Akkoc, H., & Tall, D. O. (2002). The simplicity, complexity and complication of the function concept. In Cockburn, Anne D. & Nardi, Elena (Eds.), Proceedings of the 26th Conference of the International Group for the Psychology of Mathematics Education, 2, 25–32. Norwich, UK.Google Scholar

Alcock, L. J., & Simpson, A. P. (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

Alcock, L. J., & Simpson, A. P. (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

Alcock, L., & Weber, K. (2004). Semantic and syntactic proof productions. Educational Studies in Mathematics, 56, 209–34.Google Scholar

Alexander, L., & Martray, C. (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

Argand, R. (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

Ashcraft, M. H., & Kirk, E. P. (2001). The relationships among working memory, math anxiety, and performance. Journal of Experimental Psychology, 130(2), 224–37.CrossRefGoogle Scholar

Asiala, M., Brown, A., DeVries, D., Dubinsky, E., Mathews, D., & Thomas, K. (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

Askew, M., Brown, M., Rhodes, V., Johnson, D., & Wiliam, D. (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

Atiyah, M. F. (2004). Collected Works. Vol. 6. Oxford: Clarendon Press.Google Scholar

Ausubel, D. P., Novak, J., & Hanesian, H. (1978). Educational Psychology: A Cognitive View (2nd ed.). New York: Holt, Rinehart & Winston.Google Scholar

Bachelard, G. (1938, reprinted 1983). La formation de l’esprit scientifique. Paris: J. Vrin.Google Scholar

Bakar, M. N., & Tall, D. O. (1992). Students’ mental prototypes for functions and graphs. International Journal of Mathematics Education in Science & Technology, 23(1), 39–50.Google Scholar

Baron, R., Earhard, B., & Ozier, M. (1995). Psychology (Canadian edition). Scarborough, ON: Allyn & Bacon.Google Scholar

Baroody, A. J., & Costlick, R. T. (1998). Fostering Children’s Mathematical power: An Investigative Approach to K–8 Mathematics Instruction. Mahwah, NJ: Lawrence Erlbaum.Google Scholar

Barrow, I. (1670). Lectiones Geometricae, translated by Child (1916). The Geometrical Lectures of Isaac Barrow. Chicago and London: OpenCourt.Google Scholar

Bartlett, F. C. (1932). Remembering. Cambridge: Cambridge University Press.Google Scholar

Bayazit, I. (2006). The Relationship between Teaching and Learning the Function Concept. PhD thesis, University of Warwick.Google Scholar

Berkeley, G. (1734). The Analyst (ed. D. R. Wilkins, 2002). Retrieved from (Accessed April 21, 2012).Google Scholar

Bernstein, A., & Robinson, A. (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

Beth, E. W., & Piaget, J. (1966). Mathematical Epistemology and Psychology, trans. by Mays, W.. Dordrecht, The Netherlands: Reidel.Google Scholar

Betz, N. (1978). Prevalence, distribution, and correlates of math anxiety in college students. Journal of Counseling Psychology, 25(5), 441–8.CrossRefGoogle Scholar

Biggs, J., & Collis, K. (1982). Evaluating the Quality of Learning: The SOLO Taxonomy. New York: Academic Press.Google Scholar

Birkhoff, G., & MacLane, S. (1999). Algebra (3rd ed.). Providence RI: Chelsea Publishing Co., American Mathematical Society.Google Scholar

Bitner, J., Austin, S., & Wadlington, E. (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

Blokland, P., & Giessen, C. (2000). Graphic Calculus for Windows. Retrieved from: (Accessed February 19, 2013).Google Scholar

Boyer, C. B. (1923/1939). The History of the Calculus and Its Conceptual Development. Reprinted by Dover, New York.Google Scholar

Breidenbach, D., Dubinsky, E., Hawks, J., & Nichols, D. (1992). Development of the process conception of function. Educational Studies in Mathematics, 23, 247–85.CrossRefGoogle Scholar

Bruner, J. S. (1966). Towards a Theory of Instruction. Cambridge, MA: Harvard University Press.Google Scholar

Bruner, J. S. (1977). The Process of Education (2nd ed.). Cambridge, MA: Harvard University Press.Google Scholar

Burn, R. P. (1992). Numbers and Functions: Steps into Analysis. Cambridge: Cambridge University Press.Google Scholar

Burns, M. (1998). Math: Facing an American Phobia. Sausalito, CA: Math Solutions Publications.Google Scholar

Burton, L. (2002). Recognising commonalities and reconciling differences in mathematics education. Educational Studies in Mathematics, 50(2), 157–75.CrossRefGoogle Scholar

Byers, W. (2007). How Mathematicians Think. Princeton, NJ: Princeton University Press.Google Scholar

Campbell, K., & Evans, C. (1997). Gender issues in the classroom: A comparison of mathematics anxiety. Education, 117(3), 332–9.Google Scholar

Cantor, G. (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

Cardano, G. (1545). Ars magna. Translated and published as Ars Magna or The Rules of Algebra (1993). New York: Dover.Google Scholar

Chace, A. B. (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

Challenger, M. (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

Chin, E. T. (2002). Building and Using Concepts of Equivalence Class and Partition. PhD thesis, University of Warwick.Google Scholar

Chin, E. T., & Tall, D. O. (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

Chin, E. T., & Tall, D. O. (2002). University students embodiment of quantifier. In Cockburn, Anne D. & Nardi, Elena (Eds.), Proceedings of the 26th Conference of the International Group for the Psychology of Mathematics Education, 4, 273–80. Norwich, UK.Google Scholar

Chin, K. E., & Tall, D. O. (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

Clement, J., Lochhead, J., & Monk, G. S. (1981). Translation difficulties in learning mathematics. American Mathematics Monthly, 4, 286–90.CrossRefGoogle Scholar

Clements, D. H., & Battista, M. T. (1992). Geometry and spatial reasoning. In Grouws, D. (Ed.) Handbook of Research on Teaching and Learning Mathematics (pp. 420–64). New York: Macmillan.Google Scholar

Collis, K. F. (1978). Operational thinking in elementary mathematics. In Keats, J. A., Collis, K. F., & Halford, G. S. (Eds.), Cognitive Development: Research Based on a Neo-Piagetian approach. New York: John Wiley & Sons.Google Scholar

Cornu, B. (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

Cornu, B. (1991). Limits. In Tall, D. O. (Ed.), Advanced Mathematical Thinking (pp. 153–66). Dordrecht, The Netherlands: Kluwer.Google Scholar

Cottrill, J., Dubinsky, E., Nichols, D., Schwingendorf, K., Thomas, K., & Vidakovic, D. (1996). Understanding the limit concept: Beginning with a coordinated process scheme. Journal of Mathematical Behavior, 15(2), 167–92.CrossRefGoogle Scholar

Courant, R. (1937). Differential and Integral Calculus. Vol. I. Translated from the German by McShane, E. J.. Reprint of the second edition (1988). Wiley Classics Library. New York: Wiley-Interscience.Google Scholar

Craats, J. van de (2007). Contexten en eindexamens. Euclides 82(7), 261–6.Google Scholar

Crick, F. (1994). The Astonishing Hypothesis. London: Simon & Schuster.Google Scholar

Davis, P. J., & Hersh, R. (1981). The Mathematical Experience. Boston: Houghton Mifflin.Google Scholar

Davis, R. B. (1984). Learning Mathematics: The Cognitive Science Approach to Mathematics Education. Norwood, NJ: Ablex.Google Scholar

Deacon, T. (1997). The Symbolic Species: The Co-evolution of Language and the Human Brain. London: Penguin.Google Scholar

Dedekind, R. (1872). Stetigkeit und irrationale Zahlen. Braunschweig: Vieweg. Reproduced in R. Dedekind, Gesammelte mathematische Werke, eds. Fricke, R., Noether, E., & Ore, O.. Braunschweig: Vieweg, 1930–1932.Google Scholar

DeMarois, P. (1998). Facets and Layers of the Function Concept: The Case of College Algebra. PhD thesis, University of Warwick.Google Scholar

De Morgan, A. (1831). On the Study and Difficulties of Mathematics. London: Society for the Diffusion of Useful Knowledge.Google Scholar

Descartes, R. (1641). Meditations on First Philosophy In The Philosophical Writings of René Descartes, trans. by Cottingham, J., Stoothoff, R., & Murdoch, D., Cambridge: Cambridge University Press, 1984.Google Scholar

Descartes, R. (1954). The Geometry of René Descartes, trans. by Smith, D. E. & Latham, M. L.. New York: Dover.Google Scholar

Dienes, Z. P. (1960). Building Up Mathematics. London: Hutchinson.Google Scholar

Donald, M. (2001). A Mind So Rare. New York: W. W. Norton.Google Scholar

Duffin, J. M., & Simpson, A. P. (1993). Natural, conflicting and alien. Journal of Mathematical Behaviour, 12(4), 313–28.Google Scholar

Duval, R. (1995). Sémiosis et pensée humaine. Bern, Switzerland: Peter Lang.Google Scholar

Edelman, G. M. (1992). Bright air, brilliant fire. New York: Basic Books.Google Scholar

Ernest, P. (1998). Social Constructivism as a Philosophy of Mathematics. Albany, NY: State University of New York Press.Google Scholar

Fauconnier, G., & Turner, M. (2002). The Way We Think: Conceptual Blending and the Mind’s Hidden Complexities. New York: Basic Books.Google Scholar

Feynman, R. (1985). Surely You’re joking Mr Feynman. New York: W. W. Norton. Reprinted 1992, London: Vintage.Google Scholar

Filloy, E., & Rojano, T. (1989). Solving equations: The transition from arithmetic to algebra. For the Learning of Mathematics, 9(2), 19–25.Google Scholar

Fischbein, E. (1987). Intuition in Science and Mathematics: An Educational Approach. Dordrecht, The Netherlands: Kluwer.Google Scholar

Foster, R. (2001). Children’s Use of Apparatus in the Development of the Concept of Number. PhD thesis, University of Warwick.Google Scholar

Freudenthal, H. (1983). Didactic Phenomenology of Mathematical Structures. Dordrecht, The Netherlands: Reidel.Google Scholar

Furner, J. M., & Berman, B. T. (2003). Math anxiety: Overcoming a major obstacle to the improvement of student math performance. Childhood Education, Spring, 170–4.Google Scholar

Gauss, K. (1831). Theory of biquadratic residues, part 2. lecture presented to the Royal Society, Gottingen, April 23, 1831.Google Scholar

Glasersfeld, E. von (1995). Radical Constructivism. London: Routledge Falmer.Google Scholar

Gleason, A. M., & Hughes-Hallett, D. (1994). Calculus. New York: John Wiley & Sons.Google Scholar

Gödel, K. (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

Gray, E. M. (1993). Qualitatively Different Approaches to Simple Arithmetic. PhD thesis, University of Warwick.Google Scholar

Gray, E. M., Pitta, D., Pinto, M. M. F., & Tall, D. O. (1999). Knowledge construction and diverging thinking in elementary and advanced mathematics. Educational Studies in Mathematics, 38(1–3), 111–33.CrossRefGoogle Scholar

Gray, E. M., & Tall, D. O. (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

Gray, E. M., & Tall, D. O. (1994). Duality, ambiguity and flexibility: A proceptual view of simple arithmetic. Journal for Research in Mathematics Education, 26(2), 115–41.Google Scholar

Gray, E. M., & Tall, D. O. (2001). Relationships between embodied objects and symbolic procepts: An explanatory theory of success and failure in mathematics. In Heuvel-Panhuizen, Marja van den (Ed.), Proceedings of the 25th Conference of the International Group for the Psychology of Mathematics Education, 3, 65–72. Utrecht, The Netherlands.Google Scholar

Grice, H. P. (1989). Studies in the Way of Words. Cambridge, MA: Harvard University Press.Google Scholar

Gutiérrez, A., Jaime, A., & Fortuny, J. (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

Hadamard, J. (1945). The Psychology of Invention in the Mathematical Field. Princeton, NJ: Princeton University Press. Dover edition, New York, 1954.Google Scholar

Halmos, P. (1966). Invariant subspaces for polynomially compact operators. Pacific Journal of Mathematics, 16(3), 433–7.CrossRefGoogle Scholar

Hart, K. M., Johnson, D. C., Brown, M., Dickson, L., & Clarkson, R. (1989). Children’s Mathematical Frameworks 8–13: A Study of Classroom Teaching. London: Routledge (formerly NFER Nelson).Google Scholar

Heath, T. L. (1921). History of Greek Mathematics. Vol. 1. Oxford: Oxford University Press. Reprinted Dover Publications, New York, 1963.Google Scholar

Hembree, R. (1990). The nature, effects, and relief of mathematics anxiety. Journal for Research in Mathematics Education, 21(1), 33–46.CrossRefGoogle Scholar

Herrmann, E., Call, J., Hernàndez-Lloreda, M. V., Hare, B., & Tomasello, M. (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

Heuvel-Panhuizen, M. van den (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

Hiebert, J., & Lefevre, P. (1986). Procedural and conceptual knowledge. In Hiebert, J. (Ed.), Conceptual and Procedural Knowledge: The Case of Mathematics (pp. 1–27). Hillsdale, NJ: Lawrence Erlbaum.Google Scholar

Hilbert, D. (1900). Mathematische Probleme. Göttingen Nachrichten, 253–97.Google Scholar

Hilbert, D. (1926). Über das Unendliche. Mathematische Annalen (95), 161–90.CrossRefGoogle Scholar

Hoffer, A. (1981). Geometry is more than proof. Mathematics Teacher, 74, 11–18.Google Scholar

Horgan, J. (1994). Profile: Andre Weill, great French-born mathematician, Scientific American, 270 (6), June 1994, 33–34.CrossRefGoogle Scholar

Howat, H. (2006). Participation in Elementary Mathematics: An Analysis of Engagement, Attainment and Intervention. PhD thesis, University of Warwick.Google Scholar

Howson, G. C. (1982). A History of Mathematics Education in England. Cambridge: Cambridge University Press.CrossRefGoogle Scholar

Hughes-Hallett, D. (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

Inglis, M., Mejia-Ramos, J. P., & Simpson, A. P. (2007). Modelling mathematical argumentation: The importance of qualification. Educational Studies in Mathematics, 66(7), 3–21.CrossRefGoogle Scholar

Inoue, S., & Matsuzawa, T. (2007). Working memory of numerals in chimpanzees. Current Biology, 17(23), R1004–R1005.CrossRefGoogle ScholarPubMed

Jackson, C., & Leffingwell, R. (1999). The role of instructors in creating math anxiety in students from kindergarten through college. Mathematics Teacher, 92(7), 583–7.Google Scholar

James, W. (1890). The Principles of Psychology. Vols. I & II. New York: Henry Holt.Google Scholar

Johnson, D. C. (Ed.) (1989): Children’s Mathematical Frameworks 8–13: A Study of Classroom Teaching. Windsor, UK: NFER-Nelson.Google Scholar

Jones, W. (2001). Applying psychology to the teaching of basic math: A case study. Inquiry, 6(2), 60–5.Google Scholar

Jowett, B. (1871). Plato’s The Republic. New York: Scribner’s Sons.Google Scholar

Joyce, D. E. (1998). Euclid’s Elements. Retrieved from on 26th March 2012.Google Scholar

Kant, E. (1781). Kritik der reinen Vernunft (Critique of Pure Reason). Königsberg, Germany.Google Scholar

Katz, M., & Tall, D. O. (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

Keisler, H. J. (1976). Foundations of Infinitesimal Calculus. Boston: Prindle, Weber & Schmidt.Google Scholar

Kerslake, D. (1986). Fractions: Children’s Strategies and Errors. London: NFER-Nelson.Google Scholar

Koichu, B. (2008). On considerations of parsimony in mathematical problem solving. In Figueras, O., Cortina, J. L., Alatorre, S., Rojano, T., & Sepulova, A. (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

Koichu, B., & Berman, A. (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

Kollar, D. (2000). Article in the Sacramento Bee (California), December 11, 2000.Google Scholar

Krutetskii, V. A. (1976). The Psychology of Mathematical Abilities in Schoolchildren. Chicago: University of Chicago Press.Google Scholar

Kuhn, T. (1962). The Structure of Scientific Revolutions. Chicago: University of Chicago Press.Google Scholar

Lakoff, G. (1987). Women, Fire, and Dangerous Things: What Categories Reveal About the Mind. Chicago: University of Chicago Press.CrossRefGoogle Scholar

Lakoff, G., & Johnson, M. (1980). Metaphors We Live By. Chicago: University of Chicago Press.Google Scholar

Lakoff, G., & Johnson, M. (1999). Philosophy in the Flesh: The Embodied Mind and Its Challenge to Western Thought. New York: Basic Books.Google Scholar

Lakoff, G. & Nùñez, R. (2000). Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being. New York: Basic Books.Google Scholar

Lave, J., & Wenger, E. (1991). Situated Learning: Legitimate Peripheral Participation. Cambridge: Cambridge University Press.CrossRefGoogle Scholar

Lean, G., & Clements, K. (1981). Spatial ability, visual imagery, and mathematical performance. Educational Studies in Mathematics, 12(3), 267–99.CrossRefGoogle Scholar

Leibniz, G. W. (1920). The Early Mathematical Manuscripts of Leibniz, ed. and trans. by Child, J. M.. Chicago: University of Chicago Press.Google Scholar

Li, L., & Tall, D. O. (1993). Constructing different concept images of sequences and limits by programming. In Proceedings of PME 17, Japan, 2, 41–8.Google Scholar

Lima, R. N. de, & Tall, D. O. (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

Lima, R. N. de, & Tall, D. O. (2008). Procedural embodiment and magic in linear equations. Educational Studies in Mathematics, 67(1), 3–18.CrossRefGoogle Scholar

Ma, L. (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

Ma, L. (1999b). Knowing and Teaching Elementary Mathematics. Mahwah, NJ: Lawrence Erlbaum.Google Scholar

MacLane, S. (1994). Responses to theoretical mathematics. Bulletin (new series) of the American Mathematical Society, 30(2), 190–1.Google Scholar

Mason, J. (1989). Mathematical abstraction as the result of a delicate shift of attention. For the Learning of Mathematics, 9(2), 2–8.Google Scholar

Mason, J. (2002). Researching Your Own Practice: The Discipline of Noticing. London: Routledge Falmer.CrossRefGoogle Scholar

Mason, J., Burton, L., & Stacey, K. (1982). Thinking Mathematically. London: Addison-Wesley.Google Scholar

Matthews, G. (1964). Calculus. London: John Murray.Google Scholar

McGowen, M. A. (1998). Cognitive Units, Concept Images, and Cognitive Collages: An Examination of the Process of Knowledge Construction. PhD thesis, University of Warwick.Google Scholar

McGowen, M. A., & Tall, D. O. (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

Md Ali, R. (2006). Teachers’ Indications and Pupils’ Construal and Knowledge of Fractions: The Case of Malaysia. PhD thesis, University of Warwick.Google Scholar

Mejia-Ramos, J. P. (2008). The Construction and Evaluation of Arguments in Undergraduate Mathematics. PhD thesis, University of Warwick.Google Scholar

Miller, G. A. (1956). The magic number seven plus or minus two: Some limits on our capacity for processing information. Psychological Review, 63, 81–97.CrossRefGoogle ScholarPubMed

Monaghan, J. D. (1986). Adolescent’s Understanding of Limits and Infinity. PhD thesis, University of Warwick.Google Scholar

National Council of Teachers of Mathematics. (1989). Curriculum and Evaluation Standards for School Mathematics. Reston, VA: National Council of Teachers of Mathematics.Google Scholar

Neill, H., & Shuard, H. (1982). Teaching Calculus. London: Blackie & Son.Google Scholar

Neugebauer, O. (1969). The Exact Sciences in Antiquity (2nd ed.). Reprinted by Dover, New York.Google Scholar

Núñez, R., Edwards, L. D., & Matos, J. P. (1999). Embodied cognition as grounding for situatedness and context in mathematics education. Educational Studies in Mathematics, 39(1–3), 45–65.CrossRefGoogle Scholar

Nunokawa, K. (2005). Mathematical problem solving and learning mathematics: What we expect students to obtain. Journal of Mathematical Behavior, 24, 325–40.CrossRefGoogle Scholar

Pegg, J. (1991). Editorial. Australian Senior Mathematics Journal, 5(2), 70.Google Scholar

Pegg, J., & Tall, D. O. (2005). The fundamental cycle of concept construction underlying various theoretical frameworks. International Reviews on Mathematical Education (ZDM), 37(6), 468–75.Google Scholar

Peirce, C. S. (1991). Peirce on Signs: Writings on Semiotic (ed. Hoopes, J.). University of North Carolina Press.Google Scholar

Piaget, J. (1926). The Language and Thought of the Child. New York: Harcourt, Brace, Jovanovich.Google Scholar

Piaget, J. (1952). The Child’s Conception of Number. London: Routledge & Kegan Paul.Google Scholar

Piaget, J., & Inhelder, B. (1958). Growth of Logical Thinking. London: Routledge & Kegan Paul.Google Scholar

Pinto, M. M. F. (1998). Students’ Understanding of Real Analysis. PhD thesis, University of Warwick.Google Scholar

Pinto, M. M. F., & Tall, D. O. (1999). Student constructions of formal theory: Giving and extracting meaning. In Zaslavsky, O. (Ed.), Proceedings of the 23rd Conference of PME, Haifa, Israel, 4, 65–73.Google Scholar

Pinto, M. M. F., & Tall, D. O. (2001). Following students’ development in a traditional university classroom. In Heuvel-Panhuizen, Marja van den (Ed.), Proceedings of the 25th Conference of the International Group for the Psychology of Mathematics Education 4, 57–64. Utrecht, The Netherlands.Google Scholar

Pinto, M. M. F., & Tall, D. O. (2002). Building formal mathematics on visual imagery: A theory and a case study. For the Learning of Mathematics, 22(1), 2–10.Google Scholar

Pitta, D. (1998). Beyond the Obvious: Mental Representations and Elementary Arithmetic. PhD thesis, University of Warwick.Google Scholar

Pitta, D., & Gray, E. M. (1997). In the mind: What can imagery tell us about success and failure in arithmetic? In Makrides, G. A. (Ed.), Proceedings of the First Mediterranean Conference on Mathematics, Nicosia: Cyprus, 29–41.Google Scholar

Pitta, D., & Gray, E. M. (1999). Changing Emily’s images. In Pinel, A. (Ed.), Teaching, Learning and Primary Mathematics (pp. 56–60). Derby, UK: Association of Teachers of Mathematics.Google Scholar

Plake, B. S., & Parker, C. S. (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

Plato, (360 BC). The Republic. Book VII, trans. by Jowett, Benjamin (1871). New York : Scribner’s Sons. Reprinted 1941, New York: The Modern Library.Google Scholar

Playfair, J. (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

Poincaré, H. (1913). The Foundations of Science, trans. by Halsted, G. B.. New York: The Science Press.Google Scholar

Pólya, G. (1945). How to Solve It. Princeton, NJ: Princeton University Press. Reprinted 1957, Garden City, NY: Doubleday.CrossRefGoogle Scholar

Poynter, A. (2004). Effect as a Pivot between Actions and Symbols: The case of Vector. PhD thesis, University of Warwick.Google Scholar

Presmeg, N. C. (1986). Visualisation and mathematical giftedness. Educational Studies in Mathematics, 17(3), 297–311.CrossRefGoogle Scholar

Reid, C. (1996). Hilbert. New York: Springer.CrossRefGoogle Scholar

Richardson, F. C., & Suinn, R. M. (1972). The Mathematics Anxiety Rating Scale: Psychometric data. Journal of Counseling Psychology, 19(6), 551–4.CrossRefGoogle Scholar

Robinson, A. (1966). Non-Standard Analysis. Amsterdam: North Holland.Google Scholar

Rodd, M. M. (2000). On mathematical warrants. Mathematical Thinking and Learning, 2 (3), 221–44.CrossRefGoogle Scholar

Rosch, E., Mervis, C. B., Gray, W. D., Johnson, D. M., & Boyes-Barem, P. (1976). Basic objects in natural categories. Cognitive Psychology, 8, 382–439.CrossRefGoogle Scholar

Rosnick, P. (1981). Some misconceptions concerning the concept of variable. Are you careful about defining your variables? Mathematics Teacher, 74(6), 418–20, 450.Google Scholar

Sangwin, C. J. (2004). Assessing mathematics automatically using computer algebra and the internet. Teaching Mathematics and Its Applications, 23(1), 1–14.CrossRefGoogle Scholar

Saussure, F. (1916). Cours de linguistique génerale (ed. Bally, C. & Séchehaye, A.). Paris: Payot.Google Scholar

Schools Mathematics Project (1982). Advanced Mathematics Book 1. Cambridge: Cambridge University Press.Google Scholar

Schwarzenberger, R. L. E., & Tall, D. O. (1978). Conflicts in the learning of real numbers and limits. Mathematics Teaching, 82, 44–9.Google Scholar

Sfard, A. (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

Sfard, A. (1992). Operational origins of mathematical objects and the quandary of reification – the case of function. In Harel, Guershon & Dubinsky, Ed (Eds.), The Concept of Function: Aspects of Epistemology and Pedagogy, MAA Notes 25 (pp. 59–84). Washington, DC: Mathematical Association of America.Google Scholar

Sfard, A. (2008). Thinking as Communicating. New York: Cambridge University Press.CrossRefGoogle Scholar

Sheffield, D., & Hunt, T. (2006). How does anxiety influence maths performance and what can we do about it? MSOR Connections, 6(4), 19–23.CrossRefGoogle Scholar

Skemp, R. R. (1971). The Psychology of Learning Mathematics. London: Penguin.Google Scholar

Skemp, R. R. (1976). Relational understanding and instrumental understanding. Mathematics Teaching, 77, 20–6.Google Scholar

Skemp, R. R. (1979). Intelligence, Learning, and Action. London: John Wiley & Sons.Google Scholar

Snapper, E. (1979). The three crises in mathematics: Logicism, intuitionism and formalism. Mathematics Magazine, 52(4), 207–16.CrossRefGoogle Scholar

Steele, E., & Arth, A. (1998). Lowering anxiety in the math curriculum. Education Digest, 63(7), 18–24.Google Scholar

Stewart, I. N., & Tall, D. O. (1983). Complex Analysis. Cambridge: Cambridge University Press.CrossRefGoogle Scholar

Stewart, I. N., & Tall, D. O. (2000). Algebraic Number Theory and Fermat’s Last Theorem (3rd ed.). Natick, MA: A. K. Peters.Google Scholar

Strauss, A., & Corbin, J. (1990). Basics of Qualitative Research: Grounded Theory Procedures and Techniques. London: SAGE.Google Scholar

Stroyan, K. D. (1972). Uniform continuity and rates of growth of meromorphic functions. In Luxemburg, W. J. & Robinson, A. (Eds.), Contributions to Non-Standard Analysis (pp. 47–64). Amsterdam: North-Holland.Google Scholar

Struik, D. J. (1969). A Source Book in Mathematics, 1200–1800. Cambridge, MA: Harvard University Press.Google Scholar

Sullivan, K. (1976). The teaching of elementary calculus: An approach using infinitesimals, American Mathematical Monthly, 83, 370–5.CrossRefGoogle Scholar

Tall, D. O. (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

Tall, D. O. (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

Tall, D. O. (1980a). Looking at graphs through infinitesimal microscopes, windows and telescopes. Mathematical Gazette, 64, 22–49.CrossRefGoogle Scholar

Tall, D. O. (1980b). The anatomy of a discovery in mathematical research. For the Learning of Mathematics, 1(2), 25–30.Google Scholar

Tall, D. O. (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

Tall, D. O. (1985). Understanding the calculus. Mathematics Teaching, 10, 49–53.Google Scholar

Tall, D. O. (1986a). Building and Testing a Cognitive Approach to the Calculus Using Interactive Computer Graphics. PhD thesis, University of Warwick.Google Scholar

Tall, D. O. (1986b). Constructing the concept image of a tangent. In Proceedings of the Eleventh International Conference of PME, Montreal, III, 69–75.Google Scholar

Tall, D. O. (1986c). Talking about fractions. Micromath, 2(2), 8–10.Google Scholar

Tall, D. O. (1986d). A graphical approach to integration and the fundamental theorem. Mathematics Teaching, 113, 48–51.Google Scholar

Tall, D. O. (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

Tall, D. O. (1991b). Real Functions and Graphs (for the BBC computer and Nimbus PC). Cambridge: Cambridge University Press.Google Scholar

Tall, D. O. (1991c). Advanced Mathematical Thinking. Dordrecht, The Netherlands: Kluwer.CrossRefGoogle Scholar

Tall, D. O. (1992). Visualizing differentials in two and three dimensions. Teaching Mathematics and Its Applications, 11(1), 1–7.CrossRefGoogle Scholar

Tall, D. O. (2001). A child thinking about infinity. Journal of Mathematical Behavior, 20, 7–19.CrossRefGoogle Scholar

Tall, D. O. (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

Tall, D. O. (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

Tall, D. O., Davis, G. E., & Thomas, M. O. J. (1997). What is the object of the encapsulation of a process? In Biddulph, F. & Carr, K. (Eds.), People in Mathematics Education, MERGA 20, Aotearoa, Rotarua, New Zealand, 2,132–9.Google Scholar

Tall, D. O., Lima, R. N. de, & Healy, L. (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

Tall, D. O., Thomas, M. O. J., Davis, G. E., Gray, E. M., & Simpson, A. P. (2000). What is the object of the encapsulation of a process? Journal of Mathematical Behavior, 18(2), 1–19.Google Scholar

Tall, D. O., & Vinner, S. (1981). Concept image and concept definition in mathematics, with special reference to limits and continuity. Educational Studies in Mathematics, 12, 151–69.CrossRefGoogle Scholar

Tall, D. O., Yevdokimov, O., Koichu, B., Whiteley, W., Kondratieva, M., & 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

Tarski, A. (1930). Une contribution á la théorie de la mesure. Fundamenta Mathematicae. 15, 42–50.CrossRefGoogle Scholar

Tempelaar, D., & Caspers, W. (2008). De rol van de instaptoets. Nieuw Archief voor Wiskunde, 5/9(1), 66–71.Google Scholar

Thurston, W. P. (1990). Mathematical education. Notices of the American Mathematical Society, 37(7), 844–50.Google Scholar

Thurston, W. P. (1994). On proof and progress in mathematics. Bulletin of the American Mathematical Society, 30(2), 161–77.CrossRefGoogle Scholar

Tobias, S. (1990). Mathematics anxiety: An update. NACADA Journal, 10, 47–50.CrossRefGoogle Scholar

Tomasello, M. (1999). The Cultural Origins of Human Cognition. Cambridge, MA: Harvard University Press.Google Scholar

Toulmin, S. E. (1958). The Uses of Argument. Cambridge: Cambridge University Press.Google Scholar

Van der Waerden, B. L. (1980). A History of Algebra: From al Khwarizmi to Emmy Noether. New York: Springer-Verlag.Google Scholar

Van Hiele, P. M. (1957). The child’s thought and geometry, trans. into English and reproduced in Carpenter, T. P., Dossey, J. A., & Koehler, J. L. (Eds.), Classics in Mathematics Education (pp. 61–55). Reston, VA: National Council of Teachers of Mathematics.Google Scholar

Van Hiele, P. M. (1959). Development and the learning process. Acta Paedagogica Ultrajectina (pp. 1–31). Gröningen: J. B. Wolters.Google Scholar

Van Hiele, P. M. (1986). Structure and Insight. Orlando, FL: Academic Press.Google Scholar

Van Hiele, P. M. (2002). Similarities and differences between the theory of learning and teaching of Skemp and the Van Hiele levels of thinking. In Tall, D. O. & Thomas, M. O. J. (Eds.), Intelligence, Learning and Understanding – A Tribute to Richard Skemp (pp. 27–47). Flaxton, Australia: Post Pressed.Google Scholar

Van Hiele Geldof, D. (1984). The didactics of geometry in the lowest class of secondary school. In Fuys, D., Geddes, D., & Tischler, R. (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

Vinner, S., & Hershkowitz, R. (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

Vlassis, J. (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

Watson, A. (subsequently Poynter, A.), Spyrou, P., & Tall, D. O. (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

Weber, H. (1893). Leopold Kronecker. Mathematische Annalen, 43, 1–25.CrossRefGoogle Scholar

Weber, K. (2001). Student difficulty in constructing proofs: The need for strategic knowledge. Educational Studies in Mathematics, 48(1), 101–19.CrossRefGoogle Scholar

Weber, K. (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

Werkgroep 3TU. (2006). Aansluiting vwo en technische univrsiteiten. Euclides, 81(5), 242–7.Google Scholar

Wessel, C. (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

Weyl, H. (1918). Das Continuum. trans. by Pollard, S. & Hole, T. (1987) as The Continuum: A Critical Examination of the Foundation of Analysis. New York: Dover.Google Scholar

Wilensky, U. (1993). Connected Mathematics: Building Concrete Relationships with Mathematical Knowledge. PhD thesis, M.I.T. Retrieved from: (Accessed July 11, 2012).Google Scholar

Wilensky, U. (1998). What is normal anyway? Therapy for epistemological anxiety. Educational Studies in Mathematics, 33(2), 171–202.CrossRefGoogle Scholar

Wilkins, D. R. (2002). The Analyst by George Berkeley. Retrieved from (Accessed April 21, 2012).Google Scholar

Wood, N. G. (1992). Mathematical Analysis: A Comparison of student development and historical development. PhD thesis, Cambridge University.Google Scholar

Woodard, T. (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

Zeeman, E. C. (1960). Unknotting spheres in five dimensions. Bulletin of the American Mathematical Society, 66, 198.CrossRefGoogle Scholar

Zeeman, E. C. (1977). Catastrophe Theory. London: Addison-Wesley.Google Scholar