Abstract
The purpose of the present research was to investigate how the effectiveness of learning-by-invention activities may be influenced by the composition of the small groups that engage in them in terms of the mathematical skills of their members. Undergraduates engaged in an “inventing standard deviation” activity. Groups that included both high- and low-skill members generated a broader range of solution attempts and more high-quality solution attempts during the activity. Both the range and quality of solution attempts that were generated related to better uptake of the standard deviation formula from a later lesson. These results suggest that the composition of the small groups that work together may have an impact on the effectiveness of learning-by-invention activities.
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In the literature, there are several similar approaches that explore the learning opportunities that can result from having students engage in problem solving before having received instruction. VanLehn referred to this as impasse-driven learning (1988). Schwartz and Martin (2004) have argued that an initial invention phase can provide “preparation for future learning.” Kapur frames generation and exploration (Kapur 2012), or elicitation phases (Kapur and Bielaczyc 2011), as ways to encourage productive failure. In the present paper, we adopt the terminology and instructional sequence of Schwartz and Martin (2004) where an invention phase is followed by instructional support in the form of a worked example.
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Acknowledgments
This research and collaboration was supported by a Studienstiftung des Deutschen Volkes [German National Academic Foundation] Fellowship to Michael Wiedmann and by a Humboldt Research Fellowship to Jennifer Wiley. The data reported here were collected as part of a thesis project submitted by Michael Wiedmann in partial fulfillment of the requirements for the Diploma in Psychology at the University of Freiburg. The authors thank Kelly Currier, Pat Cushen, Olga Goldenberg, Thomas Griffin, Robert Hickson, Allison Jaeger, and Andy Jarosz for their assistance with coding, data collection, and discussions on this project.
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Wiedmann, M., Leach, R.C., Rummel, N., Wiley, J. (2015). Mathematical Skills and Learning by Invention in Small Groups. In: Cho, Y., Caleon, I., Kapur, M. (eds) Authentic Problem Solving and Learning in the 21st Century. Education Innovation Series. Springer, Singapore. https://doi.org/10.1007/978-981-287-521-1_14
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