Skip to main content

Advertisement

SpringerLink
Log in

Learning to teach ambitiously: a multiple case study of practicing teachers’ experimentation with enablers and extenders

  • Published:
Journal of Mathematics Teacher Education Aims and scope Submit manuscript

Abstract

Over the past decade, teaching mathematics ambitiously has received increased attention. In this paper, we argue that to materialize this vision in contemporary classes we need to understand how practicing teachers experiment with certain aspects of this teaching and what challenges they encounter. Toward this end, we focus on an aspect of teaching ambitiously—designing and using enablers and extenders—and examine how four elementary schoolteachers experimented with it in their practice while also participating in a video-club setting. Drawing on a corpus of data including lesson plans, videotaped lessons, pre- and post-lesson interviews, end-of-program interviews, and videotaped video-club sessions, and looking across the four cases, we sketch how these teachers worked with enablers and extenders and the challenges they faced. Our analysis helped identify certain components entailed in working with enablers and extenders during the phases of lesson planning and enactment; it also yielded a classification of observed and reported challenges encountered as teachers engage with this work. This mapping of work associated with designing and using enablers and extenders, along with the classification of challenges generated, can inform professional learning development attempts aiming to support teachers enact ambitious teaching by identifying and naming separate components of practice that merit consideration and by providing insights into the types of scaffolds needed to support teachers in teaching ambitiously.

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

Similar content being viewed by others

Notes

  1. We use the term “challenges” instead of the closely related terms “problems” or “difficulties”, since this was the most frequently used term in the pertinent literature reviewed (e.g., Cheeseman et al., 2017; Hodgson, 2019; Sullivan & Mornane, 2014).

  2. These four teachers submitted full lesson plans; although doing so was not mandatory for the purposes of the program, it was considered critical for this study, given its focus on both planning and enactment.

  3. The strategies used to develop enablers and extenders represents an example of a top-down theme; an example of a bottom-up theme pertains to teacher–student interactions after administering enablers and extenders.

  4. Following Grossman et al. (2009), we preferred to name the overarching categories resulting from grouping the themes that emerged in our analysis as “components” to highlight the idea of decomposing teaching into different elements.

  5. Main tasks are the tasks which were intended to surface important mathematical ideas to be considered in the lesson and which, in most cases, pertained to a “doing mathematics” task (Stein & Smith, 1998).

  6. Flexible grouping implies that student grouping can change during a lesson based on students’ interests and needs, specific learning goals, or the type of work in which students engage (Tomlinson, 2014).

Abbreviations

EPI:

End-of-program interview

VL:

Videotaped lesson

LP:

Lesson plan

PLD:

Professional learning and development

POI:

Post-lesson interview

PRI:

Pre-lesson interview

VCS:

Video-club session

References

  • Australian Government Department of Education (2014). Review of the Australian curriculum: Final report. Department of Education.

  • Ball, D.L. (1992). Magical hopes: Manipulatives and the reform of math education. American Educator: The Professional Journal of the American Federation of Teachers, 16(2), 14–18, 46–47. Retrieved January 10, 2022 from https://www.aft.org/sites/default/files/periodicals/ae_summer1992_ball.pdf

  • Ball, D. L., & Forzani, F. M. (2009). The work of teaching and the challenge for teacher education. Journal of Teacher Education, 60(5), 497–511. https://doi.org/10.1177/0022487109348479

    Article  Google Scholar 

  • Baumert, J., Kunter, M., Blum, W., Brunner, M., Voss, T., Jordan, A., Klusmann, U., Krauss, S., Neubrand, M., & Tsai, Y.-M. (2010). Teachers’ mathematical knowledge, cognitive activation in the classroom, and student progress. American Educational Research Journal, 47(1), 133–180. https://doi.org/10.3102/0002831209345157

    Article  Google Scholar 

  • Berliner, D. C. (2001). Learning about and learning from expert teachers. International Journal of Educational Research, 35(5), 463–482. https://doi.org/10.1016/S0883-0355(02)00004-6

    Article  Google Scholar 

  • Boaler, J. (2008). Promoting ‘relational equity’ and high mathematics achievement through an innovative mixed-ability approach. British Educational Research Journal, 34(2), 167–194. https://doi.org/10.1080/01411920701532145

    Article  Google Scholar 

  • Bobis, J., Russo, J., Downton, A., Feng, M., Livy, S., McCormick, M., & Sullivan, P. (2021). Instructional moves that increase chances of engaging all students in learning mathematics. Mathematics, 9(6), 582. https://doi.org/10.3390/math906058210.3390/math9060582

    Article  Google Scholar 

  • Boerst, T., Sleep, L., Ball, D., & Bass, H. (2011). Preparing teachers to lead mathematics discussions. Teachers College Record, 113(12), 2844–2877. https://doi.org/10.1177/016146811111301207

    Article  Google Scholar 

  • Braun, V., & Clarke, V. (2006). Using thematic analysis in psychology. Qualitative Research in Psychology, 3(2), 77–101. https://doi.org/10.1191/1478088706qp063oa

    Article  Google Scholar 

  • Carter, K., Cushing, K., Sabers, D., Stein, P., & Berliner, D. (1988). Expert-novice differences in perceiving and processing visual classroom information. Journal of Teacher Education, 39(3), 25–31. https://doi.org/10.1177/002248718803900306

    Article  Google Scholar 

  • Charalambous, C. Y. (2015). Working at the intersection of teacher knowledge, productive dispositions, and teaching practice: A multiple-case study. Journal of Mathematics Teacher Education, 18(5), 427–445. https://doi.org/10.1007/s10857-015-9318-7

    Article  Google Scholar 

  • Charmaz, K. (2006). Constructing grounded theory: A practical guide through qualitative analysis. Sage Publications.

  • Cheeseman, J., Downton, A., & Livy, S. (2017). Investigating teachers' perceptions of enabling and extending prompts. In A. Downton, S. Livy & J. Hall (Eds.), 40 years on: We are still learning! Proceedings of the 40th Annual Conference of the Mathematics Education Research Group of Australasia (pp. 141–148). MERGA.

  • Clarke, D., Roche, A., Cheeseman, J., & Van Der Schans, S. (2014). Teaching strategies for building student persistence on challenging tasks: Insights emerging from two approaches to teacher professional learning. Mathematics Teacher Education and Development, 16(2), 46–70. https://doi.org/10.3316/aeipt.214061

    Article  Google Scholar 

  • Cohen, D. K. (1990). A revolution in one classroom: The case of Mrs. Oublier. Educational Evaluation and Policy Analysis, 12, 311–330. https://doi.org/10.3102/01623737012003311

    Article  Google Scholar 

  • Cohen, D. (2011). Teaching and its predicaments. Harvard University Press.

    Book  Google Scholar 

  • Cohen, J., Schuldt, L. C., Brown, L., & Grossman, P. (2016). Leveraging observation tools for instructional improvement: Exploring variability in uptake of ambitious instructional practices. Teachers College Record, 118(11), 1–36. https://doi.org/10.1177/016146811611801105

    Article  Google Scholar 

  • Darling-Hammond, L., Hyler, M. E., & Gardner, M. (2017). Effective teacher professional development. Learning Policy Institute.

  • Delaney, S. (2016). Become the primary teacher everyone wants to have: A guide to career success. Routledge.

  • Delaney, S., & Gurhy, A.M. (2019). Combining differentiation and challenge in mathematics instruction: A case from practice. In U. T. Jankvist, M. Van den Heuvel-Panhuizen, & M. Veldhuis, (Eds.). Proceedings of the Eleventh Congress of the European Society for Research in Mathematics Education (pp. 4194–4201). Freudenthal Group & Freudenthal Institute, Utrecht University and ERME.

  • Fogo, B. (2014). Core practices for teaching history: The results of a Delphi panel survey. Theory & Research in Social Education, 42(2), 151–196. https://doi.org/10.1080/00933104.2014.902781

    Article  Google Scholar 

  • Forzani, F. M. (2014). Understanding “core practices” and “practice-based” teacher education: Learning from the past. Journal of Teacher Education, 65(4), 357–368. https://doi.org/10.1177/0022487114533800

    Article  Google Scholar 

  • Franke, M. L., Kazemi, E., & Battey, D. (2007). Mathematics teaching and classroom practice. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 225–256). Information Age Publishing.

    Google Scholar 

  • Grbich, C. (2007). Qualitative data analysis: An introduction. Sage Publications.

  • Grossman, P., Hammerness, K., & McDonald, M. (2009). Redefining teaching, re-imagining teacher education. Teachers and Teaching: Theory and Practice, 15(2), 273–289. https://doi.org/10.1080/13540600902875340

    Article  Google Scholar 

  • Grossman, P., & McDonald, M. (2008). Back to the future: Directions for research in teaching and teacher education. American Educational Research Journal, 45(1), 184–205. https://doi.org/10.3102/0002831207312906

    Article  Google Scholar 

  • Hatch, T., & Grossman, P. (2009). Learning to look beyond the boundaries of representation: Using technology to examine teaching (Overview for a digital exhibition: Learning from the practice of teaching). Journal of Teacher Education, 60(1), 70–85. https://doi.org/10.1177/0022487108328533

    Article  Google Scholar 

  • Herbst, P., & Chazan, D. (2011). Research on practical rationality: Studying the justification of actions in mathematics teaching. The Mathematics Enthusiast, 8(3), 405–462. https://doi.org/10.54870/1551-3440.1225

  • Hiebert, J., Gallimore, R., Garnier, H., Givvin, K. B., Hollingsworth, H., Jacobs, J., Chui, A., Wearne, D., Smith, M., Kersting, N., Manaster, A., Tseng, E., Etterbeek, W., Manaster, C., Gonzales, P., & Stigler, J. (2003). Teaching mathematics in seven countries: Results from the TIMSS 1999 video study. U.S. Department of Education, National Center for Education Statistics.

  • Hill, H. C., Blunk, M. L., Charalambous, C. Y., Lewis, J. M., Phelps, G. C., Sleep, L., & Ball, D. L. (2008). Mathematical knowledge for teaching and the mathematical quality of instruction: An exploratory study. Cognition and Instruction, 26(4), 430–511. https://doi.org/10.1080/07370000802177235

    Article  Google Scholar 

  • Hodgson, L. (2019). Seeing is not enough for believing: Building mathematical knowledge for teaching through observing, deconstructing, and enacting particular pedagogies. [Unpublished doctoral dissertation.] Monash University. https://doi.org/10.26180/5c39d4c85b020

  • Holstein, A., Weber, K.E., Prilop, C.N. & Kleinknecht, M. (2021). The promise and pitfalls of a blended, video- and coaching-based professional development program in Germany. In E. Langran & L. Archambault (Eds.), Proceedings of Society for Information Technology & Teacher Education International Conference (pp. 953–958). Association for the Advancement of Computing in Education (AACE). Retrieved December 15, 2021 from https://www.learntechlib.org/primary/p/219240/.

  • 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, 497–522. https://doi.org/10.1007/s13394-019-00266-1

    Article  Google Scholar 

  • Kane, T. J., & Staiger, D. O. (2012). Gathering feedback for teaching: Combining high-quality observations with student surveys and achievement gains. Measures of Effective Teaching Project, Bill & Melinda Gates Foundation.

  • Kraft, M. A., & Hill., H.C. (2020). Developing ambitious mathematics instruction through web-based coaching: A randomized field trial. American Educational Research Journal, 57(6), 2378–2414. https://doi.org/10.3102/0002831220916840

    Article  Google Scholar 

  • Kunter, M., Baumert, J., Blum, W., Klusmann, U., Neubrand, M., & Krauss, S. (Eds.) (2013). Cognitive activation in the mathematics classroom and professional competence of teachers: Results from the COACTIV project. Springer.

  • Lampert, M. (1992). The practice and problems of teaching and learning authentic mathematics in school. In F. Oser, A. Dick, & J. Patry (Eds.), Effective and responsible teaching: The new synthesis (pp. 295–314). Jossey-Bass.

    Google Scholar 

  • Lampert, M. (2001). Teaching problems and the problems of teaching. Yale University Press.

    Google Scholar 

  • Lampert, M., Beasley, H., Ghousseini, H., Kazemi, E., & Franke, M. (2010). Using designed instructional activities to enable novices to manage ambitious mathematics teaching. In M. K. Stein & L. Kucan (Eds.), Instructional explanations in the disciplines (pp. 129–141). Springer. https://doi.org/10.1007/978-1-4419-0594-9_9

  • Lampert, M., Boerst, T., & Graziani, F. (2011). Organizational resources in the service of school-wide ambitious teaching practice. Teachers College Record, 113(7), 1361–1400. https://doi.org/10.1177/016146811111300706

    Article  Google Scholar 

  • Le Donné, N., Fraser, P., & Bousquet, G. (2016). Teaching strategies for instructional quality: Insights from the TALIS-PISA Link data. OECD Education Working Papers No. 148. https://doi.org/10.1787/5jln1hlsr0lr-en

  • Mertens, D. M. (2005). Research and evaluation in education and psychology: Integrating diversity with quantitative, qualitative, and mixed methods (2nd ed.). Sage Publications.

  • Mesiti, C., Hollingsworth, H., & Clarke, D. (2021). Naming aspects of teaching practice: Describing and analysing a lexicon of mathematics teachers in Australia. In C. Mesiti, M. Artigue, H. Hollingsworth, Y. Cao, & D. Clarke (Eds.), Teachers talking about their classrooms: Learning from the professional lexicons of mathematics teachers around the world (pp. 17–38). Routledge.

    Chapter  Google Scholar 

  • Minas, M. (2019). Using enabling prompts to effectively support teaching with challenging tasks. Australian Primary Mathematics Classroom, 24(4), 12–16. https://doi.org/10.3316/informit.913897045651860

    Article  Google Scholar 

  • National Council of Teachers of Mathematics (2014). Principles to actions: Ensuring mathematical success for all. Author.

  • OECD. (2016). PISA 2015 results (volume I): Excellence and equity in education. PISA, OECD Publishing. https://doi.org/10.1787/9789264266490-en

    Book  Google Scholar 

  • OECD. (2020). Global teaching inSights: A video study of teaching. OECD Publishing. https://doi.org/10.1787/20d6f36b-en

    Article  Google Scholar 

  • Patton, M. Q. (2015). Qualitative research and evaluation methods (4th ed.). Sage Publications.

  • Pfister, M., Opitz, E. M., & Pauli, C. (2015). Scaffolding for mathematics teaching in inclusive primary classrooms: A video study. ZDM Mathematics Education, 47(7), 1079–1092. https://doi.org/10.1007/s11858-015-0713-4

    Article  Google Scholar 

  • Reis, S. M., McCoach, D. B., Little, C. A., Muller, L. M., & Kaniskan, R. B. (2011). The effects of differentiated instruction and enrichment pedagogy on reading achievement in five elementary schools. American Educational Research Journal, 48(2), 462–501. https://doi.org/10.3102/0002831210382891

    Article  Google Scholar 

  • Russo, J. (2020). Designing and scaffolding rich mathematical learning experiences with challenging tasks. Australian Primary Mathematics Classroom, 25(1), 3–10. https://doi.org/10.3316/informit.118524348799862

    Article  Google Scholar 

  • Sleep, L., & Eskelson, S. L. (2012). MKT and curriculum materials are only part of the story: Insights from a lesson on fractions. Journal of Curriculum Studies, 44(4), 537–558. https://doi.org/10.1080/00220272.2012.716977

    Article  Google Scholar 

  • Stein, M. K., & Smith, M. S. (1998). Mathematical tasks as a framework for reflection: From research to practice. Mathematics Teaching in the Middle School, 3(4), 268–275. https://doi.org/10.5951/MTMS.3.4.0268

    Article  Google Scholar 

  • Stein, M. K., Remillard, J., & Smith, M. S. (2007). How curriculum influences student learning. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 319–369). Information Age Publishing.

    Google Scholar 

  • Stein, M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A. (2000). Implementing Standards-based mathematics instruction: A casebook for professional development. Teachers College Press.

  • Sullivan, P. (2011). Teaching mathematics: Using research-informed strategies. Australian Council for Educational Research (ACER).

  • Sullivan, P., Askew, M., Cheeseman, J., Clarke, D., 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

    Article  Google Scholar 

  • Sullivan, P., Borcek, C., Walker, N., & Rennie, M. (2016). Exploring a structure for mathematics lessons that initiate learning by activating cognition on challenging tasks. The Journal of Mathematical Behavior, 41, 159–170. https://doi.org/10.1016/j.jmathb.2015.12.002

    Article  Google Scholar 

  • Sullivan, P., Clarke, D., Michaels, D., & Mornane, A. (2012). Supporting teachers in choosing and using challenging mathematics tasks. In J. Dindyal, L. P. Cheng & S. F. Ng (Eds.), Mathematics education: Expanding horizons-Proceedings of the 35th annual conference of the Mathematics Education Research Group of Australasia (pp. 688–695). MERGA.

  • Sullivan, P., & Mornane, A. (2014). Exploring teachers’ use of, and students’ reactions to, challenging mathematics tasks. Mathematics Education Research Journal, 26(2), 193–213. https://doi.org/10.1007/s13394-013-0089-0

    Article  Google Scholar 

  • Sullivan, P., Mousley, J., & Jorgensen, R. (2009). Tasks and pedagogies that facilitate mathematical problem solving. In B. Kaur, Y. B. Har, & M. Kapur (Eds.), Mathematical problem solving: Yearbook 2009 Association of Mathematics Educators (pp. 17–42). Association of Mathematics Educators. https://doi.org/10.1142/9789814277228_0002

  • Sullivan, P., Mousley, J., & Zevenbergen, R. (2006). Teacher actions to maximize mathematics learning opportunities in heterogeneous classrooms. International Journal of Science and Mathematics Education, 4(1), 117–143. https://doi.org/10.1007/s10763-005-9002-y

    Article  Google Scholar 

  • Sztajn, P., Heck, D. J., Malzahn, K. A., & Dick, L. K. (2020). Decomposing practice in teacher professional development: Examining sequences of learning activities. Teaching and Teacher Education, 91, 103039. https://doi.org/10.1016/j.tate.2020.103039

    Article  Google Scholar 

  • Tekkumru-Kisa, M., Stein, M. K., & Doyle, W. (2020). Theory and research on tasks revisited: Task as a context for students’ thinking in the era of ambitious reforms in mathematics and science. Educational Researcher, 49(8), 606–617. https://doi.org/10.3102/0013189X20932480

    Article  Google Scholar 

  • Tomlinson, C. A. (2014). The differentiated classroom: Responding to the needs of all learners (2nd ed.). ASCD.

  • Tomlinson, C. A., Brighton, C., Hertberg, H., Callahan, C. M., Moon, T. R., Brimijoin, K., Conover, L.A., & Reynolds, T. (2003). Differentiating instruction in response to student readiness, interest, and learning profile in academically diverse classrooms: A review of literature. Journal for the Education of the Gifted, 27(2–3), 119–145. https://doi.org/10.1177/016235320302700203

    Article  Google Scholar 

  • Vale, C., Widjaja, W., Doig, B., & Groves, S. (2019). Anticipating students’ reasoning and planning prompts in structured problem-solving lessons. Mathematics Education Research Journal, 31(1), 1–25. https://doi.org/10.1007/s13394-018-0239-5

    Article  Google Scholar 

  • van Es, E. A., Cashen, M., Barnhart, T., & Auger, A. (2017). Learning to notice mathematics instruction: Using video to develop preservice teachers’ vision of ambitious pedagogy. Cognition and Instruction, 35(3), 1–23. https://doi.org/10.1080/07370008.2017.1317125

    Article  Google Scholar 

  • van Es, E. A., & Sherin, G. (2008). Mathematics teachers’ ‘“learning to notice”’ in the context of a video club. Teaching and Teacher Education, 24(2), 244–276. https://doi.org/10.1016/j.tate.2006.11.005

    Article  Google Scholar 

  • Wæge, K., & Fauskanger, J. (2020). Teacher time outs in rehearsals: In-service teachers learning ambitious mathematics teaching practices. Journal of Mathematics Teacher Education, 24, 563–586. https://doi.org/10.1007/s10857-020-09474-0

    Article  Google Scholar 

Download references

Acknowledgements

This work was supported by the ERASMUS+ Key Action 2—Cooperation for innovation and the exchange of good practices (Field: School Education) project entitled “Enhancing Differentiated Instruction and Cognitive Activation in Mathematics Lessons by Supporting Teacher Learning” (EDUCATE, 2017-1-CY01-KA201-026749) funded by the European Union. We would like to thank the teachers who, by participating, made their practice available for study, thus enabling us learn with and from them. We would also like to thank our partners in the ERASMUS+ KA2 EDUCATE project for our collective work in designing the video clubs and related materials.

Author information

Authors and Affiliations

  1. Department of Education, University of Cyprus, Theophanides Building, Rm 508, 11-13 Dramas St, 1077, Nicosia, Cyprus

    Charalambos Y. Charalambous & Evridiki Kasapi

  2. Cyprus Pedagogical Institute, 40 Macedonia Avenue, P.O. Box 12720, 2238, Nicosia, Latsia, Cyprus

    Sofia A. Agathangelou & Elena Christofidou

  3. Department of Education, University of Cyprus, Nicosia, Cyprus

    Evridiki Kasapi

Authors
  1. Charalambos Y. Charalambous
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sofia A. Agathangelou
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Evridiki Kasapi
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Elena Christofidou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Charalambos Y. Charalambous.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary Information

Below is the link to the electronic supplementary material.

Supplementary file1 (PDF 216 kb)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Charalambous, C.Y., Agathangelou, S.A., Kasapi, E. et al. Learning to teach ambitiously: a multiple case study of practicing teachers’ experimentation with enablers and extenders. J Math Teacher Educ 26, 363–394 (2023). https://doi.org/10.1007/s10857-022-09532-9

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10857-022-09532-9

Keywords

Search

Navigation