Differentiating Mathematics Instruction in Remote Learning Environments: Exploring Teachers’ Challenges and Supports
Keywords:differentiation, mathematics instruction, remote learning, documentational approach to didactics, schemes
This report examines one teacher’s attempts to differentiate instruction every day, in a dynamic teaching and learning environment. The report is part of a larger study that utilised a network of theories approach, coordinating the documentational approach to didactics (DAD) and Thompson and Harel’s theory of meanings, to examine teachers’ understandings as they engaged in and discussed their attempts to differentiate instruction in remote and hybrid learning environments. Analyses involved building models of teachers’ understandings of differentiated instruction and the resources they utilised to support differentiation and exploring how these models persisted or changed throughout the study. The report’s findings highlight the importance of teachers’ meanings in their attempts to differentiate instruction and the role digital resources play in supporting or hindering such practices. Finally, this report adds to a growing body of research on teacher’s work with and on digital resources as a means to support differentiated instruction, particularly as remote and hybrid teaching and learning continues throughout the U.S. and across the globe.
Adler, J. (2000). Conceptualizing resources as a theme for teacher education. Journal of Mathematics Teacher Education, 3(3), 205–224.
Anderson, L. W., Krathwohl, D. R., Airasian, P. W., Cruikshank, K. A., Mayer, R. E., Pintrich, P. R., Raths, J., & Wittrock, M. C. (Eds.). (2001). A taxonomy for learning, teaching, and assessing: A revision of Bloom’s Taxonomy of Educational Objectives. Longman.
Anthony, G., Hunter, J., & Hunter, R. (2019). Working towards equity in mathematics education: Is differentiation the answer? In G. Hine, S.
Blackley, & A. Cooke (Eds.). Mathematics education research: Impacting practice, Proceedings of the 42nd annual conference of the Mathematics Education Research Group of Australasia (pp. 117-124). MERGA.
Bikner-Ahsbahs, A., & Prediger, S. (2010). Networking of theories—an approach for exploiting the diversity of theoretical approaches. In B. Sriraman & L. English (Eds.), Theories of mathematics education: Seeking new frontiers (pp. 483–506). Springer.
Brousseau G. (1997). Theory of didactical situation in mathematics. Kluwer.
Chamberlin, M., & Powers, R. (2010). The promise of differentiated instruction for enhancing the mathematical understandings of college students. Teaching Mathematics and Its Applications, 29, 113–139.
Courtney, S. A. (2010). Exploring teachers' capacity to reflect on their practice: An investigation into the development of mathematical knowledge for teaching [Doctoral dissertation, Arizona State University].
Dosch, M., & Zidon, M. (2014). “The course fit us”: Differentiated instruction in the college classroom. International Journal of Teaching and Learning in Higher, 26, 343–357.
Frank, K. M. (2017). Examining the development of students’ covariational reasoning in the context of graphing [Doctoral dissertation, Arizona State University].
Frayer, D., Frederick, W. C., & Klausmeier, H. J. (1969). A schema for testing the level of cognitive mastery. Wisconsin Center for Education Research.
Gueudet, G., Pepin, B., Courtney, S., Kock, Z.-J., Misfeldt, M., & Tamborg, A. L. (2021). Digital platforms for mathematics teacher curriculum design: affordances and constraints. In A. Clark-Wilson, A. Donevska-Todorova, E. Faggiano, J. Trgalová & H. -G. Weigand (Eds.), Mathematics education in the digital age: Learning, practice and theory (pp. 84-98). Routledge.
Gueudet, G., Pepin, B., Sabra, H., & Trouche, L. (2016). Collective design of an e-textbook: teachers’ collective documentation. Journal of Mathematics Teacher Education, 19, 187-203.
Gueudet, G., Pepin, B., & Trouche, L. (Eds.). (2012). From textbooks to ‘lived’ resources: mathematics curriculum materials and teacher documentation. Springer.
Gueudet, G., & Trouche, L. (2009). Towards new documentation systems for mathematics teachers? Educational Studies in Mathematics, 71(3), 199–218.
Hackenberg, A. (2010). Students’ reasoning with reversible multiplicative relationships. Cognition and Instruction, 28(4), 383-432. https://doi.org/10.1080/07370008.2010.511565
Halsey, J. (2018). Independent review into regional rural and remote education—Final report. Commonwealth of Australia.
Hersh, M. (2020). Technology for inclusion (Background paper prepared for the 2020 global education monitoring report: Inclusion and education). United Nations Educational, Scientific and Cultural Organization (UNESCO), Global Education Monitoring Report. https://unesdoc.unesco.org/ark:/48223/pf0000373655
Hess, K. K., Carlock, D., Jones, B., & Walkup, J. R. (2009). What exactly do “fewer, clearer, and higher standards” really look like in the classroom? Using a cognitive rigor matrix to analyze curriculum, plan lessons, and implement assessments. Paper presented the Council of Chief State School Officers (CCSSO). CCSSO.
Hunter, T. (2019). Five reminders about Webb’s depth of knowledge. For the Teachers. https://www.fortheteachers.org/friday-five-reminders-about-webbs-depth-of-knowledge/
Interglot Translation Dictionary. (2021). Confrontation. In Interglot. https://www.interglot.com/dictionary/fr/en/search?q=confrontation&m=
Joseph, S., Thomas, M., Simonette, G., & Ramsook, L. (2013). The impact of differentiated instruction in a teacher education setting: Successes and challenges. International Journal of Higher Education, 2(3), 28–40.
Lange, K. (2009). Lessons learned in an inclusive classroom: A case study of differentiated instruction [Doctoral dissertation, Colorado State University]. Academia.
Moon, T., Callahan, C., Tomlinson, C. A., & Miller, E. (2002). Middle school classrooms: Teachers’ reported practices and student perceptions. The National Research Center on the Gifted and Talented, University of Connecticut.
Moore, K. C., & Thompson, P. W. (2015). Shape thinking and students' graphing activity. In T. Fukawa-Connelly, N. E. Infante, K. Keene & M. Zandieh (Eds.), Proceedings of the 18th Meeting of the MAA Special Interest Group on Research in Undergraduate Mathematics Education (pp. 782-789). Pittsburgh, PA: RUME.
Moosa, V., & Shareefa, M. (2019). The impact of teachers’ experience and qualification on efficacy, knowledge and implementation of differentiated instruction. International Journal of Instruction, 12(2), 587-604.
Morgan, H. (2014). Maximizing student success with differentiated learning. The Clearing House, 87, 34–38.
Office of the United Nations High Commissioner for Human Rights. (1960). Convention against discrimination in education. United Nations Educational, Scientific and Cultural Organization. https://unesdoc.unesco.org/ark:/48223/pf0000114583.page=118
Opalka, A. Gable, A., Nicola, T., & Ash, J. (2020, August 10). Rural school districts can be creative in solving the internet connectivity gap—but they need support. Brookings. https://www.brookings.edu/blog/brown-center-chalkboard/2020/08/10/rural-school-districts-can-be-creative-in-solving-the-internet-connectivity-gap-but-they-need-support/
The Organisation for Economic Co-operation and Development (OECD). (2020, April 3). Learning remotely when schools close: How well are students and schools prepared? Insights from PISA (OECD Policy Responses to Coronavirus, COVID-19). OECD. https://www.oecd.org/coronavirus/policy-responses/learning-remotely-when-schools-close-how-well-are-students-and-schools-prepared-insights-from-pisa-3bfda1f7/
Pepin, B., & Gueudet, G. (2020a). Curriculum resources and textbooks in mathematics education. In S. Lerman (Ed.), Encyclopedia of Mathematics Education (pp.172-176). Springer.
Pepin, B., & Gueudet, G. (2020b). Studying teacher collaboration with the documentational approach: From shared resources to common schemes? In H. Borko & D. Portari (Eds.), ICMI Study 25 conference proceedings: Teachers of mathematics working and learning in collaborative groups (pp.158-165). University of Lisbon.
Pepin, B., Gueudet, G., & Trouche, L. (2013). Re-sourcing teachers’ work and interactions: a collective perspective on resources, their use and transformation. ZDM – Mathematics Education, 45, 929-934.
Pozas, M., Letzel, V., & Schneider, C. (2020). Teachers and differentiated instruction: exploring differentiation practices to address student diversity. Journal of Research in Special Educational Needs, 20(3), 217-230. https://doi.org/10.1111/1471-3802.12481
Prast, E. J., van de Weijer-Bergsma, E., Kroesbergena, E. H., & Van Luit, J. (2015). Readiness-based differentiation in primary school mathematics: expert recommendations and teacher self-assessment. Frontline Learning Research, 3(2), pp. 90-116. https://doi.org/10.14786/flr.v3i2.163
Prediger, S., Bikner-Ahsbahs, A., & Arzarello, F. (2008). Networking strategies and methods for connecting theoretical approaches: First steps towards a conceptual framework. ZDM – The International Journal on Mathematics Education, 40, 165-178. https://doi.org/10.1007/s11858-008-0086-z
Preszler, J. (2006). On target: Strategies that differentiate instruction grades 4-12. ESA 6 & 7, South Dakota Department of Education.
Rabardel, P. (2002). People and technology: A cognitive approach to contemporary instruments (H. Wood, Trans.). Armand Colin. (Original work published 1995).
Rezat, S., Hénaff, C., Visnovska, J., Kim, O-K., Leroyer, L., Sabra, H., El Hage, S., & Wang, C. (2019). Documentation work, design capacity, and teachers’ expertise in designing instruction. In L. Trouche, G. Gueudet, & B. Pepin (Eds.), The ‘resource’ approach to mathematics education (pp. 323–388). Springer.
Rodriguez, A. (2012). An analysis of elementary school teachers' knowledge and use of differentiated instruction [Doctoral dissertation, Olivet Nazarene University]. Digital Commons at Olivet.
Rose, T. (2013, June 19). The myth of average [Video]. TEDxSonomaCounty. https://www.youtube.com/watch?v=4eBmyttcfU4
Rose, T., Rouhani, P., & Fischer, K. W. (2013). The science of the individual. Mind, Brain, and Education, 7(3), 152-158.
Roy, A., Guay, F., & Valois, P. (2013). Teaching to address diverse learning needs: development and validation of a differentiated instruction scale. International Journal of Inclusive Education, 17(11), 1186–1204. https://doi.org/10.1080/13603116.2012.743604
Siam, K., & Al-Natour, M. (2016). Teacher’s differentiated instruction practices and implementation challenges for learning disabilities in Jordan. International Education Studies, 9(12), 167-181.
Simon, M. A., Kara, M., Norton, A., & Placa, N. (2018). Fostering construction of a meaning for multiplication that subsumes whole-number and fraction multiplication: A study of the Learning Through Activity research program. Journal of Mathematical Behavior, 52, 151-173. https://doi.org/10.1016/j.jmathb.2018.03.002
Skemp, R. (1961). Reflective intelligence and mathematics. The British Journal of Educational Psychology, 31, 44–55.
Skemp, R. (1979). Intelligence, learning, and action. John Wiley & Sons.
Slavit, D., & Roth McDuffie, A. (2013). Self-directed teacher learning in collaborative contexts. School Science and Mathematics, 113(2), 94-105.
Small, M. (2017). Good questions: Great ways to differentiate mathematics instruction in the standards-based classroom (3rd ed.). Teachers College Press.
Small, M., & Lin, A. (2010). More good questions: Great ways to differentiate mathematics instruction in the standards-based classroom. Teachers College Press.
Smit, R., & Humpert, W. (2012). Differentiated instruction in small schools. Teaching and Teacher Education, 28, 1152-1162. https://doi.org/10.1016/j.tate.2012.07.003
Smith, M. S., & Stein, M. K. (1989). Selecting and creating mathematical tasks from research to practice. Mathematics Teaching in the Middle School, 3(5), 344-350.
Steffe, L. P. (1991). The learning paradox. In L. P. Steffe (Ed.), Epistemological foundations of mathematical experience (pp. 26–44). Springer-Verlag.
Steffe, L. P. (1995). Alternative epistemologies: An educator’s perspective. In L. P. Steffe, & J. Gale (Eds.). Constructivism in education (pp. 489–523). Erlbaum.
Suprayogi, M. N., Vakcke, M., & Godwin, R. (2017). Teachers and their implementation of differentiated instruction in the classroom. Teaching and Teacher Education, 67, 291-301. http://dx.doi.org/10.1016/j.tate.2017.06.020
Thomas, J., Barraket, J., Wilson, C. K., Holcombe-James, I., Kennedy, J., Rennie, E., Ewing, S., & MacDonald, T. (2020). Measuring Australia’s digital divide: The Australian Digital Inclusion Index 2020. RMIT and Swinburne University of Technology, Melbourne, for Telstra.
Thompson, P. W. (1994). The development of the concept of speed and its relationship to concepts of rate. In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 181-234). SUNY Press.
Thompson, P. W. (2008). Conceptual analysis of mathematical ideas: Some spadework at the foundations of mathematics education. In O. Figueras, J. L. Cortina, S. Alatorre, T. Rojano & A. Sépulveda (Eds.), Proceedings of the Annual Meeting of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 31- 49). PME.
Thompson, P. W. (2013). In the absence of meaning. In K. Leatham (Ed.), Vital directions for research in mathematics education (pp. 57–93). Springer.
Thompson, P. W. (2016). Researching mathematical meanings for teaching. In L. English & D. Kirshner, (Eds.), Handbook of international research in mathematics education (pp. 435-461). Taylor and Francis.
Thompson, P. W., Carlson, M. P., Byerley, C., & Hatfield, N. (2014). Schemes for thinking with magnitudes: An hypothesis about foundational reasoning abilities in algebra. In K. C. Moore, L. P. Steffe & L. L. Hatfield (Eds.), Epistemic algebra students: Emerging models of students' algebraic knowing, WISDOMe Monographs (Vol. 4, pp. 1–24). University of Wyoming.
Thompson, P. W., & Saldanha, L. A. (2003). Fractions and multiplicative reasoning. In J. Kilpatrick, G. Martin & D. Schifter (Eds.), Research companion to the Principles and Standards for School Mathematics (pp. 95-114). National Council of Teachers of Mathematics.
Tillema, E., & Gatza, A. (2017). The processes and products of students’ generalizing activity. In Galindo, E., & Newton, J., (Eds.), Proceedings of the 39th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. Hoosier Association of Mathematics Teacher Educators (HAMTE).
Tomlinson, C. A. (2014). The differentiated classroom: Responding to the needs of all learners. ASCD.
Tomlinson, C. A. (2017). How to differentiate instruction in academically diverse classrooms. ASCD.
Tomlinson, C. A., & Imbeau, M. B. (2010). Leading and managing a differentiated classroom. ASCD.
Tomlinson, C. A., & Jarvis, J. (2006). Teaching beyond the book. Educational Leadership, 64(1), 16-21.
Trouche, L., Gitirana, V., Miyakawa, T., Pepin, B., & Wang, C. (2019). Studying mathematics teachers’ interactions with curriculum materials through different lenses: Towards a deeper understanding of the processes at stake. International Journal of Educational Research, 93, 53-67. https://doi.org/10.1016/j.ijer.2018.09.002
Trouche, L., Gueudet, G., & Pepin, B. (2018a). Documentational approach to didactics. In S. Lerman (Ed.), Encyclopedia of mathematics education. Springer. https://doi.org/10.1007/978-3-319-77487-9_100011-1
Trouche, L. Gueudet, G, & Pepin, B. (2018b). Open educational resources: A chance for opening mathematics teachers’ resource systems? In L. Fan, L. Trouche, C. Qi, S. Rezat, & J. Visnovska (Eds.), Research on mathematics textbooks and teachers’ resources: Advances and issues (ICME-13 Monographs) (pp. 3—26). Springer.
Trouche, L., Gueudet, G., & Pepin, B. (Eds). (2019). The ‘resource’ approach to mathematics education. Springer.
Trust, T. (2012). Professional learning networks designed for teacher learning. Journal of Digital Learning in Teacher Education, 28(4), 133-138.
Tunç-Pekkan, Z. (2007). Analysis of a learning case: Jasmine. In J. H. Woo, H. C. Lew, K. S. Park, & D. Y. Seo (Eds.), Proceedings of the 31st Conference of the International Group for the Psychology of Mathematics Education, Vol. 4 (pp. 225-232). PME.
United Nations Children’s Fund (UNICEF). (2020). Covid-19: Are children able to continue learning during school closures? A global analysis of the potential reach of remote learning policies using data from 100 countries. UNICEF. https://data.unicef.org/wp-content/uploads/2020/08/COVID-19-Remote-Learning-Factsheet_English_2020.pdf
United Nations General Assembly. (1948). The universal declaration of human rights. United Nations. https://www.un.org/sites/un2.un.org/files/udhr.pdf
United Nations General Assembly. (1989). Convention on the rights of the child (Resolution No. A/RES/44/70), December 8th, 1989. United Nations. https://www.unhcr.org/uk/4aa76b319.pdf
United Nations General Assembly. (2006). Convention on the rights of persons with disabilities (Resolution No. A/RES/61/106, Annex I), December 13, 2006. United Nations. https://www.un.org/disabilities/documents/convention/convoptprot-e.pdf
Valiandes, S., & Neophytou, L. (2018). Teachers’ professional development for differentiated instruction in mixed-ability classrooms: investigating the impact of a development program on teachers’ professional learning and on students’ achievement. Teacher Development, 22(1), 123-138. https://doi.org/10.1080/13664530.2017.1338196
van Geel, M., Keuning, T., Frèrejean, J., Dolmans, D., van Merriënboer, J., & Visscher, A. J. (2019). Capturing the complexity of differentiated instruction. School Effectiveness and School Improvement, 30(1), 51-67. https://doi.org/10.1080/09243453.2018.1539013
Vergnaud, G. (2013). Conceptual development and learning. Revista Qurriculum, 26, 39-59.
von Glasersfeld, E. (1995). Radical constructivism: A way of knowing and learning. Falmer Press.
Wagner, S. R. (2018). The self-directed learning practices of elementary teachers. International Journal of Self-Directed Learning, 15(2), 18-33.
Wang, C. (2018). Mathematics teachers’ expertise in resources work and its development in collectives: A French and a Chinese cases. In L. Fan, L. Trouche, C. Qi, S. Rezat, & J. Visnovska (Eds.), Research on mathematics textbooks and teachers’ resources: Advances and issues (ICME-13 Monographs) (pp. 193—213). Springer.
Webb, N. (1997). Criteria for alignment of expectations and assessments on mathematics and science education (Research Monograph No. 6). The Council of Chief State School Officers.
Whipple, K. A. (2012). Differentiated instruction: A survey study of teacher understanding and implementation in a southeast Massachusetts school district [Doctoral dissertation, Northeastern University].
Yoon, H., Byerley, C., & Thompson, P. W. (2015). Teachers’ meanings for average rate of change in U.S.A. and Korea. In T. Fukawa-Connelly (Ed.), Proceedings of the 18th Meeting of the MAA Special Interest Group on Research in Undergraduate Mathematics Education. RUME.