Learning from worked-examples in mathematics: students relate procedures to principles
Tóm tắt
This article discusses the relevance of the worked-example effect for mathematics education. This effect refers to the finding that, in initial cognitive skill acquisition, students profit more from studying worked examples as compared to solving problems. One reason for the effectiveness of worked examples is that the students get the opportunity to interrelate principles (e.g., mathematical theorems) with multiple problem cases. A knowledge structure in which principles and multiple problem cases are represented in an interconnected way is a very good pre-condition for transfer. Several potential objections against worked examples from a mathematics education perspective are discussed (e.g., worked examples do not address typical student errors) by referring primarily to our own studies. We also propose further research exploring whether the powerful method of learning from worked-examples can be adapted to and used in other, presently intensively discussed areas of mathematics education (e.g., fostering skills of mathematical modelling). Finally, recommendations for mathematics education are outlined.
Tài liệu tham khảo
Aleven, V., McLaughlin, Glenn, R. A., & Koedinger, K. R. (2017). Instruction based on adaptive learning technologies. In R. E. Mayer & P. A. Alexander (Eds.), Handbook of research on learning and instruction (2nd ed, pp. 460–482.). New York, NY: Routledge.
Atkinson, R. K., Renkl, A., & Merrill, M. M. (2003). Transitioning from studying examples to solving problems: Combining fading with prompting fosters learning. Journal of Educational Psychology, 95, 774–783.
Berthold, K., Eysink, T. H., & Renkl, A. (2009). Assisting self-explanation prompts are more effective than open prompts when learning with multiple representations. Instructional Science, 37, 345–363.
Berthold, K., & Renkl, A. (2009). Instructional aids to support a conceptual understanding of multiple representations. Journal of Educational Psychology, 101, 70–87.
Berthold, K., Röder, H., Knörzer, D., Kessler, W., & Renkl, A. (2011). The double-edged effects of explanation prompts. Computers in Human Behavior, 27, 69–75.
Bokosmaty, S., Sweller, J., & Kalyuga, S. (2015). Learning geometry problem solving by studying worked examples: Effects of learner guidance and expertise. American Educational Research Journal, 52, 307–333.
Booth, J. L., Lange, K. E., Koedinger, K. R., & Newton, K. J. (2013). Using example problems to improve student learning in algebra: Differentiating between correct and incorrect examples. Learning & Instruction, 25, 24–34.
Booth, J. L., Oyer, M. H., Paré-Blagoev, E. J., Elliot, A. J., Barbieri, C., Augustine, A., & Koedinger, K. R. (2015). Learning algebra by example in real-world classrooms. Journal of Research on Educational Effectiveness, 8, 530–551.
Carroll, W. M. (1994). Using worked examples as an instructional support in the algebra classroom. Journal of Educational Psychology, 86, 360–367.
Chi, M. T., Bassok, H., Lewis, M., Reimann, M. W., P., &, Glaser, R (1989). Self-explanations: How students study and use examples in learning to solve problems. Cognitive Science, 13, 145–182.
Chi, M. T., & VanLehn, K. A. (2012). Seeing deep structure from the interactions of surface features. Educational Psychologist, 47, 177–188.
Cooper, G., & Sweller, J. (1987). Effects of schema acquisition and rule automation on mathematical problem-solving transfer. Journal of Educational Psychology, 79, 347–362.
Day, S. B., & Goldstone, R. L. (2012). The import of knowledge export: Connecting findings and theories of transfer of learning. Educational Psychologist, 47, 153–176.
Derry, S. J., Sherin, M. G., & Sherin, B. L. (2014). Multimedia learning with video. In R. E. Mayer (Ed.), The Cambridge handbook of multimedia learning (2nd edn., pp. 785–812). New York: Cambridge University Press.
Didierjean, A., & Cauzinille-Marmèche, E. (1998). Reasoning by analogy: Is it schema-mediated or case-based? European Journal of Psychology of Education, 13, 385–398.
Durkin, K., & Rittle-Johnson, B. (2012). The effectiveness of using incorrect examples to support learning about decimal magnitude. Learning & Instruction, 22, 206–214.
Gaudin, C., & Chaliès, S. (2015). Video viewing in teacher education and professional development: A literature review. Educational Research Review, 16, 41–67.
Große, C. S. (2015). Fostering modeling competencies: Benefits of worked examples, problems to be solved, and fading procedures. European Journal of Science and Mathematics Education, 3, 364–375.
Große, C. S., & Renkl, A. (2007). Finding and fixing errors in worked examples: Can this foster learning outcomes? Learning & Instruction, 17, 612–634.
Hefter, M. H., Berthold, K., Renkl, A., Riess, W., Schmid, S., & Fries, S. (2014). Effects of a training intervention to foster argumentation skills while processing conflicting scientific positions. Instructional Science, 42, 929–947.
Hefter, M. H., Renkl, A., Riess, W., Schmid, S., Fries, S., & Berthold, K. (2015). Effects of a training intervention to foster precursors of evaluativist epistemological understanding and intellectual values. Learning & Instruction, 39, 11–22.
Hilbert, T. S., & Renkl, A. (2009). Learning how to use a computer-based concept-mapping tool: Self-explaining examples helps. Computers in Human Behavior, 25, 267–274.
Hilbert, T. S., Renkl, A., Kessler, S., & Reiss, K. (2008). Learning to prove in geometry: Learning from heuristic examples and how it can be supported. Learning & Instruction, 18, 54–65.
Kalyuga, S., Chandler, P., Tuovinen, J., & Sweller, J. (2001). When problem solving is superior to studying worked examples. Journal of Educational Psychology, 93, 579–588.
Kersting, N. B., Givvin, K. B., Sotelo, F. L., & Stigler, J. W. (2010). Teachers’ analyses of classroom video predict student learning of mathematics: Further explorations of a novel measure of teacher knowledge. Journal of Teacher Education, 61, 172–181.
Koedinger, K. R., & Corbett, A. T. (2006). Cognitive tutors: Technology bringing learning sciences to the classroom. In R. K. Sawyer (Ed.), The Cambridge handbook of the learning sciences (pp. 61–78). New York: Cambridge University Press.
Mayer, R. E. (2014a). Cognitive theory of multimedia learning. In R. E. Mayer (Ed.), The Cambridge handbook of multimedia learning (2nd edn., pp. 43–71). New York: Cambridge University Press.
Mayer, R. E. (Ed.). (2014b). The Cambridge handbook of multimedia learning (2nd edn.). New York: Cambridge University Press.
McLaren, B. M., van Gog, T., Ganoe, C., Karabinos, M., & Yaron, D. (2016). The efficiency of worked examples compared to erroneous examples, tutored problem solving, and problem solving in computer-based learning environments. Computers in Human Behavior, 55, 87–99.
Nievelstein, F., van Gog, T., van Dijck, G., & Boshuizen, H. P. (2013). The worked example and expertise reversal effect in less structured tasks: Learning to reason about legal cases. Contemporary Educational Psychology, 38, 118–125.
Nokes-Malach, T. J., VanLehn, K., Belenky, D., Lichtenstein, M., & Cox, G. (2013). Coordinating principles and examples through analogy and self-explanation. European Journal of Education of Psychology, 28, 1237–1263.
Paas, F. G. W. C., & van Merriënboer, J. J. (1994). Variability of worked examples and transfer of geometrical problem-solving skills: A cognitive-load approach. Journal of Educational Psychology, 86, 122–133.
Perez-Felkner, L., McDonald, S. K., Schneider, B., & Grogan, E. (2012). Female and male adolescents’ subjective orientations to mathematics and the influence of those orientations on postsecondary majors. Developmental Psychology, 48, 1658.
Reed, S. K., Corbett, A., Hoffman, B., Wagner, A., & MacLaren, B. (2013). Effect of worked examples and Cognitive Tutor training on constructing equations. Instructional Science, 41, 1–24.
Reiss, K., Heinze, A., Renkl, A., & Groß, C. (2008). Reasoning and proof in geometry: Effects of a learning environment based on heuristic worked-out examples. ZDM–The International Journal on Mathematics Education, 40, 455–467.
Reiss, K., & Renkl, A. (2002). Learning to prove: The idea of heuristic examples. ZDM–The International Journal on Mathematics Education, 34, 29–35.
Renkl, A. (1997). Learning from worked-out examples: A study on individual differences. Cognitive Science, 21, 1–29.
Renkl, A. (2012). How to avoid inert knowledge—or different roads lead to Rome: The case of principle-based transfer. Paper presented at the Annual Meeting of the American Educational Research Association, Vancouver. (April 2012).
Renkl, A. (2014a). Theoretische Konzepte und Prinzipien auf den Schulalltag beziehen: Ein wenig Theorie and darauf begründete Vorschläge für die Referendariatsausbildung [Relating theoretical concepts and principles to classroom practice: A bit of theory and delineated recommendations for teacher education]. Seminar, 2/2014, 9–16.
Renkl, A. (2014b). Towards an instructionally-oriented theory of example-based learning. Cognitive Science, 38, 1–37.
Renkl, A. (2015a). Different roads lead to Rome: The case of principle-based cognitive skills. Learning: Research & Practice, 1, 79–90.
Renkl, A. (2015b). Drei Dogmen guten Lehrens: Warum sie falsch sind [Three dogmas about learning and instruction: Why they are wrong]. Psychologische Rundschau, 66, 211–220.
Renkl, A. (2017). Instruction based on examples. In R. E. Mayer & P. A. Alexander (Eds.), Handbook of research on learning and instruction (2nd edn., pp. 325–348). New York, NY: Routledge.
Renkl, A., & Atkinson, R. K. (2003). Structuring the transition from example study to problem solving in cognitive skills acquisition: A cognitive load perspective. Educational Psychologist, 38, 15–22.
Renkl, A., Atkinson, R. K., Maier, U. H., & Staley, R. (2002). From example study to problem solving: Smooth transitions help learning. Journal of Experimental Education, 70, 293–315.
Renkl, A., & Scheiter, K. (2017). Studying visual displays: How to instructionally support learning. Educational Psychology Review. doi:10.1007/s10648-015-9340-4.
Renkl, A., Solymosi, J., Erdmann, M., & Aleven, V. (2013). Training principle-based self-explanations: Transfer to new learning contents. In M. Knauff, M. Pauen, N. Sebanz & I. Wachsmuth (Eds.), Proceedings of the 35th Annual Conference of the Cognitive Science Society (pp. 1205–1210). Austin, TX: Cognitive Science Society.
Renkl, A., Stark, R., Gruber, H., & Mandl, H. (1998). Learning from worked-out examples: The effects of example variability and elicited self-explanations. Contemporary Educational Psychology, 23, 90–108.
Rittle-Johnson, B. (2006). Promoting transfer: The effects of direct instruction and self-explanation. Child Development, 77, 1–15.
Rittle-Johnson, B., Loehr, A. M., & Durkin, K. (2017). Promoting self-explanation to improve mathematics learning: A meta-analysis and instructional design principles. ZDM Mathematics Education. doi:10.1007/s11858-017-0834-z.
Ross, B. H. (1989). Remindings in learning and instruction. In S. Vosniadou & A. Ortony (Eds.), Similarity and analogical reasoning (pp. 438–469). Cambridge: Cambridge University Press.
Rourke, A. J., & Sweller, J. (2009). The worked-example effect using ill-defined problems: Learning to recognise designers’ styles. Learning & Instruction, 19, 185–199.
Rummel, N., Spada, H., & Hauser, S. (2009). Learning to collaborate while being scripted or by observing a model. International Journal of Computer-Supported Collaborative Learning, 4, 69–92.
Salden, R., Aleven, V., Renkl, A., & Schwonke, R. (2009). Worked examples and tutored problem solving: Redundant or synergistic forms of support? Topics in Cognitive Science, 1, 203–213.
Salden, R., Koedinger, K. R., Renkl, A., Aleven, V., & McLaren, B. M. (2010). Accounting for beneficial effects of worked examples in tutored problem solving. Educational Psychology Review, 22, 379–392.
Salomon, G., & Perkins, D. N. (1989). Rocky roads to transfer: Rethinking mechanisms of a neglected phenomenon. Educational Psychologist, 24, 113–142.
Schoenfeld, A. H. (1985). Mathematical problem solving. San Diego, CA: Academic Press.
Schwonke, R., Renkl, A., Krieg, K., Wittwer, J., Aleven, V., & Salden, R. (2009). The worked-example effect: Not an artefact of lousy control conditions. Computers in Human Behavior, 25, 258–266.
Schworm, S., & Renkl, A. (2006). Computer-supported example-based learning: When instructional explanations reduce self-explanations. Computers & Education, 46, 426–445.
Seidel, T., Blomberg, G., & Renkl, A. (2013). Instructional strategies for using video in teacher education. Teaching & Teacher Education, 34, 56–65.
Siegler, R. S., & Chen, Z. (2008). Differentiation and integration: Guiding principles for analyzing cognitive change. Developmental Science, 11, 433–448.
Stark, R., Mandl, H., Gruber, H., & Renkl, A. (2002). Conditions and effects of example elaboration. Learning & Instruction, 12, 39–60.
Stillman, G. A., Kaiser, G., Blum, W., & Brown, J. P. (Eds.), (2013). Teaching mathematical modelling: Connecting to research and practice. New York: Springer Science & Business Media
Sweller, J., & Cooper, G. A. (1985). The use of worked examples as a substitute for problem solving in learning algebra. Cognition & Instruction, 2, 59–89.
Tarmizi, R. A., & Sweller, J. (1988). Guidance during mathematical problem solving. Journal of Educational Psychology, 80, 424–436.
Tropper, N., Leiss, D., & Hänze, M. (2015). Teachers’ temporary support and worked-out examples as elements of scaffolding in mathematical modeling. ZDM Mathematics Education, 47, 1225–1240.
Tuovinen, J., & Sweller, J. (1999). A comparison of cognitive load associated with discovery learning and worked examples. Journal of Educational Psychology, 91, 334–341.
Vamvakoussi, X., & Vosniadou, S. (2010). How many decimals are there between two fractions? Aspects of secondary school students’ understanding about rational numbers and their notation. Cognition & Instruction, 28, 181–209.
Van Gog, T., Paas, F., & van Merriënboer, J. J. (2008). Effects of studying sequences of process-oriented and product-oriented worked examples on troubleshooting transfer efficiency. Learning & Instruction, 18, 211–222.
Van Gog, T., & Rummel, N. (2010). Example-based learning: Integrating cognitive and social-cognitive research perspectives. Educational Psychology Review, 22, 155–174.
Van Loon-Hillen, N., van Gog, T., & Brand-Gruwel, S. (2010). Effects of worked examples in a primary school mathematics curriculum. Interactive Learning Environments, 18, 1–11.
Wittwer, J., & Renkl, A. (2010). How effective are instructional explanations in example-based learning? A meta-analytic review. Educational Psychology Review, 22, 393–409.
Zhu, X., & Simon, H. A. (1987). Learning mathematics from examples and by doing. Cognition & Instruction, 4, 137–166.
Zöttl, L., Ufer, S., & Reiss, K. (2010). Modelling with heuristic worked examples in the KOMMA learning environment. Journal für Mathematik-Didaktik, 31, 143–165.