Educational Psychology Review

, Volume 27, Issue 3, pp 371–389 | Cite as

The Enactive Roots of STEM: Rethinking Educational Design in Mathematics

  • Daniel D. Hutto
  • Michael D. Kirchhoff
  • Dor Abrahamson
Review Article

Abstract

New and radically reformative thinking about the enactive and embodied basis of cognition holds out the promise of moving forward age-old debates about whether we learn and how we learn. The radical enactive, embodied view of cognition (REC) poses a direct, and unmitigated, challenge to the trademark assumptions of traditional cognitivist theories of mind—those that characterize cognition as always and everywhere grounded in the manipulation of contentful representations of some kind. REC has had some success in understanding how sports skills and expertise are acquired. But, REC approaches appear to encounter a natural obstacle when it comes to understanding skill acquisition in knowledge-rich, conceptually based domains like the hard sciences and mathematics. This paper offers a proof of concept that REC’s reach can be usefully extended into the domain of science, technology, engineering, and mathematics (STEM) learning, especially when it comes to understanding the deep roots of such learning. In making this case, this paper has five main parts. The section “Ancient Intellectualism and the REC Challenge” briefly introduces REC and situates it with respect to rival views about the cognitive basis of learning. The “Learning REConceived: from Sports to STEM?” section outlines the substantive contribution REC makes to understanding skill acquisition in the domain of sports and identifies reasons for doubting that it will be possible to apply the same approach to knowledge-rich STEM domains. The “Mathematics as Embodied Practice” section gives the general layout for how to understand mathematics as an embodied practice. The section “The Importance of Attentional Anchors” introduces the concept “attentional anchor” and establishes why attentional anchors are important to educational design in STEM domains like mathematics. Finally, drawing on some exciting new empirical studies, the section “Seeing Attentional Anchors” demonstrates how REC can contribute to understanding the roots of STEM learning and inform its learning design, focusing on the case of mathematics.

Keywords

Enactivism Ecological dynamics Attentional anchor Mathematics 

References

  1. Abrahamson, D. (2009). Embodied design: constructing means for constructing meaning. Educational Studies in Mathematics, 70(1), 27–47. [Electronic supplementary material at http://edrl.berkeley.edu/publications/journals/ESM/Abrahamson-ESM/ ]. doi:  10.1007/s10649-008-9137-1
  2. Abrahamson, D. (2012a). Discovery reconceived: product before process. For the Learning of Mathematics, 32(1), 8–15.Google Scholar
  3. Abrahamson, D. (2012c). Mathematical Imagery Trainer - Proportion (MIT-P) IPhone/iPad application (Terasoft): iTunes. Retrieved from https://itunes.apple.com/au/app/mathematical-imagery-trainer/id563185943.
  4. Abrahamson, D., & Lindgren, R. (2014). Embodiment and embodied design. In R. K. Sawyer (Ed.), The Cambridge handbook of the learning sciences (2nd ed., pp. 358–376). Cambridge: Cambridge University Press.Google Scholar
  5. Abrahamson, D., & Sánchez–García, R. (2014). Learning is moving in new ways: an ecological dynamics view on learning across the disciplines. Paper presented at the “Embodied cognition in education” symposium (A. Bakker, M. F. van der Schaaf, S. Shayan, & P. Leseman, Chairs), Freudenthal Institute for Science and Mathematics Education, University of Utrecht, The Netherlands.Google Scholar
  6. Abrahamson, D., & Trninic, D. (2015). Bringing forth mathematical concepts: signifying sensorimotor enactment in fields of promoted action. In D. Reid, L. Brown, A. Coles, & M.-D. Lozano (Eds.), Enactivist methodology in mathematics education research [Special issue]. ZDM, 47(2), 295–306.Google Scholar
  7. Abrahamson, D., Trninic, D., Gutiérrez, J. F., Huth, J., & Lee, R. G. (2011). Hooks and shifts: a dialectical study of mediated discovery. Technology, Knowledge and Learning, 16(1), 55–85.Google Scholar
  8. Abrahamson, D., Gutiérrez, J. F., Charoenying, T., Negrete, A. G., & Bumbacher, E. (2012). Fostering hooks and shifts: tutorial tactics for guided mathematical discovery. Technology, Knowledge and Learning, 17(1–2), 61–86.CrossRefGoogle Scholar
  9. Abrahamson, D., Lee, R. G., Negrete, A. G., Gutiérrez, J. F. (2014). Coordinating visualizations of polysemous action: values added for grounding proportion. In F. Rivera, H. Steinbring, & A. Arcavi (Eds.), Visualization as an epistemological learning tool [Special issue]. ZDM–The international Journal on Mathematics Education, 46(1), 79–93.Google Scholar
  10. Aguilera, M., Bedia, M. G., Santos, B. A., & Barandiaran, X. E. (2013). The situated HKB model: how sensorimotor spatial coupling can alter oscillatory brain dynamics. Frontiers in Computational Neuroscience, 7(117), 1–15.Google Scholar
  11. Bamberger, J. (2011). The collaborative invention of meaning: a short history of evolving ideas. Psychology of Music, 39(1), 82–101.CrossRefGoogle Scholar
  12. Barab, S., Zuiker, S., Warren, S., Hickey, D., Ingram-Goble, A., Kwon, E.-J., & Herring, S. C. (2007). Situationally embodied curriculum: relating formalisms and contexts. Science Education, 91, 750–782.CrossRefGoogle Scholar
  13. Bartolini Bussi, M. G., & Mariotti, M. A. (2008). Semiotic mediation in the mathematics classroom: artefacts and signs after a Vygotskian perspective. In L. D. English, M. G. Bartolini Bussi, G. A. Jones, R. Lesh, & D. Tirosh (Eds.), Handbook of international research in mathematics education (2nd ed., pp. 720–749). Mahwah: Lawrence Erlbaum Associates.Google Scholar
  14. Behr, M. J., Harel, G., Post, T., & Lesh, R. (1993). Rational number, ratio, and proportion. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 296–333). NYC: Macmillan.Google Scholar
  15. Boutin, A., Blandin, Y., Massen, C., Heuer, H., & Badets, A. (2014). Conscious awareness of action potentiates sensorimotor learning. Cognition, 133(1), 1–9.CrossRefGoogle Scholar
  16. Brown, M. C., McNeil, N. M., & Glenberg, A. M. (2009). Using concreteness in education: real problems, potential solutions. Child Development Perspectives, 3, 160–164.CrossRefGoogle Scholar
  17. Bruner, J. S. (1960). The process of education: a searching discussion of school education opening new paths to learning and teaching. New York: Vintage.Google Scholar
  18. Carey, S. (2011). Précis of The origin of concepts. Behavioral and Brain Sciences, 34, 113–167.CrossRefGoogle Scholar
  19. Carruthers, P. (2011). Opacity of mind. New York: Oxford University Press.Google Scholar
  20. Chahine, I. C. (2013). The impact of using multiple modalities on students’ acquisition of fractional knowledge: An international study in embodied mathematics across semiotic cultures. The Journal of Mathematical Behavior, 32(3), 434–449.CrossRefGoogle Scholar
  21. Chemero, T. (2009). Radical Embodied Cognitive Science. Cambridge, MA: The MIT Press.Google Scholar
  22. Chow, J. Y., Davids, K., Button, C., Shuttleworth, R., Renshaw, I., & Araújo, D. (2007). The role of nonlinear pedagogy in physical education. Review of Educational Research, 77(3), 251–278.CrossRefGoogle Scholar
  23. Davids, K. (2012). Learning design for nonlinear dynamical movement systems. The Open Sports Sciences Journal, 5(Suppl. 1), 9–16.CrossRefGoogle Scholar
  24. Davids, K., Button, C., & Bennett, S. (2008). Dynamics of skill acquisition. Campaign: Human Kinetics.Google Scholar
  25. Davids, K., Araújo, D., Vilar, L., Renshaw, I., & Pinder, R. A. (2013). An ecological dynamics approach to skill acquisition: Implications for development of talent in sport. Talent Development and Excellence, 5, 21–34.Google Scholar
  26. Davis, P. J., & Hersh, R. (1981). The mathematical experience. Boston: Birkhauser.Google Scholar
  27. de Freitas, E., & Sinclair, N. (2012). Diagram, gesture, agency: theorizing embodiment in the mathematics classroom. Educational Studies in Mathematics, 80(1–2), 133–152.Google Scholar
  28. diSessa, A. A., & Cobb, P. (2004). Ontological innovation and the role of theory in design experiments. The Journal of the Learning Sciences, 13(1), 77–103.CrossRefGoogle Scholar
  29. Dreyfus, H., & Dreyfus, S. (1986). Mind over machine. New York: The Free Press.Google Scholar
  30. Dreyfus, H. L., & Dreyfus, S. E. (1999). The challenge of Merleau-Ponty’s phenomenology of embodiment for cognitive science. In G. Weiss & H. F. Haber (Eds.), Perspectives on embodiment: the intersections of nature and culture (pp. 103–120). New York: Routledge.Google Scholar
  31. Duijzer, C. (2015). How perception guides cognition: Insights from embodied interaction with a tablet application for proportions – an eye-tracking study. Utrecht: Utrecht University.Google Scholar
  32. Ferrara, F. (2014). How multimodality works in mathematical activity: young children graphing motion. International Journal of Science and Mathematics Education, 12(4), 917–939.CrossRefGoogle Scholar
  33. Fischer, U., Moeller, K., Bientzle, M., Cress, U., & Nuerk, H.-C. (2011). Sensori-motor spatial training of number magnitude representation. Psychonomic Bulletin & Review, 18(1), 177–183.CrossRefGoogle Scholar
  34. Fodor, J. A. (1975). The language of thought. Cambridge: Harvard University Press.Google Scholar
  35. Fodor, J. A. (2003). Hume variations. Oxford: Oxford University Press.Google Scholar
  36. Fodor, J. A. (2007). The revenge of the given. In B. McLaughlin & J. Cohen (Eds.), Contemporary debates in philosophy of mind (pp. 105–116). Oxford: Blackwell.Google Scholar
  37. Fodor, J. A., & Pylyshyn, Z. (2015). Minds without meanings. Cambridge: MIT Press.Google Scholar
  38. Gallagher, S. (2005). How the body shapes the mind. Oxford: Oxford University Press.Google Scholar
  39. Gerofsky, S. (2011). Seeing the graph vs. being the graph: gesture, engagement and awareness in school mathematics. In G. Stam & M. Ishino (Eds.), Integrating gestures (pp. 245–256). Amsterdam: John Benjamins.Google Scholar
  40. Gibson, J. J. (1977). The theory of affordances. In R. Shaw & J. Bransford (Eds.), Perceiving, acting and knowing: toward an ecological psychology (pp. 67–82). Hillsdale: Lawrence Erlbaum Associates.Google Scholar
  41. Ginsburg, C. (2010). The intelligence of moving bodies: a somatic view of life and its consequences. Santa Fe: AWAREing Press.Google Scholar
  42. Goldstone, R. L., Landy, D. H., & Son, J. Y. (2009). The education of perception. Topics in Cognitive Science, 2(2), 265–284.CrossRefGoogle Scholar
  43. Howison, M., Trninic, D., Reinholz, D., & Abrahamson, D. (2011). The mathematical imagery trainer: from embodied interaction to conceptual learning. In G. Fitzpatrick, C. Gutwin, B. Begole, W. A. Kellogg, & D. Tan (Eds.), Proceedings of the annual meeting of The association for computer machinery special interest group on computer human interaction: “human factors in computing systems” (CHI 2011) (pp. 1989–1998). Vancouver: ACM Press.Google Scholar
  44. Hutto, D. D., & Myin, E. (2013). Radical enactivism. Cambridge: The MIT Press.Google Scholar
  45. Hutto, D. D., & Sánchez-García, R. (2014). Choking RECtified: embodied expertise beyond Dreyfus. Phenomenology and the Cognitive Sciences, 1–23. doi:  10.1007/s11097-014-9380-0
  46. Hutto, D. D., Kirchhoff, M. D., Myin, E. (2014). Extensive enactivism: Why keep it all in? Frontiers in Human Neuroscience, 1–11. DOI:  10.3389/fnhum.2014. 00706.
  47. Ingold, T. (2000). The perception of the environment: essays on livelihood, dwelling, and skill (2nd ed.). London: Routledge.CrossRefGoogle Scholar
  48. Karplus, R., Pulos, S., & Stage, E. K. (1983). Proportional reasoning of early adolescents. In R. Lesh & M. Landau (Eds.), Acquisition of mathematics concepts and processes (pp. 45–89). New York: Academic Press.Google Scholar
  49. Kelso, J. A. S. (1984). Phase transitions and critical behavior in human bimanual coordination. American Journal of Physiology: Regulatory, Integrative and Comparative, 246(6), R1000–R1004.Google Scholar
  50. Kelso, J. A. S. (1995). Dynamic patterns: the self-organization of brain and behavior. Cambridge: MIT Press.Google Scholar
  51. Kelso, J. A. S., & Engstrøm, D. A. (2006). The complementary nature. Cambridge: MIT Press.Google Scholar
  52. Kirsh, D. (2010). Thinking with external representations. AI & Society, 25, 441–454.CrossRefGoogle Scholar
  53. Kirsh, D., & Maglio, P. (1994). On distinguishing epistemic from pragmatic action. Cognitive Science, 18(4), 513–549.CrossRefGoogle Scholar
  54. Kostrubiec, V., Zanone, P.-G., Fuchs, A., Kelso, J. A. S. (2012). Beyond the blank slate: routes to learning new coordination patterns depend on the intrinsic dynamics of the learner -- experimental evidence and theoretical model. Frontiers in Human Neuroscience, 6. doi:  10.3389/fnhum.2012.00222.
  55. Laurence, S., & Margolis, E. (2007). Linguistic Determinism and the Innate Basis of Number. In P. Carruthers et al. (eds.), The Innate Mind, vol. 3: Foundations and the Future (Oxford University Press), pp. 139–169.Google Scholar
  56. Lee, R. G., Hung, M., Negrete, A. G., Abrahamson, D. (2013). Rationale for a ratio-based conceptualization of slope: results from a design-oriented embodied-cognition domain analysis. Paper presented at the annual meeting of the American Educational Research Association (Special Interest Group on Research in Mathematics Education), San Francisco, April 27 - May 1.Google Scholar
  57. Liao, C., & Masters, R. S. (2001). Analogy learning: a means to implicit motor learning. Journal of Sports Sciences, 19, 307–319.CrossRefGoogle Scholar
  58. Loader, P. (2012). The epistemic/pragmatic dichotomy. Philosophical Explorations: An International Journal for the Philosophy of Mind and Action, 15(2), 219–232.CrossRefGoogle Scholar
  59. Marghetis, T., & Núñez, R. l. (2013). The motion behind the symbols: a vital role for dynamism in the conceptualization of limits and continuity in expert mathematics. Topics in Cognitive Science. doi: 10.1111/tops.12013.Google Scholar
  60. Nemirovsky, R. (2003). Three conjectures concerning the relationship between body activity and understanding mathematics. In R. Nemirovsky, M. Borba, N. A. Pateman, B. J. Dougherty, & J. T. Zilliox (Eds.), Proceedings of the twenty seventh annual meeting of the international group for the psychology of mathematics education (Vol. 1 (pp. 105–109). Honolulu: OH: Eric Clearinghouse for Science, Mathematics, and Environmental Education.Google Scholar
  61. Newell, K. M., & Ranganathan, R. (2010). Instructions as constraints in motor skill acquisition. In I. Renshaw, K. Davids, & G. J. P. Savelsbergh (Eds.), Motor learning in practice: a constraints-led approach (pp. 17–32). Florence: Routledge.Google Scholar
  62. Núñez, R. E., Edwards, L. D., & Matos, J. F. (1999). Embodied cognition as grounding for situatedness and context in mathematics education. Educational Studies in Mathematics, 39, 45–65.CrossRefGoogle Scholar
  63. Olson, D. R. (1994). The world on paper. Cambridge: Cambridge University Press.Google Scholar
  64. Petrick, C. J., & Martin, T. (2011). Hands up, know body move: learning mathematics through embodied actions. Austin: University of Texas at Austin.Google Scholar
  65. Piaget, J. (1968). Genetic epistemology (E. Duckworth, trans.). New York: Columbia University Press.Google Scholar
  66. Piaget, J. (1970). Structuralism (C. Maschler, trans.). New York: Basic Books.Google Scholar
  67. Piaget, J., Inhelder, B., & Szeminska, A. (1960). The child’s conception of geometry (E. A. Lunzer, trans.) (E. A. Lunzer, trans.). New York: Basic Books.Google Scholar
  68. Presmeg, N. C. (2006). Research on visualization in learning and teaching mathematics: emergence from psychology. In A. Gutiérrez & P. Boero (Eds.), Handbook of research on the psychology of mathematics education: past, present, and future (pp. 205–235). Rotterdam: Sense Publishers.Google Scholar
  69. Prinz, J. (2002). Furnishing the mind: concepts and their perceptual basis. Cambridge: MIT Press.Google Scholar
  70. Radford, L. (2013). Three key concepts of the theory of objectification: knowledge, knowing, and learning. In L. Radford (Ed.), Theory of objectification: knowledge, knowing, and learning [Special issue]. REDIMAT - Journal of Research in Mathematics Education, 2(1), 7–44.Google Scholar
  71. Reed, E. S., & Bril, B. (1996). The primacy of action in development. In M. L. Latash & M. T. Turvey (Eds.), Dexterity and its development (pp. 431–451). Mahwah: Lawrence Erlbaum Associates.Google Scholar
  72. Reinholz, D., Trninic, D., Howison, M., & Abrahamson, D. (2010). It’s not easy being green: embodied artifacts and the guided emergence of mathematical meaning. In P. Brosnan & D. Erchick (Eds.), Proceedings of the thirty-second annual meeting of the north-american chapter of the international group for the psychology of mathematics education (PME-NA 32) (Vol. VI, Ch. 18: technology (pp. 1488–1496). Columbus: PME-NA.Google Scholar
  73. Renshaw, I., Chow, J. Y., Davids, K., & Hammond, J. (2010). A constraints-led perspective for understanding skill acquisition and game play. Physical Education & Sport Pedagogy, 15(2), 117–137.CrossRefGoogle Scholar
  74. Rick, J. (2012). Proportion: a tablet app for collaborative learning. In H. Schelhowe (Ed.), Proceedings of the 11th Annual Interaction Design and Children Conference (IDC 2012) (Vol. “Demo Papers”, pp. 316–319). Bremen, Germany: ACM-IDC.Google Scholar
  75. Roth, W.-M. (2010). Incarnation: radicalizing the embodiment of mathematics. For the Learning of Mathematics, 30(2), 8–17.Google Scholar
  76. Roth, W.-M. (2015). Excess of graphical thinking: movement, mathematics and flow. For the Learning of Mathematics, 35, 1–7.Google Scholar
  77. Roth, W-M., & Thom, J. S. (2009). Bodily experience and mathematical conceptions: from classical views to a phenomenological reconceptualization. In L. Radford, L. Edwards, & F. Arzarello (Eds.), Gestures and multimodality in the construction of mathematical meaning [Special issue]. Educational Studies in Mathematics, 70(2), 175–189.Google Scholar
  78. Rotman, B. (2000). Mathematics as sign: writing, imagining, counting. Stanford: Stanford University Press.Google Scholar
  79. Schmandt-Besserat, D. E. (1992). Before writing Vol. 1: from counting to cuneiform. Austin: University of Texas Press.Google Scholar
  80. Schneider, B., Bumbacher, E., & Blikstein, P. (2015). Discovery versus direct instruction: learning outcomes of two pedagogical models using tangible interface. In T. Koschmann, P. Häkkinen, & P. Tchounikine (Eds.), Exploring the material conditions of learning: opportunities and challenges for CSCL,“ the proceedings of the Computer Supported Collaborative Learning (CSCL) conference. Gothenburg: ISLS.Google Scholar
  81. Schwartz, D. L., & Martin, T. (2006). Distributed learning and mutual adaptation. In Harnad, S., & Dror, I. E. (Eds.), Distributed cognition [Special issue]. Pragmatics & Cognition, 14(2), 313–332.Google Scholar
  82. Sfard, A. (1994). Reification as the birth of metaphor. For the Learning of Mathematics, 14(1), 44–55.Google Scholar
  83. Sfard, A. (2002). The interplay of intimations and implementations: generating new discourse with new symbolic tools. The Journal of the Learning Sciences, 11(2&3), 319–357.CrossRefGoogle Scholar
  84. Shayan, S., Abrahamson, D., Bakker, A., Duijzer, C., & van der Schaaf, M. (2015). The emergence of proportional reasoning from embodied interaction with a tablet application: an eye-tracking study. In L. Gómez Chova, A. López Martínez, & I. Candel Torres (Eds.), Proceedings of the 9th international technology, education, and development conference (INTED 2015) (pp. 5732–5741). Madrid: IATED.Google Scholar
  85. Shea, N. (2011). Acquiring a new concept is not explicable-by-content. Behavioral and Brain Sciences, 34(3), 148–149.CrossRefGoogle Scholar
  86. Siegler, R. S. (2006). Microgenetic analyses of learning. In D. Kuhn & R. S. Siegler (Eds.), Handbook of child psychology (6 ed., Vol. 2, cognition, perception, and language (pp. 464–510). Hoboken: Wiley.Google Scholar
  87. Sinclair, N., & Gol Tabaghi, S. (2010). Drawing space: mathematicians’ kinetic conceptions of eigenvectors. Educational Studies in Mathematics, 74(3), 223–240.CrossRefGoogle Scholar
  88. Smith, L. B., & Gasser, M. (2005). The development of embodied cognition: six lessons from babies. Artificial Life, 11, 13–30.CrossRefGoogle Scholar
  89. Starra, A., Libertus, M. E., & Brannona, E. M. (2013). Number sense in infancy predicts mathematical abilities in childhood. Proceedings of the National Academy of Sciences of the United States of America, 110(45), 18116–18120.CrossRefGoogle Scholar
  90. Stevens, R., & Hall, R. (1998). Disciplined perception: learning to see in technoscience. In M. Lampert & M. L. Blunk (Eds.), Talking mathematics in school: studies of teaching and learning (pp. 107–149). New York: Cambridge University Press.Google Scholar
  91. Stoate, I., & Wulf, G. (2011). Does the attentional focus adapted by swimmers affect their performance? Journal of Sport science & Coaching, 6(1), 99–108.CrossRefGoogle Scholar
  92. Strauss, A. L., & Corbin, J. (1990). Basics of qualitative research: grounded theory procedures and techniques. Newbury Park: Sage Publications.Google Scholar
  93. Thelen, E., & Smith, L. B. (1994). A dynamic systems approach to the development of cognition and action. Cambridge: MIT Press.Google Scholar
  94. Thelen, E., & Smith, L. B. (2006). Dynamic systems theories. In R. M. Lerner (Ed.), Handbook of child psychology (Theoretical models of human development, Vol. 1, pp. 258–312). Hoboken: Wiley.Google Scholar
  95. Thompson, E. (2007). Mind in life: biology, phenomenology, and the sciences of mind. Cambridge: Harvard University Press.Google Scholar
  96. Trninic, D., & Abrahamson, D. (2012). Embodied artifacts and conceptual performances. In K. Thompson, M. J. Jacobson, & P. Reimann (Eds.), Proceedings of the 10th international conference of the learning sciences: future of learning (ICLS 2012) (Vol. 1 (pp. 283–290). Sydney: University of Sydney / ISLS.Google Scholar
  97. Uttal, D. H., & O’Doherty, K. (2008). Comprehending and learning from visual representations: a developmental approach. In J. Gilbert, M. Reiner, & M. Nakhleh (Eds.), Visualization: theory and practice in science education (pp. 53–72). New York: Springer.CrossRefGoogle Scholar
  98. van Dijk, L., Withagen, R., & Bongers, R. M. (2015). Information without content: a Gibsonian reply to enactivists’ worries. Cognition, 134, 210–214.CrossRefGoogle Scholar
  99. Van Dooren, W., De Bock, D., & Verschaffel, L. (2010). From addition to multiplication … and back. The development of students’ additive and multiplicative reasoning skills. Cognition and Instruction, 28(3), 360–381.CrossRefGoogle Scholar
  100. Varela, F., Thompson, E. & Rosch, E. (1991). The embodied mind: cognitive science and human experience. Cambridge, MA: The MIT Press.Google Scholar
  101. von Glasersfeld, E. (1983). Learning as constructive activity. In J. C. Bergeron & N. Herscovics (Eds.), Proceedings of the 5th annual meeting of the North American group for the psychology of mathematics education (Vol. 1, pp. 41–69). Montreal: PME-NA.Google Scholar
  102. Wittmann, M. C., Flood, V. J., & Black, K. E. (2013). Algebraic manipulation as motion within a landscape. Educational Studies in Mathematics, 82(2), 169–181.CrossRefGoogle Scholar
  103. Wulf, G., & Su, J. (2007). An external focus of attention enhances golf shot accuracy in beginners and experts. Research Quarterly for Exercise and Sport, 78, 384–389.CrossRefGoogle Scholar
  104. Zarghami, M., Saemi, E., & Fathi, I. (2012). External focus of attention enhances discus throwing performance. Kinesiology, 44(1), 47–51.Google Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Daniel D. Hutto
    • 1
  • Michael D. Kirchhoff
    • 1
  • Dor Abrahamson
    • 2
  1. 1.PhilosophyUniversity of WollongongWollongongAustralia
  2. 2.Graduate School of EducationBerkeley University of CaliforniaBerkeleyUSA

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