Flexibility in strategy use: Adaptation of numerosity judgement strategies to task characteristics

In previous investigations we documented that people use several strategies to determine numerosities of blocks that are presented in a square grid. One of these strategies is the clever subtraction strategy, wherein the number of empty squares in the grid is subtracted from the total number of squares in the grid. In the present study we investigated participants' flexibility in strategy use when varying the size of the grids. Results are described in terms of the theoretical framework of Lemaire and Siegler (1995) regarding strategic change, and show that this contextual variable affected the frequency, execution time, and accuracy of subjects' use of the subtraction strategy. The usefulness of this framework for analysing the nature of the adaptation to contextual variations is discussed. From a methodological point of view, this study documents the potential of Beem's (1993, 1999) segmented linear regression models for assessing subjects' strategy use in cognitive tasks.     

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.     

Talking about walking: biomechanics and the language of locomotion

What drives humans around the world to converge in certain ways in their naming while diverging dramatically in others? We studied how naming patterns are constrained by investigating whether labeling of human locomotion reflects the biomechanical discontinuity between walking and running gaits. Similarity judgments of a student locomoting on a treadmill at different slopes and speeds revealed perception of this discontinuity. Naming judgments of the same clips by speakers of English, Japanese, Spanish, and Dutch showed lexical distinctions between walking and running consistent with the perceived discontinuity. Typicality judgments showed that major gait terms of the four languages share goodness-of-example gradients. These data demonstrate that naming reflects the biomechanical discontinuity between walking and running and that shared elements of naming can arise from correlations among stimulus properties that are dynamic and fleeting. The results support the proposal that converging naming patterns reflect structure in the world, not only acts of construction by observers.     

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.     

Exemplar by feature applicability matrices and other Dutch normative data for semantic concepts

Features are at the core of many empirical and modeling endeavors in the study of semantic concepts. This article is concerned with the delineation of features that are important in natural language concepts and the use of these features in the study of semantic concept representation. The results of a feature generation task in which the exemplars and labels of 15 semantic categories served as cues are described. The importance of the generated features was assessed by tallying the frequency with which they were generated and by obtaining judgments of their relevance. The generated attributes also featured in extensive exemplar by feature applicability matrices covering the 15 different categories, as well as two large semantic domains (that of animals and artifacts). For all exemplars of the 15 semantic categories, typicality ratings, goodness ratings, goodness rank order, generation frequency, exemplar associative strength, category associative strength, estimated age of acquisition, word frequency, familiarity ratings, imageability ratings, and pairwise similarity ratings are described as well. By making these data easily available to other researchers in the field, we hope to provide ample opportunities for continued investigations into the nature of semantic concept representation. These data may be downloaded from the Psychonomic Society’s Archive of Norms, Stimuli, and Data, www.psychonomic.org/archive.     

Dutch norm data for 13 semantic categories and 338 exemplars.

A data set is described that includes eight variables gathered for 13 common superordinate natural language categories and a representative set of 338 exemplars in Dutch. The category set contains 6 animal categories (reptiles, amphibians, mammals, birds, fish, and insects), 3 artifact categories (musical instruments, tools, and vehicles), 2 borderline artifact-natural-kind categories (vegetables and fruit), and 2 activity categories (sports and professions). In an exemplar and a feature generation task for the category nouns, frequency data were collected. For each of the 13 categories, a representative sample of 5-30 exemplars was selected. For all exemplars, feature generation frequencies, typicality ratings, pairwise similarity ratings, age-of-acquisition ratings, word frequencies, and word associations were gathered. Reliability estimates and some additional measures are presented. The full set of these norms is available in Excel format at the Psychonomic Society Web archive, www.psychonomic.org/archive/.