Mind in Motion
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Paivio, A. (1978). Mental comparisons involving abstract attributes. Memory and Cognition, 6, 199–208.
Symbolic distance in other species
D’Amato, M. R., & Colombo, M. (1990). The symbolic distance effect in monkeys (Cebus paella). Animal Learning & Behavior, 18, 133–140.
Gelman, R., & Gallistel, C. R. (2004). Language and the origin of numerical concepts. Science, 306(5695), 441–443.
Transitive inference in other species
Bond, A. B, Kamil, A. C, & Balda, R. P. (2003). Social complexity and transitive inference in corvids. Animal Behavior, 65, 479–487.
Byrne, R. W., & Bates, L. A. (2007). Sociality, evolution and cognition. Current Biology, 17, 714–723.
Byrne, R. W. & Whiten, A. (1988). Machiavellian intelligence: Social expertise and the evolution of intellect in monkeys, apes, and humans. Oxford, England: Clarendon Press.
Davis, H. (1992). Transitive inference in rats (Rattus norvegicus). Journal of Comparative Psychology, 106, 342–349.
Grosenick, L., Clement, T. S., & Fernald, R. D. (2007). Fish can infer social rank by observation alone. Nature, 445, 429–432.
MacLean, E. L., Merritt, D. J., & Brannon, E. M. (2008). Social complexity predicts transitive reasoning in Prosimian primates. Animal Behavior, 76, 479–486.
Von Fersen, L., Wynee, C. D. L., Delius, J. D., & Staddon, J. E. R. (1991). Transitive inference formation in pigeons. Journal of Experimental Psychology: Animal Behavior Processes, 17, 334–341.
Approximate number system in children and other species
Brannon, E. M., & Terrace, H. S. (1998). Ordering of the numerosities 1 to 9 by monkeys. Science, 282(5389), 746–749.
Brannon, E. M., Wusthoff, C. J., Gallistel, C. R., & Gibbon, J. (2001). Numerical subtraction in the pigeon: Evidence for a linear subjective number scale. Psychological Science, 12(3), 238–243.
Cantlon, J. F., Platt, M. L., & Brannon, E. M. (2009). Beyond the number domain. Trends in Cognitive Sciences, 13(2), 83–91.
Gallistel, C. R., Gelman, R., & Cordes, S. (2006). The cultural and evolutionary history of the real numbers. Evolution and Culture, 247.
Henik, A., Leibovich, T., Naparstek, S., Diesendruck, L., & Rubinsten, O. (2012). Quantities, amounts, and the numerical core system. Frontiers in Human Neuroscience, 5, 186.
McCrink, K., & Spelke, E. S. (2010). Core multiplication in childhood. Cognition, 116(2), 204–216.
McCrink, K., & Spelke, E. S. (2016). Non-symbolic division in childhood. Journal of Experimental Child Psychology, 142, 66–82.
McCrink, K., Spelke, E. S., Dehaene, S., & Pica, P. (2013). Non-symbolic halving in an Amazonian indigene group. Developmental Science, 16(3), 451–462.
Scarf, D., Hayne, H., & Colombo, M. (2011). Pigeons on par with primates in numerical competence. Science, 334(6063), 1664–1664.
Brain substrates for approximate and exact number systems
Cohen Kadosh, R., Henik, A., Rubinsten, O., Mohr, H., Dori, H., van de ven, V.,… Linden, D. E. J. (2005). Are numbers special? The comparison systems of the human brain investigated by fMRI. Neuropsychologia, 43, 1238–1248.
Spatial-numerical associations of response (SNARC)
Dehaene, S., Bossini, S., & Giraux, P. (1993). The mental representation of parity and number magnitude. Journal of Experimental Psychology: General, 122(3), 371–396.
Tversky, B., Kugelmass, S., & Winter, A. (1991). Cross-cultural and developmental trends in graphic productions. Cognitive Psychology, 23(4), 515–557.
Greater sensitivity to smaller values (Weber-Fechner effect)
Cantlon, J. F., Platt, M. L., & Brannon, E. M. (2009). Beyond the number domain. Trends in Cognitive Sciences, 13(2), 83–91.
Greater sensitivity to smaller values in language
Talmy, L. (1983). How language structures space. In Spatial orientation (pp. 225–282). Boston, MA: Springer.
Numerical reasoning in cultures lacking names for numbers greater than three
Frank, M. C., Everett, D. L., Fedorenko, E., & Gibson, E. (2008). Number as a cognitive technology: Evidence from Pirahã language and cognition. Cognition, 108(3), 819–824.
Gordon, P. (2004). Numerical cognition without words: Evidence from Amazonia. Science, 306(5695), 496–499.
Pica, P., Lemer, C., Izard, V., & Dehaene, S. (2004). Exact and approximate arithmetic in an Amazonian indigene group. Science, 306(5695), 499–503.
Brain damage can selectively disrupt exact and approximate number systems
Dehaene, S. (2011). The number sense: How the mind creates mathematics. New York, NY: Oxford University Press.
Lemer, C., Dehaene, S., Spelke, E., & Cohen, L. (2003). Approximate quantities and exact number words: Dissociable systems. Neuropsychologia, 41(14), 1942–1958.
Exact and approximate number systems interact in intact brains
Gallistel, C. R., & Gelman, R. (1992). Preverbal and verbal counting and computation. Cognition, 44, 43–74.
Holloway, I. D., & Ansari, D. (2009). Mapping numerical magnitudes onto symbols: The numerical distance effect and individual differences in children’s mathematics achievement. Journal of Experimental Child Psychology, 103(1), 17–29.
Lonnemann, J., Linkersdörfer, J., Hasselhorn, M., & Lindberg, S. (2011). Symbolic and non-symbolic distance effects in children and their connection with arithmetic skills. Journal of Neurolinguistics, 24(5), 583–591.
Mazzocco, M. M., Feigenson, L., & Halberda, J. (2011). Preschoolers’ precision of the approximate number system predicts later school mathematics performance. PLoS One, 6(9), e23749.
Training approximate number system helps exact number system
Libertus, M. E., Feigenson, L., & Halberda, J. (2013). Is approximate number precision a stable predictor of math ability? Learning and Individual Differences, 25, 126–133.
Lyons, I. M., & Beilock, S. L. (2011). Numerical ordering ability mediates the relation between number-sense and arithmetic competence. Cognition, 121(2), 256–261.
Park, J., Bermudez, V., Roberts, R. C., & Brannon, E. M. (2016). Non-symbolic approximate arithmetic training improves math performance in preschoolers. Journal of Experimental Child Psychology, 152, 278–293.
Wang, J. J., Odic, D., Halberda, J., & Feigenson, L. (2016). Changing the precision of preschoolers’ approximate number system representations changes their symbolic math performance. Journal of Experimental Child Psychology, 147, 82–99.
History of number notation
Aczel, A. D. (2016) Finding zero. New York, NY: St. Martin’s Griffin.
Cajori, F. (1928). A history of mathematical notations. Vol. I, Notations in elementary mathematics. North Chelmsford, MA: Courier Corporation.
Cajori, F. (1928). A history of mathematical notations. Vol. II, Notations mainly in higher mathematics. Chicago, IL: Open Court Publishing.
Ifrah, G. (2000). The universal history of numbers: From prehistory to the invention of the computer. Translated by D. Vellos, E. F. Harding, S. Wood, & I. Monk. Toronto, Canada: Wiley.
Mazur, J. (2014). Enlightening symbols: A short history of mathematical notation and its hidden powers. Princeton, NJ: Princeton University Press.
Notation and writing began with accounting in the West
Schmandt-Besserat, D. (1992). Before writing, Vol. I: From counting to cuneiform. Austin: University of Texas Press.
Space is crucial to math notation
Dehaene, S. (2011). The number sense: How the mind creates mathematics. New York, NY: Oxford University Press.
Gelman, R., & Gallistel, C. R. (1978). The child’s understanding of number. Cambridge, MA: Harvard University Press.
Lakoff, G., & Núñez, R. (2000). Where mathematics comes from: How the embodied mind brings mathematics into being. New York, NY: Basic Books.
Eye movements track absent places
Kahneman, D. (1973). Attention and effort. Englewood Cliffs, NJ: Prentice Hall.
Imagined spatial distance affects reading times
Bar-Anan, Y., Liberman,
N., Trope, Y., & Algom, D. (2007). Automatic processing of psychological distance: Evidence from a Stroop task. Journal of Experimental Psychology: General, 136(4), 610.
Imagined spatial distance affects personality judgments
Liberman, N., Trope, Y., & Stephan, E. (2007). Psychological distance. In A. W. Kruglanski & E. T. Higgins (Eds.), Social psychology: Handbook of basic principles (2nd ed., pp. 353–383). New York, NY: Guilford Press.
Ross, L. (1977). The intuitive psychologist and his shortcomings: Distortions in the attribution process. In L. Berkowitz (Ed.), Advances in experimental social psychology (Vol. 10, pp. 173–220). New York, NY: Academic Press.
Trope, Y., & Liberman, N. (2010). Construal-level theory of psychological distance. Psychological Review, 117(2), 440.
Greater distance evokes more abstract language and more abstract thought
Förster, J., Friedman, R. S., & Liberman, N. (2004). Temporal construal effects on abstract and concrete thinking: consequences for insight and creative cognition. Journal of Personality and Social Psychology, 87(2), 177.
Jia, L., Hirt, E. R., & Karpen, S. C. (2009). Lessons from a faraway land: The effect of spatial distance on creative cognition. Journal of Experimental Social Psychology, 45(5), 1127–1131.
Liberman, N., Polack, O., Hameiri, B., & Blumenfeld, M. (2012). Priming of spatial distance enhances children’s creative performance. Journal of Experimental Child Psychology, 111(4), 663–670.
Semin, G. R., & Smith, E. R. (1999). Revisiting the past and back to the future: Memory systems and the linguistic representation of social events. Journal of Personality and Social Psychology, 76(6), 877.
Cognitive reference points expand close distances and shrink far ones (Weber-Fechner)
Holyoak, K. J., & Mah, W. A. (1982). Cognitive reference points in judgements of symbolic magnitude. Cognitive Psychology, 14, 328–352.
Social perspectives: Within or above
Keltner, D., Gruenfeld, D. H., & Anderson, C. (2003). Power, approach, and inhibition. Psychological Review, 110(2), 265.
Keltner, D., Van Kleef, G. A., Chen, S., & Kraus, M. W. (2008). A reciprocal influence model of social power: Emerging principles and lines of inquiry. Advances in Experimental Social Psychology, 40, 151–192.
Van Kleef, G. A., Oveis, C., Van Der Löwe, I., Luo Kogan, A., Goetz, J., & Keltner, D. (2008). Power, distress, and compassion: Turning a blind eye to the suffering of others. Psychological Science, 19(12), 1315–1322.
Language points to percepts
Arnheim, R. (1974). Art and visual perception. Berkeley: University of California Press.
Spatial language in a child who is blind
Landau, B., Gleitman, L. R., & Landau, B. (2009). Language and experience: Evidence from the blind child (Vol. 8). Cambridge, MA: Harvard University Press.
Landau, B., Spelke, E., & Gleitman, H. (1984). Spatial knowledge in a young blind child. Cognition, 16(3), 225–260.
Claiming propositions as minimal units of thought
Anderson, J. R. (2013). The architecture of cognition. New York, NY: Psychology Press.
Pylyshyn, Z. W. (1973). What the mind’s eye tells the mind’s brain: A critique of mental imagery. Psychological Bulletin, 80(1), 1.
Spatial thinking as foundation for language
Fauconnier, G. (1994). Mental spaces: Aspects of meaning construction in natural language. Cambridge, England: Cambridge University Press.
Fauconnier, G., & Sweetser, E. (Eds.). (1996). Spaces, worlds, and grammar. Chicago, IL: University of Chicago Press.
Lakoff, G., & Johnson, M. (2008). Metaphors we live by. Chicago, IL: University of Chicago Press.
Talmy, L. (1983). How language structures space. In H. L. Pick & L. P. Acredolo (Eds.), Spatial orientation (pp. 225–282). Boston, MA: Springer.
CHAPTER EIGHT: SPACES WE CREATE: MAPS, DIAGRAMS, SKETCHES, EXPLANATIONS, COMICS
Paraphrased Pessoa quote
Art proves that life is not enough. (n.d.). AZ Quotes. Retrieved from https://www.azquotes.com/author/11564-Fernando_Pessoa?p=3
Neanderthal cave paintings in Spain
Hoffmann, D. L., Standish, C. D., García-Diez, M., Pettitt, P. B., Milton, J. A., Zilhão, J.,… Lorblanchet, M. (2018). U-Th dating of carbonate crusts reveals Neanderthal origin of Iberian cave art. Science, 359(6378), 912–915.
Cognitive design principles
Adapted from Tversky, B., Morrison, J. B., & Betrancourt, M. (2002). Animation: Can it facilitate? International Journal of Human-Computer Studies, 57(4), 247–262.
Also see Norman, D. (2013). The design of everyday things: Revised and expanded edition. New York, NY: Basic Books.
History of writing
Gelb, I. J. (1952). A study of writing. Chicago, IL: University of Chicago Press.
Oldest map (so far)
Utrilla, P., Mazo, C., Sopena, M. C., Martínez-Bea, M., & Domingo, R. (2009). A Paleolithic map from 13,660 calBP: Engraved stone blocks from the Late Magdalenian in Abauntz Cave (Navarra, Spain). Journal of Human Evolution, 57(2), 99–111.
History of calendars
Boorstin, D. J. (1985). The discoverers: A history of man’s search to know his world and himself. New York, NY: Vintage.
Maps of constellations in ancient caves
Rappenglück, M. (1997). The Pleiades in the “Salle des Taureaux,” grotte de Lascaux. Does a rock picture in the cave of Lascaux show the open star cluster of the Pleiades at the Magdalénien era (ca 15.300 BC)? In C. Jaschek & F. Atrio Barendela (Eds.), Proceedings of the IVth SEAC Meeting “Astronomy and Culture” (pp. 217–225). Salamanca, Spain: University of Salamanca.
Wikipedia. (n.d.). Star chart. Retrieved from https://en.wikipedia.org/wiki/Star_chart
Native American hand maps, annotated with gestures
Finney, B. (1998). Nautical cartography and traditional navigation in Oceania. In D. Woodward & G. M. Lewis (Eds.), The history of cartography. Vol. 2, Book Three: Cartography in the traditional African, American, Arctic, Australian, and Pacific societies (pp. 443–492). Chicago, IL: University of Chicago Press.
Lewis, G. M. (1998). Maps, mapmaking, and map use by native North Americans. In D. Woodward & G. M. Lewis (Eds.), The history of cartography. Vol. 2, Book Three: Cartography in the traditional African, American, Arctic, Australian, and Pacific societies (pp. 51–182). Chicago, IL: University of Chicago Press.
Smethurst, G. (1905). A narrative of an extraordinary escape out of the hands of the Indians, in the gulph of St. Lawrence. Edited by W. F. Ganong. Whitefish, MT: Kessinger Publishing. (Original work published London, 1774).
Depictive maps in Aztec codices
Boone, E. H. (2010). Stories in red and black: Pictorial histories of the Aztecs and Mixtecs. Austin: University of Texas Press.
Syntax and semantics of sketch maps
Denis, M. (1997). The description of routes: A cognitive approach to the production of spatial discourse. Cahiers de Psychologie, 16, 409–458.
Tversky, B., & Lee, P. U. (1998). How space structures language. In C. Freksa, W. Brauer, C. Habel, & K. F. Wender (Eds.), Spatial cognition III [Lecture Notes in Computer Science] (Vol. 1404, pp. 157–175). Berlin, Germany: Springer, Berlin, Heidelberg.
Tversky, B., & Lee, P. U. (1999). Pictorial and verbal tools for conveying routes. In International Conference on Spatial Information Theory (pp. 51–64). Berlin, Germany: Springer, Berlin, Heidelberg.
Empirically establishing cognitive guidelines for map design
Agrawala, M., & Stolte, C. (2001, August). Rendering effective route maps: Improving usability through generalization. Proceedings of the 28th Annual Conference on Computer Graphics and Interactive Techniques, 241–249.
Tversky, B., Agrawala, M., Heiser, J., Lee, P., Hanrahan, P., Phan, D.,… Daniel, M.-P. (2006). Cognitive design principles for automated generation of visualizations. In G. L. Allen (Ed.), Applied spatial cognition: From research to cognitive technology (pp. 53–75). New York, NY: Psychology Press.
Three Ps
(production, preference, and performance) for designing designs
Kessell, A., & Tversky, B. (2011). Visualizing space, time, and agents: Production, performance, and preference. Cognitive Processing, 12(1), 43–52.
Interpreting the Ishango rod
Pletser, V., & Huylebrouck, D. (1999). The Ishango artefact: The missing base 12 link. FORMA-TOKYO, 14(4), 339–346.
Pletser, V., & Huylebrouck, D. (2008, January). An interpretation of the Ishango rods. In Proceedings of the Conference Ishango, 22000 and 50 Years Later: The Cradle of Mathematics (pp. 139–170). Brussels, Belgium: Royal Flemish Academy of Belgium, KVAB.
Development of understanding of number
Gelman, R., & Gallistel, C. R. (1986). The child’s understanding of number. Cambridge, MA: Harvard University Press.
Formal notations are diagrams
Landy, D., & Goldstone, R. L. (2007). Formal notations are diagrams: Evidence from a production task. Memory & Cognition, 35(8), 2033–2040.
People use space to solve math problems; proofs are stories
Landy, D., & Goldstone, R. L. (2007). How abstract is symbolic thought? Journal of Experimental Psychology: Learning, Memory, and Cognition, 33(4), 720.
Eastern environments more complex than Western in eyes of both
Miyamoto, Y., Nisbett, R. E., & Masuda, T. (2006). Culture and the physical environment: Holistic versus analytic perceptual affordances. Psychological Science, 17(2), 113–119.
Chinese arithmetic diagrams rated more complex than US
Wang, E. (2011). Culture and math visualization: Comparing American and Chinese math images (Unpublished master’s thesis). Columbia Teachers College, New York, NY.
Zheng, F. (2015). Math visualizations across cultures: Comparing Chinese and American math images. (Unpublished master’s thesis). Columbia Teachers College, New York, NY.
Measurement and calculation can reduce some biases and error
Kahneman, D., & Tversky, A. (2013). Choices, values, and frames. In W. Ziemba & L. C. MacLean (Eds.), Handbook of the fundamentals of financial decision making: Part I (pp. 269–278). Hackensack, NJ: World Scientific Publishing Co.
Tversky, A., & Kahneman, D. (1974). Judgment under uncertainty: Heuristics and biases. Science, 185(4157), 1124–1131.