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Scaffolding mathematical modelling with a solution plan

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Abstract

In the study presented in this paper, we examined the possibility to scaffold mathematical modelling with strategies. The strategies were prompted using an instrument called “solution plan” as a scaffold. The effects of this step by step instrument on mathematical modelling competency and on self-reported strategies were tested using nineth grade students (N = 91) at a German middle track school (Realschule) in a quasi-experimental design. Six classes were randomly assigned to the experimental group, in which students used the solution plan, or to the control group. The quantitative data analysis using ANOVAs reveals that (1) in the posttest the experimental group students reported more frequently about planning, rehearsal, elaboration and organizing strategies while solving modelling problems than the control group; (2) the “solution plan” as a scaffold supports the development of students’ modelling competency, including its sub-competencies. The students who used the solution plan outperformed the other students in solving modelling problems concerning the topic “Pythagorean theorem”.

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Notes

  1. IMPROVE: Introducing the new concepts, Metacognitive questioning, Practicing, Reviewing and reducing difficulties, Obtaining mastery, Verification, Enrichment.

  2. CRIME: Careful reading, Recall possible strategies, Implement strategy, Monitor, Evaluation.

  3. DISUM: Didaktische Interventionsformen für einen selbständigkeitsorientierten aufgabengesteuerten Unterricht am Beispiel Mathematik. (In English: Didactical intervention modes for mathematics teaching oriented towards self-regulation and directed by tasks). The project has been in operation since 2002 and was 2005–2012 sponsored by the German Research Foundation (Deutsche Forschungs-Gemeinschaft). Directors: W. Blum, R. Messner (both University of Kassel), R. Pekrun (University of Munich).

References

  • Anderson, J. R., Corbett, A. T., Koedinger, K. R., & Pelletier, R. (1995). Cognitive tutors: lessons learned. The journal of the learning Sciences, 4(2), 167–207.

    Article  Google Scholar 

  • Atkinson, R. K., Derry, S. J., Renkl, A., & Wortham, D. (2000). Learning from examples: instructional principles from the worked examples research. Review of Educational Research, 70, 181–214.

    Article  Google Scholar 

  • Azevedo, R., & Hadwin, A. F. (2005). Scaffolding self-regulated learning and metacognition—implications for the design of computer-based scaffolds. Instructional Science, 33(5), 367–379.

    Article  Google Scholar 

  • Blum, W. (2011). Can modelling be taught and learnt? Some answers from empirical research. In G. Kaiser, W. Blum, R. Borromeo Ferri, & G. Stillman (Eds.), Trends in the teaching and learning of mathematical modelling—Proceedings of ICTMA14 (pp. 15–30). New York: Springer.

  • Blum, W., Galbraith, P. L., Henn, H.-W., & Niss, M. (2007). Modelling and applications in mathematics education. The 14th ICMI study. New York: Springer.

  • Blum, W., & Leiss, D. (2006). “Filling up”—the problem of independence-preserving teacher interventions in lessons with demanding modelling tasks. In M. Bosch (Ed.), CERME-4—Proceedings of the 4th conference of the european society for research in mathematics education (pp. 1623–1633): Guixol.

  • Blum, W., & Leiss, D. (2007a). 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). Chichester: Horwood.

    Chapter  Google Scholar 

  • Blum, W., & Leiss, D. (2007b). Investigating quality mathematics teaching. The DISUM projekt. In C. Bergsten, & B. Grevholm (Eds.), Developing and researching quality in mathematics teaching and learning (pp. 3–16). Linköping: SMDF.

  • Cohen, J. (1960). A coefficient of agreement for nominal scales. Educational and Psychological Measurement.

  • Collins, A., Brown, J. S., & Newman, S. E. (1989). Cognitive apprenticeship: teaching the crafts of reading, writing, and mathematics. In L. B. Resnick (Ed.), Knowing, learning and instruction: essays in honor of Robert Glaser (pp. 453–492). Hillsdale: Erlbaum.

    Google Scholar 

  • De Corte, E. (2007). Learning from instruction: the case of mathematics. Learning Inquiry, 1, 19–30.

    Article  Google Scholar 

  • De Corte, E., Verschaffel, L., & O’pt Eynde, P. (2000). Self-regulation: a characteristic and a goal of mathematics education. In M. Boekaerts, P. R. Pintrich, & M. Zeidner (Eds.), Handbook of self-regulation (pp. 681–726). San Diego: Academic Press.

    Google Scholar 

  • Greer, B., & Verschaffel, L. (2007). Characterizing modeling competencies. In W. Blum, P. L. Galbraith, H.-W. Henn, & M. Niss (Eds.), Applications and modelling in mathematics education: the 14th ICMI Study (pp. 219–224). New York: Springer.

    Chapter  Google Scholar 

  • Hadwin, A. F., & Winne, P. (2001). CoNoteS2: A software tool for promoting selfregulation. Educational Research and Evaluation, 7(2/3), 313–334.

  • Hadwin, A. F., Wozney, L., & Pantin, O. (2005). Scaffolding the appropriation of selfregulatory activity; a socio-cultural analysis of changes in teacher-student discourse about a graduate research portfolio. Instructional Science, 33, 413–450.

    Article  Google Scholar 

  • Heinze, A., Star, J. R., & Verschaffel, L. (2009). Flexible and adaptive use of strategies and representations in mathematics education. ZDM The International Journal on Mathematics Education, 41(5), 535–540.

    Article  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.

    Article  Google Scholar 

  • Kirschner, P. A., Sweller, J., & Clark, R. E. (2006). Why minimal guidance during instruction does not work: an analysis of the failure of constructivist, discovery, problem-based, experiential, and inquiry-based teaching. Educational Psychologist, 41(2), 75–86.

    Article  Google Scholar 

  • Leiss, D., Schukajlow, S., Blum, W., Messner, R., & Pekrun, R. (2010). The role of the situation model in mathematical modelling—task analyses, student competencies, and teacher interventions. Journal für Mathematikdidaktik, 31(1), 119–141.

    Article  Google Scholar 

  • Leopold, C., & Leutner, D. (2012). Science text comprehension: drawing, main idea selection, and summarizing as learning strategies. Learning and Instruction, 22, 16–26.

    Article  Google Scholar 

  • Leutner, D., Leopold, C., & Elzen-Rump, Vd. (2007). Self-regulated learning with a text-highlighting strategy. Zeitschrift für Psychologie, 215(3), 174–182.

    Article  Google Scholar 

  • Mevarech, Z. R., & Kramarski, B. (1997). IMPROVE: a multidimensional method for teaching mathematics in heterogeneous classrooms. American Educational Research Journal, 34(2), 365–394.

    Article  Google Scholar 

  • Mevarech, Z. R., Terkieltaub, S., Vinberger, T., & Nevet, V. (2010). The effects of meta-cognitive instruction on third and sixth graders solving word problems. ZDM The International Journal on Mathematics Education, 42(2), 195–203.

    Article  Google Scholar 

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

    Google Scholar 

  • Pekrun, R., Jullien, S., Zirngibl, A., Hofe, R. v., & Blum, W. (2002). Skalenhandbuch PALMA 1. Messzeitpunkt (5. Klassenstufe). [Questionnaires]. München: Institut Pädagogische Psychologie.

  • Pólya, G. (1948). How to solve it: A new aspect of mathematical method. Princeton: Princeton University Press.

    Google Scholar 

  • 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.

    Article  Google Scholar 

  • Rakoczy, K., Buff, A., & Lipowsky, F. (2005). Befragungsinstrumente. [Questionnaires]. In E. Klieme, C. Pauli, & K. Reusser (Eds.), Dokumentation der Erhebungs- und Auswertungsinstrumente zur schweizerisch-deutschen Videostudie “Unterrichtsqualität, Lernverhalten und mathematisches Verständnis”. Frankfurt am Main: DIPF.

  • Salden, R. J., Aleven, V. A., Renkl, A., & Schwonke, R. (2009). Worked examples and tutored problem solving: redundant or synergistic forms of support? Topics in Cognitive Science, 1(1), 203–213.

    Article  Google Scholar 

  • Schneider, W., & Artelt, C. (2010). Metacognition and mathematics education. ZDM The International Journal on Mathematics Education, 42(2), 149–161.

    Article  Google Scholar 

  • Schoenfeld, A. H. (1992). Learning to think mathematically: problem solving, metacognition and sense making in mathematics. In D. A. Grouws (Ed.), Handbook of research on mathamatics teaching and learning (pp. 334–370). NY: Macmillan Publishing Company.

    Google Scholar 

  • Schukajlow, S., Blum, W., & Krämer, J. (2011). Förderung der Modellierungskompetenz durch selbständiges Arbeiten im Unterricht mit und ohne Lösungsplan. [Improving modelling competency in self-regulated learning environments with and without solution plan]. Praxis der Mathematik in der Schule, 53(38), 40–45.

  • Schukajlow, S., Blum, W., Messner, R., Pekrun, R., Leiss, D., & Müller, M. (2009). Unterrichtsformen, erlebte Selbständigkeit, Emotionen und Anstrengung als Prädiktoren von Schüler-Leistungen bei anspruchsvollen mathematischen Modellierungsaufgaben [Teaching methods, perceived self-regulation, emotions, and effort as predictors for students’ performance while solving mathematical modelling tasks]. Unterrichtswissenschaft, 37(2), 164–186.

    Google Scholar 

  • Schukajlow, S., & Krug, A. (2014). Do multiple solutions matter? Prompting multiple solutions, interest, competence, and autonomy. Journal for Research in Mathematics Education, 45(4), 497–533.

    Article  Google Scholar 

  • Schukajlow, S., Krug, A., & Rakoczy, K. (2015). Effects of prompting multiple solutions for modelling problems on students’ performance. Educational Studies in Mathematics. 89(3).

  • Schukajlow, S., & Leiss, D. (2011). Selbstberichtete Strategienutzung und mathematische Modellierungskompetenz. [Self-reported use of strategies and mathematical modelling]. Journal für Mathematikdidaktik, 32(1), 53–77.

  • Schukajlow, S., Leiss, D., Pekrun, R., Blum, W., Müller, M., & Messner, R. (2012). Teaching methods for modelling problems and students’ task-specific enjoyment, value, interest and self-efficacy expectations. Educational Studies in Mathematics, 79(2), 215–237.

    Article  Google Scholar 

  • Smit, J., van Eerde, H. A. A., & Bakker, A. (2013). A conceptualisation of whole-class scaffolding. British Educational Research Journal, 39(5), 817–834.

    Article  Google Scholar 

  • Staub, F. C., & Reusser, K. (1995). The role of presentational structures in understanding and solving mathematical word problems. In C. A. Weaver III, S. Mannes, & C. R. Fletcher (Eds.), Discourse comprehension: Essays in honor of Walter Kintsch (pp. 285–305). Hillsdale: Lawrence Erlbaum.

    Google Scholar 

  • Stillman, G. A., & Galbraith, P. L. (1998). Applying mathematics with real world connections: metacognitive characteristics of secondary students. Educational Studies in Mathematics, 36(2), 157–195.

    Article  Google Scholar 

  • Sweller, J., van Merrienboer, J. J. G., & Paas, F. (1998). Cognitive architecture and instructional design. Educational Psychology Review, 10(3), 251–296.

    Article  Google Scholar 

  • Teong, S. K. (2003). The effect of metacognitive training on the mathematical word problem solving of low achievers. Journal of Computer Assisted Learning, 19(1), 46–55.

    Article  Google Scholar 

  • Tharp, R., & Gallimore, R. (1988). Rousing minds to life: teaching, learning, and schooling in social context. Cambridge: Cambridge University Press.

    Google Scholar 

  • Van Merriënboer, J. J. (1997). Training complex cognitive skills: a four-component instructional design model for technical training. Educational Technology.

  • Verschaffel, L., Greer, B., & De Corte, E. (2000). Making sense of word problems. Lisse: Swets and Zeitlinger.

    Google Scholar 

  • Warm, T. A. (1989). Weighted likelihood estimation of ability in item response theory. Psychometrika, 54, 427–450.

    Article  Google Scholar 

  • Weinstein, C. E., Husman, J., & Dierking, D. R. (2000). Self-regulation interventions with a focus on learning strategies. In M. Boekaerts, P. R. Pintrich, & M. Zeidner (Eds.), Handbook self-regulation (pp. 728–747). San Diego: Academic press.

    Google Scholar 

  • Weinstein, C. E., & Mayer, R. E. (1986). The teaching of learning strategies. In M. C. Wittrock (Ed.), Handbook of research on teaching (3rd ed., pp. 315–327). New York/London: Collier-Macmillan.

    Google Scholar 

  • Wood, D., Bruner, J. S., & Ross, G. (1976). The role of tutoring in problem solving. Journal of Child Psychology and Psychiatry, 17(2), 89–100.

    Article  Google Scholar 

  • Wu, M. L., Adams, R. J., & Wilson, M. R. (1998). ACER conquest. Melbourne: The Australian Council for Educational Research.

    Google Scholar 

  • Zöttl, L., Ufer, S., & Reiss, K. (2010). Modelling with heuristic worked examples in the KOMMA learning environment. Journal für Mathematikdidaktik, 31(1), 143–165.

    Article  Google Scholar 

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  1. University of Münster, Münster, Germany

    Stanislaw Schukajlow

  2. University of Kassel, Kassel, Germany

    Jana Kolter & Werner Blum

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Schukajlow, S., Kolter, J. & Blum, W. Scaffolding mathematical modelling with a solution plan. ZDM Mathematics Education 47, 1241–1254 (2015). https://doi.org/10.1007/s11858-015-0707-2

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