From Addition to Multiplication … and Back: The Development of Students’ Additive and Multiplicative Reasoning Skills

This study builds on two lines of research that have so far developed largely separately: the use of additive methods to solve proportional word problems and the use of proportional methods to solve additive word problems. We investigated the development with age of both kinds of erroneous solution methods. We gave a test containing missing-value problems to 325 third, fourth, fifth, and sixth graders. Half of the problems had an additive structure and half had a proportional structure. Moreover, in half of the problems the internal and external ratios between the given numbers were integer, while in the other cases numbers were chosen so that these ratios were noninteger. The results indicate a development from applying additive methods “anywhere” in the early years of primary school to applying proportional methods “anywhere” in the later years. Between these two stages many students went through an intermediate stage where they simultaneously applied additive methods to proportional problems and proportional methods to additive problems, switching between them based on the numbers given in the problem.     

Children's Criteria for Representational Adequacy in the Perception of Simple Sonic Stimuli

This article describes how children build increasingly abstract knowledge of linear measurement, emphasizing ways they relate space and number. Assessments indicate children struggle to understand measurement, especially concepts related to complex paths as in perimeter tasks. This article draws on developmental accounts of children's knowledge of measurement to describe the coordination of cognitive processes as a progression through increasingly abstract layers of strategy (Clements, 2003; Lehrer, 2003) within a constructivist perspective (Steffe &; Cobb, 1988; Steffe &; Thompson, 2000). This article reports 4 case studies from a 6-month teaching experiment with Grade 4 students to examine ways of promoting their strategy development. Attending to children's understanding of length along straight objects alone proved inadequate in predicting and validating models of children's developing strategies and concepts for perimeter. Setting perimeter tasks that emphasized integral relations among unit of length, sides of polygons and perimeter, and prompting children to integrate partitive, iterative, and counting schemes appeared to promote increasingly abstract length measurement strategies. The researchers came to distinguish between a naïve use of units to find length (Level 2a) and a more abstract strategy for unitizing and iterating (Level 2b), extending an earlier framework (Clements, Battista, Sarama, Swaminathan, &; McMillen, 1997). Finally, implications for instruction are drawn from the development of tasks and questioning sequences.     

Do First Graders Make Efficient Use of External Number Representations? The Case of the Twenty-Frame

External number representations are commonly used throughout the first years of instruction. The twenty-frame is a grid that contains two rows of 10 dots each, and within each row, dots are organized in two groups of five. The assumption is that children can make use of these structures for enumerating the dots, rather than relying on one-by-one counting. We compared first-grade children's performance on two types of computerized enumeration tasks, in which between one and 20 dots were presented in random arrangements or on a twenty-frame. The number of dots was a strong predictor of response times and accuracy rates in the enumeration task with random arrangements but not in the twenty-frame task. Performance on the twenty-frame task was correlated with performance on a number and arithmetic test, even when other cognitive variables were statistically controlled. We discuss these findings in the light of theories on utilizing external representations to support numerical learning.     

Vagueness: A Conceptual Spaces Approach

The conceptual spaces approach has recently emerged as a novel account of concepts. Its guiding idea is that concepts can be represented geometrically, by means of metrical spaces. While it is generally recognized that many of our concepts are vague, the question of how to model vagueness in the conceptual spaces approach has not been addressed so far, even though the answer is far from straightforward. The present paper aims to fill this lacuna.     

Identity and similarity

The standard approach to the so-called paradoxes of identity has been to argue that these paradoxes do not essentially concern the notion of identity but rather betray misconceptions on our part regarding other metaphysical notions, like that of an object or a property. This paper proposes a different approach by pointing to an ambiguity in the identity predicate and arguing that the concept of identity that figures in many ordinary identity claims, including those that appear in the paradoxes, is not the traditional philosophical concept but one that can be defined in terms of relevant similarity. This approach to the paradoxes will be argued to be superior to the standard one.     

The relationship between the shape of the mental number line and familiarity with numbers in 5- to 9-year old children: evidence for a segmented linear model

This experiment aimed to expand previous findings on the development of mental number representation. We tested the hypothesis that children's familiarity with numbers is directly reflected by the shape of their mental number line. This mental number line was expected to be linear as long as numbers lay within the range of numbers children were familiar with. Five- to 9-year-olds (N=78) estimated the positions of numbers on an external number line and additionally completed a counting assessment mirroring their familiarity with numbers. A segmented regression model consisting of two linear segments described number line estimations significantly better than a logarithmic or a simple linear model. Moreover, the change point between the two linear segments, indicating a change of discriminability between numbers, was significantly correlated with children's familiar number range. Findings are discussed in terms of the accumulator model, assuming a linear mental representation with scalar variability.     

Maternal speech to infants at 1 and 3 months of age

The goal of this study was to assess maternal speech and in relation to changes in infant social behavior occurring around the second month post birth. Sixty infants interacted with their mother at 1 and 3 months of age in a face-to-face context. At 3 months, infants gazed, smiled, and positively vocalized significantly more than at 1 month. These findings point to a transition in infant social behavior at around the second month post birth. In addition, maternal speech to infants increased between these times in both amount and complexity, possibly in response to an increase in infant social behavior. Maternal speech was related to infant positive vocalizing at 3 months, suggesting mothers especially monitored infant vocalizing at 3 months. Individual differences in maternal speech were stable across visits.     

Decreasing overt discrimination increases covert discrimination: Adverse effects of equal opportunities policies

Building on the moral licensing literature, this paper examines whether highlighting the successful implementation of an equal opportunities policy in a company leads to covert forms of discrimination in hiring decisions (i.e., expressing a preference for a white candidate over an equally qualified black/Moroccan candidate in an ambiguous context). Furthermore, moral self‐image is indirectly tested as a possible underlying mechanism. Two scenario studies first revealed that covert discrimination is more likely after highlighting a successful implementation of an equal opportunities policy in the company (study 1) and that elevated levels of moral self‐image are related to covert discrimination (study 2). Subsequently, a field study revealed that the presence of successful equal opportunities policies positively related to employees’ moral self‐image (study 3).     

Training 2;6-year-olds to produce the transitive construction: the role of frequency, semantic similarity and shared syntactic distribution

Childers and Tomasello (2001) found that training 2 1/2-year-olds on the English transitive construction greatly improves their performance on a post-test in which they must use novel verbs in that construction. In the current study, we replicated Childers and Tomasello's finding, but using a much lower frequency of transitive verbs and models in training. We also used novel verbs that were of a different semantic class to our training verbs, demonstrating that semantic homogeneity is not crucial for generalization. We also replicated the finding that 4-year-olds are significantly more productive than 2 1/2-year-olds with the transitive construction, with the new finding that this is also true for verbs of emission. In addition, 'shared syntactic distribution' of novel verb and training verbs was found to have no observable effect on the number of 2 1/2-year-olds who were productive in the post-test.