This article responds to the reply by Cooper and Shallice to Botvinick and Plaut's previous comment . In the authors' view, the reply by Cooper and Shallice left their original responses more or less intact. The present exchange has strongly highlighted the degree to which any account of the habit system will interact with how the goal-directed…

In this article, the author argues that there is a big conceptual difference between number lines and number strips and that number lines should be consistently used as early as Y1. Her interest in the use of the number line as a powerful image of the real numbers, and how this is introduced in the early years, comes from her own experience as a…

If the goal is to promote mathematical thinking and help children become flexible problem solvers, then it is important to show students multiple representations of a problem. Because it is important to help students develop both counting-based and collections-based conceptions of number, teachers should be showing students both number line…

Structural equation modeling allows several methods of estimating the disattenuated association between 2 or more latent variables (i.e., the measurement model). In one common approach, measurement models are specified using item parcels as indicators of latent constructs. Item parcels versus original items are often used as indicators in these…

The computation and reporting of effect size estimates is becoming the norm in many journals in psychology and related disciplines. Despite the increased importance of effect sizes, researchers may not report them or may report inaccurate values because of a lack of appropriate computational tools. For instance, Pierce, Block, and Aguinis (2004)…

A computational experiment is devised for advanced inorganic laboratory course that allows the students to explore the structure and bonding patterns of ethene and some heavier analogues. The HOMO-LUMO gaps, double bond dissociation energetics, and optimized geometries of ethene, disilene, and digermene are explored.

Students in the classroom should be provided with engaging activities for improving their computational fluency, making effective use of time, and as a medium of self-motivation. Games such as Contig, the 24 Game, and Number Jumbler are useful for practicing basic facts and computational fluency of mathematics.

Establishing an instrument's factorial invariance provides the empirical foundation to compare an individual's score across time or to examine the pattern of correlations between variables in differentiated age groups. In the recently published Woodcock-Johnson Tests of Cognitive Ability (WJ COG) and Achievement (WJ ACH) Third Edition (III) the…

This article presents a formula for weighted kappa in terms of rater means, rater variances, and the rater covariance that is particularly helpful in emphasizing that weighted kappa is an absolute agreement measure in the sense that it is sensitive to differences in rater's marginal distributions. Specifically, rater mean differences will decrease…

We focus on a major difficulty in solving combinatorial problems, namely, on the verification of a solution. Our study aimed at identifying undergraduate students' tendencies to verify their solutions, and the verification strategies that they employ when solving these problems. In addition, an attempt was made to evaluate the level of efficiency…

This study investigated the development of the many-to-one counting strategy in 4- year-old children. In the first experiment, 52 children participated. Their development with respect to two kinds of tasks, a hidden-items task and a needed-items task, was studied over four sessions. Children (n = 28) who accurately used the many-to-one strategy in…

The paper presents a study of novice difficulties with range conceptions in loop design. CS2 students were asked to solve four related enumeration tasks, which required various loop boundary specifications. The student solutions varied considerably in conciseness and efficiency. The solution diversity reveals significant differences in range…

This study compared nonverbal numerical processing in 6-year-olds with that in nonhuman animals using a numerical bisection task. In the study, 16 children were trained on a delayed match-to-sample paradigm to match exemplars of two anchor numerosities. Children were then required to indicate whether a sample intermediate to the anchor values was…

Recent studies using a violation-of-expectation task suggest that preverbal infants are capable of recognizing basic arithmetical operations involving visual objects. There is still debate, however, over whether their performance is based on any expectation of the arithmetical operations, or on a general perceptual tendency to prefer visually…

In this study, we use children's prior knowledge to support their development of cardinality understanding, based on Bermejo's (1996) model of cardinality understanding and on the distinction between cardinality (the number of items in a set is represented by a number-word) and the cardinality principle (the last number-word used in counting…