# Hooks and Shifts: A Dialectical Study of Mediated Discovery

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## 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

- 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.CrossRefGoogle Scholar - 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 - 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. - 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. - 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. - 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 - 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 - 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 - 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 - 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. - 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 - 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 - Barsalou, L. W. (1999). Perceptual symbol systems.
*Behavioral and Brain Sciences,**22*, 577–660.Google Scholar - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - Collins, A., & Ferguson, W. (1993). Epistemic forms and epistemic games: Structures and strategies to guide inquiry.
*Educational Psychologist,**28*(1), 25–42.CrossRefGoogle Scholar - 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 - 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 - 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 - 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 - diSessa, A. A. (2007). An interactional analysis of clinical interviewing.
*Cognition and Instruction,**25*(4), 523–565.CrossRefGoogle Scholar - diSessa, A. A. (2008). A note from the editor.
*Cognition and Instruction,**26*(4), 427–429.CrossRefGoogle Scholar - 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 - 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 - 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 - 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 - 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 - Freudenthal, H. (1968). Why to teach mathematics so as to be useful.
*Educational Studies in Mathematics,**1*(1/2), 3–8.CrossRefGoogle Scholar - Freudenthal, H. (1971). Geometry between the devil and the deep sea.
*Educational Studies in Mathematics,**3*(3/4), 413–435.CrossRefGoogle Scholar - Freudenthal, H. (1986).
*Didactical phenomenology of mathematical structures*. Dordrecht, The Netherlands: Kluwer Academic Publishers.Google Scholar - 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 - 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 - 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 - Gigerenzer, G., & Brighton, H. (2009). Homo Heuristicus: Why biased minds make better inferences.
*Topics in Cognitive Science,**1*(1), 107–144.CrossRefGoogle Scholar - Ginsburg, H. P. (1997).
*Entering the child’s mind*. New York: Cambridge University Press.CrossRefGoogle Scholar - Glaser, B. G., & Strauss, A. L. (1967).
*The discovery of grounded theory: Strategies for qualitative research*. Chicago: Aldine Publishing Company.Google Scholar - 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 - 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 - Goodwin, C. (1994). Professional vision.
*American Anthropologist,**96*(3), 603–633.CrossRefGoogle Scholar - Goody, J. (1977).
*The domestication of the savage mind*. Cambridge: Cambridge University Press.Google Scholar - 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 - 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 - 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. - 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 - 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 - Harel, G. (in press). Intellectual need. In K. Leatham (Ed.),
*Vital directions for mathematics education research*. New York: Springer.Google Scholar - 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 - 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 - Hutchins, E. (1995). How a cockpit remembers its speeds.
*Cognitive Science,**19*, 265–288.CrossRefGoogle Scholar - Karmiloff-Smith, A. (1988). The child is a theoretician, not an inductivist.
*Mind & Language,**3*(3), 183–195.CrossRefGoogle Scholar - 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 - 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 - 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 - Lakoff, G., & Núñez, R. E. (2000).
*Where mathematics comes from: How the embodied mind brings mathematics into being*. New York: Basic Books.Google Scholar - 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.CrossRefGoogle Scholar - McLuhan, M. (1964).
*Understanding media: The extensions of man*. New York: The New American Library.Google Scholar - 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 - 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 - 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 - 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 - 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 - Newman, D., Griffin, P., & Cole, M. (1989).
*The construction zone: Working for cognitive change in school*. New York: Cambridge University Press.Google Scholar - 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 - 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 - 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 - 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 - Olson, D. R. (1994).
*The world on paper*. Cambridge, UK: Cambridge University Press.Google Scholar - Papert, S. (1980).
*Mindstorms: Children, computers, and powerful ideas*. NY: Basic Books.Google Scholar - Petrick, C., & Martin, T. (2011).
*Hands up, know body move: Learning mathematics through embodied actions*. Manuscript in progress.Google Scholar - 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 - 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 - Prawat, R. S. (1999). Dewey, Peirce, and the learning paradox.
*American Educational Research Journal,**36*, 47–76.Google Scholar - 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 - Radford, L. (2010). The eye as a theoretician: Seeing structures in generalizing activities.
*For the Learning of Mathematics,**30*(2), 2–7.Google Scholar - 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 - 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 - 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 - 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 - Sandoval, W. A., & Bell, P. (Eds.). (2004). Design-based research methods for studying learning in context [Special issue].
*Educational Psychologist, 39*(4).Google Scholar - Sarama, J., & Clements, D. H. (2009). “Concrete” computer manipulatives in mathematics education.
*Child Development Perspectives,**3*, 145–150.CrossRefGoogle Scholar - 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 - 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 - 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 - 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 - Schön, D. A. (1992). Designing as reflective conversation with the materials of a design situation.
*Research in Engineering Design,**3*, 131–147.CrossRefGoogle Scholar - Sebanz, N., & Knoblich, G. (2009). Prediction in joint action: What, when, and where.
*Topics in Cognitive Science,**1*(2), 353–367.CrossRefGoogle Scholar - 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 - 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 - Shank, G. (1987). Abductive strategies in educational research.
*American Journal of Semiotics,**5*, 275–290.Google Scholar - Shank, G. (1998). The extraordinary ordinary powers of abductive reasoning.
*Theory & Psychology,**8*(6), 841–860.CrossRefGoogle Scholar - 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 - 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 - 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 - 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 - 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 - Stigler, J. W. (1984). “Mental abacus”: The effect of abacus training on Chinese children’s mental calculation.
*Cognitive Psychology,**16*, 145–176.CrossRefGoogle Scholar - 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 - Tirosh, D., & Stavy, R. (1999). Intuitive rules: A way to explain and predict students’ reasoning.
*Educational Studies in Mathematics,**38*, 51–66.CrossRefGoogle Scholar - 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 - 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 - 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. - 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 - 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 - 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 - 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 - 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 - 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 - 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 - Vygotsky, L. S. (1934/1962).
*Thought and language*. Cambridge, MA: MIT Press.Google Scholar - 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 - 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 - 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 - Wilensky, U. (1997). What is normal anyway? Therapy for epistemological anxiety.
*Educational Studies in Mathematics,**33*(2), 171–202.CrossRefGoogle Scholar - 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 - Xu, F., & Denison, S. (2009). Statistical inference and sensitivity to sampling in 11-month-old infants.
*Cognition,**112*, 97–104.CrossRefGoogle Scholar - Yerushalmy, M. (1997). Designing representations: reasoning about functions of two variables.
*Journal for Research in Mathematics Education,**28*(4), 431–466.CrossRefGoogle Scholar - Zhang, J., & Norman, D. A. (1994). Representations in distributed cognitive tasks.
*Cognitive Science,**18*, 87–122.CrossRefGoogle Scholar