Graeme S. Halford

Children's Understanding : The Development of Mental Models

Contents: Preface. Part I: Introduction. The Nature of Understanding. Part II: General Processes. Mental Models and Representations. Capacity and Complexity. Basic Learning Processes in Cognitive Development. Analogies and Structure-Mapping Processes. Part III: Theory. How Understanding Develops: A Cognitive Developmental Theory. Part IV: Domain-Specific Processes and Concepts. Inferences and Hypothesis Testing. Classification, Quantification, and Conservation. Scientific Concepts. Part V: Conclusions. Concluding Comments.

How Many Variables Can Humans Process

The conceptual complexity of problems was manipulated to probe the limits of human information processing capacity. Participants were asked to interpret graphically displayed statistical interactions. In such problems, all independent variables need to be considered together, so that decomposition into smaller subtasks is constrained, and thus the order of the interaction directly determines conceptual complexity. As the order of the interaction increases, the number of variables increases. Results showed a significant decline in accuracy and speed of solution from three-way to four-way interactions. Furthermore, performance on a five-way interaction was at chance level. These findings suggest that a structure defined on four variables is at the limit of human processing capacity.

Separating Cognitive Capacity from Knowledge: A New Hypothesis

We propose that working memory and reasoning share related capacity limits. These limits are quantified in terms of the number of items that can be kept active in working memory, and the number of interrelationships between elements that can be kept active in reasoning. The latter defines the complexity of reasoning problems and the processing loads they impose. Principled procedures for measuring, controlling or limiting recoding and other strategies for reducing memory and processing loads have opened up new research opportunities, and yielded orderly quantification of capacity limits in both memory and reasoning. We argue that both types of limit might be based on the limited ability to form and preserve bindings between elements in memory.

A Cognitive Complexity Metric Applied to Cognitive Development.

Two experiments tested predictions from a theory in which processing load depends on relational complexity (RC), the number of variables related in a single decision. Tasks from six domains (transitivity, hierarchical classification, class inclusion, cardinality, relative-clause sentence comprehension, and hypothesis testing) were administered to children aged 3-8 years. Complexity analyses indicated that the domains entailed ternary relations (three variables). Simpler binary-relation (two variables) items were included for each domain. Thus RC was manipulated with other factors tightly controlled. Results indicated that (i) ternary-relation items were more difficult than comparable binary-relation items, (ii) the RC manipulation was sensitive to age-related changes, (iii) ternary relations were processed at a median age of 5 years, (iv) cross-task correlations were positive, with all tasks loading on a single factor (RC), (v) RC factor scores accounted for 80% (88%) of age-related variance in fluid intelligence (compositionality of sets), (vi) binary- and ternary-relation items formed separate complexity classes, and (vii) the RC approach to defining cognitive complexity is applicable to different content domains.

Analogical Reasoning and Conceptual Complexity in Cognitive Development

It is proposed that much human inference is basically analogical, and the implications of this theory for cognitive development are examined. Analogy entails mapping the problem representation (target

Relational knowledge: the foundation of higher cognition

Accumulating evidence on the nature, function and acquisition of relational knowledge indicates a crucial role of such knowledge in higher cognitive processes. In this review, we specify the essential properties of relational knowledge, together with the role it plays in reasoning, categorisation, planning, quantification and language. Furthermore, we discuss the processes involved in its acquisition and how these processes have been implemented in contemporary neural network models. We present evidence demonstrating that relational knowledge integrates heuristic and analytic cognition, is important for symbolic processes and the creation of novelty, activates specific regions of the prefrontal cortex, and is the most recently evolved and slowest-developing cognitive process. Arguably, relational knowledge represents the core of higher cognition.

A category theory approach to cognitive development

The category theory concept of a commutative diagram is used to construct a model of the way in which symbolic processes are applied to problem solving. The model provides for a relationship between symbolic processes and the problem which depends on structural isomorphism and consistency, but is independent of similarity between symbol elements and problem elements. It is then shown that several different levels of thought can be distinguished within the basic model. More information is needed to assign symbolic processes to a problem in a consistent way with higher-level thought processes than with lower-level processes. These information-processing requirements permit the approximate age of mastery of each level to be predicted, thereby offering an alternate theory of cognitive developmental stages. Two experiments designed to test the theory are reported.

Information-Processing Demands of Transitive Inference

The information-processing demands of transitive inference problems were investigated with a probe reaction-time (RT) secondary task. Two versions of a primary task were used: the standard three-term inference problem and a matched verification task that did not require premise integration. In the first two experiments the premise and target-matching components of the primary task were presented sequentially. For the transitive inference task, probe RT was especially slow when the probe occurred during the second premise phase, but no such effect was found with the matched verification task. This implies that premise integration imposed an increased load on processing resources. A third experiment showed that the processing demand associated with premise integration also occurred with simultaneous presentation. Other variations in problem form (e.g., premise markedness, negation, and pivot search) did not influence probe RT, although they are known to affect solution time. Thus, solution time and measures of processing load may be independent.

Theory of mind and relational complexity.

Cognitive complexity and control theory and relational complexity theory attribute developmental changes in theory of mind (TOM) to complexity. In 3 studies, 3-, 4-, and 5-year-olds performed TOM tasks (false belief, appearance-reality), less complex connections (Level 1 perspective-taking) tasks, and transformations tasks (understanding the effects of location changes and colored filters) with content similar to TOM. There were also predictor tasks at binary-relational and ternary-relational complexity levels, with different content. Consistent with complexity theories: (a) connections and transformations were easier and mastered earlier than TOM; (b) predictor tasks accounted for more than 80% of age-related variance in TOM; and (c) ternary-relational items accounted for TOM variance, before and after controlling for age and binary-relational items. Prediction did not require hierarchically structured predictor tasks.

Can young children integrate premises in transitivity and serial order tasks

An analysis is presented of the relational information needed for premise integration in transitivity and simple serial order tasks. The tasks are divided into those where an ordering decision can be made by considering a single binary relation, and those where two binary relations must be considered. Four experiments are reported, the principal purpose of which is to manipulate the number of relations which must be considered in making decisions about the order of a small set of elements. In every test it was found that preschool children succeeded if decisions could be made by considering one relation, but failed if two relations had to be considered. Children over 5 almost always succeeded in both cases. It is concluded that preschool children cannot integrate relational premise information, and therefore cannot understand transitivity or serial order. This would impose limitations on their understanding of quantification and a number of other concepts. It is also suggested that the amount of information required to make a single decision may be an important factor determining cognitive complexity generally.