Non-symbolic arithmetic in adults and young children

Five experiments investigated whether adults and preschool children can perform simple arithmetic calculations on non-symbolic numerosities. Previous research has demonstrated that human adults, human infants, and non-human animals can process numerical quantities through approximate representations of their magnitudes. Here we consider whether these non-symbolic numerical representations might serve as a building block of uniquely human, learned mathematics. Both adults and children with no training in arithmetic successfully performed approximate arithmetic on large sets of elements. Success at these tasks did not depend on non-numerical continuous quantities, modality-specific quantity information, the adoption of alternative non-arithmetic strategies, or learned symbolic arithmetic knowledge. Abstract numerical quantity representations therefore are computationally functional and may provide a foundation for formal mathematics.     

The impact of misleading questions on eyewitness memory in children and adults

The study examined developmental patterns of the negative effect of misleading post‐event information in two different kinds of eyewitness interviews. A total of 284 subjects aged between five and 64 years were shown a short video about a theft and three weeks as well as four weeks later questioned about it. The social pressure in the interview after three weeks was manipulated by asking half of the subjects suggestive and misleading questions. The other half was asked open‐ended and unbiased questions. In the neutral interview four weeks later, all subjects were asked the same set of recognition questions about the event. The results revealed that preschoolers in particular had problems with the interplay between cognitive and social factors (social pressure induced through the wording of the misleading questions) in the interview setting after three weeks. In the neutral recognition test, all age groups were shown to suffer from prior misinformation to about the same extent. However, with an exception in the group of 6‐year‐old children the negative effect of prior misinformation on the accuracy of recognition proved to be due to items that were peripheral to the observed event. Copyright © 2000 John Wiley & Sons, Ltd.     

The role of fingers in the development of counting and arithmetic skills

Interactions between fingers and numbers have been reported in the existing literature on numerical cognition. The aim of the present research was to test whether hand interference movements might have an impact on children performance in counting and basic arithmetic problem solving. In Experiment 1, 5-year-old children had to perform both a one-target and a two-target counting task in three different conditions: with no constraints, while making interfering hand movements or while making interfering foot movements. In Experiment 2, first and fourth graders were required to perform addition problems under the same control and sensori-motor interfering conditions. In both tasks, the hand movements caused more disruption than the foot movements, suggesting that finger-counting plays a functional role in the development of counting and arithmetic.     

High working memory capacity favours the use of finger counting in six-year-old children

In this study, we show that 6-year-old children with high working memory capacity are more likely to use their fingers in an addition task than children with lower capacity. Moreover and as attested by a strong correlation between finger counting and accuracy in the arithmetic task, finger counting appears to be a very efficient strategy. Therefore, discovering the finger counting strategy seems to require a large amount of working memory resources, which could lack in low-span children. Furthermore, when children with low working memory capacities use their fingers to solve addition problems, they more often use the laborious counting-all strategy than children with higher capacities who use more elaborated procedures such as the Min strategy. Consequently, we suggest that explicit teaching of finger counting during the first years of schooling should be promoted because it could help less gifted children to overcome their difficulties in arithmetic.     

Auditory-motor coupling of bilateral finger tapping in children with and without DCD compared to adults

The ability to modulate bilateral finger tapping in time to different frequencies of an auditory beat was studied. Twenty children, 7 years of age, 10 with and 10 without developmental coordination disorder (DCD), and 10 adults tapped their left index and right middle fingers in an alternating pattern in time with an auditory signal for 15s (four trials each, randomly, at 0.8, 1.6, 2.4, 3.2 Hz per finger). Dominant and non-dominant finger data were collapsed since no differences emerged. All three groups were able to modulate their finger frequency across trials to closely approximate the signal frequency but children with DCD were unable to slow down to the lowest frequency. Children with DCD were more variable in tap accuracy (SD of relative phase) and between finger coordination than typically developing children who were respectively more variable than the adults. Children with DCD were unable to consistently synchronize their finger with the beat. Adults were tightly synchronized and often ahead of the beat while children without DCD tended to be behind the beat. Overall, these results indicated that children with DCD can only broadly match their finger movements to an auditory signal with variability and poor synchronicity as key features of their auditory-fine-motor control. Individual inspection of the data revealed that five children with DCD had difficulty matching the slowest frequencies and that these children also had higher variability and lower percentile MABC scores from the movement assessment battery for children (MABC) than other children with DCD. Three children with DCD were more variable only at higher frequencies and two performed like typically developing children.     

From algorithmic computing to direct retrieval: Evidence from number and alphabetic arithmetic in children and adults

A number of theories of mental arithmetic suggest that the ability to solve simple addition and subtraction problems develops from an algorithmic strategy toward a strategy based on the direct retrieval of the result from memory. In the experiment presented here, 2nd and 12th graders were asked to solve two tasks of number and alphabet arithmetic. The subjects transformed series of 1 to 4 numbers or letters (item span) by adding or subtracting an operand varying from 1 to 4 (operation span). Although both the item and operation span were associated with major and identical effects in the case of both numbers and letters at 2nd grade, such effects were clearly observable only in the case of letters for the adult subjects. This suggests the use of an algorithmic strategy for both types of material in the case of the children and for the letters only in the case of the adults, who retrieved numerical results directly from memory.     

The development of many-to-one counting in 4-year-old children

This study investigated the development of children's knowledge of many-to-one counting. Four-year-olds (n = 51) either participated in a training or in a control group. The training taught children how to allot Y physically not present items to each of X objects and how to count these items in the process. It was found that most children could be taught how to perform a many-to-one count. Counting performance was related to insight into the concept of multiplication. More specifically, learning of the many-to-one strategy was related to a better insight into aggregation over sets of items.     

Frequency of finger looking during finger counting is related to children's working memory capacities

Finger counting can be useful in solving arithmetic problems, noticeably because it reduces the working memory demand of mental calculations. However, proprioceptive information might not be sufficient to keep track of the number of fingers raised during problem solving, and visual input may play an important role in this process. The present study was designed to address this question and shows that 8-year-old children look at their fingers in 60% of the trials during finger counting when solving additive problems. Moreover, our results reveal that the frequency of finger looking is negatively correlated with working memory capacities and is higher for more difficult problems. These findings suggest that finger looking is recruited in managing the cognitive demand of the arithmetic task, probably by providing additional external cues to monitor the number of steps that have to be incremented during finger counting.     

Counting on working memory in simple arithmetic when counting is used for problem solving

A concurrent-task methodology was used to investigate relations between the availability of aspects of working memory resources and both strategy selection and execution while simple addition equations (e.g., 4 + 3 = 8) were being verified. Consistent with prior research in which production trials have been used, undergraduates selected a variety of procedures other than retrieval. Availability of working memory resources did not generally affect strategy selection. Disrupting central executive and phonological aspects of memory affected strategy execution, but only when min counting was used to solve the problems. These and other features of the results suggest that availability of working memory resources does not contribute to individual differences in strategy selection and time to execute retrieval processes.     

Subitizing and counting processes in young children

Children attending the first grade in school were instructed to tell the number of dots (1–9) presented on a screen. The response latencies were related to the number of dots by two different linear relations for each subject. The first of these had a slope of about 0.1 sec/dot, was applicable for the encoding of 1–3 dots, and was taken as an indication of a subitizing process. The second linear relation was applicable for 5–9 dots and had a slope of about 1.0 sec/dot reflecting the speed of a counting process. The average intersection between the functions was located at 3.22 dots. The results were compared with earlier investigations of adult subjects who on the average subitize 6 dots and count 1 dot in about 0.4 sec. It was subitizing process being higher for adults. If the encoding of a stimulus has not been terminated within about 1.5 sec for children and adults the stimulus has instead to be identified in either a counting or an estimating process.     

The relationship between phonological processing and arithmetic in children with learning disabilities

Phonological processing skills have not only been shown to be important for reading skills, but also for arithmetic skills. Specifically, previous research in typically developing children has suggested that phonological processing skills may be more closely related to arithmetic problems that are solved through fact retrieval (e.g., remembering the solution from memory) than procedural computation (e.g., counting). However, the relationship between phonological processing and arithmetic in children with learning disabilities (LDs) has not been investigated. Yet, understanding these relationships in children with LDs is especially important because it can help elucidate the cognitive underpinnings of math difficulties, explain why reading and math disabilities frequently co-occur, and provide information on which cognitive skills to target for interventions. In 63 children with LDs, we examined the relationship between different phonological processing skills (phonemic awareness, phonological memory, and rapid serial naming) and arithmetic. We distinguished between arithmetic problems that tend to be solved with fact retrieval versus procedural computation to determine whether phonological processing skills are differentially related to these two arithmetic processes. We found that phonemic awareness, but not phonological memory or rapid serial naming, was related to arithmetic fact retrieval. We also found no association between any phonological processing skills and procedural computation. These results converge with prior research in typically developing children and suggest that phonemic awareness is also related to arithmetic fact retrieval in children with LD. These results raise the possibility that phonemic awareness training might improve both reading and arithmetic fact retrieval skills.

Research Highlights

• Relationships between phonological processing and various arithmetic skills were investigated in children with learning disabilities (LDs) for the first time.
• We found phonemic awareness was related to arithmetic involving fact retrieval, but not to arithmetic involving procedural computation in LDs.
• The results suggest that phonemic awareness is not only important to skilled reading, but also to some aspects of arithmetic.
• These results raise the question of whether intervention in phonemic awareness might improve arithmetic fact retrieval skills.

     

The role of phonological and visual working memory in complex arithmetic for Chinese- and Canadian-educated adults

Two experiments were conducted to test cultural differences in the role of phonological and visual working memory in complex arithmetic. Canadian- and Chinese-educated students solved complex subtraction problems (e.g., 85 − 27; Experiment 1) and complex multiplication problems (e.g., 6 − 13; Experiment 2) under phonological and visual working memory loads. Problem complexity (i.e., borrow or carry operations) and presentation format (i.e., horizontal vs. vertical) were also manipulated. The results showed that both Chinese- and Canadian-educated participants relied on both phonological and visual working memory resources when solving complex subtraction and multiplication problems. Selective involvement of phonological and visual working memory as a function of operation (Lee & Kang, 2002) or presentation format (Trbovich & LeFev re, 2003) was found only for Chinese-educated participants and not for Canadian-educated participants, calling into question the generalizability of these findings across arithmetic operations and cultural groups.     

The impact of homelessness on children.

This article reviews and critiques community-based research on the effects of homelessness on children. Homeless children confront serious threats to their ability to succeed and their future well-being. Of particular concern are health problems, hunger, poor nutrition, developmental delays, anxiety, depression, behavioral problems, and educational underachievement. Factors that may mediate the observed outcomes include inadequate shelter conditions, instability in residences and shelters, inadequate services, and barriers to accessing services that are available. Public policy initiatives are needed to meet the needs of homeless children.     

Aspects of situated cognition in embodied numerosity: the case of finger counting

Numerical cognitions such as spatial-numerical associations have been observed to be influenced by grounded, embodied and situated factors. For the case of finger counting, grounded and embodied influences have been reported. However, situated influences, e.g., that reported counting habits change with perception and action within a given situation, have not been systematically examined. To pursue the issue of situatedness of reported finger-counting habits, 458 participants were tested in three separate groups: (1) spontaneous condition: counting with both hands available, (2) perceptual condition: counting with horizontal (left-to-right) perceptual arrangement of fingers (3) perceptual and proprioceptive condition: counting with horizontal (left-to-right) perceptual arrangement of fingers and with busy dominant hand. Report of typical counting habits differed strongly between the three conditions. 28 % reported to start counting with the left hand in the spontaneous counting condition (1), 54 % in the perceptual condition (2) and 62 % in the perceptual and proprioceptive condition (3). Additionally, all participants in the spontaneous counting group showed a symmetry-based counting pattern (with the thumb as number 6), while in the two other groups, a considerable number of participants exhibited a spatially continuous counting pattern (with the pinkie as number 6). Taken together, the study shows that reported finger-counting habits depend on the perceptual and proprioceptive situation and thus are strongly influenced by situated cognition. We suggest that this account reconciles apparently contradictory previous findings of different counting preferences regarding the starting hand in different examination situations.     

The effects of encoding distinctiveness on retrieval variability in children and adults

To what extent are developmental differences in encoding distinctiveness responsible for differences in retrieval variability? This study examined this question by comparing the effects of different kinds of encoding distinctiveness on the ability of children and adults to reinstate the input environment at retrieval. The critical manipulations involved the use of semantic orienting questions at both encoding and retrieval. Second and fourth (Experiment 1) or fifth (Experiment 2) graders and college adults were given moderately associated word pairs (Knife-Axe) at input. Encoding was free or constrained at input and retrieval. The retrieval questions biased the Same interpretation of the cue as at input (weapon), a uniquely Different interpretation (utensil), or an inappropriate Negative interpretation. Encoding distinctiveness was varied by crossing these manipulations with either picture or word input (Experiment 1) or general or distinctive orienting questions (Experiment 2). The results suggested that encoding distinctiveness and retrieval variability contribute independently to developmental differences in recall.     

Metacognition for strategy selection during arithmetic problem-solving in young and older adults

We examined participants’ strategy choices and metacognitive judgments during arithmetic problem-solving. Metacognitive judgments were collected either prospectively or retrospectively. We tested whether metacognitive judgments are related to strategy choices on the current problems and on the immediately following problems, and age-related differences in relations between metacognition and strategy choices. Data showed that both young and older adults were able to make accurate retrospective, but not prospective, judgments. Moreover, the accuracy of retrospective judgments was comparable in young and older adults when participants had to select and execute the better strategy. Metacognitive accuracy was even higher in older adults when participants had to only select the better strategy. Finally, low-confidence judgments on current items were more frequently followed by better strategy selection on immediately succeeding items than high-confidence judgments in both young and older adults. Implications of these findings to further our understanding of age-related differences and similarities in adults’ metacognitive monitoring and metacognitive regulation for strategy selection in the context of arithmetic problem solving are discussed.     

The own-age face recognition bias in children and adults

Children recognize children's faces more accurately than adult faces, and adults recognize adult faces more accurately than children's faces (e.g., Anastasi & Rhodes, 2005 Anastasi, J. S. and Rhodes, M. G. 2005. An own-age bias in face recognition for children and older adults. Psychonomic Bulletin and Review, 12: 1043–1047. [Crossref], [PubMed], [Web of Science ®] , [Google Scholar]). This is the own-age bias. Research has shown that this bias is at least partially based on experience since trainee teachers show less of an own-age bias than do other adults (Harrison & Hole, 2009 Harrison, V. and Hole, G. J. 2009. Evidence for a contact-based explanation of the own-age bias in face recognition. Psychonomic Bulletin & Review, 16: 264–269. [Crossref], [PubMed], [Web of Science ®] , [Google Scholar]). The present research tested the own-age bias in three groups of children (age 4–6, 7–9, 10–12 years) and a group of adults in the recognition of three age groups of faces (age 7–9, 20–22, and 65–90 years). Results showed an own-age bias for 7- to 9-year-old children and adults. Specifically, children could recognize faces more accurately if they were less than two years different from their own age than if they were more than two years older or younger. These results are discussed in terms of short-term experience with faces creating biases, and this rapidly changes with age.