Complex mental arithmetic: The contribution of the number sense.

Young adults were asked to solve two-digit addition problems and to say aloud the result of each calculation step to allow the identification of computation strategies. We manipulated the position of the largest addend (e.g., 25 + 48 vs. 48 + 25) to assess whether strategies are modulated by magnitude characteristics. With some strategies, participants demonstrated a clear preference to take the largest addend as the starting point for the calculation. Hence, rather than applying strategies in an inflexible manner, participants evaluated and compared the operands before proceeding to calculation. Further, mathematically skilled participants tended to use those magnitude-based strategies more often than less skilled ones. The findings demonstrate that magnitude information plays a role in complex arithmetic by guiding the process of strategy selection, and possibly more so for mathematically skilled participants. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Crossing the divide: Infants discriminate small from large numerosities.

Although young infants have repeatedly demonstrated successful numerosity discrimination across large sets when the number of items in the sets changes twofold (E. M. Brannon, S. Abbott, & D. J. Lutz, 2004; J. N. Wood & E. S. Spelke, 2005; F. Xu & E. S. Spelke, 2000), they consistently fail to discriminate a twofold change in number when one set is large and the other is small (     

The development of language and abstract concepts: The case of natural number.

[Correction Notice: An erratum for this article was reported in Vol 137(2) of Journal of Experimental Psychology: General (see record 2008-05019-011). The DOI for the supplemental materials was printed incorrectly. The correct DOI is as follows: http://dx.doi.org/10.1037/0096-3445.137.1.22.supp] What are the origins of abstract concepts such as seven, and what role does language play in their development? These experiments probed the natural number words and concepts of 3-year-old children who can recite number words to ten but who can comprehend only one or two. Children correctly judged that a set labeled eight retains this label if it is unchanged, that it is not also four, and that eight is more than two. In contrast, children failed to judge that a set of 8 objects is better labeled by eight than by four, that eight is more than four, that eight continues to apply to a set whose members are rearranged, or that eight ceases to apply if the set is increased by 1, doubled, or halved. The latter errors contrast with children's correct application of words for the smallest numbers. These findings suggest that children interpret number words by relating them to 2 distinct preverbal systems that capture only limited numerical information. Children construct the system of abstract, natural number concepts from these foundations. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Grey parrot (Psittacus erithacus) numerical abilities: Addition and further experiments on a zero-like concept.

A Grey parrot (Psittacus erithacus), able to quantify 6 or fewer item sets (including heterogeneous subsets) by using English labels (I. M. Pepperberg, 1994), was tested on addition of quantities involving 0-6. He was, without explicit training, asked, "How many total X?" for 2 sequentially presented collections (e.g., of variously sized jelly beans or nuts) and required to answer with a vocal English number label. His accuracy suggested (a) that his addition abilities are comparable to those of nonhuman primates and young children, (b) some limits as to his correlation of "none" and the concept of zero, and (c) a possible counting-like strategy for the quantity 5. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Calculator use need not undermine direct-access ability: The roles of retrieval, calculation, and calculator use in the acquisition of arithmetic facts.

Why is subsequent recall sometimes better for self-generated answers than for answers obtained from an external source (e.g., calculator)? In this study, we explore the relative contribution of 2 processes, recall attempts and self-computation, to this generation effect (i.e., enhanced answer recall relative to when problems are practiced with a calculator). Adults (N = 36) practiced unfamiliar alphabet arithmetic problems (A + 4 = ?; answer E), in 3 learning conditions: self-generating answers through recall or counting (self-generate learning), obtaining answers with a customized calculator (calculator-only learning), or using a calculator after first attempting answer recall (retrieve-else-calculator learning). Subsequently, participants were tested without a calculator. Retrieve-else-calculator learning was expected to produce an intermediate level of performance because it captures the benefits of recall attempts but excludes any benefits of the other self-generation process (mental computation). However, retrieve-else-calculator learning proved as effective as self-generation learning (latency, accuracy, and recall rates on test), and both led to a test performance that was superior to calculator-only learning. We suggest that mental generation processes that involve the retrieval of intermediate products may introduce retrieval interference that precludes a strong contribution to the generation effect. However, tedious mental computation may contribute to superior access indirectly, by inciting automatic recall attempts. This effect can be mimicked by imposing a delay prior to calculator use. Our results are compatible with the view that learning contexts that promote recall activities produce a customized type of knowledge to support cued-recall (Rickard & Bajic, 2006). (PsycINFO Database Record (c) 2011 APA, all rights reserved)     

Temporal Construal Effects on Abstract and Concrete Thinking: Consequences for Insight and Creative Cognition.

Six studies investigate whether and how distant future time perspective facilitates abstract thinking and impedes concrete thinking by altering the level at which mental representations are construed. In Experiments 1-3, participants who envisioned their lives and imagined themselves engaging in a task 1 year later as opposed to the next day subsequently performed better on a series of insight tasks. In Experiments 4 and 5 a distal perspective was found to improve creative generation of abstract solutions. Moreover, Experiment 5 demonstrated a similar effect with temporal distance manipulated indirectly, by making participants imagine their lives in general a year from now versus tomorrow prior to performance. In Experiment 6, distant time perspective undermined rather than enhanced analytical problem solving. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Schizophrenic responses to the Proverbs Test: Abstract, concrete, or autistic.

Psychopathology was more effective in predicting autistic responses to proverbs of acutely ill schizophrenic patients than was intelligence as measured by the Wechsler Vocabulary subtest. Autistic responses decreased significantly from pre- to post-5 wk. of hospital treatment. There was no difference in the frequency of these responses between high and low verbal IQ groups, nor did level of IQ affect the decrease in autistic responses. These findings were contrasted with those of a previous study which showed that the traditional abstract and concrete scores of responses to proverbs were highly correlated with intelligence even with the effects of pathology statistically controlled, and did not improve with treatment except when IQ was above average. The present study further demonstrated that while autistic responses were negatively related to abstract scores, they were relatively unrelated to concrete scores. The overall findings were consistent with both earlier theoretical formulations and more recent empirical investigations regarding cognitive deficit in schizophrenia. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

An Encoding-Complex Approach to Numerical Cognition in Chinese-English Bilinguals.

We present a model of the cognitive architecture of basic numerical skills in adult Chinese-English bilinguals. The model is based on data reported by Campbell, Kanz, and Xue (1999) and combines Dehaene and Cohen's triple-code theory with Campbell and Clark's encoding-complex approach to modeling number processing. Participants were required to name, add or multiply Arabic or Mandarin numerals and to respond in English or Chinese. They also performed magnitude comparisons on pairs of Arabic or Mandarin numerals. The proposed model of their performance on this set of tasks assumes 1) that number processing is modular with respect to representational code (e.g., visual, visuo-spatial, verbal) rather than with respect to numerical function, 2) task-specific communication between representational codes is interactive rather than additive, and 3) memory for arithmetic facts is at least partially language- based and our Chinese-English bilinguals possessed both Chinese and English-language number-fact representations. We provide new analyses of the arithmetic data and a review of research on the role of language in simple arithmetic to substantiate our claims about linguistic codes for number-fact memory. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Mental subtraction in high- and lower skilled arithmetic problem solvers: Verbal report versus operand-recognition paradigms.

The authors used the operand-recognition paradigm (C. Thevenot, M. Fanget, & M. Fayol, 2007) in order to study the strategies used by adults to solve subtraction problems. This paradigm capitalizes on the fact that algorithmic procedures degrade the memory traces of the operands. Therefore, greater difficulty in recognizing them is expected when calculations have been solved by reconstructive strategies rather than by retrieval of number facts from long-term memory. The present results suggest that low- and high-skilled individuals differ in their strategy when they solve problems involving minuends from 11 to 18. Whereas high-skilled individuals retrieve the results of such subtractions from long-term memory, lower skilled individuals have to resort to reconstructive strategies. Moreover, the authors directly confront the results obtained with the operand-recognition paradigm and those obtained with the more classical method of verbal report collection and show clearly that this second method of investigation fails to reveal this differential pattern. The rationale behind the operand-recognition paradigm is then discussed. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Embodiment of abstract concepts: Good and bad in right- and left-handers.

Do people with different kinds of bodies think differently? According to the body-specificity hypothesis, people who interact with their physical environments in systematically different ways should form correspondingly different mental representations. In a test of this hypothesis, 5 experiments investigated links between handedness and the mental representation of abstract concepts with positive or negative valence (e.g., honesty, sadness, intelligence). Mappings from spatial location to emotional valence differed between right- and left-handed participants. Right-handers tended to associate rightward space with positive ideas and leftward space with negative ideas, but left-handers showed the opposite pattern, associating rightward space with negative ideas and leftward with positive ideas. These contrasting mental metaphors for valence cannot be attributed to linguistic experience, because idioms in English associate good with right but not with left. Rather, right- and left-handers implicitly associated positive valence more strongly with the side of space on which they could act more fluently with their dominant hands. These results support the body-specificity hypothesis and provide evidence for the perceptuomotor basis of even the most abstract ideas. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

The cognitive correlates of third-grade skill in arithmetic, algorithmic computation, and arithmetic word problems.

The purpose of this study was to examine the cognitive correlates of 3rd-grade skill in arithmetic, algorithmic computation, and arithmetic word problems. Third graders (N = 312) were measured on language, nonverbal problem solving, concept formation, processing speed, long-term memory, working memory, phonological decoding, and sight word efficiency as well as on arithmetic, algorithmic computation, and arithmetic word problems. Teacher ratings of inattentive behavior also were collected. Path analysis indicated that arithmetic was linked to algorithmic computation and to arithmetic word problems and that inattentive behavior independently predicted all 3 aspects of mathematics performance. Other independent predictors of arithmetic were phonological decoding and processing speed. Other independent predictors of arithmetic word problems were nonverbal problem solving, concept formation, sight word efficiency, and language. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Evidence for direct retrieval of relative quantity information in a quantity judgment task: Decimals, integers, and the role of physical similarity.

Participants' reaction times (RTs) in numerical judgment tasks in which one must determine which of 2 numbers is greater generally follow a monotonically decreasing function of the numerical distance between the two presented numbers. Here, I present 3 experiments in which the relative influences of numerical distance and physical similarity are assessed in just such a task using integers and decimals as stimuli. The data reveal that numerical distance is the primary feature controlling participants' RTs when integers are presented. However, the physical similarity between the decade place of the standard and the probe is the primary feature controlling participants' RTs when decimals are presented. I conclude that the unique qualities of decimals do not lend themselves to share the place-coding representation of integers, thus a direct retrieval mechanism for judging the relative quantity of decimals has developed. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Memory for individuals: Hamsters (Mesocricetus auratus) require contact to develop multicomponent representations (concepts) of others.

In humans, individuals recognize other individuals by numerous types of independent information, such as the quality of the voice, appearance of the face, smell, gait, and posture. Humans also have integrated memories of others--that is, in response to a face or a voice the individual is recognized by name and other information about that individual is remembered. In many nonhuman species, individual recognition also occurs. Although observational studies suggest that individuals of some nonhuman species may be able to use several different cues for individual recognition, little experimental proof for this is available. Golden hamsters have at least 5 individually distinctive odors and they develop integrated, multi-odor memories (concepts) of familiar individuals, as shown by across-odor habituation experiments. Little is known, however, about the conditions that are necessary to develop such integrated memories. In these experiments we investigated what kinds of experiences were necessary for male hamsters to develop multiodor memories of females. The results show that exposure to all of the odors of another individual was not sufficient to develop such multiodor memories but that physical contact between the subjects and stimulus animals was necessary. Multiodor representations were developed after interactions with anesthetized individuals, confirming the finding that physical contact was important but also showing that interaction with an awake, behaving individual was not necessary to form multiodor representations of other individuals. We are not aware of experimental proof for integrated, multicomponant memories in any other nonhuman species. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Reading and arithmetic in adolescents with autism spectrum disorders: Peaks and dips in attainment.

In describing academic attainment in autism spectrum disorders (ASD), results are typically reported at the group mean level. This may mask subgroups of individuals for whom academic achievement is incommensurate with intellectual ability. The authors tested the IQ, literacy, and mathematical abilities of a large group (N = 100) of adolescents (14–16 years old) with ASD. Seventy-three percent of the sample had at least one area of literacy or mathematical achievement that was highly discrepant (approximately 14 standard score points) from full-scale IQ (FSIQ). The authors focused on four subgroups with either word reading (“Reading Peak” and “Reading Dip”) or arithmetic (“Arithmetic Peak” and “Arithmetic Dip”) higher or lower than FSIQ. These subgroups were largely mutually exclusive and were characterized by distinct intellectual profiles. The largest was the “Arithmetic Peak” subgroup of participants, who presented with average intellectual ability alongside superior arithmetic skills and who were predominantly in a mainstream educational setting. Overall, the most pervasive profile was discrepantly poor reading comprehension, which associated with severity of social and communication difficulties. The high rate of uneven academic attainment in ASD has implications for educational practice. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

The use of open-end data as an aid in writing multiple-choice distracters: An evaluation with arithmetic reasoning and computation items.

Data gathered during the development of the Navy Arithmetic Test were analyzed to evaluate the utility of using distracters derived by administering the items in open-ended (OE) form. For both Computation items and Reasoning items moderately high correlations (about .50-.60) were found between frequency with which responses were written in by Ss in OE format and frequency with which the same responses were chosen in multiple-choice format. That OE-derived responses tend to retain their relative popularity in multiple-choice format appears to provide some support for the use of the relatively expensive OE technique in arithmetic test construction. Caution is urged in applying these findings to other test types. From Psyc Abstracts 36:05:5KJ31R. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Empty sets as part of the numerical continuum: Conceptual precursors to the zero concept in rhesus monkeys.

The goal of the current research was to explore whether monkeys possess conceptual precursors necessary for understanding zero. We trained rhesus monkeys on a nonsymbolic numerical matching-to-sample task, and on a numerical ordering task. We then introduced nondifferentially reinforced trials that contained empty sets to determine whether monkeys would treat empty sets as numerical values. All monkeys successfully matched and ordered the empty sets without any training. Accuracy showed distance effects, indicating that they treated empty sets as values on a numerical continuum. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Cultural differences in complex addition: Efficient Chinese versus adaptive Belgians and Canadians.

In the present study, the authors tested the effects of working-memory load on math problem solving in 3 different cultures: Flemish-speaking Belgians, English-speaking Canadians, and Chinese-speaking Chinese currently living in Canada. Participants solved complex addition problems (e.g., 58 + 76) in no-load and working-memory load conditions, in which either the central executive or the phonological loop was loaded. The authors used the choice/no-choice method to obtain unbiased measures of strategy selection and strategy efficiency. The Chinese participants were faster than the Belgians, who were faster and more accurate than the Canadians. The Chinese also required fewer working-memory resources than did the Belgians and Canadians. However, the Chinese chose less adaptively from the available strategies than did the Belgians and Canadians. These cultural differences in math problem solving are likely the result of different instructional approaches during elementary school (practice and training in Asian countries vs. exploration and flexibility in non-Asian countries), differences in the number language, and informal cultural norms and standards. The relevance of being adaptive is discussed as well as the implications of the results in regards to the strategy choice and discovery simulation model of strategy selection (J. Shrager & R. S. Siegler, 1998). (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Mental number line, number line estimation, and mathematical achievement: Their interrelations in grades 5 and 6.

As indicated by the distance effect and the spatial–numerical association of response codes (SNARC) effect, natural numbers are mentally represented on a number line. Purportedly, this number line underlies children's number sense, which supports the acquisition of more advanced mathematical competencies. In 3 studies with a total of 429 fifth and sixth graders, the authors compared the influences of each child's distance effect, SNARC effect, conceptual knowledge about decimal fractions, and numerical intelligence on mathematical school achievement. Additionally, they tested using decimal fractions whether number line estimation competence mediates the influence of the internal number line. In all, the results, found with path models, revealed that domain-specific conceptual knowledge, numerical intelligence, and number line estimation each were good predictors of achievement, while distance and SNARC effects were virtually unrelated to all other variables. Individual differences in the use of the internal number line, as assessed by these 2 effects, seem to be of little importance when it comes to the acquisition of the content of 5th- and 6th-grade mathematics lessons. The results instead highlight the importance of conceptual understanding and estimation competence. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

"I know, you know": Epistemic egocentrism in children and adults.

This article reviews the evidence and theory pertaining to a form of perspective-taking failure--a difficulty in setting aside the privileged information that one knows to be unavailable to another party. The authors argue that this bias (epistemic egocentrism, or EE) is a general feature of human cognition and has been tapped by 2 independent and largely uncommunicating research traditions: the theory-of-mind tradition in developmental psychology and, with more sensitive probes, the "heuristics and biases" tradition in the psychology of human judgment. This article sets the stage for facilitating communication between these traditions as well as for the recognition of EE's breadth and potential interdisciplinary significance. The authors propose a life-span account and a tentative taxonomy of EE; and they highlight the interdisciplinary significance of EE by discussing its implications for normative ethics. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Developmental change in the acuity of the number sense: The approximate number system in 3-, 4-, 5-, and 6-year-olds and adults.

Behavioral, neuropsychological, and brain imaging research points to a dedicated system for processing number that is shared across development and across species. This foundational Approximate Number System (ANS) operates over multiple modalities, forming representations of the number of objects, sounds, or events in a scene. This system is imprecise and hence differs from exact counting. Evidence suggests that the resolution of the ANS, as specified by a Weber fraction, increases with age such that adults can discriminate numerosities that infants cannot. However, the Weber fraction has yet to be determined for participants of any age between 9 months and adulthood, leaving its developmental trajectory unclear. Here we identify the Weber fraction of the ANS in 3-, 4-, 5-, and 6-year-old children and in adults. We show that the resolution of this system continues to increase throughout childhood, with adultlike levels of acuity attained surprisingly late in development. (PsycINFO Database Record (c) 2010 APA, all rights reserved)