Bridging Principles and Examples through Analogy and Explanation

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Bridging Principles and Examples through Analogy and Explanation

Timothy J. Nokes and Kurt VanLehn

Summary Table

Study 1 (In Vivo)

PIs Timothy Nokes and Kurt VanLehn
Study Start Date October, 2007
Study End Date December, 2007
LearnLab Site United States Naval Academy
Number of Students 78
Total Participant Hours 312
Data Shop Expected Spring, 2008; Analysis on-going


Abstract

The purpose of the current work is to test the hypothesis that learning the relations between principles and examples is critical to deep understanding and transfer. It is proposed that there are at least two paths to acquiring these relations. The first path is through explaining how worked examples are related to the principles. The second path is learning a schema through analogical comparison of two examples and then relating that schema to the principle. These hypotheses are tested in both a in vivo experiment in the Physics LearnLab as well as laboratory studies.

Research Question

The central problem addressed in this work is how to facilitate students’ deep learning of new concepts. Of particular interest is to determine what learning paths lead to a deep understanding of new concepts that enables the reliable retrieval and use of those concepts to solve novel problems and accelerated future learning.

Background and Significance

Much research in cognitive science has shown that when students first learn a new domain such as statistics or physics they rely heavily on prior examples to solve new problems (Anderson, Greeno, Kline, & Neves, 1981; Ross, 1984; VanLehn, 1998). Furthermore, laboratory studies indicate that students prefer to use examples even when they have access to written instructions or principles (LeFerve & Dixon, 1986; Ross, 1987). For example, LeFerve and Dixon (1986) showed that when learning to solve induction problems, students preferred to use the solution procedure illustrated in the example over the one described in the written instructions. Although using examples enables novices to make progress when solving new problems they are often only able to apply such knowledge to near transfer problems with similar surface features (see Reeves & Weissberg, 1994 for a review). It is principally through extended practice in the domain that students begin to develop more ‘expert-like’ abilities such as being able to ‘perceive’ and use the deep structural features of the problem (Chi, Feltovich, & Glaser, 1981) or use a forwards-working problem solving strategy (Sweller, Mawer, & Ward, 1983).

One reason that students may rely so heavily on prior examples to solve new problems is that they lack a deep understanding for how the principles are instantiated in the examples. That is, they may lack the knowledge and skills required for relating the principle components to the problem features. Some prior research by Nisbett and colleagues (Fong, Krantz, & Nisbett, 1986; Fong & Nisbett, 1991) has shown that when students are given brief training on an abstract rule (the statistical principle for the Law of Large Numbers) with illustrating examples they perform better than students trained on the rule or examples alone. This result was shown in a domain where the students were hypothesized to have an intuitive understanding of the principle prior to training. One plausible interpretation of this result is that the students used their intuitive understanding of the principle to relate the abstract rule to the illustrating examples. This possibility is intriguing and suggests that a training procedure designed to facilitate understanding of the relations between principles and examples may result in deep learning.

The current research builds on this result by postulating that learning activities designed to focus students on learning the relations between examples and principles should improve their conceptual understanding and lead to robust learning. We examine two learning paths to acquiring these relations: self-explanation and analogical comparison. Self-explanation has been shown to facilitate both procedural and conceptual learning and transfer of that knowledge to new contexts. Prior work by Chi, Bassok, Lewis, Reimann, and Glaser (1989) showed that good learners were more likely than poor learners to generate inferences relating the worked examples to the principles and concepts of the problem. This result suggests that prompting students to self-explain the relations between principles and worked examples will further facilitate learning. Of central interest to the current work is to understand how students learn to coordinate the knowledge representations of principles and examples through explanation. The second path is learning a schema through analogical comparison. Prior work has shown that analogical comparison can facilitate schema abstraction and transfer to new problems (Gentner, Lowenstein, & Thompson, 2003; Kurtz, Miao, & Gentner, 2001). However, this work has not examined how learning from problem comparison impacts understanding of an abstract principle. The current work examines how analogical comparison may help bridge students’ learning of the relations between principles and examples.

Independent Variables

Type of instruction

  • Problem solving
    • Participants read through a principle description and two worked examples. After reading through the learning materials they solve practice problems.
  • Explanation
    • Participants read the principle. Next they read the first example problem and are instructed to explain how each solution step relates to the principle / concepts. After completing the first example they perform the same task for the second example.
  • Analogy+explanation
    • Participants first read the principle and then perform the analogical comparison task. They are given the two worked examples and are instructed to compare each part of the examples writing a summary of the similarities and differences between the two (e.g., goals, concepts, and solution procedures). Next, participants are asked to explain how each component of their written summary relates to the principle.

Dependent Variables

Learning Measures (manipulation check)

  • Control group: Performance on practice problems
  • Explanation group: Content of explanations
  • Analogy+explanation group: Comparison summaries and content of explanations

Test Measures

  • Normal post-test
    • Problem solving both with equations given (articulating the solution) and without (determine the correct principle, then solve)
  • Transfer
    • Judgment task
      • The similarity judgment task consists of a target word problem and three comparison problems (similar to those used by Dufresne, Gerace, Hardiamnn, & Mestre, 1992). The students’ goal in this task is to determine which of the three comparison problems can be solved most similarly to the target problem. The comparison problems will vary in their similarity to the target problem and will have similar surface features (e.g., inclined planes), deep features (e.g., Newton’s Second Law), both surface and deep features, or neither.
    • Problem posing
      • The problem posing task consists of a problem principle to be tested, set-up, and diagram (adapted from Mestre, 2002). The students’ goal is to generate a statement or question that correctly completes the problem and then explain how their problem tests the basic principle.
  • Performance on Andes problems
    • Learning curves
    • Solution times
    • Error rates

Hypotheses

  • Learning the relations between principles and examples is critical to deep understanding and transfer.
    • Generating explanations can serve as one mechanism to facilitate this learning.
    • Problem schemas may help bridge the student's understanding between principles and examples.
    • Analogical comparison can serve as one mechanism to facilitate schema acquisition.

Expected Findings

  • If learning the relations is critical for deep understanding and transfer then the groups prompted to explain relations should perform better on the test tasks than the unprompted group.
  • If schema acquisition helps bridge this understanding then the Analogy+explanation group should perform best.
  • Variety of test tasks will help identify what knowledge components are learned:
    • Judgment task: Analogy+explanation > Explanation > Control; more likely to choose problems that match on deep features than surface features.
    • Problem solving with equations: Analogy+explanation = Explanation = Control; accuracy
    • Problem solving without equations: Analogy+explanation > Explanation > Control; accuracy
    • Problem posing: Analogy+explanation > Explanation > Control; accuracy and justifications
  • Andes performance: Analogy+explanation > Explanation > Control; errors rates

Explanation

Prompting students to explain how each step of a worked example is related to the principles facilitates the generation of inferences connecting the physics principles and concepts to the procedures and equations in the problem. These inferences serve to highlight the importance of the concepts in problem solving and increase the likelihood of future activation when solving novel problems. Furthermore, they serve as the critical links integrating and coordinating the principle knowledge components with the problem features.

By comparing similarities and differences of worked examples students have an opportunity to identify the important features of the problems. After having identified the important features they can be related to the principle description through explanation.

Descendents

None

Annotated Bibliography

  • Anderson, J. R., Greeno, J. G., Kline, P. J., & Neves, D. M. (1981). Acquisition of problem-solving skill. In J. R. Anderson (Ed.), Cognitive skills and their acquisition (pp. 191-230). Hillsdale, NJ: Erlbaum.
  • Chi, M. T. H., Bassok, M., Lewis, M. W., Reimann, 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. H., De Leeuw, N., Chiu, M. H., & LaVancher, C. (1994). Eliciting self-explanations improves understanding. Cognitive Science, 18, 439-477.
  • Chi, M. T. H., Feltovich, P. J., & Glaser, R. (1981). Categorization and representation of physics problems by experts and novices. Cognitive Science, 5, 121-152.
  • Dufresne, R. J., Gerace, W. J., Hardiman, P. T., & Mestre, J. P. (1992). Constraining novices to perform expertlike analyses: effects on schema acquisition. Journal of the Learning Sciences, 2, 307-331.
  • Fong, G. T., & Nisbett, R. E. (1991). Immediate and delayed transfer of training effects in statistical reasoning. Journal of Experimental Psychology: General, 120, 34-45.
  • Fong, G. T., Krantz, D. H., & Nisbett, R. E. (1986). The effects of statistical training on thinking about everyday problems. Cognitive Psychology, 18, 253-292.
  • Gentner, D., Loewenstein, J., & Thompson, L. (2003). Learning and transfer: A general role for analogical encoding. Journal of Educational Psychology, 95, 393-408.
  • Kurtz, K. J., Miao, C. H., & Gentner, D. (2001). Learning by analogical bootstrapping. Journal of the Learning Sciences, 10, 417-446.
  • LeFerve, J., & Dixon, P. (1986). Do written instructions need examples? Cognition and Instruction, 3, 1-30.
  • Mestre, J. P. (2002). Probing adults’ conceptual understanding and transfer of learning via problem posing. Applied Developmental Psychology, 23, 9-50.
  • Reeves, L. M., & Weissberg, W. R. (1994). The role of content and abstract information in analogical transfer. Psychological Bulletin, 115, 381-400.
  • Ross, B. H. (1984). Remindings and their effects in learning a cognitive skill. Cognitive Psychology, 16, 371-416.
  • Sweller, Mawer, & Ward (1983). Development of expertise in mathematical problem solving. Journal of Experimental Psychological: General, 112, 639-661.
  • VanLehn, K. (1998). Analogy events: How examples are used during problem solving. Cognitive Science, 22, 347-388.

Further Information

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