Moving attention along the mental number line in preschool age: Study of the operational momentum in 3‐ to 5‐year‐old children's non‐symbolic arithmetic

People tend to underestimate subtraction and overestimate addition outcomes and to associate subtraction with the left side and addition with the right side. These two phenomena are collectively labeled 'operational momentum' (OM) and thought to have their origins in the same mechanism of 'moving attention along the mental number line'. OM in arithmetic has never been tested in children at the preschool age, which is critical for numerical development. In this study, 3–5 years old were tested with non‐symbolic addition and subtraction tasks. Their level of understanding of counting principles (CP) was assessed using the give‐a‐number task. When the second operand's cardinality was 5 or 6 (Experiment 1), the child's reaction time was shorter in addition/subtraction tasks after cuing attention appropriately to the right/left. Adding/subtracting one element (Experiment 2) revealed a more complex developmental pattern. Before acquiring CP, the children showed generalized overestimation bias. Underestimation in addition and overestimation in subtraction emerged only after mastering CP. No clear spatial‐directional OM pattern was found, however, the response time to rightward/leftward cues in addition/subtraction again depended on stage of mastering CP. Although the results support the hypothesis about engagement of spatial attention in early numerical processing, they point to at least partial independence of the spatial‐directional and magnitude OM. This undermines the canonical version of the number line‐based hypothesis. Mapping numerical magnitudes to space may be a complex process that undergoes reorganization during the period of acquisition of symbolic representations of numbers. Some hypotheses concerning the role of spatial‐numerical associations in numerical development are proposed.     

Toddler subtraction with large sets: further evidence for an analog-magnitude representation of number

Two experiments were conducted to test the hypothesis that toddlers have access to an analog-magnitude number representation that supports numerical reasoning about relatively large numbers. Three-year-olds were presented with subtraction problems in which initial set size and proportions subtracted were systematically varied. Two sets of cookies were presented and then covered. The experimenter visibly subtracted cookies from the hidden sets, and the children were asked to choose which of the resulting sets had more. In Experiment 1, performance was above chance when high proportions of objects (3 versus 6) were subtracted from large sets (of 9) and for the subset of older participants (older than 3 years, 5 months; n = 15), performance was also above chance when high proportions (10 versus 20) were subtracted from the very large sets (of 30). In Experiment 2, which was conducted exclusively with older 3-year-olds and incorporated an important methodological control, the pattern of results for the subtraction tasks was replicated. In both experiments, success on the tasks was not related to counting ability. The results of these experiments support the hypothesis that young children have access to an analog-magnitude system for representing large approximate quantities, as performance on these subtraction tasks showed a Weber's Law signature, and was independent of conventional number knowledge.     

Subtraction by addition

University students’ self-reports indicate that they often solve basic subtraction problems (13?6=?) by reference to the corresponding addition problem (6+7=13; therefore, 13?6=7). In this case, solution latency should be faster with subtraction problems presented in addition format (6+_=13) than in standard subtraction format (13+6=_). In Experiment 1, the addition format resembled the standard layout for addition with the sum on the right (6+_=13), whereas in Experiment 2, the addition format resembled subtraction with the minuend on the left (13=6+_). Both experiments demonstrated a latency advantage for large problems (minuend > 10) in the addition format as compared with the subtraction format (13+6=_), although the effect was larger in Experiment 1 (254 msec) than in Experiment 2 (125 msec). Small subtractions (minuend ≤ 10) in Experiment 1 were solved equally quickly in the subtraction or addition format, but in Experiment 2, performance on small problems was faster in the standard format (5?3=_) than in the addition format (5=3+_). The results indicate that educated adults often use addition reference to solve large simple subtraction problems, but that they rely on direct memory retrieval for small subtractions.     

Children's number processing is context dependent

The aim of this paper was to test the hypothesis of a context dependence of number processing in children. Fifth-graders were given two numbers presented successively on screen through a self-presentation procedure after being asked either to add or subtract or compare them. We considered the self-presentation time of the first number as reflecting the complexity of the encoding for a given planned processing. In line with Dehaene's triple-code model, self-presentation times were longer for additions and subtractions than for comparisons with two-digit numbers. Alternative interpretations of these results in terms of more cognitive effort or more mental preparation in the case of addition and subtraction than comparison are discussed and ruled out.     

Influences of problem format and SES on preschoolers’ understanding of approximate addition

Recent studies suggest that 5-year-olds can add and compare large numerical quantities through approximate representations of number. However, the nature of this understanding and its susceptibility to environmental influences remain unclear. We examined whether children's early competence depends on the canonical problem format (i.e., arithmetic operations presented on the left side). Sixty children from middle-to-high-SES backgrounds (Experiment 1) and 47 children from low-SES backgrounds (Experiment 2) viewed events that required them to add and compare large numbers. Events were shown in a canonical or noncanonical format. Children from both SES backgrounds performed above chance on the approximate addition tasks, but children from middle-to-high-SES backgrounds performed significantly better. Moreover, children from middle-to-high SES backgrounds performed better when problems were presented in the canonical format, whereas children from low-SES backgrounds did not. These results suggest that children's understanding of approximate number is affected by some of the same environmental factors that affect performance on exact arithmetic tasks.     

The distributed learning effect for children's acquisition of an abstract syntactic construction

In many cognitive domains, learning is more effective when exemplars are distributed over a number of sessions than when they are all presented within one session. The present study investigated this distributed learning effect with respect to English-speaking children's acquisition of a complex grammatical construction. Forty-eight children aged 3;6–5;10 (Experiment 1) and 72 children aged 4;0–5;0 (Experiment 2) were given 10 exposures to the construction all in one session (massed), or on a schedule of two trials per day for 5 days (distributed-pairs), or one trial per day for 10 days (distributed). Children in both the distributed-pairs and distributed conditions learnt the construction better than children in the massed condition, as evidenced by productive use of this construction with a verb that had not been presented during training. Methodological and theoretical implications of this finding are discussed, with particular reference to single-process accounts of language acquisition.     

Trick or treat: Children's understanding of surprise

This study examines the hypothesis that an understanding of false belief would lead to a radical change in young children's understanding of surprise. In Experiment 1, children aged 3 to 8 years were asked to assess the knowledge state of another person and to then choose an object that would surprise that person. The results showed that whereas the 3-year-olds' choice of surprising object varied with the object, the 5-year-olds' choice of object varied with their assessment of the other's knowledge state. Hence, understanding surprise depends on an understanding of false belief. In Experiment 2, the number of questions was reduced and children were required to match a schematized facial expression to the object judged to be surprising. Again, older children, unlike their younger counterparts, pointed out that surprised faces are made when another's expectations are violated. Once children begin to ascribe belief states to others they begin to understand that surprise depends upon the unexpected. The results help resolve the differences in the findings of Wellman and Banerjee (1991) and Hadwin and Perner (1991) on children's understanding of surprise. In natural judgements, young children employ a principle of desirability; older children employ principles of belief violation.     

The role of relational reasoning in children's addition concepts

The study addresses the relational reasoning of different‐aged children and how addition reasoning is related to problem‐solving skills within addition and to reasoning skills outside addition. Ninety‐two 5‐ to 8‐year‐olds were asked to solve a series of conceptually related and unrelated addition problems, and the speed and accuracy of all self‐reported strategies were used to monitor their addition performance. Children were also given a series of general relational reasoning tasks to assess their ability to solve problems based on thematic, causal and visual relations. The results revealed that, while children were able to reason about commutativity relations, recognition of relations based on additive composition was rare. Furthermore, children's ability to reason about addition concepts increased with age and problem‐solving proficiency. Reasoning about addition concepts was related to performance on the thematic, causal and visual reasoning tasks for older children but not for younger children. Overall, the findings suggest that while children's early knowledge of addition relations is domain specific, as children develop in their broader reasoning abilities these developments enhance their addition reasoning.     

4～6岁儿童加减法反演律概念的发展与影响因素

儿童对加法和减法逆反关系的理解在加法概念和减法概念的学习中具有十分重要的作用。研究采用代数推理任务、给数取物任务、数量比较任务和记忆刷新任务,对83名4到6岁幼儿进行施测,考察4到6岁儿童加减法反演律概念的发展特点,探讨儿童的基数概念、数量比较、记忆刷新能力在反演律概念发展中的作用。结果发现:(1)5岁到6岁是儿童掌握和运用加减法反演律概念的快速发展时期。幼儿的加减法反演律概念表现出数量大小效应以及问题情境效应,小数反演问题的成绩优于大数反演问题的成绩,符号化数量反演的成绩优于集合数量反演的成绩。(2)基数概念掌握组儿童加减法反演律概念的发展显著优于未掌握组;但儿童基数概念的掌握情况并不显著预测儿童反演律概念的发展。(3)数量比较、记忆刷新对儿童加减法反演律概念的发展具有显著的正向预测作用。     

Children's appraisals of sex-typed behavior in their peers

Four experiments evaluated the effect of variations in sex-typed behavior in hypothetical peers on children's ratings of friendship. In all four studies, the children were heterogeneous with regard to social class, ethnicity, and race. In Experiment 1, children (71 boys, 90 girls) in Grades 3–6 read five stories about a target boy and in Experiment 2 (102 boys, 137 girls) about a target girl who displayed four sex-typed behaviors that ranged from exclusively masculine to exclusively feminine. In Experiment 1, boys preferred the exclusively masculine boy most as a friend. With each addition of a feminine behavior (and corresponding subtraction of a masculine behavior), the friendship ratings became increasingly negative. In contrast, the girls preferred the exclusively feminine boy most as a friend and, with each addition of a masculine behavior, the friendship ratings became increasingly negative. In Experiment 2, the converse was found although girls' ratings of friendship were less sharply affected by the target girl's sex-typed behavior than was observed for boys' ratings in Experiment 1. In Experiment 3, children (33 boys, 38 girls) in Grades K—2 were read three stories about a target boy, accompanied by detailed chromatic illustrations, whose four sex-typed behaviors were exclusively masculine, equally masculine and feminine, or exclusively feminine. The boys had significantly more favorable friendship ratings than the girls; however, in contrast to Experiments 1 and 2, the target boy's sex-typed behavior did not affect friendship ratings of either boys or girls. Experiment 4 (28 boys, 27 girls) repeated the procedure of Experiment 3 with children in kindergarten and Grade 1; in addition, the children made forced-choice friendship ratings for each of the three possible story pairs. In contrast to Experiment 3, boys' friendship ratings were affected by the target boy's sex-typed behavior, as observed in Experiment 1, but girls' friendship ratings were not. However, in the forced-choice situation, the boys significantly preferred the exclusively masculine boy whereas the girls significantly preferred the exclusively feminine boy. The results were discussed in relation to the influence sex-typed behavior has on modifying the effects of a peer's sex on affiliative preference and sex differences in appraisals of cross-gender behavior, including the concept of threshold effects.     

Children's Acquisition of Arithmetic Principles: The Role of Experience

The current study investigated how young learners' experiences with arithmetic equations can lead to learning of an arithmetic principle. The focus was elementary school children's acquisition of the Relation to Operands principle for subtraction (i.e., for natural numbers, the difference must be less than the minuend). In Experiment 1, children who viewed incorrect, principle-consistent equations and those who viewed a mix of incorrect, principle-consistent and principle-violation equations both showed gains in principle knowledge. However, children who viewed only principle-consistent equations did not. We hypothesized that improvements were due in part to improved encoding of relative magnitudes. In Experiment 2, children who practiced comparing numerical magnitudes increased their knowledge of the principle. Thus, experience that highlights the encoding of relative magnitude facilitates principle learning. This work shows that exposure to certain types of arithmetic equations can facilitate the learning of arithmetic principles, a fundamental aspect of early mathematical development.     

Inverse reference in subtraction performance: An analysis from arithmetic word problems

Studies of elementary calculation have shown that adults solve basic subtraction problems faster with problems presented in addition format (e.g., 6?+?_?=?13) than in standard subtraction format (e.g., 13 – 6?=?_). Therefore, it is considered that adults solve subtraction problems by reference to the inverse operation (e.g., for 13 – 6?=?7, “I know that 13 is 6?+?7”) because presenting the subtraction problem in addition format does not require the mental rearrangement of the problem elements into the addition format. In two experiments, we examine whether adults' use of addition to solve subtractions is modulated by the arrangement of minuend and subtrahend, regardless of format. To this end, we used arithmetic word problems since single-digit problems in subtraction format would not allow the subtrahend to appear before the minuend. In Experiment 1, subtractions were presented by arranging minuend and subtrahend according to previous research. In Experiment 2, operands were reversed. The overall results showed that participants benefited from word problems where the subtrahend appears before the minuend, including subtractions in standard subtraction format. These findings add to a growing body of literature that emphasizes the role of inverse reference in adults' performance on subtractions.     

Cognitive Analysis of Children's Mathematics Difficulties

The aim of the study is to investigate the informal and formal mathematical knowledge of children suffering from "mathematics difficulty" (MD). The research involves comparisons among three groups: fourth-grade children performing poorly in mathematics but normal in intelligence; fourth-grade peers matched for intelligence but experiencing no apparent difficulties in mathematics; and a randomly selected group of third graders. These children were individually presented with a large number of tasks designed to measure key mathematical concepts and skills. The findings suggest that: (1) MD children are not seriously deficient in key informal mathematical concepts and skills; (2) MD children seem to have elementary concepts of base ten notation but experience difficulty in related enumeration skills, particularly when large numbers are involved; (3) MD children's calculational errors often result from common error strategies; (4) MD children display severe difficulty in recalling common addition facts; and (5) in the area of problem solving, MD children are capable of "insightful" solutions and can solve simple forms of word problems, but experience difficulty with complex word problems. MD children are in many respects similar to normal, younger peers; an hypothesis of "essential cognitive normality" is advanced. The only and dramatic exception occurs in the area of number facts. While clinical experience corroborates this finding, its explanation is not evident.     

Do young children grasp the inverse relationship between addition and subtraction?: Evidence against early arithmetic

This research investigates young children’s reasoning about the inverse relationship between addition and subtraction. We argue that this investigation is necessary before asserting that preschoolers have a full understanding of addition and subtraction and use arithmetic principles. From the current models of quantification in infancy, we also propose that the children’s earliest ability to add and subtract is based on representations combining and separating sets of objects without arithmetical operations. In an initial study, 2- to 5-year-old children was tested on addition (2+1), subtraction (3−1) and inversion problems (2+1−1) by using Wynn’s procedure (1992b) of possible and impossible events. Only the oldest age group (4–5 years) succeeded on the inverse problem. In a follow-up study, 3- to 4-year-old children were given a brief training intervention in which they performed adding and subtracting transformations by manipulating small sets of objects without counting. The beneficial effects of the training support the claim that preschoolers respond to the inverse problem on the basis of object representations and not on the basis of numerical representations.     

Teaching Children to Add by Counting-On With One-Handed Finger Patterns

Three experiments report classroom instruction derived from cognitive research and focused on moving children across two developmental transitions: from counting-all to counting-on with entities present for the second addend and from the latter to counting-on as a number-word sequence solution procedure to solve symbolic single-digit addition problems. For the second transition, a new particularly efficient method of keeping track of number words counted-on was taught: counting-on with one-handed finger patterns. Classroom teachers were able to teach both kinds of transitions to most first and second graders, even to first graders below average in mathematics. Many first graders spontaneously transferred counting-on with finger patterns to the solution of addition word problems; some required instruction to do so. Most children were readily able to extend counting-on with finger patterns to counting up with finger patterns for subtraction. Counting-on with finger patterns was procedurally efficient enough to be used in instruction in multi digit addition problems; second graders successfully learned to use counting-on with finger patterns to add 10-digit numbers. Teaching these transitions to children permitted them to learn addition and subtraction topics from 1 to 4 years earlier than is usual in American schools.     

Cardinal limits: Evidence from language awareness and bilingualism for developing concepts of number

The study presented here investigated children's acquisition of cardinality in terms of a framework that isolates two cognitive processing components previously shown to be involved in children's metalinguistic development. These components are called analysis of knowledge and control of processing. In Study 1, children from 3 to 5 years of age were asked to solve three problems that required an understanding of cardinality. The problems were designed to place different demands on these processing components and examine their involvement in specific problems. In Study 2, bilingual children were given two of the problems to compare their performance to a new group of monolinguals. The results from both studies contribute to our knowledge of the development of cardinality by assessing children's ability across several tasks, by classifying those tasks in terms of their reliance on distinct processing components, and by providing a means for relating children's development of number concepts to their development of language. The results also contribute to our knowledge of the cognitive abilities of bilingual children. The implications of these results are that aspects of symbolic development may be a broadly based process that extends beyond domain-specific boundaries.     

The association between children's numerical magnitude processing and mental multi-digit subtraction

Children apply various strategies to mentally solve multi-digit subtraction problems and the efficient use of some of them may depend more or less on numerical magnitude processing. For example, the indirect addition strategy (solving 72–67 as “how much do I have to add up to 67 to get 72?”), which is particularly efficient when the two given numbers are close to each other, requires to determine the proximity of these two numbers, a process that may depend on numerical magnitude processing. In the present study, children completed a numerical magnitude comparison task and a number line estimation task, both in a symbolic and nonsymbolic format, to measure their numerical magnitude processing. We administered a multi-digit subtraction task, in which half of the items were specifically designed to elicit indirect addition. Partial correlational analyses, controlling for intellectual ability and motor speed, revealed significant associations between numerical magnitude processing and mental multi-digit subtraction. Additional analyses indicated that numerical magnitude processing was particularly important for those items for which the use of indirect addition is expected to be most efficient. Although this association was observed for both symbolic and nonsymbolic tasks, the strongest associations were found for the symbolic format, and they seemed to be more prominent on numerical magnitude comparison than on number line estimation.     

Identification and remediation of children's subtraction errors: A comparison of practical approaches

Three studies were conducted to determine the sources and frequency of children's subtraction errors and to examine four approaches to remediation. In Study 1, 56 third-grade children were asked to solve subtraction problems requiring borrowing (regrouping) and were questioned about their solution procedures. Ghildren who had difficulty either attempted to borrow incorrectly or made inversion errors. In Study 2, 80 third graders were promised rewards for correct solutions or were instructed to borrow on subtraction problems. Neither condition produced an increase in correct solutions, and instructions inflated incorrect attempts to borrow. In Study 3, 67 third-and fourth-grade children were assigned to one of three conditions: component skills for borrowing; feedback for performance; no training (control). Both treatment conditions resulted in a significant increase in the number of borrowing problems correctly solved, but neither condition produced a significant reduction in the number of computational, inversion, borrowing, or other errors. The results are discussed in terms of cost efficiency and constraints imposed by realities of today's classrooms. Future directions also are considered.     

L’outil « estimateur », la ligne numérique mentale et les habiletés arithmétiques

This report presents the effects of learning study based on the Estimator program to learn the addition and subtraction operations on children selected for mathematical difficulties. The Estimator is designed to link the magnitudes of the mental number line with the verbal representations of exact arithmetic. Experiment shows that using the Estimator for five 30-minute sessions increases not only the children's arithmetic capacities but also other numerical knowledge assessed with Zareki-R. By taking account of the limits of the sample, the results are discussed in terms of (re) educational implications.     

Children's profiles of addition and subtraction understanding

The current research explored children's ability to recognize and explain different concepts both with and without reference to physical objects so as to provide insight into the development of children's addition and subtraction understanding. In Study 1, 72 7- to 9-year-olds judged and explained a puppet's activities involving three conceptual relations: (a) a+b=c, b+a=c; (b) a-b=c, a-c=b; and (c) a+b=c, c-b=a. In Study 2, the self-reports and problem-solving accuracy of 60 5- to 7-year-olds were recorded for three-term inverse problems (i.e., a+b-b=?), pairs of complementary addition and subtraction problems (i.e., a+b=c, c-b=?), and unrelated addition and subtraction problems (e.g., 3-2). Both studies highlighted individual differences in the concepts that children understand and the role of concrete referents in their understanding. These differences were related to using efficient procedures to solve unrelated addition and subtraction problems in Study 2. The results suggest that a key advance in children's conceptual understanding is incorporating subtractive relations into their mental representations of how parts are added to form a whole.