Bridging Principles and Examples through Analogy and Explanation

Bridging Principles and Examples through Analogy and Explanation

Timothy J. Nokes and Kurt VanLehn

Abstract

How can we accelerate student learning and understanding of the conceptual relations between principles and examples? Previous research in cognitive science has shown that students typically have a difficult time acquiring deep conceptual understanding in domains like mathematics and physics and often rely on textbook examples to solve new problems (e.g., Ross, 1984). Although using prior examples facilitates student learning they often can only use that knowledge on very similar problems (Reeves & Weissberg, 1994).

One reason students may rely so heavily on using prior examples is that they lack a deep understanding for the relations between the principles and examples. That is, they do not understand how the principles are instantiated in the examples. 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.

Glossary

Analogy Self-explanation Strategies

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. One way to address this problem is to examine what knowledge components comprise ‘expert understanding’ and then design learning environments to help novices construct that knowledge (see Dufresne, Gerace, Hardiman, & Mestre, 1992 for a similar approach). Previous research on expertise has shown that when experts solve novel problems in domains such as chess and physics they can ‘perceive’ the deep structure or principles of the problem and then can use that knowledge to identify and execute a set of procedures appropriate for the task (Chase & Simon, 1973; Chi, Feltovich, & Glaser, 1981; Larkin, McDermott, Simon, & Simon, 1980). This work suggests that a key component of expert knowledge is one’s understanding of the relations between principles of a domain and the features of the problem solving task. If we can design learning activities to help students acquire these relations this should improve their conceptual understanding and future problem solving.

The purpose of the current proposal is to test the hypothesis that learning the relations between principles and examples is critical to deep understanding and robust transfer of new concepts. Specifically, I will test two learning paths hypothesized to facilitate the acquisition of these relations. The first is through self-explaining how worked examples relate to the principle. The second path involves learning a schema through analogical comparison and then relating that schema to the principle. These hypotheses will be tested in two in vivo studies in the Physics LearnLab. Before describing the experiments in detail I briefly describe prior work relevant to the investigation of these hypotheses.

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 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 robust 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 transfer. One such learning activity is learning from explanation. Self-explanation has been shown to facilitate both procedural and conceptual learning and transfer of that knowledge to new contexts. Of particular interest to the current project are some promising results from the Chi et al. (1989) study showing 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. In the next section I focus on a second learning path hypothesized to promote learning the relation between the principles and examples, namely the acquisition of 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.

Hypothesis

Learning the relations between principles and examples is critical to deep understanding and transfer. Self-explaining can serve as one mechanism to facilitate this understanding. Problem schemas of intermediate level abstraction may mediate the relation between the principle and the example. The acquisition of a schema through analogical comparison may help bridge students’ understanding of the relationship between principles and examples.