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Reverse-scaffolding algebra: empirical evaluation of design architecture

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Abstract

Scaffolding is the asymmetrical social co-enactment of natural or cultural practice, wherein a more able agent implements or performs for a novice elements of a challenging activity. What the novice may not learn, however, is how the expert’s co-enactments support the activity. Granted, in many cultural practices novices need not understand underlying process. But where process is content, such as mathematics, scaffolding is liable to undermine tenets of reform-oriented pedagogy. We point to tensions between traditional conceptualizations of scaffolding and discovery-based pedagogical methodology for mathematics education. Focusing on co-enactment as a critical feature of scaffolding activities, we introduce “reverse scaffolding”, wherein experts enact for novices only what they know to do rather than what they do not know to do. We demonstrate our approach by discussing a novel technological learning activity, Giant Steps for Algebra, wherein students construct models of realistic narratives. We argue for the method’s potential via reporting on findings from mixed-methods analyses of a quasi-experimental implementation with 40 students.

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References

  • Abrahamson, D. (2009). Embodied design: constructing means for constructing meaning. Educational Studies in Mathematics, 70(1), 27–47.

    Article  Google Scholar 

  • Abrahamson, D. (2014). Building educational activities for understanding: an elaboration on the embodied-design framework and its epistemic grounds. International Journal of Child-Computer Interaction, 2(1), 1–16.

    Article  Google Scholar 

  • Abrahamson, D., & Chase, K. (2015). Interfacing practices: domain theory emerges via collaborative reflection. Reflective Practice, 16(3), 372–389.

    Article  Google Scholar 

  • Abrahamson, D., Chase, K., Kumar, V., & Jain, R. (2014). Leveling transparency via situated, intermediary learning objectives. In J. L. Polman, E. A. Kyza, D. K. O’Neill, I. Tabak, W. R. Penuel, A. S. Jurow, K. O’Connor, T. Lee & L. D’Amico (Eds.), Proceedings of “Learning and Becoming in Practice”, the 11th International Conference of the Learning Sciences (ICLS) 2014 (Vol. 1, pp. 23–30). Boulder, CO: International Society of the Learning Sciences.

  • Alfieri, L., Brooks, P. J., Aldrich, N. J., & Tenenbaum, H. R. (2011). Does discovery-based instruction enhance learning? Journal of Educational Psychology, 103(1), 1–18.

    Article  Google Scholar 

  • Arcavi, A. (1994). Symbol sense: informal sense-making in formal mathematics. For the Learning of Mathematics, 14(3), 24–35.

    Google Scholar 

  • Barab, S., Zuiker, S., Warren, S., Hickey, D., Ingram-Goble, A., Kwon, E.-J., & Herring, S. C. (2007). Situationally embodied curriculum: relating formalisms and contexts. Science Education, 91, 750–782.

    Article  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 (pp. 720–749). Mahwah, NJ: Lawrence Erlbaum Associates.

  • Bruner, J. S. (1986). Actual minds: Possible worlds. Cambridge, MA: Harvard University Press.

  • Cazden, C. B. (1981). Performance before competence: assistance to child discourse in the Zone of Proximal Development. Quarterly Newsletter of the Laboratory of Comparative Human Cognition, 3(1), 5–8.

    Google Scholar 

  • Chow, J. Y., Davids, K., Button, C., Shuttleworth, R., Renshaw, I., & Araújo, D. (2007). The role of nonlinear pedagogy in physical education. Review of Educational Research, 77(3), 251–278.

    Article  Google 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: Lawrence Erlbaum Associates.

    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: Cambridge University Press.

    Chapter  Google Scholar 

  • Dickinson, P., & Eade, F. (2004). Using the number line to investigate the solving of linear equations. For the Learning of Mathematics, 24(2), 41–47.

    Google Scholar 

  • Freudenthal, H. (1971). Geometry between the devil and the deep sea. Educational Studies in Mathematics, 3(3/4), 413–435.

    Article  Google Scholar 

  • Ginsburg, H. P. (1997). Entering the child’s mind. New York: Cambridge University Press.

    Book  Google Scholar 

  • Goldstein, I., & Papert, S. (1977). Artificial intelligence, language, and the study of knowledge. Cognitive Science, 1(1), 84–123.

    Article  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.

    Article  Google Scholar 

  • Herscovics, N., & Linchevski, L. (1994). A cognitive gap between arithmetic and algebra. Educational Studies in Mathematics, 27(1), 59–78.

    Article  Google Scholar 

  • Holmes, N. G., Day, J., Park, A. K., Bonn, D. A., & Roll, I. (2014). Making the failure more productive: scaffolding the invention process to improve inquiry behaviors and outcomes in invention activities. Instructional Science, 42(4), 523–538.

    Article  Google Scholar 

  • Holton, D., & Clarke, D. (2006). Scaffolding and metacognition. International Journal of Mathematical Education in Science and Technology, 37(2), 127–143.

    Article  Google Scholar 

  • Jones, I. (2010). Presenting equality statements as diagrams. Paper presented at the Proceedings of the 6th Congress of the European Society for Research in Mathematics Education.

  • Jones, I., Inglis, M., Gilmore, C., & Dowens, M. (2012). Substitution and sameness: two components of a relational conception of the equals sign. Journal of Experimental Child Psychology, 113(1), 166–176.

    Article  Google Scholar 

  • Jordan, B., & Henderson, A. (1995). Interaction analysis: foundations and practice. Journal of the Learning Sciences, 4(1), 39–103.

    Article  Google Scholar 

  • Kamii, C. K., & Dominick, A. (1998). The harmful effects of algorithms in grades 1-4. In L. J. Morrow & M. J. Kenney (Eds.), The teaching and learning of algorithms in school mathematics, 1998 yearbook (pp. 130–140). Reston: NCTM.

    Google Scholar 

  • Kapur, M. (2014). Productive failure in learning math. Cognitive Science, 38(5), 1008–1022.

    Article  Google Scholar 

  • Kirschner, P. A., Sweller, J., & Clark, R. E. (2006). Why minimal guidance during instruction does not work: an analysis of the failure of constructivist, discovery, problem-based, experiential, and inquiry-based teaching. Educational Psychologist, 41(2), 75–86.

    Article  Google Scholar 

  • Kirsh, D. (2010). Thinking with external representations. AI & SOCIETY, 25, 441–454.

    Article  Google Scholar 

  • Martin, T., & Schwartz, D. L. (2005). Physically distributed learning: adapting and reinterpreting physical environments in the development of fraction concepts. Cognitive Science, 29(4), 587–625.

    Article  Google Scholar 

  • Meira, L. (1998). Making sense of instructional devices: the emergence of transparency in mathematical activity. Journal for Research in Mathematics Education, 29(2), 129–142.

    Article  Google Scholar 

  • Molina, M., & Ambrose, R. (2008). From an operational to a relational conception of the equal sign. Thirds graders’ developing algebraic thinking. Focus on Learning Problems in Mathematics, 30(1), 61–80.

    Google Scholar 

  • Moses, R. P., & Cobb, C. E. (2001). Radical equations: civil rights from Mississippi to the Algebra Project. Boston: Beacon Press.

  • Noss, R., & Hoyles, C. (1996). Windows on mathematical meanings: learning cultures and computers. Dordrecht: Kluwer.

    Book  Google Scholar 

  • Oakes, J., Joseph, R., & Muir, K. (2004). Access and achievement in mathematics and science: inequalities that endure and change. In J. Banks & M. Banks (Eds.), Handbook of research on multicultural education. San Francisco: Jossey-Bass.

    Google Scholar 

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

    Google Scholar 

  • Pea, R. D. (2004). The social and technological dimensions of scaffolding and related theoretical concepts for learning, education, and human activity. The Journal of the Learning Sciences, 13(3), 423–451.

    Article  Google Scholar 

  • Pedemonte, B., & Chiappini, G. (2008). Algebra on Numerical SETs: a system for teaching and learning algebra. International Journal of Continuing Engineering Education and Life-Long Learning, 18(5/6), 627–639.

    Article  Google Scholar 

  • Schneider, B., Bumbacher, E., & Blikstein, P. (2015). Discovery versus direct instruction: Learning outcomes of two pedagogical models using tangible interfaces. In T. Koschmann, P. Häkkinen, & P. Tchounikine (Eds.), “Exploring the material conditions of learning: opportunities and challenges for CSCL,” the Proceedings of the Computer Supported Collaborative Learning (CSCL) Conference (Vol. 1, pp. 364–371). Gothenburg, Sweden: ISLS.

  • Quintana, C., Reiser, B. J., Davis, E. A., Krajcik, J. S., Fretz, E., Duncan, R. G., & Soloway, E. (2004). A scaffolding design framework for software to support science inquiry. Journal of the Learning Sciences, 13(3), 337–386.

    Article  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.

    Article  Google Scholar 

  • Reed, E. S., & Bril, B. (1996). The primacy of action in development. In M. L. Latash & M. T. Turvey (Eds.), Dexterity and its development (pp. 431–451). Mahwah: Lawrence Erlbaum Associates.

    Google Scholar 

  • Reiser, B. J. (2004). Scaffolding complex learning: the mechanisms of structuring and problematizing student work. Journal of the Learning Sciences, 13(3), 273–304.

    Article  Google Scholar 

  • Schneider, B., Bumbacher, E., & Blikstein, P. (2015). Discovery versus direct instruction: Learning outcomes of two pedagogical models using tangible interfaces. In T. Koschmann, P. Häkkinen, & P. Tchounikine (Eds.), “Exploring the material conditions of learning: opportunities and challenges for CSCL”, the Proceedings of the Computer Supported Collaborative Learning (CSCL) Conference (Vol. 1, pp. 365-371). Gothenburg, Sweden: ISLS.

  • Siegler, R. S. (2006). Microgenetic analyses of learning. In D. Kuhn & R. S. Siegler (Eds.), Handbook of child psychology (6 ed., Vol. 2, Cognition, perception, and language, pp. 464–510). Hoboken, NJ: Wiley.

  • Smit, J., van Eerde, H. A. A., & Bakker, A. (2013). A conceptualisation of whole-class scaffolding. British Educational Research Journal, 39(5), 817–834.

    Article  Google Scholar 

  • Strauss, A. L., & Corbin, J. (1990). Basics of qualitative research: grounded theory procedures and techniques. Newbury Park: Sage Publications.

    Google Scholar 

  • van Reeuwijk, M. (1995). The role of realistic situations in developing tools for solving systems of equations. Paper presented at the annual conference of the American Educational Research Association, San Francisco.

  • Vlassis, J. (2002). The balance model: hindrance or support for the solving of linear equations with one unknown. Educational Studies in Mathematics, 49(3), 341–359.

    Article  Google Scholar 

  • Vygotsky, L. S. (1978). Mind in society: The development of higher psychological processes. Cambridge: Harvard University Press. (Original work published 1930).

  • Walkington, C., Petrosino, A., & Sherman, M. (2013). Supporting algebraic reasoning through personalized story scenarios: how situational understanding mediates performance and strategies. Mathematical Thinking and Learning, 15(2), 89–120.

    Article  Google Scholar 

  • Wood, D., Bruner, J. S., & Ross, G. (1976). The role of tutoring in problem solving. Journal of Child Psychology and Psychiatry, 17(2), 89–100.

    Article  Google Scholar 

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Chase, K., Abrahamson, D. Reverse-scaffolding algebra: empirical evaluation of design architecture. ZDM Mathematics Education 47, 1195–1209 (2015). https://doi.org/10.1007/s11858-015-0710-7

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