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- How Humans Learn to Think MathematicallyExploring the Three Worlds of Mathematics, pp. 433 - 446Publisher: Cambridge University PressPrint publication year: 2013
References
, & (2002). The simplicity, complexity and complication of the function concept. In & (Eds.), Proceedings of the 26th Conference of the International Group for the Psychology of Mathematics Education, 2, 25–32. Norwich, UK.Google Scholar
, & (2004). Convergence of sequences and series: Interactions between visual reasoning and the learner’s beliefs about their own role. Educational Studies in Mathematics, 57, 1–32.CrossRefGoogle Scholar
, & (2005). Convergence of sequences and series 2: Interactions between non-visual reasoning and the learner’s beliefs about their own role. Educational Studies in Mathematics, 58, 77–110.CrossRefGoogle Scholar
, & (2004). Semantic and syntactic proof productions. Educational Studies in Mathematics, 56, 209–34.Google Scholar
, & (1989). The development of an abbreviated version of the Mathematics Anxiety Rating Scale. Measurement and Evaluation in Counseling and Development, 22, 143–50.CrossRefGoogle Scholar
(1806). Essai sur une manière de représenter les quantités imaginaires dans les constructions géométriques. 1st ed., Paris. 2nd ed. reprinted, Paris: Albert Blanchard, 1971.Google Scholar
, & (2001). The relationships among working memory, math anxiety, and performance. Journal of Experimental Psychology, 130(2), 224–37.CrossRefGoogle Scholar
, , , , , & (1996). A framework for research and curriculum development in undergraduate mathematics education. Research in Collegiate Mathematics Education II, CBMS Issues in Mathematics Education, 6, 1–32.Google Scholar
, , , , & (1997). Effective Teachers of Numeracy, Final Report of a Study Carried Out for the Teacher Training Agency 1995–96 by the School of Education, King’s College, London.Google Scholar
, , & (1978). Educational Psychology: A Cognitive View (2nd ed.). New York: Holt, Rinehart & Winston.Google Scholar
, & (1992). Students’ mental prototypes for functions and graphs. International Journal of Mathematics Education in Science & Technology, 23(1), 39–50.Google Scholar
, & (1998). Fostering Children’s Mathematical power: An Investigative Approach to K–8 Mathematics Instruction. Mahwah, NJ: Lawrence Erlbaum.Google Scholar
(1670). Lectiones Geometricae, translated by Child (1916). The Geometrical Lectures of Isaac Barrow. Chicago and London: OpenCourt.Google Scholar
(2006). The Relationship between Teaching and Learning the Function Concept. PhD thesis, University of Warwick.Google Scholar
(1734). The Analyst (ed. D. R. Wilkins, 2002). Retrieved from (Accessed April 21, 2012).Google Scholar
, & (1966). Solution of an invariant subspace problem of K. T. Smith and P. R. Halmos. Pacific Journal of Mathematics, 16(3), 421–31.CrossRefGoogle Scholar
, & (1966). Mathematical Epistemology and Psychology, trans. by . Dordrecht, The Netherlands: Reidel.Google Scholar
(1978). Prevalence, distribution, and correlates of math anxiety in college students. Journal of Counseling Psychology, 25(5), 441–8.CrossRefGoogle Scholar
, & (1982). Evaluating the Quality of Learning: The SOLO Taxonomy. New York: Academic Press.Google Scholar
, & (1999). Algebra (3rd ed.). Providence RI: Chelsea Publishing Co., American Mathematical Society.Google Scholar
, , & (1994). A comparison of math anxiety in traditional and nontraditional developmental college students. Research and Teaching in Developmental Education, 10(2), 35–43.Google Scholar
, & (2000). Graphic Calculus for Windows. Retrieved from: (Accessed February 19, 2013).Google Scholar
(1923/1939). The History of the Calculus and Its Conceptual Development. Reprinted by Dover, New York.Google Scholar
, , , & (1992). Development of the process conception of function. Educational Studies in Mathematics, 23, 247–85.CrossRefGoogle Scholar
P. (1992). Numbers and Functions: Steps into Analysis. Cambridge: Cambridge University Press.Google Scholar
(2002). Recognising commonalities and reconciling differences in mathematics education. Educational Studies in Mathematics, 50(2), 157–75.CrossRefGoogle Scholar
, & (1997). Gender issues in the classroom: A comparison of mathematics anxiety. Education, 117(3), 332–9.Google Scholar
(1872). Uber die Ausdehnung eines Satzes aus der Theorie der trigonometrischen Reihen. Mathematische Annalen, 5, 123–32. Reproduced in G. Cantor, Gesammelte Abhandlungen mathematischen und philosophischen Inhalts, ed. E. Zermelo. Berlin: J. Springer, 1932, pp. 92–102. Reprinted Hildesheim: Olms.CrossRefGoogle Scholar
(1545). Ars magna. Translated and published as Ars Magna or The Rules of Algebra (1993). New York: Dover.Google Scholar
(1927–1929). The Rhind Mathematical Papyrus: Free Translation and Commentary with Selected Photographs, Translations, Transliterations and Literal Translations. Classics in Mathematics Education 8. 2 vols. Oberlin: Mathematical Association of America. (Reprinted Reston: National Council of Teachers of Mathematics, 1979).Google Scholar
(2009). From Triangles to a Concept: A Phenomenographic Study of A–level Students’ Development of the Concept of Trigonometry. PhD thesis, University of Warwick.Google Scholar
(2002). Building and Using Concepts of Equivalence Class and Partition. PhD thesis, University of Warwick.Google Scholar
, & (2001). Developing formal mathematical concepts over time. In M. van den Heuvel-Panhuizen (Ed.), Proceedings of the 25th Conference of the International Group for the Psychology of Mathematics Education, 2, 241–8. Norwich, UK.Google Scholar
, & (2002). University students embodiment of quantifier. In & (Eds.), Proceedings of the 26th Conference of the International Group for the Psychology of Mathematics Education, 4, 273–80. Norwich, UK.Google Scholar
, & (2012). Making sense of mathematics through perception, operation and reason: The case of trigonometric functions. Proceedings of the 36th Conference of the International Group for the Psychology of Mathematics Education, 4, 264. Full paper (Accessed February 18, 2013).Google Scholar
, , & (1981). Translation difficulties in learning mathematics. American Mathematics Monthly, 4, 286–90.CrossRefGoogle Scholar
, & (1992). Geometry and spatial reasoning. In (Ed.) Handbook of Research on Teaching and Learning Mathematics (pp. 420–64). New York: Macmillan.Google Scholar
(1978). Operational thinking in elementary mathematics. In , , & (Eds.), Cognitive Development: Research Based on a Neo-Piagetian approach. New York: John Wiley & Sons.Google Scholar
(1981). Apprentissage de la notion de limite: Modèles spontanées et modèles propres, Actes du Cinquième Colloque du Groupe Internationale PME, Grenoble, France, 322–6.Google Scholar
(1991). Limits. In (Ed.), Advanced Mathematical Thinking (pp. 153–66). Dordrecht, The Netherlands: Kluwer.Google Scholar
, , , , , & (1996). Understanding the limit concept: Beginning with a coordinated process scheme. Journal of Mathematical Behavior, 15(2), 167–92.CrossRefGoogle Scholar
(1937). Differential and Integral Calculus. Vol. I. Translated from the German by . Reprint of the second edition (1988). Wiley Classics Library. New York: Wiley-Interscience.Google Scholar
(1984). Learning Mathematics: The Cognitive Science Approach to Mathematics Education. Norwood, NJ: Ablex.Google Scholar
(1997). The Symbolic Species: The Co-evolution of Language and the Human Brain. London: Penguin.Google Scholar
(1872). Stetigkeit und irrationale Zahlen. Braunschweig: Vieweg. Reproduced in R. Dedekind, Gesammelte mathematische Werke, eds. , , & . Braunschweig: Vieweg, 1930–1932.Google Scholar
(1998). Facets and Layers of the Function Concept: The Case of College Algebra. PhD thesis, University of Warwick.Google Scholar
(1831). On the Study and Difficulties of Mathematics. London: Society for the Diffusion of Useful Knowledge.Google Scholar
(1641). Meditations on First Philosophy In The Philosophical Writings of René Descartes, trans. by , , & , Cambridge: Cambridge University Press, 1984.Google Scholar
, & (1993). Natural, conflicting and alien. Journal of Mathematical Behaviour, 12(4), 313–28.Google Scholar
(1998). Social Constructivism as a Philosophy of Mathematics. Albany, NY: State University of New York Press.Google Scholar
, & (2002). The Way We Think: Conceptual Blending and the Mind’s Hidden Complexities. New York: Basic Books.Google Scholar
(1985). Surely You’re joking Mr Feynman. New York: W. W. Norton. Reprinted 1992, London: Vintage.Google Scholar
, & (1989). Solving equations: The transition from arithmetic to algebra. For the Learning of Mathematics, 9(2), 19–25.Google Scholar
(1987). Intuition in Science and Mathematics: An Educational Approach. Dordrecht, The Netherlands: Kluwer.Google Scholar
(2001). Children’s Use of Apparatus in the Development of the Concept of Number. PhD thesis, University of Warwick.Google Scholar
(1983). Didactic Phenomenology of Mathematical Structures. Dordrecht, The Netherlands: Reidel.Google Scholar
, & (2003). Math anxiety: Overcoming a major obstacle to the improvement of student math performance. Childhood Education, Spring, 170–4.Google Scholar
(1831). Theory of biquadratic residues, part 2. lecture presented to the Royal Society, Gottingen, April 23, 1831.Google Scholar
(1931). Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, Monatshefte für Mathematik und Physik. Vol. 38. Available in English at (Accessed February 19, 2013).Google Scholar
(1993). Qualitatively Different Approaches to Simple Arithmetic. PhD thesis, University of Warwick.Google Scholar
, , , & (1999). Knowledge construction and diverging thinking in elementary and advanced mathematics. Educational Studies in Mathematics, 38(1–3), 111–33.CrossRefGoogle Scholar
, & (1991). Duality, ambiguity & flexibility in successful mathematical thinking In Proceedings of the 15th Conference for the International Group for the Psychology of Mathematics Education, 2, 72–9, Assisi, Italy.Google Scholar
, & (1994). Duality, ambiguity and flexibility: A proceptual view of simple arithmetic. Journal for Research in Mathematics Education, 26(2), 115–41.Google Scholar
, & (2001). Relationships between embodied objects and symbolic procepts: An explanatory theory of success and failure in mathematics. In (Ed.), Proceedings of the 25th Conference of the International Group for the Psychology of Mathematics Education, 3, 65–72. Utrecht, The Netherlands.Google Scholar
, , & (1991). An alternative paradigm to evaluate the acquisition of the Van Hiele levels. Journal for Research in Mathematics Education, 22(3), 237–51.CrossRefGoogle Scholar
(1945). The Psychology of Invention in the Mathematical Field. Princeton, NJ: Princeton University Press. Dover edition, New York, 1954.Google Scholar
(1966). Invariant subspaces for polynomially compact operators. Pacific Journal of Mathematics, 16(3), 433–7.CrossRefGoogle Scholar
, , , , & (1989). Children’s Mathematical Frameworks 8–13: A Study of Classroom Teaching. London: Routledge (formerly NFER Nelson).Google Scholar
(1921). History of Greek Mathematics. Vol. 1. Oxford: Oxford University Press. Reprinted Dover Publications, New York, 1963.Google Scholar
(1990). The nature, effects, and relief of mathematics anxiety. Journal for Research in Mathematics Education, 21(1), 33–46.CrossRefGoogle Scholar
, , , , & (2007). Humans have evolved specialized skills of social cognition: The cultural intelligence hypothesis. Science, September 7, 2007, 317, 1360–6. Retrieved from: (Accessed April 6, 2012).CrossRefGoogle ScholarPubMed
(1998). Realistic Mathematics Education. Work in progress, Text based on the NORMA-lecture held in Kristiansand, Norway on June 5–9, 1998, Freudenthal Institute. Retrieved frpm: (Accessed July 13, 2012).Google Scholar
, & (1986). Procedural and conceptual knowledge. In (Ed.), Conceptual and Procedural Knowledge: The Case of Mathematics (pp. 1–27). Hillsdale, NJ: Lawrence Erlbaum.Google Scholar
(1994). Profile: Andre Weill, great French-born mathematician, Scientific American, 270 (6), June 1994, 33–34.CrossRefGoogle Scholar
(2006). Participation in Elementary Mathematics: An Analysis of Engagement, Attainment and Intervention. PhD thesis, University of Warwick.Google Scholar
(1982). A History of Mathematics Education in England. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
(1991). Visualization and Calculus Reform. In W. Zimmermann & S. Cunningham (eds.), Visualization in Teaching and Learning Mathematics, MAA Notes No. 19, 121–126.Google Scholar
, , & (2007). Modelling mathematical argumentation: The importance of qualification. Educational Studies in Mathematics, 66(7), 3–21.CrossRefGoogle Scholar
, & (2007). Working memory of numerals in chimpanzees. Current Biology, 17(23), R1004–R1005.CrossRefGoogle ScholarPubMed
, & (1999). The role of instructors in creating math anxiety in students from kindergarten through college. Mathematics Teacher, 92(7), 583–7.Google Scholar
(Ed.) (1989): Children’s Mathematical Frameworks 8–13: A Study of Classroom Teaching. Windsor, UK: NFER-Nelson.Google Scholar
(2001). Applying psychology to the teaching of basic math: A case study. Inquiry, 6(2), 60–5.Google Scholar
(1998). Euclid’s Elements. Retrieved from on 26th March 2012.Google Scholar
, & (2012). The tension between intuitive infinitesimals and formal analysis. In Bharath Sriraman (Ed.), Crossroads in the History of Mathematics and Mathematics Education (pp. 71–90). The Montana Mathematics Enthusiast Monographs in Mathematics Education 12. Charlotte, NC: Information Age Publishing.Google Scholar
(2008). On considerations of parsimony in mathematical problem solving. In , , , , & (Eds.), Proceedings of the 32nd Conference of the International Group for the Psychology of Mathematics Education. Vol. 3 (pp. 273–80), Morelia, Mexico.Google Scholar
, & (2005). When do gifted high school students use geometry to solve geometry problems?
The Journal of Secondary Gifted Education, 16(4), 168–79.CrossRefGoogle Scholar
(1976). The Psychology of Mathematical Abilities in Schoolchildren. Chicago: University of Chicago Press.Google Scholar
(1962). The Structure of Scientific Revolutions. Chicago: University of Chicago Press.Google Scholar
(1987). Women, Fire, and Dangerous Things: What Categories Reveal About the Mind. Chicago: University of Chicago Press.CrossRefGoogle Scholar
, & (1999). Philosophy in the Flesh: The Embodied Mind and Its Challenge to Western Thought. New York: Basic Books.Google Scholar
& (2000). Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being. New York: Basic Books.Google Scholar
, & (1991). Situated Learning: Legitimate Peripheral Participation. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
, & (1981). Spatial ability, visual imagery, and mathematical performance. Educational Studies in Mathematics, 12(3), 267–99.CrossRefGoogle Scholar
(1920). The Early Mathematical Manuscripts of Leibniz, ed. and trans. by . Chicago: University of Chicago Press.Google Scholar
, & (1993). Constructing different concept images of sequences and limits by programming. In
Proceedings of PME
17, Japan, 2, 41–8.Google Scholar
, & (2006). The concept of equation: What have students met before? In Proceedings of the 30th Conference of the International Group for the Psychology of Mathematics Education, Prague, Czech Republic, 4, 233–41.Google Scholar
, & (2008). Procedural embodiment and magic in linear equations. Educational Studies in Mathematics, 67(1), 3–18.CrossRefGoogle Scholar
(1999a). A meta-analysis of the relationship between anxiety toward mathematics and achievement in mathematics. Journal for Research in Mathematics Education, 30(5), 520–40.CrossRefGoogle Scholar
(1994). Responses to theoretical mathematics. Bulletin (new series) of the American Mathematical Society, 30(2), 190–1.Google Scholar
(1989). Mathematical abstraction as the result of a delicate shift of attention. For the Learning of Mathematics, 9(2), 2–8.Google Scholar
(2002). Researching Your Own Practice: The Discipline of Noticing. London: Routledge Falmer.CrossRefGoogle Scholar
(1998). Cognitive Units, Concept Images, and Cognitive Collages: An Examination of the Process of Knowledge Construction. PhD thesis, University of Warwick.Google Scholar
, & (2013). Flexible Thinking and Met-befores: Impact on Learning Mathematics, with Particular Reference to the Minus Sign. Retrieved from: (Accessed February 19, 2013).Google Scholar
(2006). Teachers’ Indications and Pupils’ Construal and Knowledge of Fractions: The Case of Malaysia. PhD thesis, University of Warwick.Google Scholar
(2008). The Construction and Evaluation of Arguments in Undergraduate Mathematics. PhD thesis, University of Warwick.Google Scholar
(1956). The magic number seven plus or minus two: Some limits on our capacity for processing information. Psychological Review, 63, 81–97.CrossRefGoogle ScholarPubMed
(1986). Adolescent’s Understanding of Limits and Infinity. PhD thesis, University of Warwick.Google Scholar
. (1989). Curriculum and Evaluation Standards for School Mathematics. Reston, VA: National Council of Teachers of Mathematics.Google Scholar
, , & (1999). Embodied cognition as grounding for situatedness and context in mathematics education. Educational Studies in Mathematics, 39(1–3), 45–65.CrossRefGoogle Scholar
(2005). Mathematical problem solving and learning mathematics: What we expect students to obtain. Journal of Mathematical Behavior, 24, 325–40.CrossRefGoogle Scholar
, & (2005). The fundamental cycle of concept construction underlying various theoretical frameworks. International Reviews on Mathematical Education (ZDM), 37(6), 468–75.Google Scholar
(1991). Peirce on Signs: Writings on Semiotic (ed. Hoopes, J.). University of North Carolina Press.Google Scholar
(1998). Students’ Understanding of Real Analysis. PhD thesis, University of Warwick.Google Scholar
, & (1999). Student constructions of formal theory: Giving and extracting meaning. In (Ed.), Proceedings of the 23rd Conference of PME, Haifa, Israel, 4, 65–73.Google Scholar
, & (2001). Following students’ development in a traditional university classroom. In (Ed.), Proceedings of the 25th Conference of the International Group for the Psychology of Mathematics Education 4, 57–64. Utrecht, The Netherlands.Google Scholar
, & (2002). Building formal mathematics on visual imagery: A theory and a case study. For the Learning of Mathematics, 22(1), 2–10.Google Scholar
(1998). Beyond the Obvious: Mental Representations and Elementary Arithmetic. PhD thesis, University of Warwick.Google Scholar
, & (1997). In the mind: What can imagery tell us about success and failure in arithmetic? In (Ed.), Proceedings of the First Mediterranean Conference on Mathematics, Nicosia: Cyprus, 29–41.Google Scholar
, & (1999). Changing Emily’s images. In (Ed.), Teaching, Learning and Primary Mathematics (pp. 56–60). Derby, UK: Association of Teachers of Mathematics.Google Scholar
, & (1982). The development and validation of a revised version of the Mathematics Anxiety Rating Scale. Educational and Psychological Measurement, 42(2), 551–7.CrossRefGoogle Scholar
(360 BC). The Republic. Book VII, trans. by (1871). New York : Scribner’s Sons. Reprinted 1941, New York: The Modern Library.Google Scholar
(1860). Elements of geometry; containing the first six books of Euclid, with two books on the geometry of solids. To which are added, elements of plane and spherical trigonometry, Philadelphia: J. B. Lippincott & Co.Google Scholar
(1945). How to Solve It. Princeton, NJ: Princeton University Press. Reprinted 1957, Garden City, NY: Doubleday.CrossRefGoogle Scholar
(2004). Effect as a Pivot between Actions and Symbols: The case of Vector. PhD thesis, University of Warwick.Google Scholar
(1986). Visualisation and mathematical giftedness. Educational Studies in Mathematics, 17(3), 297–311.CrossRefGoogle Scholar
, & (1972). The Mathematics Anxiety Rating Scale: Psychometric data. Journal of Counseling Psychology, 19(6), 551–4.CrossRefGoogle Scholar
(2000). On mathematical warrants. Mathematical Thinking and Learning, 2 (3), 221–44.CrossRefGoogle Scholar
, , , , & (1976). Basic objects in natural categories. Cognitive Psychology, 8, 382–439.CrossRefGoogle Scholar
(1981). Some misconceptions concerning the concept of variable. Are you careful about defining your variables?
Mathematics Teacher, 74(6), 418–20, 450.Google Scholar
(2004). Assessing mathematics automatically using computer algebra and the internet. Teaching Mathematics and Its Applications, 23(1), 1–14.CrossRefGoogle Scholar
, & (1978). Conflicts in the learning of real numbers and limits. Mathematics Teaching, 82, 44–9.Google Scholar
(1991). On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin. Educational Studies in Mathematics, 22, 1–36.CrossRefGoogle Scholar
(1992). Operational origins of mathematical objects and the quandary of reification – the case of function. In & (Eds.), The Concept of Function: Aspects of Epistemology and Pedagogy, MAA Notes 25 (pp. 59–84). Washington, DC: Mathematical Association of America.Google Scholar
, & (2006). How does anxiety influence maths performance and what can we do about it?
MSOR Connections, 6(4), 19–23.CrossRefGoogle Scholar
(1976). Relational understanding and instrumental understanding. Mathematics Teaching, 77, 20–6.Google Scholar
(1979). The three crises in mathematics: Logicism, intuitionism and formalism. Mathematics Magazine, 52(4), 207–16.CrossRefGoogle Scholar
, & (2000). Algebraic Number Theory and Fermat’s Last Theorem (3rd ed.). Natick, MA: A. K. Peters.Google Scholar
, & (1990). Basics of Qualitative Research: Grounded Theory Procedures and Techniques. London: SAGE.Google Scholar
(1972). Uniform continuity and rates of growth of meromorphic functions. In & (Eds.), Contributions to Non-Standard Analysis (pp. 47–64). Amsterdam: North-Holland.Google Scholar
(1969). A Source Book in Mathematics, 1200–1800. Cambridge, MA: Harvard University Press.Google Scholar
(1976). The teaching of elementary calculus: An approach using infinitesimals, American Mathematical Monthly, 83, 370–5.CrossRefGoogle Scholar
(1977). Cognitive conflict in the learning of mathematics. Presented at the first meeting of the International Group for the Psychology of Learning Mathematics, Utrecht, The Netherlands. Retrieved from: (Accessed February 19, 2013).Google Scholar
(1979). Cognitive aspects of proof, with special reference to the irrationality of √2. In Proceedings of the Third International Conference for the Psychology of Mathematics Education, Warwick, 206–7.Google Scholar
(1980a). Looking at graphs through infinitesimal microscopes, windows and telescopes. Mathematical Gazette, 64, 22–49.CrossRefGoogle Scholar
(1980b). The anatomy of a discovery in mathematical research. For the Learning of Mathematics, 1(2), 25–30.Google Scholar
(1980c). Intuitive infinitesimals in the calculus. Abstracts of Short Communications, Fourth International Congress on Mathematical Education, Berkeley, p. C5. Full paper available from (Accessed February 19, 2013).Google Scholar
(1986a). Building and Testing a Cognitive Approach to the Calculus Using Interactive Computer Graphics. PhD thesis, University of Warwick.Google Scholar
(1986b). Constructing the concept image of a tangent. In Proceedings of the Eleventh International Conference of PME, Montreal, III, 69–75.Google Scholar
(1986d). A graphical approach to integration and the fundamental theorem. Mathematics Teaching, 113, 48–51.Google Scholar
(1991a). Recent developments in the use of the computer to visualize and symbolize calculus concepts. In The Laboratory Approach to Teaching Calculus, M.A.A. Notes 20. (pp. 15–25). Washington, DC: Mathematical Association of America.Google Scholar
(1991b). Real Functions and Graphs (for the BBC computer and Nimbus PC). Cambridge: Cambridge University Press.Google Scholar
(1992). Visualizing differentials in two and three dimensions. Teaching Mathematics and Its Applications, 11(1), 1–7.CrossRefGoogle Scholar
(2001). A child thinking about infinity. Journal of Mathematical Behavior, 20, 7–19.CrossRefGoogle Scholar
(2004). Thinking through three worlds of mathematics. In Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education, Bergen, Norway, 4, 281–8.Google Scholar
(2009). Dynamic mathematics and the blending of knowledge structures in the calculus. ZDM – The International Journal on Mathematics Education, 41(4), 481–92.CrossRefGoogle Scholar
, , & (1997). What is the object of the encapsulation of a process? In & (Eds.), People in Mathematics Education, MERGA 20, Aotearoa, Rotarua, New Zealand, 2,132–9.Google Scholar
, , & (2013). Evolving a three-world framework for solving algebraic equations in the light of what a student has met before. Available from (Accessed February 19, 2013.)Google Scholar
, , , , & (2000). What is the object of the encapsulation of a process?
Journal of Mathematical Behavior, 18(2), 1–19.Google Scholar
, & (1981). Concept image and concept definition in mathematics, with special reference to limits and continuity. Educational Studies in Mathematics, 12, 151–69.CrossRefGoogle Scholar
, , , , , & Cheng, Ying-Hao (2012). Cognitive development of proof. In G. Hanna & M. De Villiers (Eds.), ICMI 19: Proof and Proving in Mathematics Education.Google Scholar
(1930). Une contribution á la théorie de la mesure. Fundamenta Mathematicae. 15, 42–50.CrossRefGoogle Scholar
(1990). Mathematical education. Notices of the American Mathematical Society, 37(7), 844–50.Google Scholar
(1994). On proof and progress in mathematics. Bulletin of the American Mathematical Society, 30(2), 161–77.CrossRefGoogle Scholar
(1999). The Cultural Origins of Human Cognition. Cambridge, MA: Harvard University Press.Google Scholar
(1980). A History of Algebra: From al Khwarizmi to Emmy Noether. New York: Springer-Verlag.Google Scholar
(1957). The child’s thought and geometry, trans. into English and reproduced in , , & (Eds.), Classics in Mathematics Education (pp. 61–55). Reston, VA: National Council of Teachers of Mathematics.Google Scholar
(1959). Development and the learning process. Acta Paedagogica Ultrajectina (pp. 1–31). Gröningen: J. B. Wolters.Google Scholar
(2002). Similarities and differences between the theory of learning and teaching of Skemp and the Van Hiele levels of thinking. In & (Eds.), Intelligence, Learning and Understanding – A Tribute to Richard Skemp (pp. 27–47). Flaxton, Australia: Post Pressed.Google Scholar
(1984). The didactics of geometry in the lowest class of secondary school. In , , & (Eds.), English Translation of Selected Writings of Dina van Hiele-Geldof and Pierre M. van Hiele (pp. 1–214). Brooklyn, NY: Brooklyn College.Google Scholar
, & (1980). Concept images and some common cognitive paths in the development of some simple geometric concepts. In Proceedings of the Fourth International Conference of PME, Berkeley, 177–84.Google Scholar
(2002). The balance model: Hindrance or support for the solving of linear equations with one unknown. Educational Studies in Mathematics, 49, 341–59.CrossRefGoogle Scholar
(subsequently ), , & (2003). The relationship between physical embodiment and mathematical symbolism: The concept of vector. The Mediterranean Journal of Mathematics Education, 1(2), 73–97.Google Scholar
(2001). Student difficulty in constructing proofs: The need for strategic knowledge. Educational Studies in Mathematics, 48(1), 101–19.CrossRefGoogle Scholar
(2004). Traditional instruction in advanced mathematics courses: A case study of one professor’s lectures and proofs in an introductory real analysis course. Journal of Mathematical Behavior, 23, 115–33.CrossRefGoogle Scholar
(1799). Om directionens analytiske betegning, et forsøg, anvendt fornemmelig til plane og sphaeriske polygoners opløsning, Nye samling af det Kongelige Danske Videnskabernes Selskabs Skrifter, 5, 496–518.Google Scholar
(1918). Das Continuum. trans. by & (1987) as The Continuum: A Critical Examination of the Foundation of Analysis. New York: Dover.Google Scholar
(1993). Connected Mathematics: Building Concrete Relationships with Mathematical Knowledge. PhD thesis, M.I.T. Retrieved from: (Accessed July 11, 2012).Google Scholar
(1998). What is normal anyway? Therapy for epistemological anxiety. Educational Studies in Mathematics, 33(2), 171–202.CrossRefGoogle Scholar
(2002). The Analyst by George Berkeley. Retrieved from (Accessed April 21, 2012).Google Scholar
(1992). Mathematical Analysis: A Comparison of student development and historical development. PhD thesis, Cambridge University.Google Scholar
(2004). The effects of math anxiety on post-secondary developmental students as related to achievement, gender, and age. Virginia Mathematics Teacher, Fall, 7–9.Google Scholar
(1960). Unknotting spheres in five dimensions. Bulletin of the American Mathematical Society, 66, 198.CrossRefGoogle Scholar