A Student’s Synthesis of Tacit and Mathematical Knowledge as a Researcher’s Lens on Bridging Learning Theory

Dor Abrahamson

doi:10.29333/iejme/237

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Dor Abrahamson

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Abstract

What instructional materials and practices will help students make sense of probability notions? Li (11 years) participated in an interview-based implementation of a design for the binomial. The design was centered around an innovative urn-like random generator, creating opportunities to reconcile two mental constructions of anticipated outcome distributions: (a) holistic perceptual judgments based in tacit knowledge of population-to-sample relations and implicitly couched in terms of the aggregate events with no attention to permutations on these combinations; and (b) classicist-probability analytic treatment of ratios between the subset of favorable to all elemental events with attention to the permutations. We argue that constructivist and sociocultural perspectives on mathematics learning can be reconciled by revealing interactions of intuitive and formal resources in individual development of deep conceptual understanding. Learning is the guided process of blending two constructions of problematized situations: the phenomenologically immediate and the semiotically mediated.

Keywords

Design-based research
cognitive science
sociocultural theory
cultural semiotics
binomial
combinatorial analysis
sample space

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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Article Type: Research Article

INT ELECT J MATH ED, Volume 4, Issue 3, October 2009, 195-226

https://doi.org/10.29333/iejme/237

Publication date: 12 Dec 2009

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References

Abrahamson, D. (2006). The shape of things to come: The computational pictograph as a bridge from combinatorial space to outcome distribution International Journal of Computers for Mathematical Learning, 11(1), 137– 146. Available at http://www.springerlink.com/content/qq4021456um5066v/.
Abrahamson, D. (2008). Bridging theory: Activities designed to support the grounding of outcome-based combinatorial analysis in event-based intuitive judgment – A case study. In M. Borovcnik & D. Pratt (Eds.), Proc Probability Topic Study Group, International Congress on Mathematical Education (ICME 11). Monterrey: ICME. Retrieved June 8, 2009, from http://tsg.icme11.org/tsg/show/14#prog.
Abrahamson, D. (2009). Embodied design: Constructing means for constructing meaning. Educational Studies in Mathematics, 70(1), 27–47. Available at http://www.springerlink.com/content/j54720v17x646llu/.
Abrahamson, D. (2009b). 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.
Abrahamson, D., Bryant, M. J., Howison, M. L., & Relaford-Doyle, J. J. (2008). Toward a phenomenology of mathematical artifacts: A circumspective deconstruction of a design for the binomial. Paper presented at the annual conference of the American Education Research Ass., New York, March 24–28. Availabe from author.
Abrahamson, D. & Cendak, R. M. (2006). The odds of understanding the Law of Large Numbers: A design for grounding intuitive probability in combinatorial analysis. In J. Novotná, H. Moraová, M. Krátká, & N. Stehlíková (Eds.), Proc. Thirtieth Conf. International Group for the Psychology of Mathematics Education (Vol. 2, pp. 1–8). Charles University, Prague: PME. Available at http://gse.berkeley.edu/faculty/dabrahamson/publications/Abrahamson-Cendak_PME30.pdf
Abrahamson, D., Janusz, R., & Wilensky, U. (2006). There once was a 9-Block... – A middle-school design for probability and statistics. Journal of Statistics Education, 14(1). Retrieved June 8, 2009, from http://www.amstat.org/publications/jse/v14n1/abrahamson.html.
Abrahamson, D. & Wilensky, U. (2003). The quest of the bell curve: A constructionist approach to learning statistics through designing computer-based probability experiments. In M. A. Mariotti (Ed.), Proc. Third Conference of the European Society for Research in Mathematics Education. Pisa: University of Pisa. Retrieved June 8, 2009, from http://gse.berkeley.edu/faculty/dabrahamson/publications/AbrahamsonWilenskyCERME3.pdf.
Abrahamson, D. & Wilensky, U. (2007). Learning axes and bridging tools in a technology-based design for statistics. International Journal of Computers for Mathematical Learning, 12(1), 23–55. Available from http://www.springerlink.com/content/a216xl0vp026k75t/.
Amit, M. & Jan, I. (2007). A learning environment that promotes concept construction: The case of probability. Paper presented at the annual conference of the American Educational Research Association, Chicago, IL.
Bakker, A. & Hoffmann, M. H. G. (2005). Diagrammatic reasoning as the basis for developing concepts: A semiotic analysis of students' learning about statistical distribution. Educational Studies in Mathematics, 60(3), 333– 358.
Bamberger, J. & diSessa, A. A. (2003). Music as embodied mathematics: A study of a mutually informing affinity. International Journal of Computers for Mathematical Learning, 8(2), 123–160. Available at http://www.springerlink.com/content/qvp5m34445u48755/.
Bartolini Bussi, M. G., & Boni, M. (2003). Instruments for semiotic mediation in primary school classrooms. For the Learning of Mathematics, 23(2), 12–19.
Bereiter, C. (1985). Towards the solution of the learning paradox. Review of Educational Research, 55(2), 210–226.
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 .
Brown, A. (1992). Design experiments: Theoretical and methodological challenges in creating complex interventions in classroom settings. Journal of the Learning Sciences, 2(2), 141–178.
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.
Clement, J., Brown, B., & Zietsman, A. (1989). Not all preconceptions are misconceptions: Finding "anchoring conceptions" for grounding instruction on student's intuitions. International Journal of Science Education, 11, 554–565.
Cobb, P., Confrey, J., diSessa, A., Lehrer, R., & Schauble, L. (2003). Design experiments in educational research. Educational Researcher, 32(1), 9–13.
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.
Confrey, J. (1995). Student voice in examining "splitting" as an approach to ratio, proportions, and fractions. In L. Meira & D. Carraher (Eds.), Proc. Nineteenth Conference of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 3–29). Recife, Brazil: Universidade Federal de Pernambuco.
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.), Proc. 20th Annual Conf. North American Chapter of the Intern. 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. (2008). A note from the editor. Cognition and Instruction, 26(4), 427–429.
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.
Doorman, M., Drijvers, P., Dekker, T., van den Heuvel-Panhuizen, M., de Lange, J., & Wijers, M. (2007). Problem solving as a challenge for mathematics education in The Netherlands. ZDM: The International Journal on Mathematics Education, 39(5), 405–418. Available at http://www.springerlink.com/content/e13071543715515v/.
Drier, H. (2000). Children's meaning-making activity with dynamic multiple representations in a probability microworld. In M. L. Fernandez (Ed.), Proceedings of the 22nd annual meeting of the North American Chapter of the International Group for Psychology of Mathematics Education (Vol. 2, pp. 691–696). Columbus, OH: Eric Clearing House.
Edelson, D. C. (2002). Design research: What we learn when we engage in design. The Journal of the Learning Sciences, 11(1), 105–121.
Eizenberg, M. M., & Zaslavsky, O. (2003). Cooperative problem solving in combinatorics: The inter-relations between control processes and successful solutions. The Journal of Mathematical Behavior, 22(4), 389–403.
Ernest, P. (1988). Social constructivism as a philosophy of mathematics. Albany, NY: SUNY Press.
Falk, R. & Lann, A. (2008). The allure of equality: Uniformity in probabilistic and statistical judgment. Cognitive Psychology, 57(4), 293–334.
Fauconnier, G. & Turner, M. (2002). The way we think: Conceptual blending and the mind's hidden complexities. New York: Basic Books.
Fischbein, E. (1987). Intuition in science and mathematics. Dordrecht, Holland: D. Reidel.
Fischer, H. R. (2001). Abductive reasoning as a way of world making. Foundations of Science, 6(4), 361–383.
Freudenthal, H. (1971). Geometry between the devil and the deep sea. Educational Studies in Mathematics, 3(3/4), 413–435.
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.
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: John Wiley and Sons.
Gigerenzer, G. (1998). Ecological intelligence: An adaptation for frequencies. In D. D. Cummins & C. Allen (Eds.), The evolution of mind (pp. 9–29). Oxford: Oxford University Press.
Ginsburg, H. P. (1997). Entering the child’s mind. New York: Cambridge University Press.
Goodwin, C. (1994). Professional vision. American Anthropologist, 96(3), 603–633.
Gravemeijer, K. P. E. (1999). How emergent models may foster the constitution of formal mathematics. Mathematical Thinking and Learning, 1(2), 155–177.
Heidegger, M. (1962). Being and time (J. Macquarrie & E. Robinson, Trans.). New York: Harper & Row (Original work published 1927).
Hutchins, E. (2005). Material anchors for conceptual blends. Journal of Pragmatics, 37(10), 1555–1577.
Jones, G. A., Langrall, C. W., & Mooney, E. S. (2007). Research in probability: Responding to classroom realities. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 909–955). Charlotte, NC: Information Age Publishing.
Kelly, A. E. (2004). Design research in education: Yes, but is it methodological? Journal of the Learning Sciences, 13(1), 115–128.
Konold, C. (1989). Informal conceptions of probability. Cognition and Instruction, 6, 59–98.
Konold, C., Harradine, A., & Kazak, S. (2007). Understanding distributions by modeling them. International Journal of Computers for Mathematical Learning, 12(3), 217–230. Available at http://www.springerlink.com/content/lx2213721548n428/.
Koschmann, T., Kuuti, K., & Hickman, L. (1998). The concept of breakdown in Heidegger, Leont'ev, and Dewey and its implications for education. Mind, Culture, and Activity, 5(1), 25–41.
Kuhn, M., Hoppe, U., Lingnau, A., & Wichmann, A. (2006). Computational modelling and simulation fostering new approaches in learning probability. Innovations in Education and Teaching International, 43(2), 183–194.
LeCoutre, V. P. (1992). Cognitive models and problem spaces in “purely random” situations. Educational Studies in Mathematics, 23, 557–568.
Lemke, J. L. (1998). Multiplying meaning: Visual and verbal semiotics in scientific text. In J. R. Martin & R. Veel (Eds.), Reading science: Critical and functional perspectives on discourses of science (pp. 87–113). London: Routledge.
Liu, Y. & Thompson, P. (2002). Randomness: Rethinking the foundation of probability. In D. Mewborn, P. Sztajn, E. White, H. Wiegel, R. Bryant & K. Nooney (Eds.), Proc. 24th Annual Meeting North American Chapter of the Intern. Group for the Psychology of Mathematics Education, Athens, GA (Vol. 3, pp. 1331–1334). Columbus, OH: Eric Clearing House.
Norton, A. (2008). Josh's operational conjectures: Abductions of a splitting operation and the construction of new fractional schemes. Journal for Research in Mathematics Education, 39(4), 401–430.
Papert, S. (1980). Mindstorms: Children, computers, and powerful ideas. NY: Basic Books.
Peirce, C. S. (1931–1958). Collected papers of Charles Sanders Peirce. C. Hartshorne & P. Weiss (Eds.). Cambridge, MA: Harvard University Press.
Piaget, J. (1971). Biology and knowledge: An essay on the relations between organic regulations and cognitive processes. Chicago, IL: The University of Chicago Press.
Pratt, D. (2000). Making sense of the total of two dice. Journal for Research in Mathematics Education, 31(5), 602– 625.
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.
Roth, W. M. & Welzel, M. (2001). From activity to gestures and scientific language. Journal of Research in Science Teaching, 38(1), 103–136.
Sandoval, W. A. & Bell, P. E. (2004). Special issue on design-based research methods for studying learning in context. Educational Psychologist, 39(4).
Schank, R. C. (1996). Goal-based scenarios: Case-based reasoning meets learning by doing. In D. B. Leake (Ed.), Case-Based Reasoning (pp. 295–348). Cambridge, MA: The MIT Press.
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. (1981). Intuitive thinking? A metaphor underlying some ideas of educational reform (Working Paper 8): Unpublished manuscript, Division for Study and Research, M.I.T.
Sfard, A. (2007). When the rules of discourse change, but nobody tells you – making sense of mathematics learning from commognitive standpoint. Journal of Learning Sciences, 16(4), 567–615.
Shaughnessy, J. M. (1977). Misconceptions of probability: An experiment with a small-group, activity-based model building approach to introductory probability. Educational Studies in Mathematics, 8, 295–316.
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. Available at http://ctl.sri.com/publications/displayPublication.jsp?ID=113.
Stavy, R. & Tirosh, D. (1996). Intuitive rules in science and mathematics: The case of "more of A – more of B". International Journal of Science Education, 18(6), 653–668.
Steiner, G. (2001). Grammars of creation. New Haven, CO: Yale University Press.
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.
Tirosh, D., Stavy, R., & Cohen, S. (1998). Cognitive conflict and intuitive rules. International Journal of Science Education, 20(10), 1257–1269.
Tsal, Y. & Kolbet, L. (1985). Disambiguating ambiguous figures by selective attention. Quarterly Journal of Experimental Psychology, 37(1–A), 25–37.
Tversky, A. & Kahneman, D. (1974). Judgment under uncertainty: Heuristics and biases. Science, 185(4157), 1124– 1131.
Vahey, P., Enyedy, N., & Gifford, B. (2000). The Probability Inquiry Environment: Learning probability through the use of a collaborative, inquiry-based simulation environment. Journal of Interactive Learning Research, 11, 51– 84. Available at http://www.gseis.ucla.edu/faculty/enyedy/PIE.html.
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.
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.
Vygotsky, L. S. (1978/1930). Mind in society: The development of higher psychological processes. Cambridge: Harvard University Press
Wilensky, U. (1995). Paradox, programming and learning probability. Journal of Mathematical Behavior, 14(2), 231– 280. Available at http://ccl.northwestern.edu/papers/paradox/pplp/.
Wilensky, U. (1997). What is normal anyway?: Therapy for epistemological anxiety. Educational Studies in Mathematics, 33(2), 171–202. Available at http://ccl.northwestern.edu/papers/normal/.
Xu, F., & Vashti, G. (2008). Intuitive statistics by 8-month-old infants. Proceedings of the National Academy of Sciences, 105(13), 5012–5015. Availabe at http://www.pnas.org/content/105/13/5012.short.

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APA

Abrahamson, D. (2009). A Student’s Synthesis of Tacit and Mathematical Knowledge as a Researcher’s Lens on Bridging Learning Theory. International Electronic Journal of Mathematics Education, 4(3), 195-226. https://doi.org/10.29333/iejme/237

Vancouver

Abrahamson D. A Student’s Synthesis of Tacit and Mathematical Knowledge as a Researcher’s Lens on Bridging Learning Theory. INT ELECT J MATH ED. 2009;4(3):195-226. https://doi.org/10.29333/iejme/237

AMA

Abrahamson D. A Student’s Synthesis of Tacit and Mathematical Knowledge as a Researcher’s Lens on Bridging Learning Theory. INT ELECT J MATH ED. 2009;4(3), 195-226. https://doi.org/10.29333/iejme/237

Chicago

Abrahamson, Dor. "A Student’s Synthesis of Tacit and Mathematical Knowledge as a Researcher’s Lens on Bridging Learning Theory". International Electronic Journal of Mathematics Education 2009 4 no. 3 (2009): 195-226. https://doi.org/10.29333/iejme/237

Harvard

Abrahamson, D. (2009). A Student’s Synthesis of Tacit and Mathematical Knowledge as a Researcher’s Lens on Bridging Learning Theory. International Electronic Journal of Mathematics Education, 4(3), pp. 195-226. https://doi.org/10.29333/iejme/237

MLA

Abrahamson, Dor "A Student’s Synthesis of Tacit and Mathematical Knowledge as a Researcher’s Lens on Bridging Learning Theory". International Electronic Journal of Mathematics Education, vol. 4, no. 3, 2009, pp. 195-226. https://doi.org/10.29333/iejme/237