Hooks and Shifts: A Dialectical Study of Mediated Discovery
 Dor Abrahamson,
 Dragan Trninic,
 Jose F. Gutiérrez,
 Jacob Huth,
 Rosa G. Lee
 … show all 5 hide
Rent the article at a discount
Rent now* Final gross prices may vary according to local VAT.
Get AccessAbstract
Radical constructivists advocate discoverybased 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 mediateddiscovery hypothesis, we present and analyze vignettes selected from empirical data gathered in a conjecturedriven designbased research study investigating the microgenesis of proportional reasoning through guided engagement in technologybased embodied interaction. 22 Grade 4–6 students participated in individual or paired semistructured tutorial clinical interviews, in which they were tasked to remotecontrol 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 twostep guided reinvention 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 hookandshift 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.
Inside
Within this Article
 Introduction and Objectives
 Theoretical Background and Deliberations
 Data Source: A ConjectureDriven DesignBased Research Study of the Emergence of Proportional Reasoning from Guided EmbodiedInteraction ProblemSolving Activity
 Case Analyses: Progressive Mathematization Through Hooks and Shifts
 Conclusion: From Ontological Innovation to Design Heuristic
 References
 References
Other actions
 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.
 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.
 Abrahamson, D. (2009b). Embodied design: Constructing means for constructing meaning. Educational Studies in Mathematics, 70(1), 27–47. CrossRef
 Abrahamson, D. (2009c). Orchestrating semiotic leaps from tacit to cultural quantitative reasoning—The case of anticipating experimental outcomes of a quasibinomial random generator. Cognition and Instruction, 27(3), 175–224. CrossRef
 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/Abrahamsonetal.AERA2011EmbLearnSymp.pdf.
 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/AbrahamsonHowison2008_kinemathics.pdf, http://edrl.berkeley.edu/projects/kinemathics/MIT.mov.
 Abrahamson, D., & Howison, M. (2010a). Embodied artifacts: Coordinated action as an objecttothinkwith. 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/AbrahamsonHowisonAERA2010ReinholzTrninic.pdf.
 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.
 Abrahamson, D., & Trninic, D. (in press). Toward an embodiedinteraction 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.
 Bakker, A., & Derry, J. (2011). Lessons from inferentialism for statistics education. In K. Makar & D. BenZvi (Eds.), The role of context in developing students’ reasoning about informal statistical inference [Special issue]. Mathematical Thinking and Learning, 13(1&2), 5–26.
 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.
 Bamberger, J. (2010). Noting time. MinAd: Israel studies in musicology online (Vol. 8, issue 1&2), Retrieved November 9, 2010 from, http://www.biu.ac.il/hu/mu/minad/2010/2002BambergerNoting.pdf.
 Bamberger, J., & Schön, D. A. (1983). Learning as reflective conversation with materials: Notes from work in progress. Art Education, 36(2), 68–73.
 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.
 Barsalou, L. W. (1999). Perceptual symbol systems. Behavioral and Brain Sciences, 22, 577–660.
 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.
 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.
 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.
 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. CrossRef
 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. CrossRef
 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.
 Cole, M., & Wertsch, J. V. (1996). Beyond the individualsocial antinomy in discussions of Piaget and Vygotsky. Human Development, 39(5), 250–256. CrossRef
 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.
 Collins, A., & Ferguson, W. (1993). Epistemic forms and epistemic games: Structures and strategies to guide inquiry. Educational Psychologist, 28(1), 25–42. CrossRef
 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.
 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.
 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.
 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.
 diSessa, A. A. (2007). An interactional analysis of clinical interviewing. Cognition and Instruction, 25(4), 523–565. CrossRef
 diSessa, A. A. (2008). A note from the editor. Cognition and Instruction, 26(4), 427–429. CrossRef
 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. CrossRef
 diSessa, A. A., Hammer, D., Sherin, B., & Kolpakowski, T. (1991). Inventing graphing: Metarepresentational expertise in children. Journal of Mathematical Behavior, 10(2), 117–160.
 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.
 Edelson, D. C. (2002). Design research: What we learn when we engage in design. The Journal of the Learning Sciences, 11(1), 105–121. CrossRef
 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.
 Freudenthal, H. (1968). Why to teach mathematics so as to be useful. Educational Studies in Mathematics, 1(1/2), 3–8. CrossRef
 Freudenthal, H. (1971). Geometry between the devil and the deep sea. Educational Studies in Mathematics, 3(3/4), 413–435. CrossRef
 Freudenthal, H. (1986). Didactical phenomenology of mathematical structures. Dordrecht, The Netherlands: Kluwer Academic Publishers.
 Fuson, K. C., & Abrahamson, D. (2005). Understanding ratio and proportion as an example of the apprehending zone and conceptualphase problemsolving models. In J. Campbell (Ed.), Handbook of mathematical cognition (pp. 213–234). New York: Psychology Press.
 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.
 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.
 Gigerenzer, G., & Brighton, H. (2009). Homo Heuristicus: Why biased minds make better inferences. Topics in Cognitive Science, 1(1), 107–144. CrossRef
 Ginsburg, H. P. (1997). Entering the child’s mind. New York: Cambridge University Press. CrossRef
 Glaser, B. G., & Strauss, A. L. (1967). The discovery of grounded theory: Strategies for qualitative research. Chicago: Aldine Publishing Company.
 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.
 Goldin, G. A. (2000). A scientific perspective on structured, taskbased 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.
 Goodwin, C. (1994). Professional vision. American Anthropologist, 96(3), 603–633. CrossRef
 Goody, J. (1977). The domestication of the savage mind. Cambridge: Cambridge University Press.
 Gravemeijer, K. P. E. (1999). How emergent models may foster the constitution of formal mathematics. Mathematical Thinking and Learning, 1(2), 155–177. CrossRef
 Greeno, J. G., & van de Sande, C. (2007). Perspectival understanding of conceptions and conceptual growth in interaction. Educational Psychologist, 42(1), 9–23. CrossRef
 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/AERA2011HooksandShifts.pdf.
 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. CrossRef
 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.
 Harel, G. (in press). Intellectual need. In K. Leatham (Ed.), Vital directions for mathematics education research. New York: Springer.
 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.
 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).
 Hutchins, E. (1995). How a cockpit remembers its speeds. Cognitive Science, 19, 265–288. CrossRef
 KarmiloffSmith, A. (1988). The child is a theoretician, not an inductivist. Mind & Language, 3(3), 183–195. CrossRef
 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.
 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.
 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. CrossRef
 Lakoff, G., & Núñez, R. E. (2000). Where mathematics comes from: How the embodied mind brings mathematics into being. New York: Basic Books.
 Lee, J. C. (2008). Hacking the Nintendo Wii Remote. IEEE Pervasive Computing, 7(3), 39–45. http://johnnylee.net/projects/wii/.
 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. CrossRef
 McLuhan, M. (1964). Understanding media: The extensions of man. New York: The New American Library.
 McNeill, D., & Duncan, S. D. (2000). Growth points in thinkingforspeaking. In D. McNeill (Ed.), Language and gesture (pp. 141–161). New York: Cambridge University Press. CrossRef
 Meira, L. (2002). Mathematical representations as systems of notationsinuse. 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.
 MerleauPonty, M. (1964). An unpublished text by Maurice MerleauPonty: 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).
 Nemirovsky, R. (2003). Three conjectures concerning the relationship between body activity and understanding mathematics. In R. Nemirovsky, M. Borba (Coordinators), Perceptuomotor 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.
 Neuman, Y. (2001). Can the Baron von Münchausen phenomenon be solved? An activityoriented solution to the learning paradox. Mind, Culture & Activity, 8(1), 78–89. CrossRef
 Newman, D., Griffin, P., & Cole, M. (1989). The construction zone: Working for cognitive change in school. New York: Cambridge University Press.
 Norman, D. A. (1991). Cognitive artifacts. In J. M. Carroll (Ed.), Designing interaction: Psychology at the humancomputer interface (pp. 17–38). New York: Cambridge University Press.
 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. CrossRef
 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. CrossRef
 Olive, J. (2000). Computer tools for interactive mathematical activity in the elementary school. International Journal of Computers for Mathematical Learning, 5(3), 241–262. CrossRef
 Olson, D. R. (1994). The world on paper. Cambridge, UK: Cambridge University Press.
 Papert, S. (1980). Mindstorms: Children, computers, and powerful ideas. NY: Basic Books.
 Petrick, C., & Martin, T. (2011). Hands up, know body move: Learning mathematics through embodied actions. Manuscript in progress.
 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. CrossRef
 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.
 Prawat, R. S. (1999). Dewey, Peirce, and the learning paradox. American Educational Research Journal, 36, 47–76.
 Radford, L. (2003). Gestures, speech, and the sprouting of signs: A semioticcultural approach to students’ types of generalization. Mathematical Thinking and Learning, 5(1), 37–70. CrossRef
 Radford, L. (2010). The eye as a theoretician: Seeing structures in generalizing activities. For the Learning of Mathematics, 30(2), 2–7.
 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 thirtysecond annual meeting of the NorthAmerican chapter of the international group for the psychology of mathematics education (PMENA 32) (Vol. VI, Chap. 18: technology, pp. 1488–1496). Columbus, OH: PMENA.
 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.
 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.
 SáenzLudlow, A. (2003). A collective chain of signification in conceptualizing fractions: A case of a fourthgrade class. Journal of Mathematical Behavior, 222, 181–211. CrossRef
 Sandoval, W. A., & Bell, P. (Eds.). (2004). Designbased research methods for studying learning in context [Special issue]. Educational Psychologist, 39(4).
 Sarama, J., & Clements, D. H. (2009). “Concrete” computer manipulatives in mathematics education. Child Development Perspectives, 3, 145–150. CrossRef
 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.
 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.
 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.
 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.
 Schön, D. A. (1992). Designing as reflective conversation with the materials of a design situation. Research in Engineering Design, 3, 131–147. CrossRef
 Sebanz, N., & Knoblich, G. (2009). Prediction in joint action: What, when, and where. Topics in Cognitive Science, 1(2), 353–367. CrossRef
 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. CrossRef
 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. CrossRef
 Shank, G. (1987). Abductive strategies in educational research. American Journal of Semiotics, 5, 275–290.
 Shank, G. (1998). The extraordinary ordinary powers of abductive reasoning. Theory & Psychology, 8(6), 841–860. CrossRef
 Shreyar, S., Zolkower, B., & Pérez, S. (2010). Thinking aloud together: A teacher’s semiotic mediation of a wholeclass conversation about percents. Educational Studies in Mathematics, 73(1), 21–53. CrossRef
 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.
 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. CrossRef
 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.
 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.
 Stigler, J. W. (1984). “Mental abacus”: The effect of abacus training on Chinese children’s mental calculation. Cognitive Psychology, 16, 145–176. CrossRef
 Thagard, P. (2010). How brains make mental models. In L. Magnani, W. Carnielli, & C. Pizzi (Eds.), Modelbased reasoning in science and technology: Abduction, logic, and computational discovery (pp. 447–461). Berlin: Springer.
 Tirosh, D., & Stavy, R. (1999). Intuitive rules: A way to explain and predict students’ reasoning. Educational Studies in Mathematics, 38, 51–66. CrossRef
 Trninic, D., Gutiérrez, J. F., & Abrahamson, D. (in press). Virtual mathematical inquiry: problem solving at the gestural–symbolic interface of remotecontrol embodiedinteraction design. In G. Stahl, H. Spada, & N. Miyake (Eds.), Proceedings of the ninth international conference on computersupported collaborative learning (CSCL 2011) [Vol. (Full paper)]. Hong Kong, July 4–8, 2011.
 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.
 Trninic, D., Reinholz, D., Howison, M., & Abrahamson, D. (2010). Design as an objecttothinkwith: Semiotic potential emerges through collaborative reflective conversation with material. In P. Brosnan, D. Erchick, & L. Flevares (Eds.), Proceedings of the thirtysecond annual meeting of the NorthAmerican chapter of the international group for the psychology of mathematics education (PMENA 32) (Vol. VI, Chap. 18: technology, pp. 1523–1530). Columbus, OH: PMENA. http://gse.berkeley.edu/faculty/DAbrahamson/publications/TrninicReinholzHowisonAbrahamsonPMENA2010.pdf.
 van den HeuvelPanhuizen, 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. CrossRef
 Vergnaud, G. (1983). Multiplicative structures. In R. Lesh & M. Landau (Eds.), Acquisition of mathematical concepts and processes (pp. 127–174). New York: Academic Press.
 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.
 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. CrossRef
 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.
 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.
 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.
 Vygotsky, L. S. (1934/1962). Thought and language. Cambridge, MA: MIT Press.
 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. CrossRef
 White, T. (2008). Debugging an artifact, instrumenting a bug: Dialectics of instrumentation and design in technologyrich learning environments. International Journal of Computers for Mathematical Learning, 13(1), 1–26. CrossRef
 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.
 Wilensky, U. (1997). What is normal anyway? Therapy for epistemological anxiety. Educational Studies in Mathematics, 33(2), 171–202. CrossRef
 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.
 Xu, F., & Denison, S. (2009). Statistical inference and sensitivity to sampling in 11monthold infants. Cognition, 112, 97–104. CrossRef
 Yerushalmy, M. (1997). Designing representations: reasoning about functions of two variables. Journal for Research in Mathematics Education, 28(4), 431–466. CrossRef
 Zhang, J., & Norman, D. A. (1994). Representations in distributed cognitive tasks. Cognitive Science, 18, 87–122. CrossRef
 Title
 Hooks and Shifts: A Dialectical Study of Mediated Discovery
 Journal

Technology, Knowledge and Learning
Volume 16, Issue 1 , pp 5585
 Cover Date
 20110401
 DOI
 10.1007/s107580119177y
 Print ISSN
 22111662
 Online ISSN
 15731766
 Publisher
 Springer Netherlands
 Additional Links
 Topics
 Keywords

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

 Dor Abrahamson ^{(1)}
 Dragan Trninic ^{(1)}
 Jose F. Gutiérrez ^{(1)}
 Jacob Huth ^{(1)}
 Rosa G. Lee ^{(1)}
 Author Affiliations

 1. Graduate School of Education, University of California, 4649 Tolman Hall, Berkeley, CA, 947201670, USA