Technology, Knowledge and Learning

, Volume 16, Issue 1, pp 55–85 | Cite as

Hooks and Shifts: A Dialectical Study of Mediated Discovery

  • Dor Abrahamson
  • Dragan Trninic
  • Jose F. Gutiérrez
  • Jacob Huth
  • Rosa G. Lee
Article

Abstract

Radical constructivists advocate discovery-based pedagogical regimes that enable students to incrementally and continuously adapt their cognitive structures to the instrumented cultural environment. Some sociocultural theorists, however, maintain that learning implies discontinuity in conceptual development, because novices must appropriate expert analyses that are schematically incommensurate with their naive views. Adopting a conciliatory, dialectical perspective, we concur that naive and analytic schemes are operationally distinct and that cultural–historical artifacts are instrumental in schematic reconfiguration yet argue that students can be steered to bootstrap this reconfiguration in situ; moreover, students can do so without any direct modeling from persons fluent in the situated use of the artifacts. To support the plausibility of this mediated-discovery hypothesis, we present and analyze vignettes selected from empirical data gathered in a conjecture-driven design-based research study investigating the microgenesis of proportional reasoning through guided engagement in technology-based embodied interaction. 22 Grade 4–6 students participated in individual or paired semi-structured tutorial clinical interviews, in which they were tasked to remote-control the location of virtual objects on a computer display monitor so as to elicit a target feedback of making the screen green. The screen would be green only when the objects were manipulated on the screen in accord with a “mystery” rule. Once the participants had developed and articulated a successful manipulation strategy, we interpolated various symbolic artifacts onto the problem space, such as a Cartesian grid. Participants appropriated the artifacts as strategic or discursive means of accomplishing their goals. Yet, so doing, they found themselves attending to and engaging certain other embedded affordances in these artifacts that they had not initially noticed yet were supporting performance subgoals. Consequently, their operation schemas were surreptitiously modulated or reconfigured—they saw the situation anew and, moreover, acknowledged their emergent strategies as enabling advantageous interaction. We propose to characterize this two-step guided re-invention process as: (a) hooking—engaging an artifact as an enabling, enactive, enhancing, evaluative, or explanatory means of effecting and elaborating a current strategy; and (b) shifting—tacitly reconfiguring current strategy in response to the hooked artifact’s emergent affordances that are disclosed only through actively engaging the artifact. Looking closely at two cases and surveying others, we delineate mediated interaction factors enabling or impeding hook-and-shift learning. The apparent cognitive–pedagogical utility of these behaviors suggests that this ontological innovation could inform the development of a heuristic design principle for deliberately fostering similar learning experiences.

Keywords

Additive reasoning Cognition Conceptual change Design-based research Discovery Embodied interaction Functional extension Guided reinvention Mathematics education Proportion Proportional reasoning Remote control Sociocultural Symbolic artifact Virtual object 

References

  1. Abrahamson, D. (2008). The abduction of Peirce: the missing link between perceptual judgment and mathematical reasoning? Paper presented at the Townsend Working Group in Neuroscience and Philosophy (A. Rokem, J. Stazicker, & A. Noë, Organizers). UC Berkeley. Accessed June 1, 2010 at http://www.archive.org/details/ucb_neurophilosophy_2008_12_09_Dor_Abrahamson.
  2. Abrahamson, D. (2009a). A student’s synthesis of tacit and mathematical knowledge as a researcher’s lens on bridging learning theory. In M. Borovcnik & R. Kapadia (Eds.), Research and developments in probability education [Special Issue]. International Electronic Journal of Mathematics Education, 4(3), 195–226. Accessed Jan. 191, 2010 at http://www.iejme.com/032009/main.htm.
  3. Abrahamson, D. (2009b). Embodied design: Constructing means for constructing meaning. Educational Studies in Mathematics, 70(1), 27–47.CrossRefGoogle Scholar
  4. Abrahamson, D. (2009c). Orchestrating semiotic leaps from tacit to cultural quantitative reasoning—The case of anticipating experimental outcomes of a quasi-binomial random generator. Cognition and Instruction, 27(3), 175–224.CrossRefGoogle Scholar
  5. Abrahamson, D., Gutiérrez, J. F., Lee, R. G., Reinholz, D., & Trninic, D. (2011). From tacit sensorimotor coupling to articulated mathematical reasoning in an embodied design for proportional reasoning. In R. Goldman (Chair), H. Kwah & D. Abrahamson (Organizers), & R. P. Hall (Discussant), Diverse perspectives on embodied learning: what’s so hard to grasp? Paper presented at the annual meeting of the American Educational Research Association (SIG Advanced Technologies for Learning. New Orleans, LA, April 8–12, 2011, http://edrl.berkeley.edu/sites/default/files/Abrahamson-etal.AERA2011-EmbLearnSymp.pdf.
  6. Abrahamson, D., & Howison, M. (2008). Kinemathics: kinetically induced mathematical learning. Paper presented at the UC Berkeley Gesture Study Group (E. Sweetser, Director), December 5, 2008. http://edrl.berkeley.edu/projects/kinemathics/Abrahamson-Howison-2008_kinemathics.pdf, http://edrl.berkeley.edu/projects/kinemathics/MIT.mov.
  7. Abrahamson, D., & Howison, M. (2010a). Embodied artifacts: Coordinated action as an object-to-think-with. In D. L. Holton (Organizer & Chair) & J. P. Gee (Discussant), Embodied and enactive approaches to instruction: Implications and innovations. Paper presented at the annual meeting of the American Educational Research Association, April 30–May 4. http://gse.berkeley.edu/faculty/DAbrahamson/publications/Abrahamson-Howison-AERA2010-ReinholzTrninic.pdf.
  8. Abrahamson, D., & Howison, M. (2010b). Kinemathics: Exploring kinesthetically induced mathematical learning. Paper presented at the annual meeting of the American Educational Research Association, April 30–May 4.Google Scholar
  9. Abrahamson, D., & Trninic, D. (in press). Toward an embodied-interaction design framework for mathematical concepts. In P. Blikstein & P. Marshall (Eds.), Proceedings of the 10th annual interaction design and children conference (IDC 2011). Ann Arbor, MI: IDC.Google Scholar
  10. Bakker, A., & Derry, J. (2011). Lessons from inferentialism for statistics education. In K. Makar & D. Ben-Zvi (Eds.), The role of context in developing students’ reasoning about informal statistical inference [Special issue]. Mathematical Thinking and Learning, 13(1&2), 5–26.Google Scholar
  11. Bamberger, J. (1999). Action knowledge and symbolic knowledge: The computer as mediator. In D. Schön, B. Sanyal, & W. Mitchell (Eds.), High technology and low income communities (pp. 235–262). Cambridge, MA: MIT Press.Google Scholar
  12. Bamberger, J. (2010). Noting time. Min-Ad: Israel studies in musicology online (Vol. 8, issue 1&2), Retrieved November 9, 2010 from, http://www.biu.ac.il/hu/mu/min-ad/2010/2002-Bamberger-Noting.pdf.
  13. Bamberger, J., & Schön, D. A. (1983). Learning as reflective conversation with materials: Notes from work in progress. Art Education, 36(2), 68–73.Google Scholar
  14. Bamberger, J., & Schön, D. A. (1991). Learning as reflective conversation with materials. In F. Steier (Ed.), Research and reflexivity (pp. 186–209). London: SAGE Publications.Google Scholar
  15. Barsalou, L. W. (1999). Perceptual symbol systems. Behavioral and Brain Sciences, 22, 577–660.Google Scholar
  16. 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 revised edition ed., pp. 720–749). Mahwah, NG: Lawrence Erlbaum Associates.Google Scholar
  17. 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
  18. Borovcnik, M., & Bentz, H.-J. (1991). Empirical research in understanding probability. In R. Kapadia & M. Borovcnik (Eds.), Chance encounters: Probability in education (pp. 73–105). Dordrecht, Holland: Kluwer.Google Scholar
  19. Botzer, G., & Yerushalmy, M. (2008). Embodied semiotic activities and their role in the construction of mathematical meaning of motion graphs. International Journal of Computers for Mathematical Learning, 13(2), 111–134.CrossRefGoogle Scholar
  20. Brock, W. H., & Price, M. H. (1980). Squared paper in the nineteenth century: Instrument of science and engineering, and symbol of reform in mathematical education. Educational Studies in Mathematics, 11(4), 365–381.CrossRefGoogle Scholar
  21. Clement, J. (2000). Analysis of clinical interviews: Foundations and model viability. In A. E. Kelly & R. A. Lesh (Eds.), Handbook of research design in mathematics and science education (pp. 547–589). Mahwah, NJ: Lawrence Erlbaum Associates.Google Scholar
  22. Cole, M., & Wertsch, J. V. (1996). Beyond the individual-social antinomy in discussions of Piaget and Vygotsky. Human Development, 39(5), 250–256.CrossRefGoogle Scholar
  23. Collins, A. (1992). Towards a design science of education. In E. Scanlon & T. O’shea (Eds.), New directions in educational technology (pp. 15–22). Berlin: Springer.Google Scholar
  24. Collins, A., & Ferguson, W. (1993). Epistemic forms and epistemic games: Structures and strategies to guide inquiry. Educational Psychologist, 28(1), 25–42.CrossRefGoogle Scholar
  25. Confrey, J. (1998). Building mathematical structure within a conjecture driven teaching experiment on splitting. In S. B. Berenson, K. R. Dawkins, M. Blanton, W. N. Coulombe, J. Kolb, K. Norwood, & L. Stiff (Eds.), Proceedings of the twentieth annual conference of the North American chapter of the international group for the psychology of mathematics education (pp. 39–48). Columbus, OH: Eric Clearinghouse for Science, Mathematics, and Environmental Education.Google Scholar
  26. Confrey, J. (2005). The evolution of design studies as methodology. In R. K. Sawyer (Ed.), The Cambridge handbook of the learning sciences (pp. 135–151). Cambridge, MA: Cambridge University Press.Google Scholar
  27. diSessa, A. A. (1995). Designing Newton’s laws: patterns of social and representational feedback in a learning task. In R.-J. Beun, M. Baker, & M. Reiner (Eds.), Dialogue and interaction: modeling interaction in intelligent tutoring systems (pp. 105–122). Berlin: Springer.Google Scholar
  28. diSessa, A. A. (2005). A history of conceptual change research: threads and fault lines. In R. K. Sawyer (Ed.), The Cambridge handbook of the learning sciences (pp. 265–282). Cambridge, MA: Cambridge University Press.Google Scholar
  29. diSessa, A. A. (2007). An interactional analysis of clinical interviewing. Cognition and Instruction, 25(4), 523–565.CrossRefGoogle Scholar
  30. diSessa, A. A. (2008). A note from the editor. Cognition and Instruction, 26(4), 427–429.CrossRefGoogle Scholar
  31. 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
  32. diSessa, A. A., Hammer, D., Sherin, B., & Kolpakowski, T. (1991). Inventing graphing: Meta-representational expertise in children. Journal of Mathematical Behavior, 10(2), 117–160.Google Scholar
  33. diSessa, A. A., Philip, T. M., Saxe, G. B., Cole, M., & Cobb, P. (2010). Dialectical approaches to cognition (Symposium). Paper presented at the Annual Meeting of American Educational Research Association, Denver, CO, April 30–May 4.Google Scholar
  34. Edelson, D. C. (2002). Design research: What we learn when we engage in design. The Journal of the Learning Sciences, 11(1), 105–121.CrossRefGoogle Scholar
  35. Engeström, Y. (2008). From design experiments to formative interventions. In G. Kanselaar, J. V. Merriënboer, P. Kirschner, & T. D. Jong (Eds.), Proceedings of the 8th international conference of the learning sciences (Vol. 1, pp. 3–24). Utrecht, the Netherlands: ISLS.Google Scholar
  36. Freudenthal, H. (1968). Why to teach mathematics so as to be useful. Educational Studies in Mathematics, 1(1/2), 3–8.CrossRefGoogle Scholar
  37. Freudenthal, H. (1971). Geometry between the devil and the deep sea. Educational Studies in Mathematics, 3(3/4), 413–435.CrossRefGoogle Scholar
  38. Freudenthal, H. (1986). Didactical phenomenology of mathematical structures. Dordrecht, The Netherlands: Kluwer Academic Publishers.Google Scholar
  39. Fuson, K. C., & Abrahamson, D. (2005). Understanding ratio and proportion as an example of the apprehending zone and conceptual-phase problem-solving models. In J. Campbell (Ed.), Handbook of mathematical cognition (pp. 213–234). New York: Psychology Press.Google Scholar
  40. Gelman, R. (1998). Domain specificity in cognitive development: Universals and nonuniversals. In M. Sabourin, F. Craik, & M. Robert (Eds.), Advances in psychological science: (Vol. 2 biological and cognitive aspects). Hove, England: Psychology Press Ltd. Publishers.Google Scholar
  41. Gelman, R., & Williams, E. (1998). Enabling constraints for cognitive development and learning: Domain specificity and epigenesis. In D. Kuhn & R. Siegler (Eds.), Cognition, perception and language (5th ed., Vol. 2, pp. 575–630). New York: Wiley.Google Scholar
  42. Gigerenzer, G., & Brighton, H. (2009). Homo Heuristicus: Why biased minds make better inferences. Topics in Cognitive Science, 1(1), 107–144.CrossRefGoogle Scholar
  43. Ginsburg, H. P. (1997). Entering the child’s mind. New York: Cambridge University Press.CrossRefGoogle Scholar
  44. Glaser, B. G., & Strauss, A. L. (1967). The discovery of grounded theory: Strategies for qualitative research. Chicago: Aldine Publishing Company.Google Scholar
  45. Goldin, G. A. (1987). Levels of language in mathematical problem solving. In C. Janvier (Ed.), Problems of representation in the teaching and learning of mathematics (pp. 59–65). Hillsdale, NJ: Lawrence Erlbaum Associates.Google Scholar
  46. Goldin, G. A. (2000). A scientific perspective on structured, task-based interviews in mathematics education research. In A. E. Kelly & R. A. Lesh (Eds.), Handbook of research design in mathematics and science education (pp. 517–545). Mahwah, NJ: Lawrence Erlbaum Associates.Google Scholar
  47. Goodwin, C. (1994). Professional vision. American Anthropologist, 96(3), 603–633.CrossRefGoogle Scholar
  48. Goody, J. (1977). The domestication of the savage mind. Cambridge: Cambridge University Press.Google Scholar
  49. Gravemeijer, K. P. E. (1999). How emergent models may foster the constitution of formal mathematics. Mathematical Thinking and Learning, 1(2), 155–177.CrossRefGoogle Scholar
  50. Greeno, J. G., & van de Sande, C. (2007). Perspectival understanding of conceptions and conceptual growth in interaction. Educational Psychologist, 42(1), 9–23.CrossRefGoogle Scholar
  51. Gutiérrez, J. F., Trninic, D., Lee, R. G., & Abrahamson, D. (2011). Hooks and shifts in instrumented mathematics learning. Paper presented at the annual meeting of the American Educational Research Association (SIG learning sciences). New Orleans, LA, April 8–12, 2011. http://www.edrl.berkeley.edu/sites/default/files/AERA2011-Hooks-and-Shifts.pdf.
  52. Hall, R. (2001). Cultural artifacts, self regulation, and learning: Commentary on Neuman’s “Can the Baron von Munchhausen phenomenon be solved?”. Mind, Culture & Activity, 8(1), 98–108.CrossRefGoogle Scholar
  53. Halldén, O., Scheja, M., & Haglund, L. (2008). The contextuality of knowledge: An intentional approach to meaning making and conceptual change. In S. Vosniadou (Ed.), International handbook of research on conceptual change (pp. 509–532). Routledge, New York: Taylor & Francis.Google Scholar
  54. Harel, G. (in press). Intellectual need. In K. Leatham (Ed.), Vital directions for mathematics education research. New York: Springer.Google Scholar
  55. Hoffmann, M. H. G. (2003). Peirce’s ‘diagrammatic reasoning’ as a solution of the learning paradox. In G. Debrock (Ed.), Process pragmatism: Essays on a quiet philosophical revolution (pp. 121–143). Amsterdam: Rodopi.Google Scholar
  56. 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 CHI: ACM conference on human factors in computing systems (CHI 2011), Vancouver. May 7–12, 2011 (Vol. “Full Papers”, pp. 1989–1998). ACM: CHI (CD ROM).Google Scholar
  57. Hutchins, E. (1995). How a cockpit remembers its speeds. Cognitive Science, 19, 265–288.CrossRefGoogle Scholar
  58. Karmiloff-Smith, A. (1988). The child is a theoretician, not an inductivist. Mind & Language, 3(3), 183–195.CrossRefGoogle Scholar
  59. Kelly, A. E. (2003). Research as design. In A. E. Kelly (Ed.), The role of design in educational research [Special issue]. Educational Researcher, 32, 3–4.Google Scholar
  60. Kirsh, D. (2006). Distributed cognition: a methodological note. In S. Harnad & I. E. Dror (Eds.), Distributed cognition [Special issue]. Pragmatics & Cognition, 14(2), 249–262.Google Scholar
  61. Kuchinsky, S. E., Bock, K., & Irwin, D. E. (2011). Reversing the hands of time: changing the mapping from seeing to saying. Journal of Experimental Psychology: Learning, Memory, and Cognition, 37(3), 748–756.CrossRefGoogle Scholar
  62. Lakoff, G., & Núñez, R. E. (2000). Where mathematics comes from: How the embodied mind brings mathematics into being. New York: Basic Books.Google Scholar
  63. Lee, J. C. (2008). Hacking the Nintendo Wii Remote. IEEE Pervasive Computing, 7(3), 39–45. http://johnnylee.net/projects/wii/.
  64. Mariotti, M. A. (2009). Artifacts and signs after a Vygotskian perspective: The role of the teacher. ZDM: The International Journal on Mathematics Education, 41, 427–440.CrossRefGoogle Scholar
  65. McLuhan, M. (1964). Understanding media: The extensions of man. New York: The New American Library.Google Scholar
  66. McNeill, D., & Duncan, S. D. (2000). Growth points in thinking-for-speaking. In D. McNeill (Ed.), Language and gesture (pp. 141–161). New York: Cambridge University Press.CrossRefGoogle Scholar
  67. Meira, L. (2002). Mathematical representations as systems of notations-in-use. In K. Gravenmeijer, R. Lehrer, B. V. Oers, & L. Verschaffel (Eds.), Symbolizing, modeling and tool use in mathematics education (pp. 87–104). Dordrecht, The Netherlands: Kluwer.Google Scholar
  68. Merleau-Ponty, M. (1964). An unpublished text by Maurice Merleau-Ponty: prospectus of his work (trans: Dallery, A. B.). In J. M. Edie (Ed.), The primacy of perception, and other essays on phenomenological psychology, the philosophy of art, history and politics. Evanston, IL: Northwestern University Press. (Original work 1962).Google Scholar
  69. Nemirovsky, R. (2003). Three conjectures concerning the relationship between body activity and understanding mathematics. In R. Nemirovsky, M. Borba (Coordinators), Perceptuo-motor activity and imagination in mathematics learning (research forum). In N. A. Pateman, B. J. Dougherty, & J. T. Zilliox (Eds.), Twenty seventh annual meeting of the international group for the psychology of mathematics education (Vol. 1, pp. 105–109). Honolulu, Hawaii: Columbus, OH: Eric Clearinghouse for Science, Mathematics, and Environmental Education.Google Scholar
  70. Neuman, Y. (2001). Can the Baron von Münchausen phenomenon be solved? An activity-oriented solution to the learning paradox. Mind, Culture & Activity, 8(1), 78–89.CrossRefGoogle Scholar
  71. Newman, D., Griffin, P., & Cole, M. (1989). The construction zone: Working for cognitive change in school. New York: Cambridge University Press.Google Scholar
  72. Norman, D. A. (1991). Cognitive artifacts. In J. M. Carroll (Ed.), Designing interaction: Psychology at the human-computer interface (pp. 17–38). New York: Cambridge University Press.Google Scholar
  73. Noss, R., Healy, L., & Hoyles, C. (1997). The construction of mathematical meanings: Connecting the visual with the symbolic. Educational Studies in Mathematics, 33(2), 203–233.CrossRefGoogle Scholar
  74. 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
  75. Olive, J. (2000). Computer tools for interactive mathematical activity in the elementary school. International Journal of Computers for Mathematical Learning, 5(3), 241–262.CrossRefGoogle Scholar
  76. Olson, D. R. (1994). The world on paper. Cambridge, UK: Cambridge University Press.Google Scholar
  77. Papert, S. (1980). Mindstorms: Children, computers, and powerful ideas. NY: Basic Books.Google Scholar
  78. Petrick, C., & Martin, T. (2011). Hands up, know body move: Learning mathematics through embodied actions. Manuscript in progress.Google Scholar
  79. Pirie, S. E. B., & Kieren, T. E. (1994). Growth in mathematical understanding: How can we characterize it and how can we represent it? Educational Studies in Mathematics, 26, 165–190.CrossRefGoogle Scholar
  80. Pratt, D., & Kapadia, R. (2009). Shaping the experience of young and naive probabilists. Research and developments in probability education [Special Issue]. International Electronic Journal of Mathematics Education, 4(3), 213–228.Google Scholar
  81. Prawat, R. S. (1999). Dewey, Peirce, and the learning paradox. American Educational Research Journal, 36, 47–76.Google Scholar
  82. Radford, L. (2003). Gestures, speech, and the sprouting of signs: A semiotic-cultural approach to students’ types of generalization. Mathematical Thinking and Learning, 5(1), 37–70.CrossRefGoogle Scholar
  83. Radford, L. (2010). The eye as a theoretician: Seeing structures in generalizing activities. For the Learning of Mathematics, 30(2), 2–7.Google Scholar
  84. 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, & L. Flevares (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, Chap. 18: technology, pp. 1488–1496). Columbus, OH: PME-NA.Google Scholar
  85. Roth, W.-M. (2009). Embodied mathematical communication and the visibility of graphical features. In W.-M. Roth (Ed.), Mathematical representation at the interface of body and culture (pp. 95–121). Charlotte, NC: Information Age Publishing.Google Scholar
  86. 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
  87. Sáenz-Ludlow, A. (2003). A collective chain of signification in conceptualizing fractions: A case of a fourth-grade class. Journal of Mathematical Behavior, 222, 181–211.CrossRefGoogle Scholar
  88. Sandoval, W. A., & Bell, P. (Eds.). (2004). Design-based research methods for studying learning in context [Special issue]. Educational Psychologist, 39(4).Google Scholar
  89. Sarama, J., & Clements, D. H. (2009). “Concrete” computer manipulatives in mathematics education. Child Development Perspectives, 3, 145–150.CrossRefGoogle Scholar
  90. Saxe, G. B. (2004). Practices of quantification from a sociocultural perspective. In K. A. Demetriou & A. Raftopoulos (Eds.), Developmental change: Theories, models, and measurement (pp. 241–263). NY: Cambridge University Press.Google Scholar
  91. Saxe, G. B., Gearhart, M., Shaughnessy, M., Earnest, D., Cremer, S., Sitabkhan, Y., et al. (2009). A methodological framework and empirical techniques for studying the travel of ideas in classroom communities. In B. Schwarz, T. Dreyfus, & R. Hershkowitz (Eds.), Transformation of knowledge through classroom interaction (pp. 203–222). Routledge, New York: Taylor & Francis.Google Scholar
  92. Schoenfeld, A. H. (1998). Making pasta and making mathematics: From cookbook procedures to really cooking. In J. G. Greeno & S. V. Goldman (Eds.), Thinking practice in mathematics and science learning (pp. 299–319). Mahwah, NJ: LEA.Google Scholar
  93. Schoenfeld, A. H., Smith, J. P., & Arcavi, A. (1991). Learning: The microgenetic analysis of one student’s evolving understanding of a complex subject matter domain. In R. Glaser (Ed.), Advances in instructional psychology (pp. 55–175). Hillsdale, NJ: Erlbaum.Google Scholar
  94. Schön, D. A. (1992). Designing as reflective conversation with the materials of a design situation. Research in Engineering Design, 3, 131–147.CrossRefGoogle Scholar
  95. Sebanz, N., & Knoblich, G. (2009). Prediction in joint action: What, when, and where. Topics in Cognitive Science, 1(2), 353–367.CrossRefGoogle Scholar
  96. Sfard, A. (2002). The interplay of intimations and implementations: Generating new discourse with new symbolic tools. Journal of the Learning Sciences, 11(2&3), 319–357.CrossRefGoogle Scholar
  97. Sfard, A. (2007). When the rules of discourse change, but nobody tells you—Making sense of mathematics learning from a commognitive standpoint. Journal of Learning Sciences, 16(4), 567–615.CrossRefGoogle Scholar
  98. Shank, G. (1987). Abductive strategies in educational research. American Journal of Semiotics, 5, 275–290.Google Scholar
  99. Shank, G. (1998). The extraordinary ordinary powers of abductive reasoning. Theory & Psychology, 8(6), 841–860.CrossRefGoogle Scholar
  100. Shreyar, S., Zolkower, B., & Pérez, S. (2010). Thinking aloud together: A teacher’s semiotic mediation of a whole-class conversation about percents. Educational Studies in Mathematics, 73(1), 21–53.CrossRefGoogle Scholar
  101. Slobin, D. I. (1996). From “thought and language” to “thinking to speaking”. In J. Gumperz & S. C. Levinson (Eds.), Rethinking linguistic relativity (pp. 70–96). Cambridge: Cambridge University Press.Google Scholar
  102. Smith, J. P., diSessa, A. A., & Roschelle, J. (1993). Misconceptions reconceived: A constructivist analysis of knowledge in transition. Journal of the Learning Sciences, 3(2), 115–163.CrossRefGoogle Scholar
  103. Stetsenko, A. (2002). Commentary: Sociocultural activity as a unit of analysis: How Vygotsky and Piaget converge in empirical research on collaborative cognition. In D. J. Bearison & B. Dorval (Eds.), Collaborative cognition: Children negotiating ways of knowing (pp. 123–135). Westport, CN: Ablex Publishing.Google Scholar
  104. 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
  105. Stigler, J. W. (1984). “Mental abacus”: The effect of abacus training on Chinese children’s mental calculation. Cognitive Psychology, 16, 145–176.CrossRefGoogle Scholar
  106. Thagard, P. (2010). How brains make mental models. In L. Magnani, W. Carnielli, & C. Pizzi (Eds.), Model-based reasoning in science and technology: Abduction, logic, and computational discovery (pp. 447–461). Berlin: Springer.Google Scholar
  107. Tirosh, D., & Stavy, R. (1999). Intuitive rules: A way to explain and predict students’ reasoning. Educational Studies in Mathematics, 38, 51–66.CrossRefGoogle Scholar
  108. Trninic, D., Gutiérrez, J. F., & Abrahamson, D. (in press). Virtual mathematical inquiry: problem solving at the gestural–symbolic interface of remote-control embodied-interaction design. In G. Stahl, H. Spada, & N. Miyake (Eds.), Proceedings of the ninth international conference on computer-supported collaborative learning (CSCL 2011) [Vol. (Full paper)]. Hong Kong, July 4–8, 2011.Google Scholar
  109. Trninic, D., Gutiérrez, J. F., Lee, R. G., & Abrahamson, D. (2011). Generative immersion and immersive generativity in instructional design. Paper presented at the the annual meeting of the American Educational Research Association (SIG research in mathematics education). New Orleans, LA, April 8–12, 2011.Google Scholar
  110. Trninic, D., Reinholz, D., Howison, M., & Abrahamson, D. (2010). Design as an object-to-think-with: Semiotic potential emerges through collaborative reflective conversation with material. In P. Brosnan, D. Erchick, & L. Flevares (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, Chap. 18: technology, pp. 1523–1530). Columbus, OH: PME-NA. http://gse.berkeley.edu/faculty/DAbrahamson/publications/TrninicReinholzHowisonAbrahamson-PMENA2010.pdf.
  111. van den Heuvel-Panhuizen, M. (2003). The didactical use of models in realistic mathematics education: An example from a longitudinal trajectory on percentage. Educational Studies in Mathematics, 54(1), 9–35.CrossRefGoogle Scholar
  112. Vergnaud, G. (1983). Multiplicative structures. In R. Lesh & M. Landau (Eds.), Acquisition of mathematical concepts and processes (pp. 127–174). New York: Academic Press.Google Scholar
  113. Vergnaud, G. (2009). The theory of conceptual fields. In T. Nunes (Ed.), Giving meaning to mathematical signs: Psychological, pedagogical and cultural processes. Human Development [Special Issue], 52, 83–94.Google Scholar
  114. Vérillon, P., & Rabardel, P. (1995). Cognition and artifacts: A contribution to the study of thought in relation to instrumented activity. European Journal of Psychology of Education, 10(1), 77–101.CrossRefGoogle Scholar
  115. Voigt, J. (1995). Thematic patterns of interaction and sociomathematical norms. In P. Cobb & H. Bauersfeld (Eds.), The emergence of mathematical meaning: Interaction in classroom cultures (pp. 163–202). Hillsdale, NJ: Lawrence Erlbaum.Google Scholar
  116. von Glasersfeld, E. (1987). Learning as a constructive activity. In C. Janvier (Ed.), Problems of representation in the teaching and learning of mathematics (pp. 3–18). Hillsdale, NJ: Lawrence Erlbaum.Google Scholar
  117. von Glasersfeld, E. (1992). Aspects of radical constructivism and its educational recommendations (working group #4). Paper presented at the Seventh international congress on mathematics education (ICME7), Quebec.Google Scholar
  118. Vygotsky, L. S. (1934/1962). Thought and language. Cambridge, MA: MIT Press.Google Scholar
  119. Wertsch, J. V. (1979). From social interaction to higher psychological processes: A clarification and application of Vygotsky’s theory. Human Development, 22(1), 1–22.CrossRefGoogle Scholar
  120. White, T. (2008). Debugging an artifact, instrumenting a bug: Dialectics of instrumentation and design in technology-rich learning environments. International Journal of Computers for Mathematical Learning, 13(1), 1–26.CrossRefGoogle Scholar
  121. White, T., & Pea, R. (in press). Distributed by design: On the promises and pitfalls of collaborative learning with multiple representations. Journal of the Learning Sciences.Google Scholar
  122. Wilensky, U. (1997). What is normal anyway? Therapy for epistemological anxiety. Educational Studies in Mathematics, 33(2), 171–202.CrossRefGoogle Scholar
  123. Wilensky, U., & Papert, S. (2010). Restructurations: Reformulations of knowledge disciplines through new representational forms. In J. Clayson & I. Kallas (Eds.), Proceedings of the constructionism 2010 conference, Paris.Google Scholar
  124. Xu, F., & Denison, S. (2009). Statistical inference and sensitivity to sampling in 11-month-old infants. Cognition, 112, 97–104.CrossRefGoogle Scholar
  125. Yerushalmy, M. (1997). Designing representations: reasoning about functions of two variables. Journal for Research in Mathematics Education, 28(4), 431–466.CrossRefGoogle Scholar
  126. Zhang, J., & Norman, D. A. (1994). Representations in distributed cognitive tasks. Cognitive Science, 18, 87–122.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  • Dor Abrahamson
    • 1
  • Dragan Trninic
    • 1
  • Jose F. Gutiérrez
    • 1
  • Jacob Huth
    • 1
  • Rosa G. Lee
    • 1
  1. 1.Graduate School of EducationUniversity of CaliforniaBerkeleyUSA

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