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Professional development of mathematics teachers toward the facilitation of small-group collaboration

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Abstract

Collaborative work in small groups is often a suitable context for yielding substantial individual learning outcomes. Indeed, small-group collaboration has recently become an educational goal rather than a means. Yet, this goal is difficult to attain, and students must be taught how to learn together. In this paper, we focus on how to prepare teachers to become facilitators of small-group collaboration. The current case study monitors a group of six prospective teachers and their instructor during a one-semester course. The instructor was a skilled mathematics teacher with strong beliefs about what is entailed in establishing a mini-culture of learning to learn together and about how to facilitate student group work in problem-solving situations. We describe the learning path followed by the instructor, including the digital environment. The findings show that by the end of the course, the students became more competent facilitators of learning to learn together.

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Notes

  1. Chapman (2011), O’Connor (2001) and Stein and colleagues (2008) conducted studies with elementary or middle school mathematics teachers that considered the whole class as a group and were aimed at promoting teachers’ inquiry-based learning of their own practice. Nevertheless, these studies, which reported on teachers’ advancements, focused on the whole class and not on facilitating small-group collaboration.

  2. Metafora provides tools that allow students to engage in discussion and argumentation. Specifically, LASAD enables the co-elaboration of argumentation maps. Hence, LASAD is used as a space for engaging in critical peer evaluation (Loll, Pinkwart, Scheuer, & McLaren, 2012).

  3. In sociology and social psychology, breaching of experiments is well known. In these experiments, a norm or rule is intentionally violated to learn about participant attitude toward such violation. In our case, norms were breached not as a pre-planned methodical decision, but as part of what happened in the course. Hence, we believe that our interpretation is plausible.

  4. https://www.youtube.com/watch?v=7StEk9ys39w

  5. Here, we provide only a short description of the students’ work, as we focus on the teacher group. For a detailed account of the students’ actions while solving this problem, see Schwarz et al., 2015.

References

  • Andersen, E. (2016). Learning to learning. Harvard Business Review, 2016, 98–101.

    Google Scholar 

  • Andriessen, J., Baker, M., & Suthers, D. (2003). Arguing to learn: Confronting cognitions in computer-supported collaborative learning environments. Dordrecht: Kluwer.

    Book  Google Scholar 

  • Brauning, K., & Steinbring, H. (2011). Communicative characteristics of teachers’ mathematical talk with children: From knowledge transfer to knowledge investigation. ZDM Mathematics Education, 43, 927–939. https://doi.org/10.1007/s11858-011-0351-4

    Article  Google Scholar 

  • Chapman, O. (2011). Elementary school teachers’ growth in inquiry-based teaching of mathematics. ZDM Mathematics Education, 43, 951–963. https://doi.org/10.1007/s11858-011-0360-3

    Article  Google Scholar 

  • Claxton, G. (2004). Teaching children to learn: Beyond flat-packs and fine words. Burning Issues in Primary Education No. 11. Birmingham: National Primary Trust.

  • Cobb, P., & Bauersfeld, H. (Eds.). (1995). The emergence of mathematical meaning: Interaction in classroom culture. Hillsdale, NJ: Lawrence Erlbaum Associates.

    Google Scholar 

  • Cobb, P., Stephan, M., McClain, K., & Gravemeijer, K. (2001). Participating in classroom mathematical practices. The Journal of the Learning Sciences, 10(1&2), 113–164.

    Article  Google Scholar 

  • Crespo, S. (2003). Learning to pose mathematical problems: Exploring changes in preservice teachers’ practices. Educational Studies in Mathematics, 52, 243–270.

    Article  Google Scholar 

  • Engeström, Y. (2001). Expansive learning at work: Toward an activity theoretical reconceptualization. Journal of Education and Work, 14(1), 133–156.

    Article  Google Scholar 

  • Fredriksson, U., & Hoskins, B. (2007). The development of learning to learn in a European context. Curriculum Journal, 18(2), 127–134.

    Article  Google Scholar 

  • Ghousseini, H. (2009). Designing opportunities to learn to lead classroom mathematics discussions in pre-service teacher education: Focusing on enactment. In D. Mewborn & H. Lee (Eds.), Scholarly practices and inquiry in the preparation of mathematics teachers (pp. 147–158). San Diego, CA: Association of Mathematics Teacher Educators.

    Google Scholar 

  • Gillies, R. M. (2006). Teachers’ and students’ verbal behaviours during cooperative and small-group learning. British Journal of Educational Psychology, 76, 271–287.

    Article  Google Scholar 

  • Heyd-Metzuyanim, E., Tabach, M., & Nachlieli, T. (2016). Opportunities for learning given to prospective mathematics teachers – Between ritual and explorative instruction. The Journal of Mathematics Teacher Education, 16(6), 547–574. https://doi.org/10.1007/s10857-015-9311-1

    Article  Google Scholar 

  • Jansen, A. (2012). Developing productive dispositions during small-group work in two sixth-grade mathematics classrooms: Teachers’ facilitation efforts and students’ self-reported benefits. Middle Grades Research Journal, 7(1), 37–56.

    Google Scholar 

  • Karacop, A., & Doymus, K. (2013). Effects of jigsaw cooperative learning and animation techniques on students’ understanding of chemical bonding and their conceptions of the particulate nature of matter. Journal of Science Education and Technology, 22(2), 186–203.

    Article  Google Scholar 

  • Kazemi, E., Franke, M., & Lampert, M. (2009). Developing pedagogies in teacher education to support novice teachers’ ability to enact ambitious instruction. In R. Hunter, B. Bicknell, & T. Burgess (Eds.), Crossing divides: Proceedings of the 32 nd Annual Conference of the Mathematics Education Research Group of Australasia (Vol. 1, pp. 12–30). Wellington, NZ: MERGA.

  • Loll, F., Pinkwart, N., Scheuer, O., McLaren, B. M. (2012). How tough should it be? Simplifying the development of argumentation systems using a configurable platform. In N. Pinkwart & B. M. McLaren (Eds.), Educational technology teaching argumentation skills (pp. 169–197). Emirate of Sharjah: Bentham Science Publishers.

  • McMillan, J. H., & Hearn, J. (2009). Student self-assessment. The Education Digest, 74(8), 39.

    Google Scholar 

  • Mercer, N. (1996). The quality of talk in children’s collaborative activity in the classroom. Learning and Instruction, 6, 359–375.

    Article  Google Scholar 

  • Mercer, N., & Sams, C. (2006). Teaching children how to use language to solve maths problems. Language and Education, 20(6), 507–528.

    Article  Google Scholar 

  • NCTM. (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics.

    Google Scholar 

  • Nicol, C. (1998). Learning to teach mathematics: Questioning, listening, and responding. Educational Studies in Mathematics, 37(1), 45–66.

    Article  Google Scholar 

  • O’Connor, M. C. (2001). “Can any fraction be turned into a decimal?” A case study of a mathematical group discussion. Educational Studies in Mathematics, 46, 143–185.

  • OECD. (2003). Education at a Glance: OECD Indicators. OECD iLibrary. Paris: OECD.

  • OECD. (2004). Education at a glance: OECD indicators. OECD iLibrary. Paris: OECD.

  • Presmeg, N. (2014). A dance of instruction with construction in mathematics education. In U. Kortenkamp, B. Brandt, C. Benz, G. Krummheuer, S. Ladel, & R. Vogel (Eds.), Early mathematics learning (pp. 9–17). The Netherlands: Springer.

    Chapter  Google Scholar 

  • Prusak, N., Hershkowitz, R., & Schwarz, B. B. (2012). From visual reasoning to logical necessity through argumentative design. Educational Studies in Mathematics, 79, 19–40.

    Article  Google Scholar 

  • Remillard, J. T. (2005). Examining key concepts in research on teachers’ use of mathematics curricula. Review of Educational Research, 75(2), 211–246. https://doi.org/10.3102/00346543075002211

    Article  Google Scholar 

  • Rummel, N., & Spada, H. (2005). Learning to collaborate: An instructional approach to promoting collaborative problem solving in computer-mediated settings. Journal of the Learning Sciences, 14(2), 201–241.

    Article  Google Scholar 

  • Sadler, P., & Good, E. (2006). The impact of self- and peer-grading on student learning. Educational Assessment, 11(1), 1–31.

    Article  Google Scholar 

  • Scardamalia, M. (2004). CSILE/knowledge forum®. In Education and technology: An encyclopedia (pp. 183–192). Santa Barbara: ABC-CLIO.

    Google Scholar 

  • Scardamalia, M., & Bereiter, C. (1994). Computer support for knowledge building communities. Journal of the Learning Sciences, 3(3), 265–283.

    Article  Google Scholar 

  • Schwarz, B. B., & Asterhan, C. S. C. (2011). E-moderation of synchronous discussions in educational settings: A nascent practice. Journal of the Learning Sciences, 20(3), 259–282.

    Article  Google Scholar 

  • Schwarz, B. B., de Groot, R., Mavrikis, M., & Dragon, T. (2015). Learning to learn together with CSCL tools. International Journal of Computer-Supported Collaborative Learning, 10(3), 239–271.

    Article  Google Scholar 

  • Slakmon, B., & Schwarz, B. B. (2017). "You will be a polis": Political (democratic?) education, public space and CSCL discussions. The Journal of the Learning Sciences, 26(2), 184–225.

    Article  Google Scholar 

  • Sluijsmans, D. M. A., Brand-Gruwel, S., & van Merriënboer, J. J. G. (2002). Peer assessment training in teacher education: Effects on performance and perceptions. Assessment & Evaluation in Higher Education, 27(5), 443–454.

    Article  Google Scholar 

  • Springer, L., Stanne, M. E., & Donovan, S. S. (1999). Effects of small-group learning on undergraduates in science, mathematics, engineering, and technology: A meta-analysis. Review of Educational Research, 69(1), 21–51.

    Article  Google Scholar 

  • Stahl, G. (2016). Constructing dynamic triangles together: The Development of Mathematical Group Cognition. Cambridge: Cambridge University Press.

  • Stein, M. K., Engle, R. A., Smith, M. S., & Hughes, E. K. (2008). Orchestrating productive mathematical discussions: Five practices for helping teachers move beyond show and tell. Mathematical Thinking and Learning, 10(4), 313–340. https://doi.org/10.1080/10986060802229675

    Article  Google Scholar 

  • Topping, K. (2003). Self- and peer assessment in school and university: Reliability, validity and utility. In M. Segers, F. Dochy, & E. Cacallar (Eds.), Optimising new modes of assessment: In search of qualities and standards (pp. 55–87). Dordrecht: Kluwer Academic Publishers.

    Chapter  Google Scholar 

  • Topping, K. (2009). Peer assessment. Theory Into Practice, 48, 20–27.

    Article  Google Scholar 

  • Tyminski, A. M., Zambak, V. S., Drake, C., & Land, T. J. (2014). Using representations, decomposition, and approximations of practices to support prospective elementary mathematics teachers’ practice of organizing discussions. Journal of Mathematics Teacher Education, 17, 463–487. https://doi.org/10.1007/s10857-013-9261-4

    Article  Google Scholar 

  • Veenman, S., Denessen, E., van den Akker, A., & van der Rijt, J. (2005). Effects of a cooperative learning program on the elaborations of students during help-seeking and help-giving. American Educational Research Journal, 42(1), 115–153. https://doi.org/10.3102/00028312042001115

    Article  Google Scholar 

  • Webb, N. M. (2009). The teacher’s role in promoting collaborative dialogue in the classroom. British Journal of Educational Psychology, 79, 1–28.

    Article  Google Scholar 

  • Wegerif, R. (2007). Dialogic, educational and technology: Expanding the space of learning. New York, NY: Springer-Verlag.

    Book  Google Scholar 

  • Wegerif, R., & De Laat, M. F. (2010). Reframing the teaching of higher order thinking for the network society. In S. Ludvigsen, A. Lund, & R. Saljo (Eds.), Learning in social practices: ICT and new artefacts transformation of social and cultural practices. Abingdon: Routledge.

  • Yackel, E., & Cobb, P. (1996). Sociomathematical norms, argumentation and autonomy in mathematics. The Journal of Research in Mathematics Education, 27, 458–477.

    Article  Google Scholar 

  • Zhang, J., Scardamalia, M., Reeve, M., & Messina, R. (2009). Designs for collective cognitive responsibility in knowledge-building communities. The Journal of the Learning Sciences, 18, 7–44.

    Article  Google Scholar 

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Tabach, M., Schwarz, B.B. Professional development of mathematics teachers toward the facilitation of small-group collaboration. Educ Stud Math 97, 273–298 (2018). https://doi.org/10.1007/s10649-017-9796-x

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