Worked Examples in Math Ed: Theoretical and Empirical Support

WrongAnswer-isoIn preparing our Small Business Innovation Research (SBIR) proposal on the topic of developmental math games aimed at community college students, we’ve partnered with, and sought feedback from, several experts in math education, notably Dr. Michelle Chamberlin of the University of Wyoming. Dr. Chamberlin volunteered to write a short briefing on the theoretical and empirical support for worked example problems (click to see our first post on the topic).

What follows is Dr. Chamberlin’s sketch on the background of worked example problems, which she has kindly agreed to let us publish here.

Worked Example Problems: Theoretical and Empirical Support

by Michelle Chamberlin, Ph.D., Associate Professor, Department of Mathematics, University of Wyoming

Worked examples have a strong theoretical foundation in the field of cognitive science and in particular Cognitive Load Theory.  According to Cognitive Load Theory, learning, as a complex cognitive task, requires processing simultaneously a high number of interacting elements (Paas & van Gog, 2006).  Yet, working memory is limited (Miller, 1956) and especially for novice learners is taxed by processing this potentially large number of interacting elements (Sweller, 1988).  Viewing worked examples, in contrast to fully solving the problems, reduces this cognitive load (Sweller & Cooper, 1985).  Thus, students are able to focus on making sense of the steps to solving the problem without being cognitively overwhelmed.  Furthermore, students in developmental math courses often have high levels of mathematical anxiety, which has been shown to even further compromise the functioning of working memory (Ashcraft & Krause, 2007).

This theoretical basis is supported by empirical work, which has found that worked examples are beneficial for learning both within a laboratory setting (e.g., Sweller & Cooper, 1985) and within real classrooms (e.g., Carroll, 1994; Ward & Sweller, 1990).  Furthermore, worked examples have been effectively used with students of varying ages, including children in upper elementary grades (e.g., Durkin & Rittle-Johnson, 2012), middle and high school students (e.g., Booth, Lange, Koedinger, & Newton, 2013; McLaren et al., 2012), and adults (e.g., Curry, 2004).  Worked examples appear beneficial for many reasons, which align with our intended short-term outcomes of addressing misconceptions and enhancing conceptual understanding, procedural fluency, strategic competence, and adaptive reasoning.

With regard to misconceptions, well-designed worked examples that include incorrect steps or processes can be especially useful for anticipating student misconceptions and having students address them (Booth, 2011).  In other words, such worked examples require students to directly confront their misconceptions.  This in turn may lead students’ attention to what makes such procedures inappropriate, potentially leading them to replace the faulty conception with correct conceptual knowledge (Durkin & Rittle-Johnson, 2012).

Worked examples have also been shown to enhance procedural fluency (e.g., Sweller & Cooper, 1985; Siegler, 2002) and/or conceptual understanding (e.g., Rittle-Johnson & Star, 2009; Schwonke et al., 2009).  For example, Zhu and Simon (1987) found that worked examples support the same amount of procedural learning in less time than engaging in other learning tasks such as solving practice problems.  Other studies have shown that worked examples may support conceptual learning at the same time as or without compromising procedural learning.  Booth, Lange, Koedinger, and Newton (2013) implemented worked examples in an Algebra I class with high school students.  They found that the combination of worked examples (including incorrect examples) and practice problems is more beneficial for student learning than practice alone and supported students’ conceptual understanding of the features in equations without compromising procedural knowledge.  Such experiences with worked experiences were especially helpful for minority students (Booth et al. submitted for publication as cited in Booth et al., 2013).  Like Booth et al. (2013), many researchers have employed worked examples with various mathematical topics, including algebra and equality (e.g., Sweller & Cooper, 1985; Booth et al., 2013) and decimal concepts (Durkin & Rittle-Johnson, 2012; McLaren et al., 2012).  Not only do worked examples exhibit promise for procedural fluency and conceptual understanding, they potentially support strategic competence and adaptive reasoning as well.  Worked examples provide material students need to understand and solve problems, but they also engage them in a different type of active learning (McLaren et al., 2012).  This active learning includes evaluating and justifying solution procedures, providing students with experiences in analyzing, critiquing, and articulating mathematical arguments.

Finally, we expect (and an overall intent of this project is to determine the feasibility and effectiveness of this) that the promises of worked examples will be attainable through a gaming environment.  Other educators have used worked examples in various technological formats, e.g., Schwonke et al. (2009) with a Cognitive Tutor and McLaren et al. (2012) with interactive exercises on the Internet.


References:

Ashcraft, M. H., & Krause, J. A. (2007). Working memory, math performance, and math anxiety. Psychonomic Bulletin & Review, 14(2), 243-248.

Booth, J. L. (2011). Why can’t students get the concept of math? Perspectives on Language & Literacy, 37(2), 31-35.

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.

Carroll, W. M. (1994).  Using worked examples as an instructional support in the algebra classroom.  Journal of Educational Psychology, 86, 360-367.

Curry, L. (2004). The effects of self-explanations of correct and incorrect solutions on algebra problem-solving performance. In K. Forbus, D. Gentner, & T. Regier (Eds.), Proceedings of the 26th annual conference of the cognitive science society (pp. 1548). Mahwah, NJ: Erlbaum.

Durkin, K., & Rittle-Johnson, B. (2012). The effectiveness of using incorrect examples to support learning about decimal magnitude. Learning & Instruction, 22, 206-214.

McLaren, B.M., Adams, D., Durkin, K., Goguadze, G. Mayer, R.E., Rittle-Johnson, B., Sosnovsky, S., Isotani, S., & Van Velsen, M. (2012). To err is human, to explain and correct is divine: A study of interactive erroneous examples with middle school math students. In A. Ravenscroft, S. Lindstaedt, C. Delgado Kloos, & D. Hernándex-Leo (Eds.), Proceedings of ECTEL 2012: Seventh European Conference on Technology Enhanced Learning, LNCS 7563 (pp. 222-235). Springer, Berlin.

Miller, G. A. (1956). The magical number seven, plus or minus two: Some limits on our capacity to process information. Psychological Review, 63(2), 81-97.

National Research Council. (2001). Adding it up: Helping children learn mathematics. J. Kilpatrick, J. Swafford, and B. Findell (Eds.), Mathematics Learning Study Committee, Center for Education, Division of Behavioral and Social Sciences and Education. Washington, DC: National Academy Press.

Paas, F., & van Gog, T. (2006). Optimising worked example instruction: Different ways to increase germane cognitive load. Learning and Instruction, 16(2), 87-91.

Rittle-Johnson, B., & Star, J. R. (2007). Does comparing solution methods facilitate conceptual and procedural knowledge? An experimental study on learning to solve equations. Journal of Educational Psychology, 99(3), 561-574.

Schwonke, R., Renkl, A., Krieg, C., 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.

Siegler, R. S. (2002). Microgenetic studies of self-explanations. In N. Granott, & J. Parziale (Eds.), Microdevelopment: Transition processes in development and learning (pp. 31-58). New York: Cambridge University.

Sweller, J. (1988). Cognitive load during problem-solving: Effects on learning. Cognitive Science, 12, 257-285.

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.

Ward, M., & Sweller, J. (1990). Structuring effective worked examples. Cognition & Instruction, 7(1), 1-39.

Zhu, X., & Simon, H. A. (1987). Learning mathematics from examples and by doing. Cognition and Instruction, 4, 137-166.

About Bret Norwood

Bret Norwood is a staff blogger for Learning Laboratory in addition to other roles, including UI design and content development for Study Putty, our free memorization tool for chemistry and many other course topics. He is also a published writer of literary fiction–see BretNorwood.com

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