Does strategic knowledge matter? Effects of strategic knowledge about drawing on students’ modeling competencies in the domain of geometry

References

• Blomhøj, M., & Jensen, T. H. (2003). Developing mathematical modelling competence: Conceptual clarification and educational planning. Teaching Mathematics and Its Applications, 22(3), 123–139. https://doi.org/10.1093/teamat/22.3.123
   Google Scholar
• Blum, W., & Borromeo Ferri, R. (2009). Mathematical modelling: Can it be taught and learnt? Journal of Mathematical Modelling and Application, 1(1), 45–58.
   Google Scholar
• Blum, W., & Leiss, D. (2007). How do students and teachers deal with mathematical modelling problems? The example sugarloaf and the DISUM project. In C. Haines, P. L. Galbraith, W. Blum, & S. Khan (Eds.), Mathematical modelling (ICTMA 12): Education, engineering and economics (pp. 222–231). Horwood.
   Google Scholar
• Bond, T., & Fox, C. M. (2015). Applying the Rasch model. Fundamental measurement in the human sciences. Routledge.
   Google Scholar
• Borkowski, J. G., Chan, L. K., & Muthukrishna, N. (2000). A process-oriented model of metacognition: Links between motivation and executive functioning. In G. Schraw & J. C. Impara (Eds.), Issues in the measurement of metacognition (pp. 1–41). University of Nebraska Press.
   Google Scholar
• Borkowski, J. G., Milstead, M., & Hale, C. (1988). Components of children’s metamemory: Implications for strategy generalization. In F. E. Weinert & M. Perlmutter (Eds.), Memory development: Universal changes and individual differences (pp. 73–100). Lawrence Erlbaum.
   Google Scholar
• Bräuer, V., Leiss, D., & Schukajlow, S. (2021). Skizzen zeichnen zu Modellierungsaufgaben – Eine Analyse themenspezifischer Differenzen einer Visualisierungsstrategie beim mathematischen Modellieren [Drawing for modelling problems. An analysis of content-specific differences of a visualazation strategy to solve modelling problems]. Journal für Mathematik-Didaktik 42 , 1–33 doi:10.1007/s13138-021-00182-7.
   Google Scholar
• Brown, A. L. (1987). Metacognition, executive control, self-regulation, and other mysterious mechanisms. In F. E. Weinert & R. H. Kluwe (Eds.), Metacognition, motivation, and understanding (pp. 65–116). Erlbaum.
   Google Scholar
• Cox, R. (1999). Representation construction, externalised cognition and individual differences. Learning and Instruction, 9(4), 343–363. https://doi.org/10.1016/S0959-4752(98)00051-6
   Web of Science ®Google Scholar
• 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
   Web of Science ®Google Scholar
• De Bock, D., Verschaffel, L., Janssens, D., Van Dooren, W., & Claes, K. (2003). Do realistic contexts and graphical representations always have a beneficial impact on students’ performance? Negative evidence from a study on modeling non-linear geometry problems. Learning and Instruction, 13(4), 441–463. https://doi.org/10.1016/S0959-4752(02)00040-3
   Web of Science ®Google Scholar
• De Bock, D., Verschaffel, L., & Janssens, D. (1998). The predominance of the linear model in secondary school students’ solutions of word problems involving length and area of similar plane figures. Educational Studies in Mathematics, 35(1), 65–83. https://doi.org/10.1023/A:1003151011999
   Google Scholar
• Dignath, C., & Büttner, G. (2008). Components of fostering self-regulated learning among students. A meta-analysis on intervention studies at primary and secondary school level. Metacognition and Learning, 3(3), 231–264. https://doi.org/10.1007/s11409-008-9029-x
   Web of Science ®Google Scholar
• English, L., & Sriraman, B. (2010). Problem solving for the 21st century. In B. Sriraman & L. English (Eds.), Theories of mathematics education: Seeking new frontiers (pp. 263–290). Springer.
   Google Scholar
• Fiorella, L., & Zhang, Q. (2018). Drawing boundary conditions for learning by drawing. Educational Psychology Review, 30(3), 1115–1137. https://doi.org/10.1007/s10648-018-9444-8
   Web of Science ®Google Scholar
• Flavell, J. H. (1979). Metacognition and cognitive monitoring: A new area of cognitive developmental inquiry. American Psychologist, 34(10), 906–911. https://doi.org/10.1037/0003-066X.34.10.906
   Web of Science ®Google Scholar
• Galbraith, P. L., & Stillman, G. (2006). A framework for identifying student blockages during transitions in the modelling process. ZDM–The International Journal on Mathematics Education, 38(2), 143–162. https://doi.org/10.1007/BF02655886
   Google Scholar
• Geiger, V., & Frejd, P. (2015). A reflection on mathematical modelling and applications as a field of research: Theoretical orientation and diversity. In G. A. Stillman, W. Blum, & M. S. Biembengut (Eds.), Mathematical modelling in education research and practice (pp. 161–171). Springer.
   Google Scholar
• Greefrath, G., Hertleif, C., & Siller, H.-S. (2018). Mathematical modelling with digital tools—a quantitative study on mathematising with dynamic geometry software. ZDM, 50(1), 233–244. https://doi.org/10.1007/s11858-018-0924-6
   Google Scholar
• Hembree, R. (1992). Experiments and relational studies in problem solving: A meta-analysis. Journal for Research in Mathematics Education, 23(3), 242–273. https://doi.org/10.2307/749120
   Web of Science ®Google Scholar
• Hershkowitz, R., Ben-Chaim, D., Hoyles, C., Lappan, G., Mitchelmore, M., & Vinner, S. (1989). Psychological aspects of learning geometry. In P. Nesher & J. Kilpatrick (Eds.), Mathematics and cognition (ICMI study series) (pp. 70–95). University Press.
   Google Scholar
• Kaiser, G., & Sriraman, B. (2006). A global survey of international perspectives on modelling in mathematics education. Zentralblatt Für Didaktik der Mathematik, 38(3), 302–310. https://doi.org/10.1007/BF02652813
   Google Scholar
• Klieme, E., Hartig, J., & Rauch, D. (2008). The concept of competence in educational contexts. In J. Hartig, E. Klieme, & D. Leutner (Eds.), Assessment of competencies in educational contexts (pp. 3–22). Hogrefe & Huber.
   Google Scholar
• Kozhevnikov, M., Hegarty, M., & Mayer, R. (2002). Revising the visualizer-verbalizer dimension: Evidence for two types of visualizers. Cognition and Instruction, 20(1), 47–77. https://doi.org/10.1207/S1532690XCI2001_3
   Web of Science ®Google Scholar
• Krawitz, J., & Schukajlow, S. (2020). When can making a drawing hinder problem solving? Effect of the drawing strategy on linear overgeneralizations and problem solving. Frontiers in Psychology, 11, 506. https://doi.org/10.3389/fpsyg.2020.00506
   Google Scholar
• Leopold, C., & Leutner, D. (2012). Science text comprehension: Drawing, main idea selection, and summarizing as learning strategies. Learning and Instruction, 22(1), 16–26. https://doi.org/10.1016/j.learninstruc.2011.05.005
   Web of Science ®Google Scholar
• Lucangeli, D., & Cornoldi, C. (1997). Mathematics and metacognition: What is the nature of the relationship? Mathematical Cognition, 3(2), 121–139. https://doi.org/10.1080/135467997387443
   Google Scholar
• Maaß, K. (2006). What are modelling competencies? ZDM–The International Journal on Mathematics Education, 38(2), 113–142. https://doi.org/10.1007/BF02655885
   Google Scholar
• Niss, M., Blum, W., & Galbraith, P. L. (2007). Introduction. In W. Blum, P. L. Galbraith, H.-W. Henn, & M. Niss (Eds.), Modelling and applications in mathematics education: The 14th ICMI study (pp. 1–32). Springer.
   Google Scholar
• OECD. (2016). Mathematics performance among 15-year-olds PISA 2015 results (Volume I): Excellence and equity in education.
   Google Scholar
• Presmeg, N. C. (2006). Research on visualization in learning and teaching mathematics. In A. Gutiérrez & P. Boero (Eds.), Handbook of research on the psychology of mathematics education (pp. 205–235). Sense Publishers.
   Google Scholar
• Raudenbush, S., & Bryk, A. (2002). Hierarchical linear models: Applications and data analysis methods. Sage Publications.
   Google Scholar
• Rellensmann, J. (2019). Selbst erstellte Skizzen beim mathematischen Modellieren: Ergebnisse einer empirischen Untersuchung [Using self-generated drawings to solve modelling problems: Results of an empirical study]. Wiesbaden: Springer Spektrum.
   Google Scholar
• Rellensmann, J., Schukajlow, S., & Leopold, C. (2017). Make a drawing. Efects of strategic knowledge, drawing accuracy, and type of drawing on students’ mathematical modelling performance. Educational Studies in Mathematics, 95(1), 53–78. https://doi.org/10.1007/s10649-016-9736-1
   Web of Science ®Google Scholar
• Rellensmann, J., Schukajlow, S., & Leopold, C. (2020). Measuring and investigating strategic knowledge about drawing to solve geometry modelling problems. ZDM, 52(1), 97–110. https://doi.org/10.1007/s11858-019-01085-1
   Google Scholar
• Schneider, W., & Artelt, C. (2010). Metacognition and mathematics education. ZDM–The International Journal on Mathematics Education, 42(2), 149–161. https://doi.org/10.1007/s11858-010-0240-2
   Google Scholar
• Schukajlow, S., Blomberg, J., Rellensmann, J., & Leopold, C. (2021). The role of strategy-based motivation in mathematical problem solving: The case of learner-generated drawings. Learning and Instruction, Article 101561. https://doi.org/10.1016/j.learninstruc.2021.101561
   Google Scholar
• Schukajlow, S., Blomberg, J., Rellensmann, J., & Leopold, C. (2021). Do emotions and prior performance facilitate the use of the learner-generated drawing strategy? Effects of enjoyment, anxiety, and intramathematical performance on the use of the drawing strategy and modelling performance. Contemporary Educational Psychology, 65, 101967. https://doi.org/10.1016/j.cedpsych.2021.101967
   Web of Science ®Google Scholar
• Stender, P., Krosanke, N., & Kaiser, G. (2017). Scaffolding complex modelling processes: An in-depth study. In: Stillman, G., Blum, W., and Kaiser, G. Mathematical modelling and applications: Crossing and researching boundaries in mathematics education (pp. 467–477). Cham: Springer.
   Google Scholar
• Stillman, G., Blum, W., & Kaiser, G. (2017). Mathematical modelling and applications: Crossing and researching boundaries in mathematics education. Springer.
   Google Scholar
• Stylianou, D. A. (2011). An examination of middle school students’ representation practices in mathematical problem solving through the lens of expert work: Towards an organizing scheme. Educational Studies in Mathematics, 76(3), 265–280. https://doi.org/10.1007/s10649-010-9273-2
   Web of Science ®Google Scholar
• Taub, M., Azevedo, R., Bouchet, F., & Khosravifar, B. (2014). Can the use of cognitive and metacognitive self-regulated learning strategies be predicted by learners’ levels of prior knowledge in hypermedia-learning environments?. Computers in Human Behavior, 39, 356–367 doi:https://doi.org/10.1016/j.chb.2014.07.018.
   Web of Science ®Google Scholar
• Uesaka, Y., Manalo, E., & Ichikawa, S. (2010). The effects of perception of efficacy and diagram construction skills on students’ spontaneous use of diagrams when solving math word problems. In A. K. Goel, M. Jamnik, & N. H. Narayanan (Eds.), Diagrammatic representation and inference (Vol. 6170, pp. 197–211). Springer.
   Google Scholar
• Van Essen, G., & Hamaker, C. (1990). Using self-generated drawings to solve arithmetic word problems. The Journal of Educational Research, 83(6), 301–312. https://doi.org/10.1080/00220671.1990.10885976
   Web of Science ®Google Scholar
• Van Garderen, D., Scheuermann, A., & Jackson, C. (2013). Examining how students with diverse abilities use diagrams to solve mathematics word problems. Learning Disability Quarterly, 36(3), 145–160. https://doi.org/10.1177/0731948712438558
   Web of Science ®Google Scholar
• Vorhölter, K. (2018). Conceptualization and measuring of metacognitive modelling competencies: Empirical verification of theoretical assumptions. ZDM Mathematics Education, 50(1), 343–354. https://doi.org/10.1007/s11858-017-0909-x
   Google Scholar