Bridging Principles and Examples through Analogy and Explanation

Bridging Principles and Examples through Analogy and Explanation

Timothy J. Nokes and Kurt VanLehn

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 self-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 two in vivo experiments in the Physics LearnLab.

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 accelerate 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, 1990). 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 et al., 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 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 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 et al. (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 in the current work is to understand how students learn to coordinate the knowledge representations of principles and examples through explanation. The second learning path examined in the current work is learning a schema through analogical comparison. Prior work has shown that analogical comparison can facilitate schema abstraction and transfer of that knowledge to new problems (). 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.

Dependent Variables

Learning Measures (manipulation checks)

• Traditional instruction group: Performance on practice problems
• Explanation group: Content of explanations
• Analogy + Explanation Group: Comparison summaries and content of explanations

Test Measures

• Similarity judgment task
• Problem solving
  • with equations given (articulating the solution)
  • without (determine the correct principle, then solve)
• Problem posing

Performance on ANDES problems Class test performance

Independent Variables

Type of instruction

• Traditional problem solving
• Explanation
• Analogy+explanation

Hypothesis

• 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 of examples can serve as one mechanism to facilitate schema acquisition

Expected Findings

• If learning the relations is critical for deep understanding then groups prompted to explain relations should perform better than unprompted group.
• If schema acquisition helps bridge this understanding then the analogical+explanation group should perform best
• Variety of test tasks will help identify what knowledge components are learned:
  • Principle representation, access, use, and understanding

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 of the problem.

By comparing similarities and differences of worked examples students abstract beyond the surface content of the example which provides an