Teachers’ structuring of mathematical inquiry lessons: shifting from “task-first” to “scaffolded inquiry”

References

• 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. https://doi.org/10.1037/a0021017
   Web of Science ®Google Scholar
• Artigue, M., & Blomhøj, M. (2013). Conceptualising inquiry-based education in mathematics. ZDM Mathematics Education, 45(6), 797–810. https://doi.org/10.1007/s11858-013-0506-6
   Google Scholar
• Askew, M., Brown, M., Rhodes, V., Johnson, D., & William, D. (1997). Effective teachers of numeracy: Final report to the teacher training agency. King’s College.
   Google Scholar
• Ball, D. L. (2001). Teaching, with respect to mathematics and students. In T. Wood, B. S. Nelson, & J. Warfield (Eds.), Beyond classical pedagogy: Teaching elementary school mathematics (pp. 11–22). Erlbaum.
   Google Scholar
• Baxter, J. A., & Williams, S. (2010). Social and analytic scaffolding in middle school mathematics: Managing the dilemma of telling. Journal of Mathematics Teacher Education, 13(1), 7–26. https://doi.org/10.1007/s10857-009-9121-4
   Google Scholar
• BERA. (2018). British educational research association: Ethical guidelines for educational research (4th ed.). BERA.
   Google Scholar
• Bliss, J., Askew, M., & Macrae, S. (1996). Effective teaching and learning: Scaffolding revisited. Oxford Review of Education, 22(1), 37–61. https://doi.org/10.1080/0305498960220103
   Web of Science ®Google Scholar
• Boyatzis, R. (1998). Transforming qualitative information: Thematic analysis and code development. Sage.
   Google Scholar
• Brown, S., Collins, A., & Duguid, P. (1989). Situated cognition and the culture of learning. Educational Researcher, 18(1), 32–42. https://doi.org/10.3102/0013189X018001032
   Google Scholar
• Buhagiar, M. A., & Murphy, R. (2008). Teachers’ assessments of students learning of mathematics. Assessment in Education: Principles. Policy & Practice, 15(2), 169–182. https://doi.org/10.1080/09695940802164192
   Google Scholar
• Burkhardt, H. (1988). Teaching problem solving. In H. Burkhardt, S. Groves, A. Schoenfeld, & K. Stacey (Eds.), Problem solving – A world view: Proceedings of the problem-solving theme group, ICME 5 (pp. 17–42). Shell Centre.
   Google Scholar
• Calleja, J. (2016). Teaching mathematics through inquiry: A continuing professional development programme design. Educational Designer, 3(9).
   Google Scholar
• Calleja, J. (2018). Teacher participation in continuing professional development: Motivating factors and programme effectiveness. Malta Review of Educational Research, 12(1), 5–29.
   Google Scholar
• Calleja, J. (2019). Learning to teach mathematics through inquiry: A case study of continuing professional development in Malta [PhD Thesis]. Nottingham, UK: University of Nottingham. http://eprints.nottingham.ac.uk/55835
   Google Scholar
• Calleja, J. (2022). Changes in mathematics teachers’ self-reported beliefs and practices over the course of a blended continuing professional development programme. Mathematics Education Research Journal, 34(4), 835–861. https://doi.org/10.1007/s13394-021-00366-x
   Web of Science ®Google Scholar
• Calleja, J., Foster, C., & Hodgen, J. (2021). Integrating ‘just-in-time’ learning in the design of mathematics professional development. Mathematics Teacher Education and Development, 23(2), 79–101.
   Google Scholar
• Charalambous, C. Y., Agathangelou, S. A., Kasapi, E., & Christofidou, E. (2022). Learning to teach ambitiously: A multiple case study of practicing teachers’ experimentation with enablers and extenders. Journal of Mathematics Teacher Education, 1–32. https://doi.org/10.1007/s10857-022-09532-9
   Google Scholar
• Christiansen, B., & Walther, G. (1986). Task and activity. In B. Christiansen, A. G. Howson, & M. Otte (Eds.), Perspectives on mathematics education: Papers submitted by members of the bacomet group (pp. 243–307). Reidel.
   Google Scholar
• Doyle, W. (1988). Work in mathematics classes: The context of students’ thinking during instruction. Educational Psychologist, 23(2), 167–180. https://doi.org/10.1207/s15326985ep2302_6
   Web of Science ®Google Scholar
• European Commission. (2019). Education and Training Monitor 2019: Country analysis. Luxembourg: Publications Office of the European Union.
   Google Scholar
• Foster, C. (2019). The fundamental problem with teaching problem solving. Mathematics Teaching, 265, 8–10.
   Google Scholar
• Goos, M. (2004). Learning mathematics in a classroom community of inquiry. Journal for Research in Mathematics Education, 35(4), 258–291. https://doi.org/10.2307/30034810
   Web of Science ®Google Scholar
• Hargreaves, A. (1994). Changing teachers, changing times: Teachers’ work and culture in the postmodern age. Cassell.
   Google Scholar
• Hmelo-Silver, C. E., Duncan, R. G., & Chinn, C. A. (2007). Scaffolding and achievement in problem-based and inquiry learning: A response to Kirschner, Sweller, and Clark (2006). Educational Psychologist, 42(2), 99–107. https://doi.org/10.1080/00461520701263368
   Web of Science ®Google Scholar
• Ingram, N., Holmes, M., Linsell, C., Livy, S., McCormick, M., & Sullivan, P. (2019). Exploring an innovative approach to teaching mathematics through the use of challenging tasks: A New Zealand perspective. Mathematics Education Research Journal, 32(3), 497–522. https://doi.org/10.1007/s13394-019-00266-1
   Web of Science ®Google Scholar
• Jaworski, B. (2007). Theory in developmental research in mathematics teaching and learning: Social practice theory and community of inquiry as analytical tools. Proceedings of the 5th Congress of the European Society for Research in Mathematics Education. Cyprus: Department of Education, University of Cyprus.
   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. https://doi.org/10.1207/s15326985ep4102_1
   Web of Science ®Google Scholar
• Lampert, M. (2001). Teaching problems and the problems of teaching. Yale University Press.
   Google Scholar
• Lappan, G., Frey, J. T., Fitzgerald, W. M., Freil, S. N., & Phillips, E. D. (1998). The connected mathematics project (2nd ed.). Dale Seymour.
   Google Scholar
• Leikin, R., & Rota, S. (2006). Learning through teaching: A case study on the development of a mathematics teacher’s proficiency in managing an inquiry-based classroom. Mathematics Education Research Journal, 18(3), 44–68. https://doi.org/10.1007/BF03217442
   Google Scholar
• Lobato, J., Clarke, D., & Ellis, A. B. (2005). Initiating and eliciting in teaching: A reformulation of telling. Journal for Research in Mathematics Education, 36(2), 101–136. https://doi.org/10.2307/30034827
   Web of Science ®Google Scholar
• Loucks-Horsley, S., Stiles, K., Mundry, S., Love, N., & Hewson, P. (2010). Designing professional development for teachers of science and mathematics (3rd ed.). Corwin.
   Google Scholar
• Luft, J. A. (2001). Changing inquiry practices and beliefs: The impact of an inquiry-based professional development programme on beginning and experienced secondary science teachers. International Journal of Science Education, 23(5), 517–534. https://doi.org/10.1080/09500690121307
   Web of Science ®Google Scholar
• Marshall, J. C., & Horton, R. M. (2011). The relationship of teacher-facilitated, inquiry-based instruction to student higher-order thinking. School Science and Mathematics, 111(3), 93–101. https://doi.org/10.1111/j.1949-8594.2010.00066.x
   Google Scholar
• Menezes, L., Oliveira, H., & Canavarro, A. P. (2015). Inquiry-based mathematics teaching: The case of Célia. In U. Gellert, J. G. Rodriguez, C. Hahn, & S. Kafoussi (Eds.), Educational paths to mathematics (pp. 305–321). Springer International Publishing.
   Google Scholar
• Nichols, S. E., Tippins, D., & Wieseman, K. (1997). A “Toolkit” for developing critically reflective science teachers. Research in Science Education, 27(2), 175–194. https://doi.org/10.1007/BF02461315
   Google Scholar
• Patton, M. Q. (1990). Qualitative evaluation and research methods (2nd ed.). Sage.
   Google Scholar
• Piaget, J. (1985). The equilibration of cognitive structure: The central problem of intellectual development. University of Chicago Press.
   Google Scholar
• Ponte, J. P. (2011). Using video episodes to reflect on the role of the teacher in mathematical discussions. In O. Zaslavsky, & P. Sullivan (Eds.), Constructing knowledge for teaching secondary mathematics: Tasks to enhance prospective and practicing teacher learning (pp. 249–261). Springer.
   Google Scholar
• Ponte, J. P., Branco, N., & Quaresma, M. (2014). Exploratory activity in the mathematics classroom. In Y. Li, E. Silver, & S. Li (Eds.), Transforming mathematics instruction: Multiple approaches and practices (pp. 103–125). Springer International Publishing.
   Google Scholar
• Ponte, J. P., Segurado, I., & Olivier, A. (2003). A collaborative project using narratives: What happens when pupils work on mathematical investigations? In A. Perter-Koop, V. Santos-Wagner, C. Breen, & A. Begg (Eds.), Collaboration in teacher education: Examples from the context of mathematics education (pp. 85–97). Kluwer.
   Google Scholar
• Priestley, M. (2010). Curriculum for excellence: Transformational change or business as usual? Scottish Educational Review, 42(1), 23–36. https://doi.org/10.25755/int.1541
   Google Scholar
• Rocard, M., Csermely, P., Jorde, D., Lenzen, D., Walberg-Henriksson, H., & Hemmo, V. (2007). Science education now: A renewed pedagogy for the future of Europe (EU 22845). Office for Official Publications of the European Communities.
   Google Scholar
• Russo, J., & Hopkins, S. (2017). How does lesson structure shape teacher perceptions of teaching with challenging tasks? Mathematics Teacher Education and Development, 19(1), 30–46.
   Google Scholar
• Russo, J., & Hopkins, S. (2018). Teaching primary mathematics with challenging tasks: How should lessons be structured? The Journal of Educational Research, 112(1), 1–12. https://doi.org/10.1080/00220671.2018.1440369
   Web of Science ®Google Scholar
• Schoenfeld, A. H. (1985). Mathematical problem solving. Academic Press.
   Google Scholar
• Schoenfeld, A. H. (2013). Classroom observations in theory and practice. ZDM Mathematics Education, 45(4), 607–621. https://doi.org/10.1007/s11858-012-0483-1
   Google Scholar
• Shimizu, Y. (1999). Aspects of mathematical teacher education in Japan: Focusing on the teachers’ roles. Journal of Mathematics Teacher Education, 2(1), 107–116. https://doi.org/10.1023/A:1009960710624
   Google Scholar
• Silver, E. A., Clark, L. M., & Ghousseini, H. N. (2007). Where is the mathematics? Examining teachers’. Mathematical learning opportunities in practice-based professional learning tasks. Journal of Mathematics Teacher Education, 10, 261–277. https://doi.org/10.1007/s10857-007-9039-7
   Google Scholar
• 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
   Google Scholar
• Stein, M. K., Smith, M. S., Henningsen, M. A., & Silver, E. (2000). Implementing standards-based mathematics instruction: A casebook for professional development. Teacher College Press.
   Google Scholar
• Stigler, J. W., & Hiebert, J. (1999). The teaching Gap: Best ideas from the world's teachers for improving education in classrooms. The Free Press.
   Google Scholar
• Stone, C. A. (1998). Should we salvage the scaffolding metaphor? Journal of Learning Disabilities, 31(4), 409–413. https://doi.org/10.1177/002221949803100411
   PubMed Web of Science ®Google Scholar
• Sullivan, P., Askew, M., Cheeseman, J., Clarke, D., Mornane, A., Roche, A., & Walker, N. (2015). Supporting teachers in structuring mathematics lessons involving challenging tasks. Journal of Mathematics Teacher Education, 18(2), 123–140. https://doi.org/10.1007/s10857-014-9279-2
   Google Scholar
• Swan, M. (2005). Standards unit. Improving learning in mathematics: Challenges and strategies. University of Nottingham.
   Google Scholar
• Tafoya, E., Sunal, D., & Knecht, P. (1980). Assessing inquiry potential: A tool for curriculum decision makers. School Science and Mathematics, 80(1), 43–48. https://doi.org/10.1111/j.1949-8594.1980.tb09559.x
   Google Scholar
• Takahashi, A. (2006). Characteristics of Japanese mathematics lessons. Tsukuba Journal of Educational Study in Mathematics, 25(1), 37–44.
   Google Scholar
• Tanner, H., & Jones, S. (2000). Scaffolding for success: Reflective discourse and the effective teaching of mathematical thinking skills. Research in Mathematics Education, 2(1), 19–32. https://doi.org/10.1080/14794800008520065
   Google Scholar
• Van de Pol, J., Volman, M., & Beishuizen, J. (2010). Scaffolding in teacher–student interaction: A decade of research. Educational Psychology Review, 22(3), 271–296. https://doi.org/10.1007/s10648-010-9127-6
   Web of Science ®Google Scholar
• Van de Walle, J. (2004). Elementary and middle school mathematics: Teaching developmentally. Pearson.
   Google Scholar
• Vygotsky, L. (1978). Mind in society. Harvard University Press.
   Google Scholar
• Ward, M., & Sweller, J. (1990). Structuring effective worked examples. Cognition and Instruction, 7(1), 1–39. https://doi.org/10.1207/s1532690xci0701_1
   Web of Science ®Google Scholar
• Williams, G., & Clarke, D. (1997). Mathematical task complexity and task selection. In D. Clarke, P. Clarkson, D. Gronn, M. Horne, L. Lowe, & M. Mackinlay (Eds.), Mathematics: Imagine the possibilities: Proceedings of the 34th annual conference of the mathematics association of Victoria (pp. 406–415). Mathematical Association of Victoria.
   Google Scholar
• Williams, S. R., & Baxter, J. A. (1996). Dilemmas of discourse-oriented teaching in one middle school mathematics classroom. The Elementary School Journal, 97(1), 21–38. https://doi.org/10.1086/461847
   Web of Science ®Google Scholar
• Wood, D., Bruner, J. S., & Ross, G. (1976). The role of tutoring in problem-solving. Journal of Child Psychology and Psychiatry and Allied Disciplines, 17(2), 89–100. https://doi.org/10.1111/j.1469-7610.1976.tb00381.x
   PubMed Web of Science ®Google Scholar
• Wood, T. (1994). Patterns of interaction and the culture of mathematics classrooms. In S. Lennan (Ed.), Cultural perspectives on the mathematics classroom (pp. 149–168). Kluwer Academic Publishers.
   Google Scholar
• Wood, T. (1999). Creating a context for argument in mathematics class. Journal for Research in Mathematics Education, 30(2), 171–191. https://doi.org/10.2307/749609
   Web of Science ®Google Scholar
• Yin, R. K. (2003). Case study research: Design and methods. Sage.
   Google Scholar