Literature review of mathematics teaching design for problem solving and reasoning

Authors

Johan Sidenvall

Abstract

To characterize teaching designs intended to enhance students’ problem solving and reasoning skills or to develop other mathematical competencies via problem solving and reasoning, a literature review was conducted of 26 articles published in seven top-ranked journals on mathematics education from 2000 to 2016. Teaching designs were characterized by a) the educational goals of the designs, b) the claims about how to reach these goals, and c) the empirical and theoretical arguments underlying these claims. Thematic analysis was used to analyze the retrieved articles. All but two studies had goals concerned with developing students’ mathematical competencies. The overarching ideas of the identified emergent claims regarding the achievement of stipulated goals, concerned scaffolding students’ learning and letting students construct their own mathematics. Four recurring theoretical arguments were found to support emergent claims: hypothetical learning trajectories, realistic mathematics education, theory of didactical situations and zone of proximal development.

References

Abdu, R., Schwarz, B. & Mavrikis, M. (2015). Whole-class scaffolding for learning to solve mathematics problems together in a computer-supported environment. ZDM, 47 (7), 1163-1178. https://doi.org/10.1007/s11858-015-0719-y

Akker, J. van den (2010). Building bridges: how research may improve curriculum policies and classroom practice. In S. Stoney (Ed.), Beyond Lisbon 2010: perspectives from research and development for education policy in Europe (CIDREE Yearbook 2010) (pp. 175-196). Slough: National Foundation for Educational Research.

Akker, J. van den, Gravemeijer, K., McKenney, S. & Nieveen, N. (2006). Introducing educational design research. In J. van den Akker, K. Gravemeijer, S. McKenney & N. Nieveen (Eds.), Educational design research (Vol. 1, pp. 3-7). London: Routledge. https://doi.org/10.4324/9780203088364

Anderson, T. & Shattuck, J. (2012). Design-based research: A decade of progress in education Research? Educational Researcher, 41 (1), 16-25. https://doi.org/10.3102/0013189X11428813

Ball, D. & Bass, H. (2003). Making mathematics reasonable in school. In J. Kilpatrick, W. G. Martin & D. Schifter (Eds.), A research companion to Principles and standards for mathematics (pp. 27-44). Reston: NCTM.

Beer, H. de, Gravemeijer, K. & Eijck, M. van(2015). Discrete and continuous reasoning about change in primary school classrooms. ZDM, 47 (6), 981-996. https://doi.org/10.1007/s11858-015-0684-5

Black, P. & Wiliam, D. (2009). Developing the theory of formative assessment. Educational assessment, Evaluation and Accountability, 21 (1), 5-31. https://doi.org/10.1007/s11092-008-9068-5

Boesen, J., Helenius, O., Bergqvist, E., Bergqvist, T., Lithner, J. et al. (2014). Developing mathematical competence: from the intended to the enacted curriculum. The Journal of Mathematical Behavior, 33 (1), 72-87. https://doi.org/10.1016/j.jmathb.2013.10.001

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

Brousseau, G. (1997). Theory of didactical situations in mathematics: didactique des mathématiques, 1970-1990. Dordrecht: Kluwer Academic.

Brown, A. L. (1992). Design experiments: theoretical and methodological challenges in creating complex interventions in classroom settings. The Journal of the Learning Sciences, 2 (2), 141-178. https://doi.org/10.1207/s15327809jls0202_2

Carpenter, T. P., Blanton, M. L., Cobb, P., Franke, M. L., Kaput, J. & McClain, K. (2004). Scaling up innovative practices in mathematics and science. Madison: NCISLA/Mathematics & Science. Retrieved from http://greenframingham.org/stem/research/item2_scalingup_innovative_practices_math_scienceNCISLAReport1.pdf

Csíkos, C., Szitányi, J. & Kelemen, R. (2012). The effects of using drawings in developing young children's mathematical word problem solving: a design experiment with third-grade Hungarian students. Educational Studies in Mathematics, 81 (1), 47-65. https://doi.org/10.1007/s10649-011-9360-z

Clements, D. H. & Sarama, J. (2004). Learning trajectories in mathematics education. Mathematical Thinking and Learning, 6 (2), 81-89. https://doi.org/10.1207/s15327833mtl0602_1

Cobb, P., Confrey, J., diSessa, A., Lehrer, R. & Schauble, L. (2003). Design experiments in educational research. Educational Researcher, 32 (1), 9-13. https://doi.org/10.3102/0013189X032001009

Collins, A. (2012). What is the most effective way to teach problem solving? A commentary on productive failure as a method of teaching. An International Journal of the Learning Sciences, 40 (4), 731-735. https://doi.org/10.1007/s11251-012-9234-5

Doorman, L. & Gravemeijer, K. (2008). Emergent modeling: discrete graphs to support the understanding of change and velocity. ZDM, 41 (1-2), 199-211. https://doi.org/10.1007/s11858-008-0130-z

Freudenthal, H. (1973). Mathematics as an educational task. Dordrecht: Reidel. https://doi.org/10.1007/978-94-010-2903-2_2

Freudenthal, H. (1991). Revisiting mathematics education: China lectures. Dordrecht: Kluwer Academic.

Gough, D., Oliver, S. & Thomas, J. (2013). Learning from research: systematic reviews for informing policy decisions: a quick guide (A paper for the Alliance for useful evidence). London: Nesta.

Granberg, C. (2016). Discovering and addressing errors during mathematics problem-solving - A productive struggle? Journal of Mathematical Behavior, 42, 33-48. https://doi.org/10.1016/j.jmathb.2016.02.002

Gravemeijer, K. (1994). Educational development and developmental research in mathematics education. Journal for Research in Mathematics Education, 25 (5), 443-471. https://doi.org/10.2307/749485

Gravemeijer, K. (1999). How emergent models may foster the constitution of formal mathematics. Mathematical Thinking and Learning, 1 (2), 155-177. https://doi.org/10.1207/s15327833mtl0102_4

Hiebert, J. & Grouws, D. (2007). The effects of classroom mathematics teaching on students' learning. In F. Lester (Ed.), Second handbook of research on mathematics teaching and learning: (pp. 371-404). Charlotte: Information Age.

Hoffkamp, A. (2011). The use of interactive visualizations to foster the understanding of concepts of calculus: design principles and empirical results. ZDM, 43 (3), 359-372. https://doi.org/10.1007/s11858-011-0322-9

Jonsson, B., Norqvist, M., Lithner, J. & Liljekvist, Y. (2014). Learning mathematics through algorithmic and creative reasoning. Journal of Mathematical Behavior, 36, 20-32. https://doi.org/10.1016/j.jmathb.2014.08.003

Kali, Y. (2008). The design principles database as a means for promoting design-based research. In A. Kelly, J. Baek & R. Lesh, (Ed.), Handbook of design research methods in education: innovations in science, technology, engineering, and mathematics learning and teaching (pp. 423-438). New York: Routledge.

Kazak, S., Wegerif, R. & Fujita, T. (2015). Combining scaffolding for content and scaffolding for dialogue to support conceptual breakthroughs in understanding probability. ZDM, 47 (7), 1269-1283. https://doi.org/10.1007/s11858-015-0720-5

Kilpatrick, J., Swafford, J. & Findell, B. (2001). Adding it up: helping children learn mathematics. Washington: National Academy Press.

Koichu, B., Berman, A. & Moore, M. (2004). Promotion of heuristic literacy in a regular mathematics classroom. For the learning of mathematics, 24 (1), 33-39.

Kotsopoulos, D. & Lee, J. (2012). An analysis of math congress in an eighth grade classroom. Mathematical Thinking and Learning, 14 (3), 181-198. https://doi.org/10.1080/10986065.2012.682958

Lee, N., Yeo, D. & Hong, S. (2014). A metacognitive-based instruction for primary four students to approach non-routine mathematical word problems. ZDM, 46 (3), 465-480. https://doi.org/10.1007/s11858-014-0599-6

Lithner, J. (2008). A research framework for creative and imitative reasoning. Educational Studies in Mathematics, 67 (3): 255-276. https://doi.org/10.1007/s10649-007-9104-2

Lithner, J. (2017). Principles for designing mathematical tasks that enhance imitative and creative reasoning. ZDM, 1-13. https://doi.org/10.1007/s11858-017-0867-3

Lynch, K. & Star, J. (2014). Views of struggling students on instruction incorporating multiple strategies in algebra I: an exploratory study. Journal for Research in Mathematics Education, 45 (1), 6-18. https://doi.org/10.5951/jresematheduc.45.1.0006

MathEduc Database (2017). Retrieved from https://www.zentralblatt-math.org/matheduc/

McKenney, S. & Reeves, T. (2012). Conducting educational design research. New York: Routledge. https://doi.org/10.4324/9780203818183

McKenney, S. & Reeves, T. (2013). Systematic review of design-based research progress: Is a little knowledge a dangerous thing? Educational Researcher, 42 (2), 97-100. https://doi.org/10.3102/0013189X12463781

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

National Research Council (2000). How people learn: mind, brain, experience, and school (Exanded edition). Washington: The National Academies Press.

Niss, M. (2003, January). Mathematical competencies and the learning of mathematics: the Danish KOM project. Paper presented at the 3rd Mediterranean Conference on Mathematics Education, Athens, Greece.

Niss, M. (2007). Reflections on the state and trends in research on mathematics teaching and learning: from here to utopia. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (Vol. 2, pp. 1293-1312). Charlotte: Information Age.

Niss, M. & Jensen, T. H. (Eds.) (2002). Kompetencer og matematiklæring: idéer og inspiration til udvikling af matematikundervisning i Danmark. København: Undervisningsministeriets forlag.

Pol, J. van de, 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

Prediger, S., Gravemeijer, K. & Confrey, J. (2015). Design research with a focus on learning processes: an overview on achievements and challenges. ZDM, 47 (6), 877-891. https://doi.org/10.1007/s11858-015-0722-3

Prediger, S. & Pöhler, B. (2015). The interplay of micro- and macro-scaffolding: an empirical reconstruction for the case of an intervention on percentages. ZDM, 47 (7), 1179-1194. https://doi.org/10.1007/s11858-015-0723-2

Puntambekar, S. & Hubscher, R. (2005). Tools for scaffolding students in a complex learning environment: What have we gained and what have we missed? Educational Psychologist, 40 (1), 1-12. https://doi.org/10.1207/s15326985ep4001_1

Ridlon, C. L. (2009). Learning mathematics via a problem-centered approach: a two-year study. Mathematical Thinking and Learning, 11 (4), 188-225. https://doi.org/10.1080/10986060903225614

Ruthven, K., Laborde, C., Leach, J. & Tiberghien, A. (2009). Design tools in didactical research: instrumenting the epistemological and cognitive aspects of the design of teaching sequences. Educational Researcher, 38 (5), 329-342. https://doi.org/10.3102/0013189X09338513

Ryve, A., Nilsson, P., Palm, T., Steenbrugge, H. van, Andersson, C. et al., (2015). Kartläggning av forskning om formativ bedömning, klassrumsundervisning och läromedel i matematik: delrapport från skolforsk-projektet [Survey of research on formative assessment, classroom teaching and mathematics teaching materials]. Stockholm: Vetenskapsrådet. Retrieved from https://www.vr.se/analys-och-uppdrag/vi-analyserar-och-utvarderar/alla-publikationer/publikationer/2015-09-10-kartlaggning-av-forskning-om-formativ-bedomning-klassrumsundervisning-och-laromedel-i-matematik.-delrapport.html

Schoenfeld, A. (1985). Mathematical problem solving. Orlando: Academic Press.

Schoenfeld, A. (2002). Research methods in (mathematics) education. In L. D. English (Ed.), Handbook of international research in mathematics education (pp. 435-487). Mahwah: Lawrence Erlbaum.

Simon, M. (1995). Reconstructing mathematics pedagogy from a constructivist perspective. Journal for Research in Mathematics Education, 26 (2), 114-145. https://doi.org/10.2307/749205

Smit, J., Eerde, H. van & Bakker, A. (2013). A conceptualisation of whole-class scaffolding. British Educational Research Journal, 39 (5), 817-834. https://doi.org/10.1002/berj.3007

Stephan, M. (2015). Conducting classroom design research with teachers. ZDM, 47 (6), 905-917. https://doi.org/10.1007/s11858-014-0651-6

Stephan, M. & Akyuz, D. (2012). A proposed instructional theory for integer addition and subtraction. Journal for Research in Mathematics Education, 43 (4), 428-464. https://doi.org/10.5951/jresematheduc.43.4.0428

Stylianides, A. & Stylianides, G. (2013). Seeking research-grounded solutions to problems of practice: classroom-based interventions in mathematics education. ZDM, 45 (3), 333-341. https://doi.org/10.1007/s11858-013-0501-y

Tempier, F. (2016). New perspectives for didactical engineering: an example for the development of a resource for teaching decimal number system. Journal of Mathematics Teacher Education, 19 (2-3), 261-276. https://doi.org/10.1007/s10857-015-9333-8

Thomas, G. (1997). What's the use of theory? Harvard Educational Review, 67 (1), 75-104. https://doi.org/10.17763/haer.67.1.1x807532771w5u48

Toerner, G. & Arzarello, F. (2012). Grading mathematics education research journals. Newsletter of the European Mathematical Society, 86, 52-54.

Visnovska, J. & Cobb, P. (2015). Learning about whole-class scaffolding from a teacher professional development study. ZDM, 47 (7), 1133-1145. https://doi.org/10.1007/s11858-015-0739-7

Vygotsky, L. (1978). Interaction between learning and development. Readings on the development of children, 23 (3), 34-41.

Warshauer, H. K. (2015). Productive struggle in middle school mathematics classrooms. Journal of Mathematics Teacher Education, 18 (4), 375-400. https://doi.org/10.1007/s10857-014-9286-3

Wood, D., Bruner, J. & Ross, G. (1976). The role of tutoring in problem solving. Journal of Child Psychology and Psychiatry, 17 (2), 89-100. https://doi.org/10.1111/j.1469-7610.1976.tb00381.x

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Published

2024-11-19

How to Cite

Sidenvall, J. (2024). Literature review of mathematics teaching design for problem solving and reasoning. NOMAD Nordic Studies in Mathematics Education, 24(1), 51–74. Retrieved from https://tidsskrift.dk/NOMAD/article/view/148997