Skip to main content
SpringerLink
Log in

Learning axes and bridging tools in a technology-based design for statistics

  • Published:
International Journal of Computers for Mathematical Learning Aims and scope Submit manuscript

Abstract

We introduce a design-based research framework, learning axes and bridging tools, and demonstrate its application in the preparation and study of an implementation of a middle-school experimental computer-based unit on probability and statistics, ProbLab (Probability Laboratory, Abrahamson and Wilensky 2002 [Abrahamson, D., & Wilensky, U. (2002). ProbLab. Northwestern University, Evanston, IL: The Center for Connected Learning and Computer-Based Modeling, Northwestern University. http://www.ccl.northwestern.edu/curriculum/ProbLab/]). ProbLab is a mixed-media unit, which utilizes traditional tools as well as the NetLogo agent-based modeling-and-simulation environment (Wilensky 1999) [Wilensky, U. (1999). NetLogo. Northwestern University, Evanston, IL: The Center for Connected Learning and Computer-Based Modeling http://www.ccl.northwestern.edu/netlogo/] and HubNet, its technological extension for facilitating participatory simulation activities in networked classrooms (Wilensky and Stroup 1999a) [Wilensky, U., & Stroup, W. (1999a). HubNet. Evanston, IL: The Center for Connected Learning and Computer-Based Modeling, Northwestern University]. We will focus on the statistics module of the unit, Statistics As Multi-Participant Learning-Environment Resource (S.A.M.P.L.E.R.). The framework shapes the design rationale toward creating and developing learning tools, activities, and facilitation guidelines. The framework then constitutes a data-analysis lens on implementation cases of student insight into the mathematical content. Working with this methodology, a designer begins by focusing on mathematical representations associated with a target concept—the designer problematizes and deconstructs each representation into a pair of historical/cognitive antecedents (idea elements), each lying at the poles of a learning axis. Next, the designer creates bridging tools, ambiguous artifacts bearing interaction properties of each of the idea elements, and develops activities with these learning tools that evoke cognitive conflict along the axis. Students reconcile the conflict by means of articulating strategies that embrace both idea elements, thus integrating them into the target concept.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

Similar content being viewed by others

Notes

  1. In later implementations of ProbLab, students operated the NetLogo models individually.

  2. The current NetLogo environment that is in its 4.x generation does not support Macintosh computers that predate OSX.

  3. We chose to discuss two episodes that are consecutive in our video data so as to demonstrate variability in student mathematical fluency coming in to the design. Also, the episode shows the flexibility of the design in engaging and stimulating understanding at different levels.

  4. Luke’s combinatorial–statistical link could work also for populations of unequal green-to-blue ratios. For example, Luke could speak of a 73%-green population as being “from the green side” of the combinations tower.

  5. This specific episode from the implementation of ProbLab does not directly describe a S.A.M.P.L.E.R. activity but is relevant to the discussion of learning axes and bridging tools.

References

  • Abrahamson, D. (2004). Keeping meaning in proportion: The multiplication table as a case of pedagogical bridging tools. Unpublished doctoral dissertation. Northwestern University, Evanston, IL.

  • Abrahamson, D. (2006a) 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.

    Article  Google Scholar 

  • Abrahamson, D. (2006b). Bottom-up stats: Toward an agent-based “unified” probability and statistics. In D. Abrahamson (Org.), U. Wilensky (Chair), & M. Eisenberg (Discussant), Small steps for agents... giant steps for students?: Learning with agent-based models. Paper presented at the Symposium conducted at the annual meeting of the American Educational Research Association, San Francisco, CA.

  • Abrahamson, D. (2006c). Mathematical representations as conceptual composites: Implications for design. In S. Alatorre, J. L. Cortina, M. Sáiz, & A. Méndez (Eds.), Proceedings of the Twenty Eighth Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 464–466). Universidad Pedagógica Nacional.

  • Abrahamson, D., Berland, M. W., Shapiro, R. B., Unterman, J. W., & Wilensky, U. J. (2006). Leveraging epistemological diversity through computer-based argumentation in the domain of probability. For the Learning of Mathematics, 26(3), 39–45.

    Google Scholar 

  • 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.), Proceedings of the Thirtieth Conference of the International Group for the Psychology of Mathematics Education, PME (Vol. 2, pp. 1–8). Prague: Charles University.

  • Abrahamson, D., Janusz, R., & Wilensky, U. (2006). There once was a 9-Block... – A middle-school design for probability and statistics [Electronic Version]. Journal of Statistics Education, 14(1) from http://www.amstat.org/publications/jse/v14n1/abrahamson.html.

  • Abrahamson, D., & Wilensky, U. (2002). ProbLab. Northwestern University, Evanston, IL: The Center for Connected Learning and Computer-Based Modeling, Northwestern University. http://www.ccl.northwestern.edu/curriculum/ProbLab/.

  • Abrahamson, D., & Wilensky, U. (2004a) S.A.M.P.L.E.R.: Collaborative interactive computer-based statistics learning environment. Paper presented at the 10th International Congress on Mathematical Education, Copenhagen, Denmark.

  • Abrahamson, D., & Wilensky, U. (2004b). NetLogo 9-Blocks model. Center for Connected Learning and Computer-Based Modeling, Northwestern University, Evanston, IL. http://www.ccl.northwestern.edu/netlogo/models/9-Blocks.

  • Abrahamson, D., & Wilensky, U. (2005). The stratified learning zone: Examining collaborative-learning design in demographically-diverse mathematics classrooms. In D. Y. White (Chair) & E. H. Gutstein (Discussant), Equity and diversity studies in mathematics learning and instruction. Paper presented at the annual meeting of the American Educational Research Association, Montreal, Canada.

  • Artigue, M. (2002). Learning mathematics in a CAS environment: The genesis of a reflection about instrumentation and the dialectics between technical and conceptual work. International Journal of Computers for Mathematical Learning, 7(3), 245–274.

    Article  Google Scholar 

  • Bamberger, J., & Ziporyn, E. (1991). Getting it wrong. The World of Music: Ethnomusicology and Music Cognition, 34(3), 22–56.

    Google Scholar 

  • Bereiter, C., & Scardamalia, M. (1985). Cognitive coping strategies and the problem of “inert knowledge”. In S. F. Chipman, J. W. Segal, & R. Glaser (Eds.), Thinking and learning skills: Current research and open questions (Vol. 2, pp. 65–80). Hillsdale, NJ: Lawrence Erlbaum.

  • Berland, M., & Wilensky, U. (2004). Virtual robotics in a collaborative constructionist learning environment. Paper presented at the annual meeting of the American Educational Research Association, San Diego, CA.

  • Biehler, R. (1995). Probabilistic thinking, statistical reasoning, and the search of causes: Do we need a probabilistic revolution after we have taught data analysis? Newsletter of the international study group for research on learning probability and statistics, 8(1). http://www.seamonkey.ed.asu.edu/∼ behrens/teach/intstdgrp.probstat.jan95.html. Accessed Dec. 12, 2002.

  • 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.

    Article  Google Scholar 

  • Case, R., & Okamoto, Y. (1996). The role of central conceptual structures in the development of children’s thought. Monographs of the society for research in child development. Serial No. 246, Vol. 61, Nos. 1–2. Chicago: University of Chicago Press.

  • Cobb, P., Confrey, J., diSessa, A., Lehrer, R., & Schauble, L. (2003). Design experiments in educational research. Educational Researcher, 32(1), 9–13.

    Google Scholar 

  • Cobb, P., Gravemeijer, K., Yackel, E., McClain, K., & Whitenack, J. (1997). Mathematizing and symbolizing: The emergence of chains of signification in one first-grade classroom. In D. Kirshner & J. A. Whitson (Eds.), Situated cognition: Social, semiotic and psychological perspectives (pp. 151–233). Mahwah, NJ: Lawrence Erlbaum Associates.

    Google Scholar 

  • Colella, V., Borovoy, R., & Resnick, M. (1998). Participatory simulations: Using computational objects to learn about dynamic. Paper presented at the Proceedings of the Computer Human Interface (CHI) ’98 Conference, Los Angeles, CA, April 1998.

  • 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., Joseph, D., & Bielaczyc, K. (2004). Design research: Theoretical and methodological issues. Journal of the Learning Sciences: Volume in honor of Ann Brown (J Campione Ed). 13(1), 15–42.

  • 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 

  • Edelson, D. C. (2002). Design research: What we learn when we engage in design. Journal of the Learning Sciences, 11(1), 105–121.

    Google Scholar 

  • Fauconnier, G., & Turner, M. (2002). The way we think: Conceptual blending and the mind’s hidden complexities. New York: Basic Books.

    Google Scholar 

  • Finzer, W. F. (2000). Design of Fathom, a dynamic statistics environment, for the teaching of mathematics. Paper given at the Ninth International Congress on Mathematical Education, Tokyo, Japan, July/August 2000.

  • Fischbein, E. (1975). The intuitive sources of probabilistic thinking in children. London: Reidel.

    Google Scholar 

  • Forman, G., & Pufall, P. (1988). Constructionism in the computer age. Hillsdale, NJ: Lawrence Erlbaum Associates.

    Google Scholar 

  • Freudenthal, H. (1986). Didactical phenomenology of mathematical structure. 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 

  • Gal, I. (2005). Towards “probability literacy” for all citizens: Building blocks and instructional dilemmas. In G. A. Jones (Ed.), Exploring probability in school: Challenges for teaching and learning (pp. 39–64). New York: Springer.

    Chapter  Google Scholar 

  • Gibson, J. (1977). The theory of affordances. In R. Shaw & J. Bransford (Eds.), Perceiving, acting and knowing: Toward an ecological psychology (pp. 67–82). Hillsdale, NJ: Lawrence Erlbaum Associates.

    Google Scholar 

  • 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.

    Google Scholar 

  • Gravemeijer, K. P. E. (1994). Developing realistic mathematics education. Utrecht: CDbeta Press.

    Google Scholar 

  • Hacking, I. (1975). The emergence of probability. Cambridge: Cambridge University Press.

    Google Scholar 

  • Hacking, I. (2001). An introduction to probability and inductive logic. Cambridge, UK: Cambridge University Press.

    Google Scholar 

  • Heidegger, M. (1962). Being and time (J. Macquarrie & E. Robinson, Trans.). NY: Harper & Row (Original work published 1927).

  • Henry, M. (Ed.) (2001). Autour de la modelisation en probabilities. Paris: Press Universitaire Franc-Comtoises.

    Google Scholar 

  • Hoyles, C., & Noss, R. (Eds.) (1992). Learning mathematics and logo. Cambridge, MA: MIT Press.

    Google Scholar 

  • Jones, G. A., Langrall, C. W., & Mooney, E. S. (2007). Research in probability: Responding to classroom realities. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 909–955). Charlotte NC: Information Age Publishing.

    Google Scholar 

  • Klopfer, E., Yoon, S., & Perry, J. (2005). Using palm technology in participatory simulations of complex systems: A new take on ubiquitous and accessible mobile computing. Journal of Science Education and Technology, 14(3), 285–297.

    Article  Google Scholar 

  • Konold, C. (1989). Informal conceptions of probability. Cognition and Instruction, 6, 59–98.

    Article  Google Scholar 

  • Konold, C. (2004). TinkerPlots : Dynamic data exploration: Overview-empowering students as data analysts [Electronic article]. Accessed November 25, 2004, from http://www.umass.edu/srri/serg/projects/tp/tpovervw.html.

  • Levy, S. T., & Wilensky, U. (in press). Inventing a "mid-level" to make ends meet: Reasoning through the levels of complexity. Cognition and Instruction.

  • Levy, S. T., & Wilensky, U. (2004). Making sense of complexity: Patterns in forming causal connections between individual agent behaviors and aggregate group behaviors. Paper presented at the annual meeting of the American Educational Research Association, San Diego, CA, April 12–16, 2004.

  • 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.), Proceedings of the Twenty Fourth Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Athens, GA (Vol. 3, pp. 1331–1334). Columbus, OH: Eric Clearinghouse.

  • Maher, C. A., Speiser, R., Friel, S., & Konold, C. (1998). Learning to reason probabilistically. Proceedings of the Twentieth Annual Conference of the North American Group for the Psychology of Mathematics Education (pp. 82–87). Columbus, OH: Eric Clearinghouse for Science, Mathematics, and Environmental Education.

  • Merleau-Ponty, M. (1962). Phenomenology of perception (C. Smith, Trans.). New York: Humanities Press (Original work published in the French, 1945).

  • Metz, K. E. (1998). Emergent understanding and attribution of randomness: Comparative analysis of the reasoning of primary grade children and undergraduates. Cognition and Instruction, 16(3), 285–365.

    Article  Google Scholar 

  • Minsky, M. (1985). The society of mind. London, England: Hienemann.

    Google Scholar 

  • Papert, S. (1980). Mindstorms: Children, computers, and powerful ideas. NY: Basic Books.

    Google Scholar 

  • Papert, S. (1991). Situating constructionism. In I. Harel & S. Papert (Eds.), Constructionism (pp. 1–12). Norwood, NJ: Ablex Publishing.

    Google Scholar 

  • Papert, S. (1996). An exploration in the space of mathematics educations. International Journal of Computers for Mathematical Learning 1(1), 95–123.

    Google Scholar 

  • Papert, S. (2000). What’s the big idea?...IBM Systems Journal—MIT Media Laboratory, 39(3–4).[Electronic article] Accessed November 25, 2004 from http://www.research.ibm.com/journal/sj/393/part2/papert.html.

  • Piaget, J., & Inhelder, B. (1952). The child’s conception of number. London: Routledge.

    Google Scholar 

  • Piaget, J., & Inhelder, B. (1975). The origin of the Idea of chance in children. London: Routledge and Kegan Paul.

    Google Scholar 

  • Poincaré, J. H. (2003). Science and method (F. Maitland Trans.). New York: Dover (Original work published 1897).

    Google Scholar 

  • Polanyi, M. (1967). The tacit dimension. London: Routledge & Kegan Paul Ltd.

    Google Scholar 

  • Pratt, D. (2000). Making sense of the total of two dice. Journal for Research in Mathematics Education, 31(5), 602–625.

    Article  Google Scholar 

  • Pratt, D. (2004). Towards the design of tools for the organization of the stochastic. In M. Allessandra, B. C., R. Biehler, M. Henry & D. Pratt (Eds.), Proceedings of the Third Conference of the European Society for Research in Mathematics Education. Balleria, Italy.

  • Pratt, D., Jones, I., & Prodromou, T. (2006). An elaboration of the design construct of phenomenalisation. In A. Rossman & B. Chance (Eds.), Proceedings of the Seventh International Conference on Teaching Statistics, Salvador, Brazil.

  • Quine, W. V. O. (1960). Word & object. Cambridge, MA: MIT Press.

    Google Scholar 

  • Resnick, L. B. (1992). From protoquantities to operators: Building mathematical competence on a foundation of everyday knowledge. In G. Leinhardt, R. Putnam, & R. A. Hattrup (Eds.), Analysis of arithmetic for mathematics teaching (pp. 373–429). Hillsdale, NJ: Lawrence Erlbaum.

    Google Scholar 

  • Resnick, M., & Wilensky, U. (1998). Diving into complexity: Developing probabilistic decentralized thinking through role-playing activities. Journal of Learning Sciences 7(2), 153–171.

    Article  Google Scholar 

  • Schoenfeld, A., Smith, J. P., & Arcavi, A. (1991). Learning: The mircrogenetic 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. (1981). Intuitive thinking? A metaphor underlying some ideas of educational reform (Working Paper 8): Division for Study and Research, MIT.

  • Shaughnessy, M. (1992). Research in probability and statistics: Reflections and directions. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 465–494). New York: Macmillan.

    Google Scholar 

  • Simon, J. L., & Bruce, P. (1991). Resampling: A tool for everyday statistical work. Chance: New directions for statistics and computing, 4(1), 22–32.

    Google Scholar 

  • Steiner, G. (2001). Grammars of creation. New Haven, CO: Yale University Press.

    Google Scholar 

  • Surowiecki, J. (2004). The wisdom of crowds. New York: Random House, Doubleday.

    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. (1990). Environment and communication. In L. P. Steffe & T. Wood (Eds.), Transforming children’s mathematics education (pp. 357–376). Hillsdale NJ: Lawrence Erlbaum Publishers.

    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.

  • von Mises, R. (1957). Probability, statistics, and truth (Trans. H. Geiringer). New York, NY: Dover Publications Inc. (original work published 1928).

  • Whewell, W. (1989). Theory of scientific method. Indianapolis, IN: Hackett Publishing Company (Original work published 1837).

  • Wilensky, U. (1991). Abstract meditations on the concrete and concrete implications for mathematics education. In I. Harel & S. Papert (Eds.), Constructionism (pp. 193–204). Norwood, NJ: Ablex Publishing Corporation.

    Google Scholar 

  • Wilensky, U. (1993). Connected mathematics-building concrete relationships with mathematical knowledge. Unpublished manuscript. Cambridge, MA: MIT.

  • Wilensky, U. (1995). Paradox, programming and learning probability. Journal of Mathematical Behavior, 14(2), 231–280.

    Article  Google Scholar 

  • Wilensky, U. (1997). What is normal anyway?: Therapy for epistemological anxiety. Educational Studies in Mathematics, 33(2), 171–202.

    Article  Google Scholar 

  • Wilensky, U. (1999). NetLogo. Northwestern University, Evanston, IL: The Center for Connected Learning and Computer-Based Modeling. http://www.ccl.northwestern.edu/netlogo/

  • Wilensky, U., & Papert, S. (2007). Restructurations: Reformulations of knowledge disciplines through new representational forms. (Manuscript in preparation).

  • Wilensky, U., & Stroup, W. (1999a). HubNet. Evanston, IL: The Center for Connected Learning and Computer-Based Modeling, Northwestern University.

  • Wilensky, U., & Stroup, W. (1999b). Participatory simulations: Network-based design for systems learning in classrooms. Paper presented at the Conference on Computer-Supported Collaborative Learning, Stanford University, December 12–15, 1999.

  • Wilensky, U., & Stroup, W. (2000). Networked gridlock: Students enacting complex dynamic phenomena with the HubNet architecture. Paper presented at the Fourth Annual International Conference of the Learning Sciences, Ann Arbor, MI, June 14–17, 2000.

  • Wilensky, U., & Stroup, W. (2005). Computer-HubNet guide: A guide to computer-based participatory simulations activities. Retrieved 11/06/06, from http://www.ccl.northwestern.edu/ps/guide/Computer%20Part%20Sims%20Guide.pdf.

Download references

Acknowledgements

We wish to express our gratitude to members of the design-research team at The Center for Connected Learning and Computer-Based Modeling, who supported us in facilitating classroom implementations and in collecting data: Sharona T. Levy, Matthew W. Berland, and Joshua W. Unterman. Thank you to Walter Stroup for his valuable comments on the design and classroom implementation of participatory simulation activities. Thank you Eric Betzold, our Research Program Coordinator, for coordinating the many implementation details of this project. Thank you also to Ms. Ruth Janusz, the 6th-grade science-and-mathematics teacher who hosted our implementation and helped with classroom management and in facilitating the design. Thanks to Mr. Paul Brinson, the district’s Chief Information Officer, for evaluating our project and approving the implementation. Thank you Ms. Barb Hiller, the district’s Assistant Superintendent for Curriculum and Instruction, for matching us with the school. Thanks to Mr. Gordon Hood, the school principal, for providing a research site and coordinating with teachers. Also, thank you to all the students who voluntarily participated in this study. Finally, we are grateful to the two anonymous IJCML reviewers who helped us improve and shape this draft. This work was supported by National Science Foundation grant REC 0126227. Completing this paper was facilitated by the first author’s National Academy of Education/Spencer Postdoctoral Fellowship 2005–2006.

Author information

Authors and Affiliations

  1. Graduate School of Education, University of California, 4649 Tolman Hall, Berkeley, CA, 94720-1670, USA

    Dor Abrahamson

  2. The Center for Connected Learning and Computer-Based Modeling, Northwestern University, Evanston, IL, USA

    Uri Wilensky

Authors
  1. Dor Abrahamson
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Uri Wilensky
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dor Abrahamson.

Additional information

The research reported on this paper was funded by NSF ROLE Grant No. REC-0126227. The opinions expressed here are those of the authors and do not necessarily reflect those of NSF. This paper is based on the authors’ AERA 2004 paper titled S.A.M.P.L.E.R.: Statistics As Multi-Participant Learning-Environment Resource.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Abrahamson, D., Wilensky, U. Learning axes and bridging tools in a technology-based design for statistics. Int J Comput Math Learning 12, 23–55 (2007). https://doi.org/10.1007/s10758-007-9110-6

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10758-007-9110-6

Keywords

Search

Navigation