Commonalities and differences in similarity comparisons

Similarity is a central component of many cognitive processes. Current research suggests that similarity is well characterized as a comparison of structured representations. This process yields commonalities, differences related to the commonalities (alignable differences), and differences not related to the commonalities (nonalignable differences). In the first study, further evidence for this tripartite distinction is provided in a commonality and difference listing study involving pairs of pictures. This study indicates that alignable differences rather than nonalignable differences are central to the comparison process by virtue of their connection to the commonalities. The second study further demonstrates that alignable differences count more against the similarity of a pair than do nonalignable differences. We end by discussing implications of the distinction between alignable and nonalignable differences for other cognitive processes involving comparisons.     

Surface and structural similarity in analogical transfer

Two experiments investigated factors that influence the retrieval and use of analogies in problem solving, Experiment 1 demonstrated substantial spontaneous analogical transfer with a delay of several days between presentation of the source and target analogues. Experiment 2 examined the influence of different types of similarity between the analogues. A mechanism for retrieval of source analogues is proposed, based on summation of activation from features shared with a target problem. The results of Experiment 2 indicated that both structural features, which play a causal role in determining possible problem solutions, and salient surface features, which do not have a causal role, influence spontaneous selection of an analogue. Structural features, however, have a greater impact than do surface features on a problem solver’s ability to use an analogue once its relevance has been pointed out.     

Timed magnitude comparisons of numerical and nonnumerical expressions of uncertainty

Two experiments involving paired comparisons of numerical and nonnumerical expressions of uncertainty are reported. Subjects were timed under two opposing sets of instructions ("choose higher probability" vs. "choose lower probability"). Numerical comparisons were consistently faster and easier than their nonnumerical counterparts. Consistent distance and congruity effects were obtained, illustrating that both numerical and nonnumerical expressions of uncertainty contain subjective magnitude information, and suggesting that similar processes are employed in manipulating and comparing numerical and verbal terms. To account for the general pattern of results obtained, Holyoak's reference point model (1978) was generalized by explicitly including the vagueness of the nonnumerical expressions. This generalized model is based on the notion that probability expressions can be represented by membership functions (Wallsten, Budescu, Rapoport, Zwick, & Forsyth, 1986) from which measures of location for each word, and measures of overlap for each pair can be derived. A good level of fit was obtained for this model at the individual level.     

The impact of bundle comparisons on bundle preference

The bundling literature largely holds that a person's reaction to a given product bundle depends only on the characteristics of the products contained in the bundle. This paper, instead, proposes that people evaluate bundles in reference to other bundles that they have seen. Prior research indicates that people are sensitive to a bundle's “attribute inventory” or the aggregate level of comparable attributes possessed by its constituent products. We show that when people evaluate a bundle, they compare the attribute inventories that it offers to those offered by other bundles that they have seen. The resulting compositional comparisons can occur without changes to the products that comprise the target and contextual bundles, vary by attribute comparability and attentional focus, and coexist with (and at times reverse the effects of) well‐established product‐specific context effects, which are determined solely by the products and their attributes.     

Preschoolers' use of surface similarity in object comparisons: taking context into account

Previous research has emphasized the role of within-match similarity in children's comparisons. The current study investigated another potentially important contributing factor, namely the distinctiveness of the matching items relative to other items in the scene. Using a well-known relational mapping task, we found that 3- and 4-year-olds made more correct matches between identical objects when those objects were maximally distinctive from the foils. In a cross-mapping experiment, where same relative size was pitted against object similarity, 3-year-olds made more incorrect object matches when the objects were both similar to each other and distinctive from the foils. Furthermore, 3- and 4-year-olds performed the same, regardless of within-match similarity, so long as the ratios of within-match and nonmatch similarity were roughly equal. These findings suggest that children's comparisons are guided by a ratio consisting of many pairwise similarity relations, including (but not limited to) the degree of within-match similarity.     

Exploring the impact of dimensional comparisons in health contexts

“Did I exercise enough this week?” “Should I eat that last donut?” “Am I in good shape?” Due to the ambiguity and difficulty of answering such questions, people often use comparison-based information to contextualize their standing, inform self-evaluations, and guide behavior. For example, to ascertain whether they have exercised enough, people can make social comparisons to peers (e.g., “Have I exercised more than my friends?”) or temporal comparisons of the present to the past (e.g., “Have I exercised more this week than last week?”). While ample research has examined the relevance and impact of social and temporal comparisons in a range of health contexts (e.g., health behaviors, risk perceptions), the overarching goal of the present review was to explicate the potential relevance and impact of a third type of comparison standard—dimensional comparisons—in health contexts (with an emphasis on health behaviors like exercise). First, we discuss how social and temporal comparisons shape self-evaluations in both health and non-health contexts. Second, we provide an overview of research conducted on dimensional comparisons in non-health contexts and discuss how comparison sources shape self-evaluations and self-concept formation. Third, we explore the potential impact and relevance of dimensional comparisons on self-evaluations in health behavior contexts. Fourth, and finally, we highlight potential future directions and considerations for advancing our knowledge of the relevance and influence of dimensional comparisons in health contexts.     

The role of subtractions and comparisons in comparative judgments involving numerical reference points

Experiments in which subjects are asked to decide which of two digits is closer in magnitude to a third raise problems for many theories of linear orders. Holyoak (1978), for example, performed a number of these reference point experiments and concluded that they posed serious difficulties for a number of leading models. In their place, he offered the distance ratio model in which the ease of the decision in a reference point task is a function of the ratio of the distances between each stimulus and the reference point. In the present article, three experiments are presented that bear on the adequacy of Holyoak's position. In the first two studies, we present evidence that an important assumption of the distance ratio model is incorrect. In the third experiment, we compare the empirical adequacy of the distance ratio model with our own subtraction model. This model treats the reference point task as a concatenation of two subtractions and a simple digit comparison. This comparison operation is equivalent to the magnitude comparison required in standard linear order experiments. Overall, the subtraction model gives a somewhat better account than the distance ratio.     

The development of ordinal numerical knowledge in infancy

A critical question in cognitive science concerns how numerical knowledge develops. One essential component of an adult concept of number is ordinality: the greater than and less than relationships between numbers. Here it is shown in two experiments that 11-month-old infants successfully discriminated, whereas 9-month-old infants failed to discriminate, sequences of numerosities that descended in numerical value from sequences that increased in numerical value. These results suggest that by 11 months of age infants possess the ability to appreciate the greater than and less than relations between numerical values but that this ability develops between 9 and 11 months of age. In an additional experiment 9-month-old infants succeeded at discriminating the ordinal direction of sequences that varied in the size of a single square rather than in number, suggesting that a capacity for non-numerical ordinal judgments may develop before a capacity for ordinal numerical judgments. These data raise many questions about how infants represent number and what happens between 9 and 11 months to support ordinal numerical judgments.     

Quantitative numerical and lexical knowledge: evidence of double dissociation

The aim of this work was to explain the involvement of a specific brain injury in the numerical processing and calculation system. The method employed was > analysis and the administration of various cognitive neuropsychology tests. The results of this study revealed a double dissociation between quantitative numerical knowledge and qualitative or lexical numerical knowledge. Patient M.C. preserved quantitative numerical knowledge, as indicated by the results obtained in the numerical comprehension and calculation tasks. However, she showed a drastic deficit in qualitative numerical knowledge. On the other hand, patient M.L. preserved qualitative numerical knowledge, but she had serious problems in all the abilities that require internal manipulation of magnitude; that is, quantitative numerical knowledge. These results have two important implications, as conclusions: firstly, quantitative numerical knowledge may be made up of different elements susceptible to damage independently. And secondly, quantitative and qualitative numerical knowledge were functionally independent.     

The impact of training sequence and between-category similarity on unsupervised induction

Studies of supervised categorization often show better learning when examples are presented in random alternation rather than massed by category, but such interleaving impairs learning in unsupervised tasks. The exemplar comparison hypothesis explains this result by assuming that people in unsupervised tasks discover generalizations about categories by comparing individual examples, and that interleaving increases the difficulty of such within-category comparisons. The category invention hypothesis explains the interleaving effect by assuming that people are more likely to merge or aggregate potentially separable categories when they are interleaved, and this initial failure to recognize separate categories then acts as an effective barrier to further learning. The present experiments show that the interleaving effect depends on the similarity or alignability of the presented categories. This result provides evidence in favour of the category invention hypothesis, which expects that highly dissimilar (nonalignable) categories will resist aggregation and hence will not be affected by interleaving. The nonmonotonic pattern of learning, and the interaction between sequence and similarity, observed in the alignable conditions of Experiment 3 were also consistent with category invention, but not with exemplar comparison. Implications are discussed for real-world learning, especially the relationship between exposure and learning and between supervised and unsupervised learning.     

Language supports for children's understanding of numerical fractions: cross-national comparisons.

This study examined Croatian, Korean, and U.S. children's knowledge of numerical fractions prior to school instruction. The research is part of an ongoing project examining the influence of language characteristics on mathematical thinking and performance. The part-whole quantitative relation denoted by numerical fractions may be easier to understand in East Asian languages like Korean. In these languages, the concept of fractional parts is embedded in the mathematics terms used for fractions. The results from this study suggest that the Korean vocabulary of fractions may influence the meaning 6- to 7-year-old children ascribe to numerical fractions and that this results in children being able to associate numerical fractions with corresponding pictorial representations prior to formal instruction.     

Visual field asymmetries in numerical size comparisons of digits, words, and signs

Visual field asymmetries were examined in American Sign Language-English bilinguals for speeded numerical size judgments of pairs of digits, number words, and number signs. Physical size of the number pairs was either congruent or incongruent with their numerical size. The results revealed a greater left visual field (LVF) interference for numbers represented as digits and a greater right visual field (RVF) interference for numbers represented as words or signs. Subjects' performance on number words and signs was also influenced by their skill in English and ASL: interference was greater in the RVF in the subjects' better language but was greater in the LVF for the less skilled language. These findings suggest that lateralization of numerical size judgments is moderated by the mode of number presentation and by prior language experience.     

Friendship similarity during early adolescence: gender and racial patterns.

We studied the relationship of reciprocity, gender, and racial composition (Caucasian, African American, cross-race) of adolescent friendship dyads to similarity and proximity in 136 young adolescents. We found that adolescents selected friends who were of the same gender and race and that female dyads were more similar than male dyads on verbal achievement and several personality dimensions. Caucasian dyads were more similar than African American dyads on verbal achievement, mental alertness, and dominance. African American adolescents had more contact with their best friends outside school, whereas Caucasian adolescent friends had more in-school contact. African American students had fewer reciprocal relationships than the Caucasian students. Cross-race friendships were less reciprocal than same-race friendships. Race and gender were important in determining friendship patterns. Similarity and proximity were more important than reciprocity in understanding early adolescent friendships.     

Magnitude knowledge: the common core of numerical development

The integrated theory of numerical development posits that a central theme of numerical development from infancy to adulthood is progressive broadening of the types and ranges of numbers whose magnitudes are accurately represented. The process includes four overlapping trends: (1) representing increasingly precisely the magnitudes of non‐symbolic numbers, (2) connecting small symbolic numbers to their non‐symbolic referents, (3) extending understanding from smaller to larger whole numbers, and (4) accurately representing the magnitudes of rational numbers. The present review identifies substantial commonalities, as well as differences, in these four aspects of numerical development. With both whole and rational numbers, numerical magnitude knowledge is concurrently correlated with, longitudinally predictive of, and causally related to multiple aspects of mathematical understanding, including arithmetic and overall math achievement. Moreover, interventions focused on increasing numerical magnitude knowledge often generalize to other aspects of mathematics. The cognitive processes of association and analogy seem to play especially large roles in this development. Thus, acquisition of numerical magnitude knowledge can be seen as the common core of numerical development.     

Reducing the gap in numerical knowledge between low- and middle-income preschoolers

We compared the learning from playing a linear number board game of preschoolers from middle-income backgrounds to the learning of preschoolers from low-income backgrounds. Playing this game produced greater learning by both groups than engaging in other numerical activities for the same amount of time. The benefits were present on number line estimation, magnitude comparison, numeral identification, and arithmetic learning. Children with less initial knowledge generally learned more, and children from low-income backgrounds learned at least as much, and on several measures more, than preschoolers from middle-income backgrounds with comparable initial knowledge. The findings suggest a class of intervention that might be especially effective for reducing the gap between low-income and middle-income children's knowledge when they enter school.     

The early development of numerical reasoning.

Children of age 1-4 years were found capable of engaging in numerical reasoning. Children were presented with a task in which they placed a set of objects one by one into an opaque container. An experimenter then visibly performed either an addition, a subtraction, or no transformation on the screened set. Children were then instructed to remove all objects from the container. Across two experiments, children searched for and removed the correct number of objects when set numerosity was small. Knowledge of numerical identity and knowledge of the effects of addition and subtraction transformations on numerosity were present even in children who had not yet begun to count verbally. These findings provide evidence that the emergence of numerical reasoning does not depend upon the prior development of a verbal counting ability or upon cultural experience with numbers.     

Inference and exact numerical representation in early language development

How do children as young as 2 years of age know that numerals, like one, have exact interpretations, while quantifiers and words like a do not? Previous studies have argued that only numerals have exact lexical meanings. Children could not use scalar implicature to strengthen numeral meanings, it is argued, since they fail to do so for quantifiers [Papafragou, A., & Musolino, J. (2003). Scalar implicatures: Experiments at the semantics–pragmatics interface. Cognition, 86, 253–282]. Against this view, we present evidence that children’s early interpretation of numerals does rely on scalar implicature, and argue that differences between numerals and quantifiers are due to differences in the availability of the respective scales of which they are members. Evidence from previous studies establishes that (1) children can make scalar inferences when interpreting numerals, (2) children initially assign weak, non-exact interpretations to numerals when first acquiring their meanings, and (3) children can strengthen quantifier interpretations when scalar alternatives are made explicitly available.     

The effects of perceptual similarity and category membership on early word-referent identification

We investigated the impact of perceptual and categorical relatedness between a target and a distracter object on early referent identification in infants and adults. In an intermodal preferential looking (IPL) task, participants looked at a target object paired with a distracter object that could be perceptually similar or dissimilar and drawn from the same or different global category. The proportion of target looking measures revealed that infants and adults were sensitive to the interplay between category membership and perceptual similarity. Online latency measures demonstrated an advantage for perceptually dissimilar items regardless of their categorical status, indicating that different IPL measures index different processes during target identification. Results suggest that perceptual similarity and category membership of the objects lead to competition effects in word recognition and referent identification in both adults and infants and that lexical categorization and nonlinguistic categorization processes are closely related during infancy.