Order effects in learning relational structures

Much of the knowledge people acquire is structured: number systems, taxonomies; chemical structures. Learning using the individual components that compose a structured theory may be difficult due to the memory load induced by remembering the entities and their relations. Though much research has demonstrated the effects of ordering on category learning, to our knowledge, none has been conducted on the learning of relational structures. In three experiments we explore the effects of different orderings in learning different relational structures, finding that ordering affects learning, only orderings that tend to eliminate simpler alternative structures are better, and that the complexity of learning appears to be driven by the number of relations, as opposed to the number of nodes. The effects of data ordering on incremental learning are well-documented. Ordering is of obvious importance in sequence learning in which a human or a machine must learn the sequence of actions that produce a desired effect (e.g., language, planning, skill acquisition, etc) (Clegg, DiGirolamo, & Keele, 1998; Sutton & Barto, 1998). Ordering is studied in instructional design in which students must learn multiple interdependent topics. These topics could be presented in a variety of orders that may lead to different learning outcomes in different contexts (Ritter, 2007, p.19-39). Category learning researches have attempted to formalize methods for presenting data in optimal sequences. Elio and Anderson (1984) found that to facilitate learning of novel categories, it is best to start with low-variance exemplars and gradually increase the variance, Medin and Bettger (1994) showed that it is best to show successive examples that maximize the similarity between exemplars, and Mathy and Feldman (2009) suggest that it is better still to present exemplars in rule-based order in which categories are further divided into subclasses and members of subclasses are shown in succession. Category learning has been a target for fine-tuned order analysis because it is a well-formalized problem that can be readily adapted for the lab setting. However, in both intuitive experience and educational endeavors, people learn about relations among concepts. For instance, people learn about the relations among categories of living things that compose a taxonomy, the relations among elements that compose the periodic table, and sequences of events that form a causal chain. In each of these cases, the information is not just relational, but can be characterized by an abstract pattern: trees, periods, or chains (Kemp & Tenenbaum, 2008). Recent work by Kemp, Goodman, and Tenenbaum (2008b) has formalized relational theories. Kemp (2008) investigated learning relational structures based on randomly sampled examples, finding that simpler structures were faster to learn. While there has been a considerable amount of research into learning of concepts and categories (see Murphy, 2004; Smith & Medin, 1981, for reviews), considerably less work has been done to understand how people learning more rich, relational structures (see Kemp, Goodman, & Tenenbaum, 2008a, 2008b) and as a result, ordering effects in theory learning are not well understood. Models of theory acquisition predict biases toward structures that are compactly-represented in predicate notation (see Kemp et al., 2008a, 2008b)—biased toward simpler structures—but do not explicitly explore the implications of different orderings. Consider, for example, the relational structure in Figure 1a. The overall structure is a line. Each node (with the exception of the two end nodes) has a single incoming and a single outgoing link. If one attempted to learn this structure, it would require tracking 11 different names, one for each node, and 10 relations among the nodes. If these were independent bits of information, remembering them might be quite difficult (Miller, 1956). Because the relations are structured, it may be possible to learn quite quickly. For example, one might organize the information based on the structure—learning the relations from left to right. In other structures it may not be so obvious which ordering is best. When teaching relations that form a tree (see Figure 1b) is it best to order examples from root to leaves or from level to level? A more thorough analysis of which order is best is needed for these non-obvious cases. Motivated by previous research formalizing relational theories, we investigate the hypothesis that better orderings are those that rule out other abstract forms; orders that demonstrate the underlying structure of the relations or do not suggest other structures. First, to demonstrate order effects in theory learning, in Experiment 1 we teach a linear structure with linearly-ordered and random examples. In Experiment 2, we begin to speak to which orderings facilitate learning. We teach a binary tree and compare learning outcomes under different orderings inspired by graph-traversal algorithms. Lastly, in Experiment 3 we teach a more complex structure based on electron orbitals. We contrast orders based on how electron orbitals are often presented in textbooks. Across experiments we find that order affects learning but more importantly that different, intuitively-sound orderings can have vastly different effects on learning and that some orders are