Numerical competence is one of the many aspects of animal cognition that have enjoyed a resurgence of interest during the past decade. Evidence for numerical abilities in animals has followed a tortuous path to respectability, however, from Clever Hans, the counting horse, to modern experimental studies. Recent surveys of the literaturereveal theoretical as well as definitional confusion arising from inconsistent terminology for numerical processes and procedures. The term “counting” has been applied to situations having little to do with its meaning in the human literature. We propose a consistent vocabulary and theoretical framework for evaluating numerical competence. Relative numerousness judgments, subitizing, counting, and estimation may be the essential processes by which animals perform numerical discriminations. Ordinality, cardinality, and transitivity also play an important role in these processes. Our schema is applied to a variety of recent experimental situations. Some evidence of transfer is essential in demonstrating higher-order ability such as counting or “sense of number.” Those instances of numerical competence in which all viable alternatives to counting (e.g., subitizing) have been precluded, but no evidence of transfer has been demonstrated might be described as “protocounting.” To show that animals are capable of “true” counting future research will have to demonstrate generality across situations.

P3 is the most prominent of the electrical potentials of the human electroencephalogram that are sensitive to psychological variables. According to the most influential current hypothesis about its psychological significance [E. Donchin's], the “context updating” hypothesis, P3 reflects the updating of working memory. This hypothesis cannot account for relevant portions of the available evidence and it entails some basic contradictions. A more general formulation of this hypothesis is that P3 reflects the updating of expectancies. This version implies that P3-evoking stimuli are initially unexpected but later become expected. This contradiction cannot be resolved within this formulation. The alternative “context closure” hypothesis retains the concept of “strategic information processing” emphasized by the context updating hypothesis. P3s are evoked by events that are awaited when subjects deal with repetitive, highly structured tasks; P3s arise from subjects' combining successive stimuli into larger units. The tasks in which P3s are elicited can accordingly be classified in terms of their respective formal sequences of stimuli. P3 may be a physiological indicator of excess activation being released from perceptual control areas.

A set of hypotheses is formulated for a connectionist approach to cognitive modeling. These hypotheses are shown to be incompatible with the hypotheses underlying traditional cognitive models. The connectionist models considered are massively parallel numerical computational systems that are a kind of continuous dynamical system. The numerical variables in the system correspond semantically to fine-grained features below the level of the concepts consciously used to describe the task domain. The level of analysis is intermediate between those of symbolic cognitive models and neural models. The explanations of behavior provided are like those traditional in the physical sciences, unlike the explanations provided by symbolic models.

Higher-level analyses of these connectionist models reveal subtle relations to symbolic models. Parallel connectionist memory and linguistic processes are hypothesized to give rise to processes that are describable at a higher level as sequential rule application. At the lower level, computation has the character of massively parallel satisfaction of soft numerical constraints; at the higher level, this can lead to competence characterizable by hard rules. Performance will typically deviate from this competence since behavior is achieved not by interpreting hard rules but by satisfying soft constraints. The result is a picture in which traditional and connectionist theoretical constructs collaborate intimately to provide an understanding of cognition.

Several hundred thousand intellectually talented 12-to 13-year-olds have been tested nationwide over the past 16 years with the mathematics and verbal sections of the Scholastic Aptitude Test (SAT). Although no sex differences in verbal ability have been found, there have been consistent sex differences favoring males in mathematical reasoning ability, as measured by the mathematics section of the SAT (SAT-M). These differences are most pronounced at the highest levels of mathematical reasoning, they are stable over time, and they are observed in other countries as well. The sex difference in mathematical reasoning ability can predict subsequent sex differences in achievement in mathematics and science and is therefore of practical importance. To date a primarily environmental explanation for the difference in ability has not received support from the numerous studies conducted over many years by the staff of Study of Mathematically Precocious Youth (SMPY) and others. We have studied some of the classical environmental hypotheses: attitudes toward mathematics, perceived usefulness of mathematics, confidence, expectations/ encouragement from parents and others, sex-typing, and differential course-taking. In addition, several physiological correlates of extremely high mathematical reasoning ability have been identified (left-handedness, allergies, myopia, and perhaps bilateral representation of cognitive functions and prenatal hormonal exposure). It is therefore proposed that the sex difference in SAT-M scores among intellectually talented students, which may be related to greater male variability, results from both environmental and biological factors.

To understand the endogenous components of the event-related brain potential (ERP), we must use data about the components' antecedent conditions to form hypotheses about the information-processing function of the underlying brain activity. These hypotheses, in turn, generate testable predictions about the consequences of the component. We review the application of this approach to the analysis of the P300 component. The amplitude of the P300 is controlled multiplicatively by the subjective probability and the task relevance of the eliciting events, whereas its latency depends on the duration of stimulus evaluation. These and other factors suggest that the P300 is a manifestation of activity occurring whenever one's model of the environment must be revised. Tests of three predictions based on this “context updating” model are reviewed. Verleger's critique is based on a misconstrual of the model as well as a partial and misleading reading of the relevant literature.