Analogical comparison principle

Brief statement of principle

Analogical comparison can facilitate schema abstraction and transfer of that knowledge to new problem. By comparing the commonalities between two examples, students can focus on the causal structure and improve their learning about the concept.

Description of principle

A problem schema is a knowledge organization of the information associated with a particular problem category. Problem schemas typically include declarative knowledge of principles, concepts, and formulae, as well as the procedural knowledge for how to apply that knowledge to solve a problem. Schemas have been hypothesized as the underlying knowledge organization of expert knowledge (Chase & Simon, 1973; Chi et al., 1981; Larkin et al., 1980). One way in which schemas can be acquired is through analogical comparison (Gick & Holyoak, 1983). Analogical comparison operates through aligning and mapping two example problem representations to one another and then extracting their commonalities (Gentner, 1983; Gick & Holyoak, 1983; Hummel & Holyoak, 2003). This process discards the elements of the knowledge representation that do not overlap between two examples but preserves the common elements. The resulting knowledge organization typically consists of fewer superficial similarities (than the examples) but retains the deep causal structure of the problems.

Operational definition

Analogical comparison is defined as the process of extracting the commonalities between two or more example problems to help form a schema for a problem.

Examples

Students are given 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.

Experimental support

Empirical and correlational support

Research studies of mathematics classrooms show use of this principle correlates with cross-country standardized achievement results (Richland, Zur, Holyoak, 2007).

Laboratory experiment support

Analogical comparison has also been shown to improve learning even when both examples are not initially well understood (Kurtz, Miao, & Gentner, 2001; Gentner Lowenstein, & Thompson, 2003). By comparing the commonalities between two examples, students could focus on the causal structure and improve their learning about the concept. Kurtz et al. (2001) showed that students who were learning about the concept of heat transfer learned more when comparing examples than when studying each example separately. The process of analogical comparison has also been shown to aid transfer. For example, Ross (1987) found that giving learners analogical examples to illustrate a probability principle facilitated their later use of the probability formula to solve other problems.

In vivo experiment support

Nokes & VanLehn, (2007) found that when students learned to solve problems on rotational kinematics by either self-explaining worked examples or engage in analogical comparison of worked examples, they outperformed students who simply read the worked examples, on far transfer tests.

Theoretical rationale

Comparing and contrasting problems can facilitate analogical comparison. When students compare two or more similar problems, and extract a problem-solving schema from that information, they are engaging in a constructive sense making process, which they may not have the opportunity to do in simply reading the examples.

Conditions of application

Under what conditions is analogical comparison beneficial to students?

Caveats, limitations, open issues, or dissenting views

High degree of structural similarity required; Reminding of prior problems helpful

Generalizations (ascendants)

Example-rule coordination

References

• Richland, L.E., Zur, O., Holyoak, K.J. (2007). Cognitive Supports for Analogies in the Mathematics Classroom. Science, 316, pp.1128-1129.