The Emergence of the “FlexTech” Orchestration of Inferential Reasoning on Pattern Generalization
Tóm tắt
The purpose of this study is to further our understanding of orchestrating math-talk with digital technology. The technology used is common in Swedish mathematics classrooms and involves personal computers, a projector directed towards a whiteboard at the front of the class and software programs for facilitating communication and collective exploration. We use the construct of instrumental orchestration to conceptualize a teacher’s intentional and systematic organization and use of digital technology to guide math-talk in terms of a collective instrumental genesis. We consider math-talk as a matter of inferential reasoning, taking place in the Game of Giving and Asking for Reasons (GoGAR).The combination of instrumental orchestration and inferential reasoning laid the foundation of a design experiment that addressed the research question: How can collective inferential reasoning be orchestrated in a technology-enhanced learning environment? The design experiment was conducted in lower-secondary school (students 14–16 years old) and consisted of three lessons on pattern generalization. Each lesson was tested and refined twice by the research team. The design experiment resulted in the emergence of the FlexTech orchestration, which provided teachers and students with opportunities to utilize the flexibility to construct, switch and mark in the orchestration of an instrumental math-GoGAR.
Tài liệu tham khảo
Bakhurst, D. (2011). The formation of reason. London, UK: Wiley Blackwell.
Bakker, A. (2018). Design research in education: A practical guide for early career researchers. Routledge.
Barron, B. (2003). When smart groups fail. The Journal of the Learning Sciences, 12(3), 307–359.
Bauersfeld, H. (1998). Interaction, construction, and knowledge: Alternative perspectives for mathematics education. In D. Grouws, T. Cooney, & D. Jones (Eds), Perspectives on research on effective mathematics teaching (pp. 27–46). Reston, VA: National Council of Teachers of Mathematics.
Brandom, R. (1994). Making it explicit: Reasoning, representing, and discursive commitment. Harvard University Press.
Brandom, R. (1995). Knowledge and the social articulation of the space of reasons. Philosophy and Phenomenological Research, 55(4), 895–908.
Brandom, R. (2000). Articulating reasons: An introduction to inferentialism. Harvard University Press.
Bransen, J. (2002). Normativity as the key to objectivity: An exploration of Robert Brandom’s articulating reasons. Inquiry, 45(3), 373–391.
Brodie, K. (2011). Working with learners’ mathematical thinking: Towards a language of description for changing pedagogy. Teaching and Teacher Education, 27(1), 174–186.
Brodie, K. (2013). The power of professional learning communities. Education as Change, 17(1), 5–18.
Clark-Wilson, A., Hoyles, C., Noss, R., Vahey, P., &Roschelle, J. (2015). Scaling a technology-based innovation: Windows on the evolution of mathematics teachers’ practices. ZDM: The International Journal on Mathematics Education, 47(1), 79–92.
Clark-Wilson, A. (2010). Emergent pedagogies and the changing role of the teacher in the TI-Nspire Navigator-networked mathematics classroom. ZDM: The International Journal on Mathematics Education, 42(7) 747–761.
Cobb, P., Confrey, J., Lehrer, R., & Schauble, L. (2003). Design experiments in educational research. Educational Researcher, 32(1), 9–13.
Conner, A., Singletary, L., Smith, R., Wagner, P., & Francisco, R. (2014). Teacher support for collective argumentation: A framework for examining how teachers support students’ engagement in mathematical activities. Educational Studies in Mathematics, 86(3), 401–429.
Cusi, A., Morselli, F., & Sabena, C. (2017). Promoting formative assessment in a connected classroom environment: Design and implementation of digital resources. ZDM: The International Journal on Mathematics Education, 49(5), 755–767.
Derry, J. (2013). Vygotsky: Philosophy and education. London, UK: Wiley Blackwell.
diSessa, A., & Cobb, P. (2004). Ontological innovation and the role of theory in design experiments. Journal of the Learning Sciences, 13(1), 77–103.
Doerr, H., & Zangor, R. (2000). Creating meaning for and with the graphing calculator. Educational Studies in Mathematics, 41(2), 143–163.
Drijvers, P. (2011). Teachers transforming resources into orchestrations. In G. Gueudet, B. Pepin, & L. Trouche (Eds.), From text to ‘lived’ resources: Mathematics curriculum materials and teacher development (pp. 265–281). Springer.
Drijvers, P. (2015). Digital technology in mathematics education: Why it works (or doesn’t). In S. Cho (Ed.), Selected regular lectures from the 12th International Congress on Mathematical Education (pp. 135–151). Springer.
Drijvers, P., Doorman, M., Boon, P., Reed, H., & Gravemeijer, K. (2010). The teacher and the tool: Instrumental orchestrations in the technology-rich mathematics classroom. Educational Studies in Mathematics, 75(2), 213–234.
Drijvers, P. (2019). Embodied instrumentation: Combining different views on using digital technology in mathematics education. In U. Jankvist, M. van den Heuvel-Panhuizen, & M. Veldhuis (Eds), Proceedings of the eleventh congress of the European Society for Research in Mathematics Education (pp. 8–28). Utrecht, The Netherlands: Freudenthal Group &Freudenthal Institute.
Eckert, A. (2017). Theorizing the interactive nature of teaching mathematics: Contributing to develop contributions as a metaphor for teaching. Nordic Studies in Mathematics Education, 22(3), 51–69.
Eckert, A., & Nilsson, P. (2017). Introducing a symbolic interactionist approach on teaching mathematics: the case of revoicing as an interactional strategy in the teaching of probability. Journal of Mathematics Teacher Education, 20(1), 31–48. https://doi.org/10.1007/s10857-015-9313-z.
Eckert, A., & Nilsson, P. (2019). Designing for digitally enriched Math Talks: The case of pattern generalization. In Eleventh Congress of the European Society for Research in Mathematics Education (CERME11), Utrecht, the Netherlands, February 6-10, 2019. (pp. 4210-4217). Freudenthal Group & Freudenthal Institute, Utrecht University, Netherlands and ERME.
Fraivillig, J., Murphy, L., & Fuson, K. (1999). Advancing children’s mathematical thinking in everyday mathematics classrooms. Journal for Research in Mathematics Education, 30(2), 148–170.
Guin, D., &Trouche, L. (2002). Mastering by the teacher of the instrumental genesis in CAS environments: Necessity of instrumental orchestrations. ZDM: The International Journal on Mathematics Education, 34(5), 204–211.
Hegedus, S., Laborde, C., Brady, C., Dalton, S., Siller, H.-S., Tabach, M., Trgalova, J., & Moreno-Armella, L. (2017). Uses of technology in upper secondary mathematics education (pp. 1–36). Springer.
Hegedus, S., & Moreno-Armella, L. (2009). Intersecting representation and communication infrastructures. ZDM: The International Journal on Mathematics Education, 41(4), 399–412.
Hintz, A., & Tyson, K. (2015). Complex listening: Supporting students to listen as mathematical sense-makers. Mathematical Thinking and Learning, 17(4), 296–326.
Hoekstra, A., Brekelmans, M., Beijaard, D., & Korthagen, F. (2009). Experienced teachers’ informal learning: Learning activities and changes in behavior and cognition. Teaching and Teacher Education, 25(5), 663–673.
Hufferd-Ackles, K., Fuson, K., & Sherin, M. (2004). Describing levels and components of a math-talk learning community. Journal for Research in Mathematics Education, 35(2), 81–116.
Irving, K. (2006). The impact of educational technology on student achievement: Assessment of and for learning. Science Educator, 15(1), 13–20.
Jackson, K., Garrison, A., Wilson, J., Gibbons, L., & Shahan, E. (2013). Exploring relationships between setting up complex tasks and opportunities to learn in concluding whole-class discussions in middle-grades mathematics instruction. Journal for Research in Mathematics Education, 44(4), 646–682.
Kaput, J. (1992). Technology and mathematics education. In D. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 515–556). Macmillan.
Kazemi, E., & Stipek, D. (2001). Promoting conceptual thinking in four upper-elementary mathematics classrooms. The Elementary School Journal, 102(1), 59–80.
Kendal, M., & Stacey, K. (2002). Teachers in transition: Moving towards CAS-supported classrooms. ZDM: The International Journal on Mathematics Education, 34(5), 196–203.
Kosyvas, G. (2016). Levels of arithmetic reasoning in solving an open-ended problem. International Journal of Mathematical Education in Science and Technology, 47(3), 356–372.
Lagrange, J.-B. & Monaghan, J. (2010). On the adoption of a model to interpret teachers’ use of technology in mathematics lessons. In V. Durand-Guerrier, S. Soury-Lavergne & F. Arzarello (Eds), Proceedings of the sixth congress of the European Society for Research in Mathematics Education (pp. 1605–1614). Lyon, France: Institut National de Recherche Pédagogique.
Lerman, S. (2000). The social turn in mathematics education research. In J. Boaler (Ed.), Multiple perspectives on mathematics teaching and learning (pp. 19–44). Ablex.
Leung, A., Chan, Y., & Lopez-Real, F. (2006). Instrumental genesis in dynamic geometry environments. In C. Hoyles, J.-B. Lagrange, L. Son & N. Sinclair (Eds), Technology revisited: Proceedings of the seventeenth study conference of the International Commission on Mathematical Instruction (pp. 346–353). Hanoi, Vietnam: Hanoi Institute of Technology and DidiremUniversité Paris 7.
Lonchamp, J. (2012). An instrumental perspective on CSCL systems. International Journal ofComputer-Supported Collaborative Learning, 7(2), 211–237.
Manouchehri, A., & Enderson, M. (1999). Promoting mathematical discourse: Learning from classroom examples. Mathematics Teaching in the Middle School, 4(4), 216–222.
Marton, F., Tsui, A., Chik, P., Ko, P., & Lo, M. (2004). Classroom discourse and the space of learning. Lawrence Erlbaum Associates.
Mouhayar, R., & Jurdak, M. (2015). Variation in strategy use across grade level by pattern generalization types. International Journal of Mathematical Education in Science and Technology, 46(4), 553–569.
Nilsson, P. (2019). An inferentialist perspective on how note-taking can constrain the orchestration of math-talk. Scandinavian Journal of Educational Research, 67(3), 1121–1133.
Nilsson, P. (2020). A framework for investigating qualities of procedural and conceptual knowledge in mathematics: An inferentialist perspective. Journal for Research in Mathematics Education, 51(5), 574–599.
Nilsson, P., & Ryve, A. (2014). The nature and role of common ground in the learning of mathematics in small-group discussions. Scandinavian Journal of Educational Research, 58(5), 609–623.
Noorloos, R., Taylor, S., Bakker, A., & Derry, J. (2017). Inferentialism as an alternative to socioconstructivism in mathematics education. Mathematics Education Research Journal, 29(4), 437–453.
Rezat, S., &Sträßer, R. (2012). From the didactical triangle to the socio-didactical tetrahedron: Artifacts as fundamental constituents of the didactical situation. ZDM: The International Journal on Mathematics Education, 44(5), 641–651.
Rivera, F. (2010). Visual templates in pattern generalization activity. Educational Studies in Mathematics, 73(3), 297–328.
Robert, A., & Rogalski, J. (2005). A cross-analysis of the mathematics teacher’s activity: An example in a French 10th-grade class. Educational Studies in Mathematics, 59(1–3), 269–298.
Roschelle, J., Noss, R., Blikstein, P., &Jackiw, N. (2017). Technology for learning mathematics. In J. Cai (Ed.), Compendium for research in mathematics education (pp. 853–878). Reston, VA: National Council of Teachers of Mathematics.
Ruthven, K., Laborde, C., Leach, J., & Tiberghien, A. (2009). Design tools in didactical research: Instrumenting the epistemological and cognitive aspects of the design of teaching sequences. Educational Researcher, 38(5), 329–342.
Schindler, M., & Seidouvy, A. (2019). Informal inferential reasoning and the social: Understanding students’ informal inferences through an inferentialist perspective. In G. Burril & D. Ben-Zvi (Eds.), Topics and trends in current statistics education research: International perspectives (pp. 153–171). Springer.
Sellars, W., Rorty, R., & Brandom, R. (1997). Empiricism and the philosophy of mind. Harvard University Press.
Stacey, K. (1989). Finding and using patterns in linear generalising problems. Educational Studies in Mathematics, 20(2), 147–164.
Staples, M. (2007). Supporting whole-class collaborative inquiry in a secondary mathematics classroom. Cognition and Instruction, 25(2–3), 161–217.
Stein, M., Engle, R., Smith, M., & Hughes, E. (2008). Orchestrating productive math-talk: Five practices for helping teachers move beyond show and tell. Mathematical Thinking and Learning, 10(4), 313–340.
Trouche, L. (2004). Managing the complexity of human/machine interactions in computerized learning environments: Guiding students’ command process through instrumental orchestrations. International Journal of Computers for Mathematical Learning, 9(3), 281–307.
Walshaw, M., & Anthony, G. (2008). The teacher’s role in classroom discourse: A review of recent research into mathematics classrooms. Review of Educational Research, 78(3), 516–551.
White, T. (2018). Connecting levels of activity with classroom network technology. International Journal of Computer-Supported Collaborative Learning, 13(1), 93–122.