Surfaces and Essences

Gamo, Sylvie, Emmanuel Sander, and Jean-François Richard (2010). “Transfer of strategies by semantic recoding in arithmetic problem solving”. Learning and Instruction, 20, pp. 400–410.

  Gamo, Sylvie, Lynda Taabane, and Emmanuel Sander (2011). “Rôle de la nature des variables dans la résolution de problèmes additifs complexes”. L’Année psychologique, 111, pp. 613–640.

  Gick, Mary L. and Keith J. Holyoak (1983). “Schema induction and analogical transfer”. Cognitive Psychology, 15, pp. 1–38.

  Ginsburg, Herbert P. (1977). Children’s Arithmetic. New York: Van Nostrand.

  Hatano, Giyoo and Inagaki, Kayoko (1994). “Young children’s naive theory of biology”. Cognition, 50, pp. 171–188.

  Hudson, Tom (1983). “Correspondences and numerical differences between disjoint sets”. Child Development, 54, pp. 84–90.

  Keil, Frank C. (1989). Concepts, Kinds, and Cognitive Development. Cambridge, Mass.: MIT Press.

  Kieran, Carolyn (1981). “Concepts associated with the equality symbol”. Educational Studies in Mathematics, 12, pp. 317–326.

  Kintsch, Walter and James G. Greeno (1985). “Understanding and solving word arithmetic problems”. Psychological Review, 92, pp. 109–129.

  Lakoff, George and Rafael Núñez (2000). Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being. New York: Basic Books.

  Lautrey, Jacques, Sylvianne Rémi-Giraud, Emmanuel Sander, and Andrée Tiberghien (2008). Les Connaissances naïves. Paris: Armand Colin.

  Leary, David E. (1990). Metaphors in the History of Psychology. New York: Cambridge University Press.

  Linchevski, Liora and Shlomo Vinner. (1988). “The naive concept of sets in elementary teachers”. Proceedings of the Twelfth International Conference, Psychology of Mathematics Education (Veszprem, Hungary), vol. II, pp. 471–478.

  Mahajan, Sanjoy (2010). Street-Fighting Mathematics: The Art of Educated Guessing and Opportunistic Problem Solving. Cambridge, Mass.: MIT Press.

  Nesher, Pearla (1982). “Levels of description in the analysis of addition and subtraction word problems”. In Thomas P. Carpenter, James M. Moser and Thomas A. Romberg (eds.), Addition and Subtraction: A Cognitive Perspective. Hillsdale, NJ: Lawrence Erlbaum, pp. 25–38.

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  ————— (1993). Things that Make Us Smart. Reading, Mass.: Addison-Wesley.

  Novick, Laura R. and Miriam Bassok (2005). “Problem solving”. In Keith J. Holyoak and Robert G. Morrison (eds.), Cambridge Handbook of Thinking and Reasoning. New York: Cambridge University Press, pp. 321–349.

  Nunes, Terezinha and Peter Bryant (1996). Children Doing Mathematics. Oxford: Blackwell.

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  Resnick, Lauren B. (1982). “Syntax and semantics in learning to subtract”. In Thomas P. Carpenter, James M. Moser and Thomas A. Romberg (eds.), Addition and Subtraction: A Cognitive Perspective. Hillsdale, New Jersey: Lawrence Erlbaum, pp. 136–155.

  Richard, Jean-François and Emmanuel Sander (2000). “Activités d’interprétation et de recherche de solution dans la résolution de problèmes”. In Jean-Noël Foulin and Corinne Ponce (eds.), Les Apprentissages scolaires fondamentaux. Bordeaux: Éditions CRDP, pp. 91–102.

  Riley, Mary S., James G. Greeno, and Joan I. Heller (1983). “Development of children’s problem solving ability in arithmetic”. In Herbert P. Ginsburg (ed.), The Development of Mathematical Thinking. New York: Academic Press.

  Sander, Emmanuel (2001). “Solving arithmetic operations: A semantic approach”. In Proceedings of the 23rd Annual Conference of the Cognitive Science Society (Edinburgh), pp. 915–920.

  ————— (2002). “L’analogie, source de nos apprentissages”. La Recherche, 353, pp. 40–43.

  ————— (2007a). “Manipuler l’habillage d’un problème pour évaluer les apprentissages”. Bulletin de psychologie, 60, pp. 119–124.

  ————— (2007b). “Processus cognitifs, analogie et conception/évaluation de sites”. In Serban Ionescu and Alain Blanchet (eds.), Nouveau cours de psychologie. Psychologie sociale et ressources humaines (coordinated by Marcel Bromberg and Alain Trognon). Paris: Presses universitaires de France, pp. 479–488.

  ————— (2008). “En quoi Internet a-t-il changé notre façon de penser ?” In Philippe Cabin and Jean-François Dortier (eds.), La Communication. État des savoirs. Auxerre: Éditions Sciences humaines, pp. 363–369.

  ————— (2011). “Les mécanismes de la pensée dans les apprentissages”. In Nicolas Balacheff and Michel Fayol (eds.), Apprendre et transmettre. Des idées, des savoir-faire, des valeurs. Paris: Autrement.

  Schliemann, Analucia, Claudia Araujo, Maria Angela Cassunde, Suzana Macedo, and Lenice Niceas (1998). “Use of multiplicative commutativity by school children and street sellers”. Journal for Research in Mathematics Education, 29, pp. 422–435.

  Schoenfeld, Alan H. and Douglas J. Herrmann (1982). “Problem perception and knowledge structure in expert and novice mathematical problem solvers”. Journal of Experimental Psychology: Learning, Memory, and Cognition, 8, pp. 484–494.

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  Silver, Edward A. (1981). “Recall of mathematical problem information: Solving related problems”. Journal of Research in Mathematical Education, 12, pp. 54–64.

  Spalding, Thomas and Gregory L. Murphy (1996). “Effects of background knowledge on category construction”. Journal of Experimental Psychology: Learning, Memory, and Cognition, 22, pp. 525–538.

  Thagard, Paul, Keith J. Holyoak, Greg Nelson, and David Gochfeld (1990). “Analog retrieval by constraint satisfaction”. Artificial Intelligence, 46, pp. 259–310.

  Thevenot, Catherine, Michel Devidal, Pierre Barrouillet, and Michel Fayol (2007). “Why does placing the question before an arithmetic word problem improve performance? A situation model account”. Quarterly Journal of Experimental Psychology, 60 (1), pp. 43–56.

  Thevenot, Catherine and Jane Oakhill (2005). “The strategic use of alternative representations in arithmetic word problem solving”. Quarterly Journal of Experimental Psychology, A, 58 (7), pp. 1311–1323.

  Tiberghien, Andrée (2003). “Des connaissances naïves au savoir scientifique”. In Michèle Kail and Michel Fayol (eds.), Les Sciences cognitives et l’école. La question des apprentissages. Paris: PUF.

  Tirosh, Dina and Anna O. Graeber (1991). “The influence of problem type and common misconceptions on preservice elementary teachers’ thinking about division”. School Science and Mathematics, 91, pp. 157–163.

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  Verschaffel, Lieven, Brian Greer, Wim van Dooren, and Swapna Mukhopadhyay, editors (2009). Words and Worlds: Modelling Verbal Descriptions of Situations. Rotterdam: Sense Publications.

  Viennot, Laurence (1979). Le Raisonnement spontané en mécanique élémentaire. Paris: Herman.

  Vosniadou, Stella and William F. Brewer (1992). “Mental models of the earth: A study of conceptual change in childhood”. Cognitive Psychology, 24, pp. 535–585.

  Chapter 8

  The books by Bartha, Changeux and Connes, Hesse, Fischbein, Lakoff and Núñez, Nersessian, Oppenheimer, Poincaré, and Polya, which cover the process of scientific discovery from epistemological, philosophical, or psychological points of view, are relevant to the chapter as a whole. Those by De M
organ, Dunham, Kasner and Newman, Leibniz, Sawyer, Stewart, Stillwell, Timmermans, and Ulam are rich resources concerning the evolution of ideas in mathematics. The books by Born, Holton (and Brush), Miller, Pais, Pullman, Segrè, Stehle, and Tomonaga are marvelous gems documenting the history of ideas in physics in general, while those by Einstein, Hoffmann, Holton (2000), Miller, Pais (1982), Rigden, and Stachel focus on the more specific story of Albert Einstein’s ideas. McAllister, Stewart, and Wechsler explore the role of esthetics in scientific discoveries, while Weiner recounts the ever-present role of analogies in the story of his own life as a physicist. The books by David and Mendel, Ulam, and Villani are all quoted in the closing section of the chapter.

  Bartha, Paul (2010). By Parallel Reasoning: The Construction and Evaluation of Analogical Arguments. New York: Oxford University Press.

  Bernstein, Jeremy (2006). Secrets of the Old One: Einstein, 1905. New York: Copernicus Books.

  Born, Max (1936). The Restless Universe. New York: Harper and Brothers.

  Changeux, Jean-Pierre and Alain Connes (1998). Conversations on Mind, Mathematics, and Matter. Princeton: Princeton University Press.

  David, Hans T. and Arthur Mendel (1966). The Bach Reader. New York: W. W. Norton & Company.

  De Morgan, Augustus (1831). On the Study and Difficulties of Mathematics. Reprinted in 2004 by Kessinger Publishing, Whitefish, Montana.

  Dunham, William (1991). Journey through Genius: The Great Theorems of Mathematics. New York: Penguin.

  Einstein, Albert (1920). Relativity: The Special and the General Theory. Reprinted in 1961 by Crown Publishers (New York).

  Everitt, C. W. F. (1975). James Clerk Maxwell, Physicist and Natural Philosopher. New York: Charles Scribner’s Sons.

  Fischbein, Efraim. (1987). Intuition in Science and Mathematics: An Educational Approach. Dordrecht: D. Reidel.

  Hesse, Mary (1966). Models and Analogies in Science. South Bend: Notre Dame University Press.

  Hoffmann, Banesh (1972). Albert Einstein: Creator and Rebel. New York: Viking.

  ————— (1983). Relativity and its Roots. New York: Scientific American Books.

  Holton, Gerald (1988). Thematic Origins of Scientific Thought: Kepler to Einstein. Cambridge, Mass.: Harvard University Press.

  ————— (1998). The Scientific Imagination. Cambridge, Mass.: Harvard University Press.

  ————— (2000). Einstein, History, and Other Passions: The Rebellion against Science at the End of the Twentieth Century. Cambridge, Mass.: Harvard University Press.

  Holton, Gerald and Stephen G. Brush (2001). Physics, The Human Adventure. New Brunswick: Rutgers University Press.

  Kao, T. I. and Frank J. Swetz (1977). Was Pythagoras Chinese? An Examination of Right Triangle Theory in Ancient China. University Park: Pennsylvania State University Press.

  Kasner, Edward, and James Newman (1940). Mathematics and the Imagination. New York: Simon and Schuster.

  Lakoff, George and Rafael Núñez (2000). Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being. New York: Basic Books.

  Leibniz, Gottfried Wilhelm von (1702). “Specimen novum analyseos pro scientia infiniti, circa summas & quadraturas”. Reprinted in 1858 by C. I. Gerhardt as Leibnizens Mathematische Schriften, Sec. 2, I, No. XXIV. Halle: Verlag H. W. Schmidt.

  McAllister, James W. (1996). Beauty and Revolution in Science. Ithaca: Cornell University Press.

  Miller, Arthur I. (1985). Frontiers of Physics: 1900–1911. Boston: Birkhäuser.

  ————— (1986). Imagery in Scientific Thought: Creating 20th-Century Physics. Cambridge, Mass.: MIT Press.

  ————— (1997). Albert Einstein’s Special Theory of Relativity: Emergence (1905) and Early Interpretation (1905-1911). New York: Springer.

  ————— (2000). Insights of Genius: Imagery and Creativity in Science and Art. Cambridge, Mass.: MIT Press.

  Nersessian, Nancy J. (2008). Creating Scientific Concepts. Cambridge, Mass.: MIT Press (Bradford Books).

  Oppenheimer, J. Robert (1956). “Analogy in science”. American Psychologist, 11, pp. 127–135.

  Pais, Abraham (1982). Subtle Is the Lord: The Science and Life of Albert Einstein. Oxford: Clarendon Press.

  ————— (1986). Inward Bound: Of Matter and Forces in the Physical World. Oxford: Oxford University Press.

  ————— (1991). Niels Bohr’s Times, in Physics, Philosophy, and Polity. Oxford: Clarendon Press.

  Poincaré, Henri (1908). Science et méthode. Paris, Flammarion.

  ————— (1911). La Valeur de la science. Paris, Flammarion.

  Polya, George (1954). Induction and Analogy in Mathematics. Princeton: Princeton University Press.

  ————— (1954). How to Solve It. Princeton: Princeton University Press.

  Pullman, Bernard (1998). The Atom in the History of Human Thought. Oxford: Oxford University Press.

  Recorde, Robert (1557). Whetstone of Witte. New York: Da Capo Press (reprinted 1969).

  Rigden, John S. (2005). Einstein 1905: The Standard of Greatness. Cambridge, Mass.: Harvard University Press.

  Sawyer, Walter W. (1955). Prelude to Mathematics. Baltimore: Penguin Books.

  Segrè, Emilio (1980). From X-Rays to Quarks: Modern Physicists and Their Discoveries. New York: W. H. Freeman.

  ————— (1984). From Falling Bodies to Radio Waves: Classical Physicists and Their Discoveries. New York: W. H. Freeman.

  Stachel, John (2001). Einstein from ‘B’ to ‘Z’. Boston: Birkhäuser.

  Stachel, John, editor (2005). Einstein’s Miraculous Year: Five Papers that Changed the Face of Physics. Princeton: Princeton University Press.

  Stehle, Philip (1994). Order, Chaos, Order: The Transition from Classical to Quantum Physics. New York: Oxford University Press.

  Stewart, Ian (2007). Why Beauty is Truth: A History of Symmetry. New York: Basic Books.

  Stillwell, John (1989). Mathematics and its History. New York: Springer.

  ————— (2001). The Four Pillars of Geometry. New York: Springer.

  Timmermans, Benoît (2012). Histoire philosophique de l’algèbre moderne: Les origines de la pensée abstraite. Paris: Classiques Garnier.

  Tomonaga, Sin-itiro (1997). The Story of Spin. Chicago: University of Chicago Press.

  Ulam, Stanislaw M. (1976). Adventures of a Mathematician. New York: Charles Scribner’s Sons.

  Villani, Cédric (2012). Théorème vivant. Paris: Grasset.

  Wechsler, Judith, editor (1978). On Aesthetics in Science. Cambridge, Mass.: MIT Press.

  Weiner, Richard M. (2008). Analogies in Physics and Life: A Scientific Autobiography. Singapore: World Scientific.

  Epidialogue

  The contributions by Dietrich, French, Hofstadter, Mitchell, Sander, and Turner deal with the question of how analogy-making and categorizaton are related, and the possibility of finding differences between the two. Fischbein, Oppenheimer, and Polya demonstrate the ubiquity of analogy-making in science. Murphy and Ross’s article is concerned with how and when inferences can be made as a result of categorization, while Anderson and Thomson’s article looks at abstraction hierarchies, and that by Barsalou examines the possible influences of context on categorization.

  Anderson, John R. and Ross Thomson (1989). “Use of analogy in a production system architecture”. In Stella Vosniadou and Andrew Ortony (eds.), Similarity and Analogical Reasoning. New York: Cambridge University Press, pp. 267–297.

  Barsalou, Lawrence W. (1982). “Context-independent and context-dependent information in concepts”. Memory and Cognition, 10, pp. 82–93.

  Dietrich, Eric (2010). “Analogical insight: Toward unifying categorization and analogy”. Cognitive Processing, 11 (4), pp. 331–345.

  Fischbein, Efraim (1987). Intuition in Science and Mathematics: An Educational Approach. Dordrecht: D. Reidel.

  French, Robert M. (1995). The Subtlety of Sameness: A Theory and Computer Model of Analogy-Making. Cambridge, Mass.
: MIT Press.

  Gentner, Dedre and Catherine Clement (1988). “Evidence for relational selectivity in the interpretation of analogy and metaphor”. In G. H. Bower (ed.), The Psychology of Learning and Motivation: Advances in Research and Theory. New York: Academic Press, vol. 22, pp. 307–358.

  Gick, Mary L. and Keith J. Holyoak (1983). “Schema induction and analogical transfer”. Cognitive Psychology, 15, pp. 1–38.

  Hofstadter, Douglas (2001). “Analogy as the core of cognition”. In Dedre Gentner, Keith J. Holyoak, and Boicho N. Kokinov (eds.), The Analogical Mind: Perspectives from Cognitive Science. Cambridge, Mass.: MIT Press, pp. 499–538.

  Holyoak, Keith J. and Paul Thagard (1995). Mental Leaps: Analogy in Creative Thought. Cambridge, Mass.: MIT Press.

  Mitchell, Melanie (1993). Analogy-Making as Perception: A Computer Model. Cambridge, Mass.: MIT Press.

  Murphy, Gregory L. and Brian H. Ross (1994). “Prediction from uncertain categorizations”. Cognitive Psychology, 27, pp. 148–193.

  Oppenheimer, J. Robert (1956). “Analogy in science”. American Psychologist, 11, pp. 127–135.

  Plato (1950). Plato’s Phædrus. Millis, Mass.: Agora Publications (reissued in 2009).

  Polya, George (1957). How to Solve It. Princeton: Princeton University Press.

  Sander, Emmanuel (2003a). “Les analogies spontanées: analogies ou catégorisations ?” In Charles A. Tijus (ed.), Métaphores et analogies. Paris: Hermès, pp. 83–114.

  ————— (2003b). “Analogie et catégorisation”. Revue d’intelligence artificielle, 17 (5–6), pp. 719–732.

  Spalding, Thomas and Gregory L. Murphy (1996). “Effects of background knowledge on category construction”. Journal of Experimental Psychology: Learning, Memory, and Cognition, 22, pp. 525–538.

  Thagard, Paul, Keith J. Holyoak, Greg Nelson, and David Gochfeld (1990). “Analog retrieval by constraint satisfaction”. Artificial Intelligence, 46, pp. 259–310.

  Turner, Mark (1988). “Categories and analogies”. In David H. Helman (ed.), Analogical Reasoning. Amsterdam: Kluwer Academic Publishers, pp. 3–24.

  Wisniewski, Edward J. (1995). “Prior knowledge and functionally relevant features in concept learning”. Journal of Experimental Psychology: Learning, Memory, and Cognition, 21, pp. 449–468.