Adaptive Tasks as a Differentiation Strategy in the Mathematics Classroom: Features from Research and Teachers’ Views
Keywords:
Adaptivity, Differentiation Strategy, Open adaptive Tasks, Teacher Professional Development, Three-Tetrahedron-ModelAbstract
Tasks play a central role in the mathematics classroom. Especially when teaching in a heterogeneous mathematics class, teachers should be able to find, select, modify, and assign tasks adequately. The focus in this paper is on adaptivity to cognitively activate all learners at the individual level, and on teachers’ abilities to allow for such adaptivity by means of selecting appropriate tasks. More specifically, when planning lesson phases for practice and consolidation, teachers may consider which tasks have differentiation potential and can thus be completed by all students at the same time but at different levels. In order to analyse teachers’ strategies in this regard, we investigated the task features that teachers may consider when assessing the differentiation potential of exercise tasks. We first deductively constructed rating categories based on the literature on instructionally relevant features of adaptive tasks. Then, we inductively extended and refined the constructed categories by analysing teachers’ reasoning based on a sample of 78 in-service teachers at secondary schools. We validated the resulting 22 categories by determining interrater reliability. Our findings indicate that teachers consider a broad spectrum of task features when analysing the differentiation potential of tasks. However, only some of these features are directly relevant with regard to using adaptive tasks as a differentiation strategy. Our results also show that many teachers arrived at conclusions about the differentiation potential of tasks that were different from task-design experts. Based on our findings regarding teachers’ perspectives on the differentiation potential of tasks and on certain task features, we discuss how these findings may have arisen and how important the knowledge about the deep structure of adaptive tasks is for teachers’ professional development.
References
Anderson, J. (1996). Some teachers’ beliefs and perceptions of problem solving. In P. Clarkson (Ed.), Technology in mathematics education (Proceedings of the 19th Conference of the Mathematics Education Research Group of Australasia). (pp. 30–37). MERGA.
Anderson, J. (2003). Teachers’ choice of tasks: A window into beliefs about the role of problem solving in learning mathematics. In L. Bragg, C. Campbell, G. Herbert, & J. Mousley (Eds.), Mathematics educations research: Innovation, networking, opportunity (Proceedings of the 26th Conference of the Mathematics Education Research Group of Australasia). (pp. 72–79). MERGA.
Barzel, B., Leuders, T., Prediger, S., & Hußmann, S. (2013). Designing tasks for engaging students in active knowledge organization. In C. Margolinas, A. Watson, M. Ohtani, J. Ainley, J. Bolite Frant, M. Doorman, C. Kieran, A. Leung, P. Sullivan, D. Thompson, & Y. Yang (Eds.), ICMI Study 22 on Task Design:Pproceedings of the study conference (pp. 285–294). ICMI.
Beck, E., Baer, M., Guldimann, T., Bischoff, S., Brühwiler, C., Müller, P., Niedermann, R., Rogalla, M., & Vogt, F. (2008). Adaptive lehrkompetenz [Adaptive teaching skills]. Waxmann.
Blömeke, S., Risse, J., Müller, C., Eichler, D., & Schulz, W. (2006). Analyse der Qualität von Aufgaben aus didaktischer und fachlicher Sicht: Ein allgemeines Modell und seine exemplarische Umsetzung im Unterrichtsfach Mathematik [Analysis of the quality of tasks from a didactic and subject perspective: A general model and its exemplary implementation in the School subject Mathematics]. Unterrichtswissenschaft, 34(4), 330–357.
Bölsterli Bardy, K., & Wilhelm, M. (2018). Von kompetenzorientierten zu kompetenzfördernden Aufgaben im Schulbuch [From competence-oriented to competence-promoting tasks in the textbook]. Erziehung und Unterricht, 168, 1/2, 121–129.
Boston, M. D., & Smith, M. S. (2009). Transforming secondary mathematics teaching: Increasing the cognitive demands of instructional tasks used in teachers’ classrooms. Journal for Research in Mathematics Education, 40(2), 119–156.
Bromme, R., Seeger, F., & Steinbring, H. (1990). Aufgaben als Anforderungen an Lehrer und Schüler [Tasks as demands on teachers and students]. Aulis.
Brown, A. L. (1984). Metakognition, Handlungskontrolle, Selbststeuerung und andere, noch geheimnisvolle Mechanismen [Metacognition, action control, self-control, and other still mysterious mechanisms]. In F. E. Weinert & R. H. Kluwe (Eds.), Metakognition, Motivation und Lernen (pp. 60–109). Kohlhammer.
Carpenter, T. P., Fennema, E., Franke, M. L., Levi, L., & Empson, S. B. (1999). Children’s mathematics: Cognitively guided instruction. Heinemann.
Chapman, O. (2013). Mathematical-task knowledge for teaching. Journal of Mathematics Teacher Education, 16(1), 1–6. https://doi.org/10.1007/s10857-013-9234-7
Clarke, D. M., Cheeseman, J., Roche, A., & 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.
Doyle, W. (1983). Academic work. Review of Educational Research, 53(2), 159–199. https://doi.org/10.3102%2F00346543053002159
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
Glaser, B. G., & Strauss, A. L. (1967). A discovery of grounded theory. Aldine Publishing.
Hammer, S. (2016). Professionelle Kompetenz von Mathematiklehrkräften im Umgang mit Aufgaben in der Unterrichtsplanung: theoretische Grundlegung und empirische Untersuchung [Professional competence of mathematics teachers in dealing with tasks in lesson planning: theoretical grounding and empirical investigation; Doctoral dissertation, Ludwig–Maximilians–Universität München]. https://edoc.ub.uni-muenchen.de/20439/1/Hammer_Sabine.pdf
Helmke, A. (2010). Unterrichtsqualität und Lehrerprofessionalität: Diagnose, Evaluation und Verbesserung des Unterrichts [Teaching quality and teacher professionalism: diagnosis, evaluation and improvement of teaching]. Klett-Kallmeyer.
Hengartner, E., Hirt, U., Wälti, B., & Primarschulteam Lupsingen (2006). Lernumgebungen für Rechenschwache bis Hochbegabte: Natürliche Differenzierung im Mathematikunterricht [Learning environments for low- to high-achieving students in arithmetic: Natural differentiation in mathematics education]. Klett & Balmer.
Heymann, H. W. (1991). Innere Differenzierung im Mathematikunterricht [Internal differentiation in mathematics education]. Mathematik lehren, 49, 63–66.
Hiebert, J., & Wearne, D. (1997). Instructional tasks, classroom discourse and student learning in second grade arithmetic. American Educational Research Journal, 30(2), 393–425. https://doi.org/10.3102/00028312030002393
Hirt, U., & Wälti, B. (2008). Lernumgebungen im Mathematikunterricht: Natürliche Differenzierung für Rechenschwache und Hochbegabte [Learning environments in mathematics education: Natural differentiation for low- and high-achieving students]. Klett-Kallmeyer.
Holzäpfel, L., Lacher, M., Leuders, T., & Rott, B. (2018). Problemlösen lehren lernen: Wege zum mathematischen Denken [Learning to teach problem solving: pathways to mathematical reasoning]. Klett-Kallmeyer.
Holzäpfel, L., Leuders, T., & Bardy, T. (2019). Preparing in-service teachers for the differentiated classroom. In M. Graven, H. Venkat, A. Essien, & P. Vale (Eds.), Proceedings of the 43rd Conference of the International Group for the Psychology of Mathematics Education, 2, 60–109. PME.
Hußmann, S. & Prediger, S. (2007). Mit Unterschieden rechnen - Differenzieren und Individualisieren [Reckoning with differences: Differentiating and Individualizing]. Praxis der Mathematik in der Schule, 49(17), 2–8.
Jordan, A., Ross, N., Krauss, S., Baumert, J., Blum, W., Löwen, K., Brunner, M., & Kunter, M. (2006). Klassifikationsschema für Mathematikaufgaben: Dokumentation der Aufgabenklassifikation im COACTIV-Projekt [Classification scheme for mathematics tasks: Documentation of task classification in the COACTIV project]. Max-Planck-Institut für Bildungsforschung. https://pure.mpg.de/rest/items/item_2100753_3/component/file_2197661/content
Jordan, A., Krauss, S., Löwen, K., Blum, W., Neubrand, M., & Brunner, M. (2008). Aufgaben im COACTIV-Projekt: Zeugnisse des kognitiven Aktivierungspotentials im deutschen Mathematikunterricht [Tasks in the COACTIV project: testimonies of cognitive activation potential in German mathematics education]. Journal für Mathematik-Didaktik, 29(2), 83–107. https://doi.org/10.1007/BF03339055
Krauthausen, G., & Scherer, P. (2010). Umgang mit Heterogenität: Natürliche Differenzierung im Mathematikunterricht der Grundschule [Dealing with heterogeneity: Natural differentiation in elementary school mathematics classrooms]. IPN. http://www.sinus-an-grundschulen.de/fileadmin/uploads/Material_aus_SGS/ Handreichung_Krauthausen-Scherer.pdf
Kunter, M., Baumert, J., Blum, W., Klusmann, U., Krauss, S., & Neubrand, M. (2013). Cognitive activation in the mathematics classroom and professional competence of teachers: Results from the COACTIV project. Springer. https://doi.org/10.1007/978-1-4614-5149-5
Leuders, T. (2015). Aufgaben in Forschung und Praxis [Tasks in research and practice]. In R. Bruder, L. Hefendehl-Hebeker, B. Schmidt-Thieme, & H.-G. Weigand (Eds.), Handbuch Mathematikdidaktik [Handbook mathematics education] (pp. 433-458). Springer.
Leuders, T., & Holzäpfel, L. (2011). Kognitive Aktivierung im Mathematikunterricht [Cognitive activation in mathematics education]. Unterrichtswissenschaft, 39, 213–230.
Leuders, T., & Prediger, S. (2016). Flexibel differenzieren und fokussiert fördern im Mathematikunterricht [Flexible differentiation and focused support in mathematics teaching]. Cornelsen Scriptor.
Lipowsky, F. & Rzejak, D. (2015). Key features of effective professional development programmes for teachers. Ricercazione, 7(2), 27–51.
Lou, Y., Abrami, P. C., Spence, J. C., Poulsen, C., Chambers, B., & d'Apollonia, S. (1996). Within-class grouping: A meta-analysis. Review of Educational Research, 66(4), 423–458. https://doi.org/10.2307/1170650
Maier, U., Kleinknecht, M., Metz, K., & Bohl, T. (2010). Ein allgemeindidaktisches Kategoriensystem zur Analyse des kognitiven Potenzials von Aufgaben [A general didactic category system for analyzing the cognitive potential of tasks]. Beiträge zur Lehrerinnen- und Lehrerbildung, 28(1), 84–96.
McDougall, D. E. (2004). School mathematics improvement: Leadership handbook. Thomson Nelson.
Müller, G., & Wittmann, E. Ch. (1998). Das Zahlenbuch: Lehrerband [The numbers book: Teacher’s volume]. Klett.
Neubrand, J. (2002). Eine Klassifikation mathematischer Aufgaben zur Analyse von Unterrichtssituationen: Selbsttätiges Arbeiten in Schülerarbeitsphasen in den Stunden der TIMSS-Video-Studie [A classification of mathematical tasks for analyzing classroom situations: Self-acting in student work phases in lessons of the TIMSS video study; Doctoral dissertation, Freie Universität Berlin].
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
Prediger, S., & Scherres, C. (2012). Niveauangemessenheit von Arbeitsprozessen in selbstdifferenzierenden Lernumgebungen: Qualitative Fallstudie am Beispiel der Suche aller Würfelnetze [Level appropriateness of work processes in self-differentiating learning environments: Qualitative case study using the example of searching all cube nets]. Journal für Mathematik-Didaktik, 33, 143–173. https://doi.org/10.1007/s13138-012-0035-9
Prediger, S., Roesken-Winter, B., & Leuders, T. (2019). Which research can support PD facilitators? Strategies for content-related PD research in the Three-Tetrahedron Model. Journal of Mathematics Teacher Education, 22, 407–425. https://doi.org/10.1007/s10857-019-09434-3
Prediger, S., Barzel, B., Hußmann, S., & Leuders, T. (2021). Towards a research base for textbooks as teacher support: The case of engaging students in active knowledge organization in the KOSIMA project. ZDM Mathematics Education. [Online first]. https://doi.org/10.1007/s11858-021-01245-2
Russo, J., Minas, M., Hewish, T., & McCosh, J. (2020). Using prompts to empower learners: Exploring primary students’ attitudes towards enabling prompts when learning mathematics through problem solving. Mathematics Teacher Education and Development, 22(1), 48–67. https://mted.merga.net.au/index.php/mted/article/view/592
Scherres, C. (2012). Niveauangemessenes mathematisches Arbeiten in einer selbstdifferenzierenden Lernumgebung. Eine qualitative Fallstudie am Beispiel einer Würfelnetz-Lernumgebung [Level-appropriate mathematical work in a self-differentiating learning environment: A qualitative case study using the example of a cube net learning environment; Doctoral dissertation, IEEM Dortmund]. https://doi.org/10.1007/978-3-658-02083-5
Schoenfeld, A. H. (1988). When good teaching leads to bad results: The disasters of “well-taught” mathematics courses. Educational Psychologist, 23(2), 145–166.
Seidel, T., Blomberg, G., & Stürmer, K. (2010). „Observer“-Validierung eines videobasierten Instruments zur Erfassung der professionellen Wahrnehmung von Unterricht [”Observer“-validation of a video-based instrument to capture professional perceptions of teaching]. Zeitschrift für Pädagogik, 56, 296–306.
Snow, R. (1989). Aptitude-treatment interaction as a framework for research on individual differences in learning. In P. Ackerman, R. J. Sternberg, & R. Glaser (Eds.), Learning and individual differences (pp. 13–60). W. H. Freeman.
Stein, M., Grover, B., & Henningsen, M. (1996). Building student capacity for mathematical thinking and reasoning: An analysis of mathematical tasks used in reform classrooms. American Educational Research Journal, 33(2), 455–488.
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.
Sullivan, P. (1999). Seeking a rationale for particular classroom tasks and activities. In J. M. Truran & K. N. Truran (Eds.), Making the difference (Proceedings of the 21st annual conference of the Mathematics Educational Research Group of Australasia). (pp. 15–29). MERGA.
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 (pp. 17–42). World Scientific Publishing Co. https://doi.org/10.1142/9789814277228_0002
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
van Es, E., & Sherin, M. (2002). Learning to notice: Scaffolding new teachers’ interpretations of classroom Interactions. Journal of Technology and Teacher Education, 10(4), 571–596.
Vygotsky, L. S. (1978). Mind in society: The development of higher psychological processes. Harvard University Press.
Watson, A., & Mason, J. (2006). Seeing exercise as a single mathematical object: Using variation to structure sense-making. Mathematical Teaching and Learning, 8(2), 91–111. https://doi.org/10.1207/s15327833mtl0802_1
Wellenreuther, M. (2004). Lehren und Lernen – aber wie? Empirisch-experimentelle Forschung zum Lehren und Lernen im Unterricht [Teaching and learning - but how? Empirical-experimental research on teaching and learning in the classroom]. Schneider Hohengehren.
Wittmann, E. C., & Müller, G. N. (1990). Handbuch produktiver Rechenübungen [Handbook of productive arithmetic exercises]. Band II. Klett.
Zaslavsky, O., & Sullivan, P. (Eds.). (2011). Constructing knowledge for teaching secondary mathematics. Springer. https://doi.org/10.1007/978-0-387-09812-8
