Cognitive processes that underlie mathematical precociousness in young children

The working memory (WM) processes that underlie young children's (ages 6-8 years) mathematical precociousness were examined. A battery of tests that assessed components of WM (phonological loop, visual-spatial sketch pad, and central executive), naming speed, random generation, and fluency was administered to mathematically precocious and average-achieving children. The results showed that (a) precocious children performed better on executive processing, inhibition, and naming speed tasks than did average-achieving children, although the two groups were statistically comparable on measures of the phonological loop and visual-spatial sketch pad, and (b) the executive component of WM predicted mathematical accuracy independent of chronological age, reading, inhibition, and naming speed. The results support the notion that the executive system is an important predictor of children's mathematical precociousness and that this system can operate independent of individual differences in the phonological loop, inhibition, and reading in predicting mathematical accuracy.     

Stability and change in children's division strategies

Age-related changes in children's performance on simple division problems (e.g., 6/2, 72/9) were investigated by asking children in Grades 4 through 7 to solve 32 simple division problems. Differences in performance were found across grade, with younger children performing more slowly and less accurately than older children. Problem size effects were also found in that children were faster and more accurate on small problems than on large problems. Two strategies changed across age, with children in Grade 4 relying heavily on the strategy of "addition" (adding the divisor until the dividend was reached) to solve the problems and children in Grades 5 through 7 relying primarily on the strategy of "multiplication" (recasting the division problem as a multiplication problem) to solve the problems. Surprisingly, the frequency of direct retrieval (retrieving the answer directly from memory) did not increase across grade and never became the dominant strategy of choice. Reasons for why retrieval use remains infrequent and age invariant are discussed. Overall, the results suggest that division is a unique operation and that the continued study of division may have implications for further understanding of how procedural and conceptual knowledge of arithmetic develops.     

Set-Theoretic Absoluteness and the Revision Theory of Truth

We describe the solution of the Limit Rule Problem of Revision Theory and discuss the philosophical consequences of the fact that the truth set of Revision Theory is a complete 1/2 set.     

A computational model of frontal lobe dysfunction: working memory and the Tower of Hanoi task

A symbolic computer model, employing the perceptual strategy, is presented for solving Tower of Hanoi problems. The model is calibrated—in terms of the number of problems solved, time taken, and number of moves made—to the performance of 20 normal subjects. It is then "lesioned" by increasing the decay rate of elements in working memory to model the performance of 20 patients with lesions to the prefrontal cortex. The model captures both the main effects of subject groups (patients and normal controls) performance, and the subject groups (patients and normal controls) by problem difficulty interactions. This leads us to support the working memory hypothesis of frontal lobe functions, but for a narrow range of problems.     

Skeptical theism and the problem of moral <Emphasis Type="Italic">aporia</Emphasis>

Skeptical theism seeks to defend theism against the problem of evil by invoking putatively reasonable skepticism concerning human epistemic limitations in order to establish that we have no epistemological basis from which to judge that apparently gratuitous evils are not in fact justified by morally sufficient reasons beyond our ken. This paper contributes to the set of distinctively practical criticisms of skeptical theism by arguing that religious believers who accept skeptical theism and take its practical implications seriously will be forced into a position of paralysis or aporia when faced with a wide set of morally significant situations. It is argued that this consequence speaks strongly against the acceptance of skeptical theism insofar as such moral aporia is inconsistent with religious moral teaching and practice. In addition, a variety of arguments designed to show that accepting skeptical theism does not lead to this consequence are considered, and shown to be deficient.     

Spatial visualization in physics problem solving

Three studies were conducted to examine the relation of spatial visualization to solving kinematics problems that involved either predicting the two-dimensional motion of an object, translating from one frame of reference to another, or interpreting kinematics graphs. In Study 1, 60 physics-naíve students were administered kinematics problems and spatial visualization ability tests. In Study 2, 17 (8 high- and 9 low-spatial ability) additional students completed think-aloud protocols while they solved the kinematics problems. In Study 3, the eye movements of fifteen (9 high- and 6 low-spatial ability) students were recorded while the students solved kinematics problems. In contrast to high-spatial students, most low-spatial students did not combine two motion vectors, were unable to switch frames of reference, and tended to interpret graphs literally. The results of the study suggest an important relationship between spatial visualization ability and solving kinematics problems with multiple spatial parameters.     

Younger apes and human children plan their moves in a maze task

Planning defined as the predetermination of a sequence of actions towards some goal is crucial for complex problem solving. To shed light on the evolution of executive functions, we investigated the ontogenetic and phylogenetic origins of planning. Therefore, we presented all four great apes species (N = 12) as well as 4- and 5-year-old human preschoolers (N = 24) with a vertical maze task. To gain a reward placed on the uppermost level of the maze, subjects had to move the reward to the bottom through open gaps situated at each level of the maze. In total, there were ten gaps located over three of the maze’s levels, and free passage through these gaps could be flexibly blocked using multiple traps. Due to the decision tree design of the maze, the subjects had to plan their actions depending on the trap configuration up to two steps ahead to successfully retrieve the reward. We found that (1) our measure of planning was negatively correlated with age in nonhuman apes, (2) younger apes as well as 5-year-old children planned their moves up to two steps ahead whereas 4-year-olds were limited to plan one step ahead, and (3) similar performance but different underlying limitations between apes and children. Namely, while all species of nonhuman apes were limited by a lack of motor control, human children exhibited a shortage in shifting their attention across a sequence of subgoals.     

Discovering the Sequential Structure of Thought

Multi‐voxel pattern recognition techniques combined with Hidden Markov models can be used to discover the mental states that people go through in performing a task. The combined method identifies both the mental states and how their durations vary with experimental conditions. We apply this method to a task where participants solve novel mathematical problems. We identify four states in the solution of these problems: Encoding, Planning, Solving, and Respond. The method allows us to interpret what participants are doing on individual problem‐solving trials. The duration of the planning state varies on a trial‐to‐trial basis with novelty of the problem. The duration of solution stage similarly varies with the amount of computation needed to produce a solution once a plan is devised. The response stage similarly varies with the complexity of the answer produced. In addition, we identified a number of effects that ran counter to a prior model of the task. Thus, we were able to decompose the overall problem‐solving time into estimates of its components and in way that serves to guide theory.     

Productive Failure in Learning Math

When learning a new math concept, should learners be first taught the concept and its associated procedures and then solve problems, or solve problems first even if it leads to failure and then be taught the concept and the procedures? Two randomized‐controlled studies found that both methods lead to high levels of procedural knowledge. However, students who engaged in problem solving before being taught demonstrated significantly greater conceptual understanding and ability to transfer to novel problems than those who were taught first. The second study further showed that when given an opportunity to learn from the failed problem‐solving attempts of their peers, students outperformed those who were taught first, but not those who engaged in problem solving first. Process findings showed that the number of student‐generated solutions significantly predicted learning outcomes. These results challenge the conventional practice of direct instruction to teach new math concepts and procedures, and propose the possibility of learning from one's own failed problem‐solving attempts or those of others before receiving instruction as alternatives for better math learning.     

Fodor on global cognition and scientific inference

This paper addresses the extent to which quotidian cognition is like scientific inference by focusing on Jerry Fodor's famous analogy. I specifically consider and rebut a recent attempt made by Tim Fuller and Richard Samuels to deny the usefulness of Fodor's analogy. In so doing, I reveal some subtleties of Fodor's arguments overlooked by Fuller and Samuels and others. Recognizing these subtleties provides a richer appreciation of the analogy, allowing us to gain better traction on the issue concerning the extent to which everyday cognition is like scientific inference. In the end, I propose that quotidian cognition is indeed like scientific inference, but not precisely in the way Fodor claims it is.