
  mathematics education}, vol.~4, Taylor \& Francis, 1994.

\bibitem{ernest1994constructivism}
\bysame, \emph{Constructivism: Which form provides the most adequate theory of
  mathematics learning?}, Journal f{\"u}r Mathematik-Didaktik \textbf{15}
  (1994), no.~3-4, 327--342.

\bibitem{fereday2008demonstrating}
Jennifer Fereday and Eimear Muir-Cochrane, \emph{Demonstrating rigor using
  thematic analysis: A hybrid approach of inductive and deductive coding and
  theme development}, International journal of qualitative methods \textbf{5}
  (2008), no.~1, 80--92.

\bibitem{fischbein1987intuition}
H~Fischbein, \emph{Intuition in science and mathematics: An educational
  approach}, vol.~5, Springer Science \& Business Media, 1987.

\bibitem{Franklin}
J.~Franklin and A.~Daoud, \emph{Proof in mathematics: An introduction},
  \url{web.maths.unsw.edu.au/∼jim/proofsch2.pdf}, 2011.

\bibitem{fried2014single}
Itzhak Fried, Ueli Rutishauser, Moran Cerf, and Gabriel Kreiman, \emph{Single
  neuron studies of the human brain: Probing cognition}, MIT Press, 2014.

\bibitem{gibson1998students}
DAVID Gibson, \emph{Students’ use of diagrams to develop proofs in an
  introductory analysis course}, CBMS issues in mathematics education
  \textbf{7} (1998), 284--307.

\bibitem{glaser2008conceptualization}
Barney~G Glaser, \emph{Conceptualization: On theory and theorizing using
  grounded theory}, International Journal of Qualitative Methods \textbf{1}
  (2008), no.~2, 23--38.

\bibitem{glaser1970theoretical}
Barney~G Glaser and Anselm~L Strauss, \emph{Theoretical sampling}, Sociological
  methods. A sourcebook (1970), 105--114.

\bibitem{glaser1968discovery}
Barney~G Glaser, Anselm~L Strauss, and Elizabeth Strutzel, \emph{The discovery
  of grounded theory; strategies for qualitative research.}, Nursing Research
  \textbf{17} (1968), no.~4, 364.

\bibitem{gray1994duality}
Eddie~M Gray and David~O Tall, \emph{Duality, ambiguity, and flexibility: A
  "proceptual" view of simple arithmetic}, Journal for research in Mathematics
  Education (1994), 116--140.

\bibitem{hammack2013book}
Richard~H Hammack, \emph{Book of proof}, Virginia Commonwealth University,
  2013.

\bibitem{harel1998students}
Guershon Harel and Larry Sowder, \emph{Students’ proof schemes: Results from
  exploratory studies}, Research in collegiate mathematics education III
  \textbf{7} (1998), 234--282.

\bibitem{harel1991constructionism}
Idit~Ed Harel and Seymour~Ed Papert, \emph{Constructionism.}, Ablex Publishing,
  1991.

\bibitem{hasselmo1999neuromodulation}
Michael~E Hasselmo, \emph{Neuromodulation: acetylcholine and memory
  consolidation}, Trends in cognitive sciences \textbf{3} (1999), no.~9,
  351--359.

\bibitem{hazzan2003students}
Orit Hazzan, \emph{How students attempt to reduce abstraction in the learning
  of mathematics and in the learning of computer science}, Computer Science
  Education \textbf{13} (2003), no.~2, 95--122.

\bibitem{hiebert2013conceptual}
James Hiebert, \emph{Conceptual and procedural knowledge: The case of
  mathematics}, Routledge, 2013.

\bibitem{huang2015highest}
Li~Huang, Francesca Gino, and Adam~D Galinsky, \emph{The highest form of
  intelligence: Sarcasm increases creativity for both expressers and
  recipients}, Organizational Behavior and Human Decision Processes (2015).

\bibitem{huth2004logic}
Michael Huth and Mark Ryan, \emph{Logic in computer science: Modelling and
  reasoning about systems}, Cambridge University Press, 2004.

\bibitem{huttel2010transitions}
Hans H{\"u}ttel, \emph{Transitions and trees: An introduction to structural
  operational semantics}, Cambridge University Press, 2010.

\bibitem{kaput1998research}
James~J Kaput, Alan~H Schoenfeld, and Ed~Dubinsky, \emph{Research in collegiate
  mathematics education iii}, vol.~4, American Mathematical Soc., 1998.

\bibitem{kendall1999axial}
Judy Kendall, \emph{Axial coding and the grounded theory controversy}, Western
  journal of nursing research \textbf{21} (1999), no.~6, 743--757.

\bibitem{krefting1991rigor}
Laura Krefting, \emph{Rigor in qualitative research: The assessment of
  trustworthiness}, American journal of occupational therapy \textbf{45}
  (1991), no.~3, 214--222.

\bibitem{lehrer2000inter}
Richard Lehrer, Leona Schauble, Susan Carpenter, and David Penner, \emph{The
  inter-related development of inscriptions and conceptual understanding},
  Symbolizing and communicating in mathematics classrooms: Perspectives on
  discourse, tools, and instructional design (2000), 325--360.

\bibitem{lerman2012articulating}
Stephen Lerman, \emph{Articulating theories of mathematics learning}, Paul
  Ernest, Studies in Mathematics Education Series \textbf{4} (2012), 41--49.

\bibitem{leron1983structuring}
Uri Leron, \emph{Structuring mathematical proofs}, American Mathematical
  Monthly (1983), 174--185.

\bibitem{leron1995abstract}
Uri Leron and Ed~Dubinsky, \emph{An abstract algebra story}, American
  Mathematical Monthly (1995), 227--242.

\bibitem{lincoln1985naturalistic}
Yvonna~S Lincoln and Egon~G Guba, \emph{Naturalistic inquiry}, vol.~75, Sage,
  1985.

\bibitem{lincoln1986but}
\bysame, \emph{But is it rigorous? trustworthiness and authenticity in
  naturalistic evaluation}, New directions for program evaluation \textbf{1986}
  (1986), no.~30, 73--84.

\bibitem{loewenberg2003mathematical}
Deborah Loewenberg et~al., \emph{Mathematical proficiency for all students:
  Toward a strategic research and development program in mathematics
  education}, Rand Corporation, 2003.

\bibitem{marton1981phenomenography}
Ference Marton, \emph{Phenomenography—describing conceptions of the world
  around us}, Instructional science \textbf{10} (1981), no.~2, 177--200.

\bibitem{marton1997learning}
Ference Marton and Shirley~A Booth, \emph{Learning and awareness}, Routledge,
  1997.

\bibitem{marton2006some}
Ference Marton and Ming~Fai Pang, \emph{On some necessary conditions of
  learning}, The Journal of the Learning sciences \textbf{15} (2006), no.~2,
  193--220.

\bibitem{marton2013meanings}
\bysame, \emph{Meanings are acquired from experiencing differences against a
  background of sameness, rather than from experiencing sameness against a
  background of difference: Putting a conjecture to the test by embedding it in
  a pedagogical tool}, Frontline Learning Research \textbf{1} (2013), no.~1,
  24--41.

\bibitem{marton2005unit}
Ference Marton and Wing~Yan Pong, \emph{On the unit of description in
  phenomenography}, Higher Education Rresearch \& Development \textbf{24}
  (2005), no.~4, 335--348.

\bibitem{marton1976qualitative}
Ference Marton and Roger S{\"a}lj{\"o}, \emph{On qualitative differences in
  learning: I—outcome and process*}, British journal of educational
  psychology \textbf{46} (1976), no.~1, 4--11.

\bibitem{marton1997discontinuities}
Ference Marton, David Watkins, and Catherine Tang, \emph{Discontinuities and
  continuities in the experience of learning: An interview study of high-school
  students in hong kong}, Learning and instruction \textbf{7} (1997), no.~1,
  21--48.

\bibitem{mason1989mathematical}
John Mason, \emph{Mathematical abstraction as the result of a delicate shift of
  attention}, For the learning of mathematics (1989), 2--8.

\bibitem{Mattuck}
Arthur Mattuck, \emph{Introduction to analysis}, Prentice Hall, 1999.

\bibitem{mcgowen2010metaphor}
Mercedes~A McGowen and David~O Tall, \emph{Metaphor or met-before? the effects
  of previouos experience on practice and theory of learning mathematics}, The
  Journal of Mathematical Behavior \textbf{29} (2010), no.~3, 169--179.

\bibitem{merriam1995can}
S~Merriam, \emph{What can you tell from an n ofl?: Issues of validity and
  reliability in qualitative research}, PAACE Journal of Lifelong Learning
  \textbf{4} (1995), 50--60.

\bibitem{Merriam2002}
Sharan~B. Merriam, \emph{Qualitative research in practice}, Jossey-Bass, 2002.

\bibitem{merriam2009qualitative}
Sharan~B Merriam, \emph{Qualitative research: A guide to design and
  implementation}, John Wiley \& Sons, 2009.

\bibitem{Merriam2009}
Sharan~B. Merriam, \emph{Qualitative research a guide to design and
  implementation}, Jossey-Bass, 2009.

\bibitem{moore1994making}
Robert~C Moore, \emph{Making the transition to formal proof}, Educational
  Studies in Mathematics \textbf{27} (1994), no.~3, 249--266.

\bibitem{patton1990qualitative}
Michael~Quinn Patton, \emph{Qualitative evaluation and research methods .},
  SAGE Publications, inc, 1990.

\bibitem{piaget1952origins}
Jean Piaget, Margaret Cook, and WW~Norton, \emph{The origins of intelligence in
  children}, vol.~8, International Universities Press New York, 1952.

\bibitem{polya1954mathematics}
George Polya, \emph{Mathematics and plausible reasoning: vol 1: Induction and
  analogy in mathematics.}, Oxford University Press, 1954.

\bibitem{puchner2014using}
Laurel Puchner and Linda Markowitz, \emph{Using an adult development model to
  help explain pre-service teacher resistance to learning about race}, Race
  Ethnicity and Education (2014), 1--20.

\bibitem{quiroga2015concept}
Rodrigo~Quian Quiroga, \emph{Concept cells in the human brain}, Advances in
  Cognitive Neurodynamics (IV), Springer, 2015, pp.~143--146.

\bibitem{rasslan2002definitions}
Shaker Rasslan and David Tall, \emph{Definitions and images for the definite
  integral concept}, PME CONFERENCE, vol.~4, 2002, pp.~4--089.

\bibitem{reddy2015learning}
Leila Reddy, Marlene Poncet, Matthew~W Self, Judith~C Peters, Linda Douw, Edwin
  van Dellen, Steven Claus, Jaap~C Reijneveld, Johannes~C Baayen, and Pieter~R
  Roelfsema, \emph{Learning of anticipatory responses in single neurons of the
  human medial temporal lobe}, Nature communications \textbf{6} (2015).

\bibitem{richards2014handling}
Lyn Richards, \emph{Handling qualitative data: A practical guide}, Sage, 2014.

\bibitem{rosen2003}
Kenneth~H Rosen, \emph{Discrete mathematics and its applications, 5th ed.},
  McGraw Hill, 2003.

\bibitem{rosken2007integrating}
Bettina R{\"o}sken and Katrin Rolka, \emph{Integrating intuition: The role of
  concept image and concept definition for students' learning of integral
  calculus}, The Montana Mathematics Enthusiast \textbf{3} (2007), 181--204.

\bibitem{rota1997phenomenology}
Gian-Carlo Rota, \emph{The phenomenology of mathematical beauty}, Synthese
  \textbf{111} (1997), no.~2, 171--182.

\bibitem{runesson2005beyond}
Ulla Runesson, \emph{Beyond discourse and interaction. variation: a critical
  aspect for teaching and learning mathematics}, Cambridge journal of education
  \textbf{35} (2005), no.~1, 69--87.

\bibitem{rutten2012learning}
Nico Rutten, Wouter~R van Joolingen, and Jan~T van~der Veen, \emph{The learning
  effects of computer simulations in science education}, Computers \& Education
  \textbf{58} (2012), no.~1, 136--153.

\bibitem{saljo1979learning}
Roger S{\"a}lj{\"o}, \emph{Learning in the learner's perspective. {I}. some
  common-sense conceptions}, ERIC Clearinghouse, 1979.

\bibitem{schwandt2007judging}
Thomas~A Schwandt, Yvonna~S Lincoln, and Egon~G Guba, \emph{Judging
  interpretations: But is it rigorous? trustworthiness and authenticity in
  naturalistic evaluation}, New directions for evaluation \textbf{2007} (2007),
  no.~114, 11--25.

\bibitem{sillitoapp}
Jonathan Sillito, \emph{Saturate application}.

\bibitem{sipser2012introduction}
Michael Sipser, \emph{Introduction to the theory of computation}, Cengage
  Learning, 2012.

\bibitem{sipser}
Michael Sipser, \emph{Introduction to the theory of computation}, Cengage,
  2013.

\bibitem{smedslund1970circular}
Jan Smedslund, \emph{Circular relation between understanding and logic},
  Scandinavian Journal of Psychology \textbf{11} (1970), no.~1, 217--219.

\bibitem{stickgold2013sleep}
Robert Stickgold and Matthew~P Walker, \emph{Sleep-dependent memory triage:
  evolving generalization through selective processing}, Nature Neuroscience
  \textbf{16} (2013), no.~2, 139--145.

\bibitem{strauss1990basics}
Anselm~Leonard Strauss, Juliet~M Corbin, et~al., \emph{Basics of qualitative
  research}, vol.~15, Sage Newbury Park, CA, 1990.

\bibitem{svensson1997theoretical}
Lennart Svensson, \emph{Theoretical foundations of phenomenography}, Higher
  Education Research \& Development \textbf{16} (1997), no.~2, 159--171.

\bibitem{tall2012cognitive}
D.~Tall, O.~Yevdokimov, B.~Koichu, W.~Whiteley, M.~Kondratieva, and Y.H. Cheng,
  \emph{Cognitive development of proof}, Proof and Proving in Mathematics
  Education (2012), 13--49.

\bibitem{thompson1994students}
Patrick~W Thompson, \emph{Students, functions, and the undergraduate
  curriculum}, Research in collegiate mathematics education \textbf{1} (1994),
  21--44.

\bibitem{thota2014programming}
Neena Thota, \emph{Programming course design: Phenomenographic approach to
  learning and teaching}, Teaching and Learning in Computing and Engineering
  (LaTiCE), 2014 International Conference on, IEEE, 2014, pp.~125--132.

\bibitem{tversky1981framing}
Amos Tversky and Daniel Kahneman, \emph{The framing of decisions and the
  psychology of choice}, Science \textbf{211} (1981), no.~4481, 453--458.

\bibitem{valiant2000circuits}
Leslie~G Valiant, \emph{Circuits of the mind}, Oxford University Press, 2000.

\bibitem{van1992jasper}
JP~Van~Haneghan, Linda Barron, MF~Young, SM~Williams, NJ~Vye, and JD~Bransford,
  \emph{The jasper series: An experiment with new ways to enhance mathematical
  thinking}, Enhancing thinking skills in the sciences and mathematics (1992),
  15--38.

\bibitem{van2011slow}
EJ~Van~Someren, \emph{Slow brain oscillations of sleep, resting state and
  vigilance}, vol. 193, Elsevier, 2011.

\bibitem{velleman2006prove}
Daniel~J Velleman, \emph{How to prove it: a structured approach}, Cambridge
  University Press, 2006.

\bibitem{viete2006analytic}
Fran{\c{c}}ois Vi{\`e}te and T~Richard Witmer, \emph{The analytic art}, Courier
  Corporation, 2006.

\bibitem{von1987learning}
Ernst Von~Glasersfeld, \emph{Learning as a constructive activity}, In C.
  Janvier (Ed.), Problems of representation in the, Citeseer, 1987.

\bibitem{von2013radical}
\bysame, \emph{Radical constructivism}, Routledge, 2013.

\bibitem{vygotsky1987zone}
L.~Vygotsky, \emph{Zone of proximal development}, Mind in society: The
  development of higher psychological processes (1987), 52--91.

\bibitem{vygotsky1978mind}
Lev~S Vygotsky, \emph{Mind in society (m. cole, v. john-steiner, s. scribner,
  \& e. souberman, eds.)}, Cambridge, MA: Harvard UniversityPress. VygotskyMind
  in society1978 (1978).

\bibitem{vygotsky1962language}
Lev~Semenovich Vygotsky, \emph{Language and thought}, Massachusetts Institute
  of Technology Press, Ontario, Canada (1962).

\bibitem{wenger1999communities}
Etienne Wenger, \emph{Communities of practice: Learning, meaning, and
  identity}, Cambridge university press, 1999.

\bibitem{whitehead1959aims}
Alfred~North Whitehead, \emph{The aims of education}, Daedalus \textbf{88}
  (1959), no.~1, 192--205.

\bibitem{wittgenstein1989wittgenstein}
Ludwig Wittgenstein, RG~Bosanquet, N~Malcolm, R~Hrees, Y~Symthies, and Cora
  Diamond, \emph{Wittgenstein's lectures on the foundations of mathematics,
  cambridge, 1939}, Cornell University Press, 1976.

\bibitem{wolcott1994transforming}
Harry~F Wolcott, \emph{Transforming qualitative data: Description, analysis,
  and interpretation}, Sage, 1994.

\end{thebibliography}