Abstract
Mathematical argumentation skills (MAS) are considered an important outcome of mathematics learning, particularly in secondary and tertiary education. As MAS are complex, an effective way of supporting their acquisition may require combining different scaffolds. However, how to combine different scaffolds is a delicate issue, as providing learners with more than one scaffold may be overwhelming, especially when these scaffolds are presented at the same time in the learning process and when learners’ individual learning prerequisites are suboptimal. The present study therefore investigated the effects of the presentation sequence of introducing two scaffolds (collaboration script first vs. heuristic worked examples first) and the fading of the primarily presented scaffold (fading vs. no fading) on the acquisition of dialogic and dialectic MAS of participants of a preparatory mathematics course at university. In addition, we explored how prior knowledge and working memory capacity moderated the effects. Overall, 108 university freshmen worked in dyads on mathematical proof tasks in four treatment sessions. Results showed no effects of the presentation sequence of the collaboration script and heuristic worked examples on dialogic and dialectic MAS. Yet, fading of the initially introduced scaffold had a positive main effect on dialogic MAS. Concerning dialectic MAS, fading the collaboration script when it was presented first was most effective for learners with low working memory capacity. The collaboration script might be appropriate to initially support dialectic MAS, but might be overwhelming for learners with lower working memory capacity when combined with heuristic worked examples later on.
Similar content being viewed by others
References
Alexander, P. A., & Jetton, T. L. (2003). Learning from traditional and alternative texts: New conceptualization for an information age. In A. Graesser, M. Gernsbacher, & S. Goldman (Eds.), Handbook of discourse processes (pp. 199–241). Mahwah: Erlbaum.
Andriessen, J., Baker, M., & Suthers, D. D. (2003). Argumentation, computer support, and the educational context of confronting cognitions. In J. Andriessen, M. Baker & D. D. Suthers (Eds.), Arguing to learn: Confronting cognitions in computer-supported collaborative learning environments (pp. 1–25). New York: Springer.
Asterhan, C. S. C., & Schwarz, B. B. (2007). The effects of monological and dialogical argumentation on concept learning in evolutionary theory. Journal of Educational Psychology, 99, 626–639. https://doi.org/10.1037/0022-0663.99.3.626.
Asterhan, C. S. C., & Schwarz, B. B. (2009). Argumentation and explanation in conceptual change: Indications from protocol analyses of peer-to-peer dialog. Cognitive Science, 33(3), 374–400. https://doi.org/10.1111/j.1551-6709.2009.01017.x.
Baddeley, A. D., Allen, R. J., & Hitch, G. J. (2011). Binding in visual working memory: The role of the episodic buffer. Neuropsychologia, 49, 1393–1400. https://doi.org/10.1016/j.neuropsychologia.2010.12.042.
Bausch, I., Biehler, R., Bruder, R., Fischer, P. R., Hochmuth, R., Koepf, W., et al. (Eds.). (2014). Mathematische Vor- und Brückenkurse [Mathematical preparatory and bridging courses]. Wiesbaden: Springer Fachmedien Wiesbaden.
Boero, P. (1999). Argumentation and mathematical proof: A complex, productive, unavoidable relationship in mathematics and mathematics education. International Newsletter on the Teaching and Learning of Mathematical Proof, 7(8). Retrieved on http://www.lettredelapreuve.org/OldPreuve/Newsletter/990708Theme/990708ThemeUK.html [Sept. 14, 2017].
Bühner, M., Kröner, S., & Ziegler, M. (2008). Working memory, visual–spatial-intelligence and their relationship to problem-solving. Intelligence, 36(6), 672–680. https://doi.org/10.1016/j.intell.2008.03.008.
CCSSI. (2017). Common core state standards for mathematics. Retrieved from http://www.corestandards.org/Math/Practice/#CCSS.Math.Practice.MP1.
Clark, D. B., D’Angelo, C. M., & Menekse, M. (2009). Initial structuring of online discussions to improve learning and argumentation: Incorporating students’ own explanations as seed comments versus an augmented-preset approach to seeding discussions. Journal of Educational Science and Technology, 18, 321–333. https://doi.org/10.1007/s10956-009-9159-1.
Clarke, T., Ayres, P., & Sweller, J. (2005). The impact of sequencing and prior knowledge on learning mathematics through spreadsheet applications. Educational Technology Research and Development, 53(3), 15–24. https://doi.org/10.1007/BF02504794.
Daneman, M., & Merikle, P. M. (1996). Working memory and language comprehension: A meta-analysis. Psychonomic Bulletin & Review, 3(4), 422–433. https://doi.org/10.3758/BF03214546.
Dawkins, P. C., & Weber, K. (2016). Values and norms of proof for mathematicians and students. Educational Studies in Mathematics. https://doi.org/10.1007/s10649-016-9740-5.
de Jong, T. (2010). Cognitive load theory, educational research, and instructional design: Some food for thought. Instructional Science, 38(2), 105–134 https://doi.org/10.1007/s11251-009-9110-0.
Deiglmayr, A., & Spada, H. (2010). Developing adaptive collaboration support: The example of an effective training for collaborative inferences. Educational Psychology Review, 22(1), 103–113. https://doi.org/10.1007/s10648-010-9119-6.
Fischer, F., Kollar, I., Stegmann, K., & Wecker, C. (2013). Toward a script theory of guidance in computer-supported collaborative learning. Educational Psychologist, 48(1), 56–66. https://doi.org/10.1080/00461520.2012.748005.
Hailikari, T., Nevgi, A., & Komulainen, E. (2008). Academic self-beliefs and prior knowledge as predictors of student achievement in mathematics: A structural model. Educational Psychology, 28(1), 59–71 https://doi.org/10.1080/01443410701413753.
Hanna, G. (2000). Proof, explanation and exploration: An overview. Educational Studies in Mathematics, 44, 5–23. https://doi.org/10.1023/A:1012737223465.
Hayes, A. F. (2012). Process: A versatile computational tool for observed variable mediation, moderation, and conditional process modeling (white paper). Retrieved from http://www.afhayes.com/public/ process2012.pdf.
Hayes, A. F. (2013). Introduction to mediation, moderation, and conditional process analysis: A regression-based approach. New York: Guilford Press.
Heinze, A., Reiss, K., & Rudolph, F. (2005). Mathematics achievement and interest from a differential perspective. Zentralblatt für Didaktik der Mathematik, 37(3), 212–220 https://doi.org/10.1007/s11858-005-0011-7.
Hodds, M., Alcock, L., & Inglis, M. (2014). Self-explanation training improves proof comprehension. Journal for Research in Mathematics Education, 45(1), 62–101. https://doi.org/10.5951/jresematheduc.45.1.0062.
Hudson, H. T., & Rottmann, R. M. (1981). Correlation between performance in physics and prior mathematics knowledge. Journal of Research in Science Teaching, 18(4), 291–294. https://doi.org/10.1002/tea.3660180403.
Jiménez-Aleixandre, M. P., Rodríguez, A. B., & Duschl, R. A. (2000). “Doing the lesson” or “doing science”: Argument in high school genetics. Science Education, 84, 757–792. https://doi.org/10.1002/1098-237X(200011)84:6<757::AID-SCE5>3.0.CO;2-F.
Kalyuga, S. (2007). Expertise reversal effect and its implications for learner-tailored instruction. Educational Psychology Review, 19(4), 509–539. https://doi.org/10.1007/s10648-007-9054-3.
Kalyuga, S. (2013). Effects of learner prior knowledge and working memory limitations on multimedia learning. Procedia - Social and Behavioral Sciences, 83, 25–29. https://doi.org/10.1016/j.sbspro.2013.06.005.
Kalyuga, S., Rikers, R., & Paas, F. (2012). Educational implications of expertise reversal effects in learning and performance of complex cognitive and sensorimotor skills. Educational Psychology Review, 24(2), 313–337. https://doi.org/10.1007/s10648-012-9195-x.
Kane, M. J., Hambrick, D. Z., Tuholski, S. W., Wilhelm, O., Payne, T. W., & Engle, R. W. (2004). The generality of working memory capacity: A latent-variable approach to verbal and visuospatial memory span and reasoning. Journal of Experimental Psychology: General, 133(2), 189–217. https://doi.org/10.1037/0096-3445.133.2.189.
King, A. (2007). Scripting collaborative learning processes: A cognitive perspective. In F. Fischer, I. Kollar, H. Mandl, & J. M. Haake (Eds.), Scripting computer-supported collaborative learning: Cognitive, computational, and educational perspectives (pp. 13–37). New York: Springer.
Kollar, I., Ufer, S., Reichersdorfer, E., Vogel, F., Fischer, F., & Reiss, K. (2014). Effects of collaboration scripts and heuristic worked examples on the acquisition of mathematical argumentation skills of teacher students with different levels of prior achievement. Learning and Instruction, 32(1), 22–36. https://doi.org/10.1016/j.learninstruc.2014.01.003.
Leitão, S. (2000). The potential of argument in knowledge building. Human Development, 43, 332e360. https://doi.org/10.1159/000022695.
Leppink, J., Broers, N. J., Imbos, T., van der Vleuten, C. P. M., & Berger, M. P. F. (2012). Prior knowledge moderates instructional effects on conceptual understanding of statistics. Educational Research and Evaluation, 18(1), 37–51. https://doi.org/10.1080/13803611.2011.640873.
Miyake, A., & Friedman, N. P. (2012) The Nature and Organization of Individual Differences in Executive Functions. Current Directions in Psychological Science, 21(1), 8–14. https//doi.org/10.1177/0963721411429458.
Pea, R. D. (2004) The Social and Technological Dimensions of Scaffolding and Related Theoretical Concepts for Learning, Education, and Human Activity. Journal of the Learning Sciences, 13(3), 423–451. http://dx.doi.org/10.1207/s15327809jls1303_6.
Peng, P., Namkung, J., Barnes, M., & Sun, C. (2016). A meta-analysis of mathematics and working memory: Moderating effects of working memory domain, type of mathematics skill, and sample characteristics. Journal of Educational Psychology, 108(4), 455–473. https://doi.org/10.1037/edu0000079.
Rach, S., & Heinze, A. (2016). The transition from school to university in mathematics: Which influence do school-related variables have? International Journal of Science and Mathematics Education. https://doi.org/10.1007/s10763-016-9744-8.
Redick, T. S., Unsworth, N., Kelly, A. J., & Engle, R. W. (2012b). Faster, smarter? Working memory capacity and perceptual speed in relation to fluid intelligence. Journal of Cognitive Psychology, 24, 844–854. https://doi.org/10.1080/20445911.2012.704359.
Reiss, K., & Renkl, A. (2002). Learning to prove: The idea of heuristic examples. Zentralblatt für Didaktik der Mathematik, 34(1), 29–23. https://doi.org/10.1007/BF02655690.
Reiss, K., Heinze, A., Renkl, A., & Große, 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(3), 455–467. https://doi.org/10.1007/s11858-008-0105-0.
Renkl, A. (2014). Toward an instructionally oriented theory of example-based learning. Cognitive Science, 38(1), 1–37. https://doi.org/10.1111/cogs.12086.
Renkl, A., & Atkinson, R. K. (2007). An example order for cognitive skill acquisition. In F. E. Ritter, J. Nerb, E. Lehtinen, & T. M. O’Shea (Eds.), In order to learn. How the sequence of topics influences learning (pp. 95–105). New York: Oxford University Press.
Renkl, A., Hilbert, T., & Schworm, S. (2009). Example-based learning in heuristic domains: A cognitive load theory account. Educational Psychology Review, 21, 67–78. https://doi.org/10.1007/s10648-008-9093-4.
Rummel, N., Mullins, D., & Spada, H. (2012). Scripted collaborative learning with the cognitive tutor algebra. International Journal of Computer-Supported Collaborative Learning, 7(2), 307–339. https://doi.org/10.1007/s11412-012-9146-z.
Sadler, T. D. (2004). Informal reasoning regarding socioscientific issues: A critical review of research. Journal of Research in Science Teaching, 41(5), 513–536. https://doi.org/10.1002/tea.20009.
Schellens, T., Van Keer, H., De Wever, B., & Valcke, M. (2007). Scripting by assigning roles: Does it improve knowledge construction in asynchronous discussion groups? International Journal of Computer-Supported Collaborative Learning, 2(2–3), 225–246. https://doi.org/10.1007/s11412-007-9016-2.
Schwaighofer, M., Fischer, F., & Bühner, M. (2015). Does working memory training transfer? A meta-analysis including training conditions as moderators. Educational Psychologist, 50(2), 138–166. https://doi.org/10.1080/00461520.2015.1036274.
Schwaighofer, M., Bühner, M., & Fischer, F. (2016). Executive functions as moderators of the worked example effect: When shifting is more important than working memory capacity. Journal of Educational Psychology, 108(7), 982–1000. https://doi.org/10.1037/edu0000115.
Schwaighofer, M., Bühner, M., & Fischer, F. (2017). Executive functions in the context of complex learning: Malleable moderators? Frontline Learning Research, 5(1), 58–75. https://doi.org/10.1037/edu0000115.
Schwarz, B. B. (2009). Argumentation and learning. In Muller-Mirza and A-N. Perret-Clermont (Eds.), Argumentation and education – theoretical foundations and practices (pp. 91–126). Berlin: Springer Verlag.
Schwarz, B. B., & Shahar, N. (2017). Combining the dialogic and the dialectic: Putting argumentation into practice for classroom talk. Learning, Culture and Social Interaction, 12, 113–132. https://doi.org/10.1016/j.lcsi.2016.12.003.
Selden, J., Benkhalti, A., & Selden, A. (2014). An analysis of transition-to-proof course students’ proof constructions with a view towards course redesign. Proceedings of the 17th Annual Conference on Research in Undergraduate Mathematics Education.
Shipstead, Z., Lindsey, D. R. B., Marshall, R. L., & Engle, R. W. (2014). The mechanisms of working memory capacity: Primary memory, secondary memory, and attention control. Journal of Memory and Language, 72, 116–141. https://doi.org/10.1016/j.jml.2014.01.004.
Spiro, R. J., Coulson, R. L., Feltovich, P. J., & Anderson, D. K. (1988). Cognitive flexibility theory: Advanced knowledge acquisition in illstructured domains. In Proceedings of the Tenth Annual Conference of the Cognitive Science Society (pp. 375–383). Hillsdale: Erlbaum.
Stanovich, K. E. (1986). Matthew Effects in Reading: Some Consequences of Individual Differences in the Acquisition of Literacy. Reading Research Quarterly 21(4), 360–407. https://doi.org/10.1598/RRQ.21.4.1.
Sweller, J. (2010). Element interactivity and intrinsic, extraneous, and germane cognitive load. Educational Psychology Review, 22(2), 123–138. https://doi.org/10.1007/s10648-010-9128-5.
Sweller, J. (2011). Cognitive load theory. In J. Mestre & B. Ross (Eds.), The psychology of learning and motivation: Cognition in education (Vol. 55, pp. 37–76). Oxford: Academic Press.
Tabak, I. (2004). Synergy: A complement to emerging patterns of distributed scaffolding. The Journal of the Learning Sciences, 13(3), 305e335. https://doi.org/10.1207/s15327809jls1303_3.
Tchounikine, P. (2016). Contribution to a theory of CSCL scripts: Taking into account the appropriation of scripts by learners. International Journal for Computer-Supported Collaborative Learning, 11(3), 349–369. https://doi.org/10.1007/s11412-016-9240-.
Teasley, S. D. (1997). Talking about reasoning: How important is the peer in peer collaborations? In C. O’Malley (Ed.), Disourse, tools, and reasoning: Situated cognition and technologically supported environments (pp. 361–384). Berlin: Springer.
Unsworth, N., Heitz, R. P., Schrock, J. C., & Engle, R. W. (2005). An automated version of the operation span task. Behavior Research Methods, 37(3), 498–505. https://doi.org/10.3758/BF03192720.
Vogel, F., Kollar, I., Ufer, S., Reichersdorfer, E., Reiss, K., & Fischer, F. (2016). Developing argumentation skills in mathematics through computer-supported collaborative learning: The role of transactivity. Instructional Science, 44(5), 477–500. https://doi.org/10.1007/s11251-016-9380-2.
Vollstedt, M., Heinze, A., Gojdka, K. & Rach, S. (2014). Framework for examining the transformation of mathematics and mathematics learning in the transition from school to university. In S. Rezat, M. Hattermann & A. Peter-Koop (Hrsg.), Transformation - a fundamental idea of mathematics education (p. 29–50). New York: Springer.
Webb, N. M., Nemer, K. M., & Zuniga, S. (2002). Short circuits or superconductors? Effects of group composition on high-achieving students’ science assessment performance. American Educational Research Journal, 39(4), 943–989. https://doi.org/10.3102/00028312039004943.
Wecker, C., & Fischer, F. (2011). From guided to self-regulated performance of domain-general skills: The role of peer monitoring during the fading of instructional scripts. Learning and Instruction, 21(6), 746–756. https://doi.org/10.1016/j.learninstruc.2011.05.001.
Wegerif, R. (2008). Dialogic or dialectic? The significance of ontological assumptions in research on educational dialogue. British Educational Research Journal, 34(3), 347–361. https://doi.org/10.1080/01411920701532228.
Weinberger, A., Stegmann, K., & Fischer, F. (2010). Learning to argue online: Scripted groups surpass individuals (unscripted groups do not). Computers in Human Behavior, 26(4), 506–515. https://doi.org/10.1016/j.chb.2009.08.007.
Weiß, R. H. (2006). Grundintelligenztest Skala 2 revision, CFT 20-R. Göttingen: Hogrefe.
Yackel, E., & Cobb, P. (1996). Sociomathematical norms, argumentation, and autonomy in mathematics. Journal for Research in Mathematics Education, 27(4), 458–477. https://doi.org/10.2307/749877.
Acknowledgements
This research was funded by the Deutsche Forschungsgemeinschaft (DFG) under grants FI 792/7-2, KO 3462/2-2, RE 1247/9-2, and UF 59/3-2.
Author information
Authors and Affiliations
Corresponding author
Additional information
The first author of this article, Dr. Matthias Schwaighofer, lost his life in a tragic accident before the publication process was finished. The co-authors hope that this article will inspire further research to continue and extend his important and innovative work.
Appendix
Appendix
Rights and permissions
About this article
Cite this article
Schwaighofer, M., Vogel, F., Kollar, I. et al. How to combine collaboration scripts and heuristic worked examples to foster mathematical argumentation – when working memory matters. Intern. J. Comput.-Support. Collab. Learn 12, 281–305 (2017). https://doi.org/10.1007/s11412-017-9260-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11412-017-9260-z