Concept-procedure interactions in children's addition and subtraction

A 3-week problem-solving practice phase was used to investigate concept-procedure interactions in children’s addition and subtraction. A total of 72 7- and 8-year-olds completed a pretest and posttest in which their accuracy and procedures on randomly ordered problems were recorded along with their reports of using concept-based relations in problem solving and their conceptual explanations. The results revealed that conceptual sequencing of practice problems enhances children’s ability to extend their procedural learning to new unpracticed problems. They also showed that well-structured procedural practice leads to improvement in children’s ability to verbalize key concepts. Moreover, children’s conceptual advances were predicted by their initial procedural skills. These results support an iterative account of the development of basic concepts and key skills in children’s addition and subtraction.     

Children’s understanding of addition and subtraction concepts

After the onset of formal schooling, little is known about the development of children’s understanding of the arithmetic concepts of inversion and associativity. On problems of the form a + b − b (e.g., 3 + 26 − 26), if children understand the inversion concept (i.e., that addition and subtraction are inverse operations), then no calculations are needed to solve the problem. On problems of the form a + b − c (e.g., 3 + 27 − 23), if children understand the associativity concept (i.e., that the addition and subtraction can be solved in any order), then the second part of the problem can be solved first. Children in Grades 2, 3, and 4 solved both types of problems and then were given a demonstration of how to apply both concepts. Approval of each concept and preference of a conceptual approach versus an algorithmic approach were measured. Few grade differences were found on either task. Conceptual understanding was greater for inversion than for associativity on both tasks. Clusters of participants in all grades showed that some had strong understanding of both concepts, some had strong understanding of the inversion concept only, and others had weak understanding of both concepts. The findings highlight the lack of developmental increases and the large individual differences in conceptual understanding on two arithmetic concepts during the early school years.     

Number words in young children's conceptual and procedural knowledge of addition, subtraction and inversion

Three studies addressed children's arithmetic. First, 50 3- to 5-year-olds judged physical demonstrations of addition, subtraction and inversion, with and without number words. Second, 20 3- to 4-year-olds made equivalence judgments of additions and subtractions. Third, 60 4- to 6-year-olds solved addition, subtraction and inversion problems that varied according to the inclusion of concrete referents and number words. The results indicate that number words play a different role in conceptual and procedural development. Children have strong addition and subtraction concepts before they can translate the physical effects of these operations into number words. However, using number words does not detract from their calculation procedures. Moreover, consistent with iterative relations between conceptual and procedural development, the results suggest that inversion acquisition depends on children's calculation procedures and that inversion understanding influences these procedures.     

Children's understanding of the relationship between addition and subtraction

In learning mathematics, children must master fundamental logical relationships, including the inverse relationship between addition and subtraction. At the start of elementary school, children lack generalized understanding of this relationship in the context of exact arithmetic problems: they fail to judge, for example, that 12 + 9 − 9 yields 12. Here, we investigate whether preschool children’s approximate number knowledge nevertheless supports understanding of this relationship. Five-year-old children were more accurate on approximate large-number arithmetic problems that involved an inverse transformation than those that did not, when problems were presented in either non-symbolic or symbolic form. In contrast they showed no advantage for problems involving an inverse transformation when exact arithmetic was involved. Prior to formal schooling, children therefore show generalized understanding of at least one logical principle of arithmetic. The teaching of mathematics may be enhanced by building on this understanding.     

Children's understanding of ownership transfers

An understanding of ownership entails the recognition that ownership can be transferred permanently and the ability to differentiate legitimate from illegitimate transfers. Two experiments explored the development of this understanding in 2-, 3-, 4- and 5-year olds, using stories about gift-giving and stealing. The possibility that children use simple biases to identify owners, such as a first possessor, current possessor or a loan bias, was also investigated. Five-year olds appropriately acknowledged a permanent transfer of ownership in the case of giving but not stealing. Four-year olds allowed permanent transfers but struggled to differentiate legitimate from illegitimate transfers. Many 4-year olds allowed adults, but not children, to keep property that had been stolen. Two- and 3-year olds exhibited a first possessor bias for both stories. We conclude that, by 5 years of age, children possess a mature understanding of ownership transfer whereas younger children are prone to biases.     

Strategies in subtraction problem solving in children

The aim of this study was to investigate the strategies used by third graders in solving the 81 elementary subtractions that are the inverses of the one-digit additions with addends from 1 to 9 recently studied by Barrouillet and Lépine. Although the pattern of relationship between individual differences in working memory, on the one hand, and strategy choices and response times, on the other, was the same in both operations, subtraction and addition differed in two important ways. First, the strategy of direct retrieval was less frequent in subtraction than in addition and was even less frequent in subtraction solving than the recourse to the corresponding additive fact. Second, contrary to addition, the retrieval of subtractive answers is confined to some peculiar problems involving 1 as the subtrahend or the remainder. The implications of these findings for developmental theories of mental arithmetic are discussed.     

Working memory and children's use of retrieval to solve addition problems

This study tested the hypothesis that children with high working memory capacities solve single-digit additions by direct retrieval of the answers from long-term memory more often than do children with low working memory capacities. Counting and reading letter span tasks were administered to groups of third-grade (mean age=107 months) and fourth-grade (mean age=118 months) children who were also asked to solve 40 single-digit additions. High working memory capacity was associated with more frequent use of retrieval and faster responses in solving additions. The effect of span on the use of retrieval increased with the size of the minimum addend. The relation between working memory measures and use and speed of retrieval did not depend on the numerical or verbal nature of the working memory task. Implications for developmental theories of cognitive arithmetic and theories of working memory are discussed.     

Operational momentum in large-number addition and subtraction by 9-month-olds

Recent studies on nonsymbolic arithmetic have illustrated that under conditions that prevent exact calculation, adults display a systematic tendency to overestimate the answers to addition problems and underestimate the answers to subtraction problems. It has been suggested that this operational momentum results from exposure to a culture-specific practice of representing numbers spatially; alternatively, the mind may represent numbers in spatial terms from early in development. In the current study, we asked whether operational momentum is present during infancy, prior to exposure to culture-specific representations of numbers. Infants (9-month-olds) were shown videos of events involving the addition or subtraction of objects with three different types of outcomes: numerically correct, too large, and too small. Infants looked significantly longer only at those incorrect outcomes that violated the momentum of the arithmetic operation (i.e., at too-large outcomes in subtraction events and too-small outcomes in addition events). The presence of operational momentum during infancy indicates developmental continuity in the underlying mechanisms used when operating over numerical representations.     

Children's understanding of the relation between addition and subtraction: inversion, identity, and decomposition.

In order to understand addition and subtraction fully, children have to know about the relation between these two operations. We looked at this knowledge in two studies. In one we asked whether 5- and 6-year-old children understand that addition and subtraction cancel each other out and whether this understanding is based on the identity of the addend and subtrahend or on their quantity. We showed that children at this age use the inversion principle even when the addend and subtrahend are the same in quantity but involve different material. In our second study we showed that 6- to 8-year-old children also use the inversion in combination with decomposition to solve a + b - (b + 1) problems. In both studies, factor analyses suggested that the children were using different strategies in the control problems, which require computation, than in the inversion problems, which do not. We conclude that young children understand the relations between addition and subtraction and that this understanding may not be based on their computational skills.     

Children's understanding of the arithmetic concepts of inversion and associativity

Previous studies have shown that even preschoolers can solve inversion problems of the form a+b-b by using the knowledge that addition and subtraction are inverse operations. In this study, a new type of inversion problem of the form d x e/e was also examined. Grade 6 and 8 students solved inversion problems of both types as well as standard problems of the form a+b-c and d x e/f. Students in both grades used the inversion concept on both types of inversion problems, although older students used inversion more frequently and inversion was used most frequently on the addition/subtraction problems. No transfer effects were found from one type of inversion problem to the other. Students who used the concept of associativity on the addition/subtraction standard problems (e.g., a+b-c=[b-c]+a) were more likely to use the concept of inversion on the inversion problems, although overall implementation of the associativity concept was infrequent. The findings suggest that further study of inversion and associativity is important for understanding conceptual development in arithmetic.     

Children's Developing Intuitions About the Truth Conditions and Implications of Novel Generics Versus Quantified Statements

Generic statements express generalizations about categories and present a unique semantic profile that is distinct from quantified statements. This paper reports two studies examining the development of children's intuitions about the semantics of generics and how they differ from statements quantified by all, most, and some. Results reveal that, like adults, preschoolers (a) recognize that generics have flexible truth conditions and are capable of representing a wide range of prevalence levels; and (b) interpret novel generics as having near‐universal prevalence implications. Results further show that by age 4, children are beginning to differentiate the meaning of generics and quantified statements; however, even 7‐ to 11‐year‐olds are not adultlike in their intuitions about the meaning of most‐quantified statements. Overall, these studies suggest that by preschool, children interpret generics in much the same way that adults do; however, mastery of the semantics of quantified statements follows a more protracted course.     

Where cognitive development and aging meet: face learning ability peaks after age 30

Research on age-related cognitive change traditionally focuses on either development or aging, where development ends with adulthood and aging begins around 55 years. This approach ignores age-related changes during the 35 years in-between, implying that this period is uninformative. Here we investigated face recognition as an ability that may mature late relative to other abilities. Using data from over 60,000 participants, we traced the ability to learn new faces from pre-adolescence through middle age. In three separate experiments, we show that face learning ability improves until just after age 30 – even though other putatively related abilities (inverted face recognition and name recognition) stop showing age-related improvements years earlier. Our data provide the first behavioral evidence for late maturation of face processing and the dissociation of face recognition from other abilities over time demonstrates that studies on adult age development can provide insight into the organization and development of cognitive systems.     

Operator and operand preview effects in simple addition and multiplication: A comparison of Canadian and Chinese adults

Using an arithmetic-based retrieval-induced forgetting (RIF) paradigm, researchers have found evidence that participants with very high arithmetic proficiency (Chinese adults), but not less-skilled participants (Canadian adults), solved some simple additions (e.g. 3 + 2) using fast procedural skills. Here we sought converging evidence for this using the operator-priming paradigm. Previous research testing simple addition and multiplication found that a 150-ms preview of the operator (+ or ×) facilitated only addition performance. This was taken as evidence that addition, but not multiplication, was solved by procedural algorithms that could be primed by presentation of the plus sign. In the present study, Chinese and Canadian adults (N = 144) were tested in the operator-priming paradigm but, in contrast to the RIF results, there was little evidence that operator-priming effects differed between the groups and robust operator priming was observed in both addition and multiplication. Thus, the operator preview results did not reinforce the results of previous research but the experiment revealed robust group differences in operand preview effects: For the Chinese, but not the Canadians, a preview of the numerical operands produced much greater facilitation for multiplication than addition. The fact that CN obtained a mean 103-ms gain for multiplication from the 150-ms preview of the operands strongly suggests that multiplication was their default operation in this paradigm. This result adds a potentially important new phenomenon to the behavioural distinctions between Chinese and North American adults' arithmetic skills.     

Temperamental profiles and linguistic development: Differences in the quality of linguistic production in relation to temperament in children of 28 months

The temperamental constellations that can be found in the infant population may influence the development trajectories of single domains of knowledge, such as that relative to language. The main objective of this study is to identify temperamental profiles to which one associates different levels of linguistic competence and to identify the profile associated with the highest risk for language acquisition. The temperamental characteristics of a sample of 106 children of 28 months attending day-care centres were surveyed and three temperamental profiles were highlighted: a profile typical of the Italian population which grouped most of the children; another made up of easily distractible and not very persistent children, who show a poor capacity to modulate motor activity and finally, the third with children inhibited in new situations. A comparison of the three groups on the basis of the level of linguistic competence revealed important differences regarding certain indices such as the vocabulary size and composition: in particular, the group of “inattentive” children has a more “immature” vocabulary composition, characterised by the presence of more primitive components of the lexical repertory.     

Children's understanding of the inverse relation between multiplication and division

Children's understanding of the inversion concept in multiplication and division problems (i.e., that on problems of the form d * e/e no calculations are required) was investigated. Children in Grades 6, 7, and 8 completed an inversion problem-solving task, an assessment of procedures task, and a factual knowledge task of simple multiplication and division. Application of the inversion concept in the problem-solving task was low and constant across grades. Most participants approved of the inversion-based shortcut but only a slight majority preferred it. Three clusters of children were identified based on their performance on the three tasks. The inversion cluster used and approved of the inversion shortcut the most and had high factual knowledge. The negation cluster used the negation strategy, had lower approval of the inversion shortcut, and had medium factual knowledge. The computation cluster used computation and had the lowest approval and the weakest factual knowledge. The findings highlight the importance of addressing the multiplication and division inversion concept in theories of children's mathematical competence.     

Integrating working memory capacity and context-processing views of cognitive control

Individuals low in working memory capacity (WMC) exhibit impaired performance on a variety of cognitive control tasks. The executive-attention theory of WMC (Engle & Kane, [2004[) accounts for these findings as failures of goal maintenance and response conflict resolution. Similarly, the context-processing view (Braver et al., [2001]) provides an explanation of cognitive control deficits observed in schizophrenia patients and older adults that is based on the ability to maintain context information. Instead of maintenance deficits, the inhibition view (Hasher, Lustig, & Zacks, [2007]) states that older adults and individuals low in WMC primarily have an impairment in the ability to inhibit information. In the current experiment, we explored the relationships among these theories. Individuals differing in performance on complex span measures of WMC performed the AX-Continuous Performance Test to measure context-processing performance. High-WMC individuals were predicted to maintain the context afforded by the cue, whereas low-WMC individuals were predicted to fail to maintain the context information. Low-WMC individuals made more errors on AX and BX trials and were slower to respond correctly on AX, BX, and BY trials. The overall pattern of results is most consistent with both the executive-attention and context-processing theories of cognitive control.     

Children's understanding of death as the cessation of agency: a test using sleep versus death

An important problem faced by children is discriminating between entities capable of goal-directed action, i.e. intentional agents, and non-agents. In the case of discriminating between living and dead animals, including humans, this problem is particularly difficult, because of the large number of perceptual cues that living and dead animals share. However, there are potential costs of failing to discriminate between living and dead animals, including unnecessary vigilance and lost opportunities from failing to realize that an animal, such as an animal killed for food, is dead. This might have led to the evolution of mechanisms specifically for distinguishing between living and dead animals in terms of their ability to act. Here we test this hypothesis by examining patterns of inferences about sleeping and dead organisms by Shuar and German children between 3 and 5-years old. The results show that by age 4, causal cues to death block agency attributions to animals and people, whereas cues to sleep do not. The developmental trajectory of this pattern of inferences is identical across cultures, consistent with the hypothesis of a living/dead discrimination mechanism as a reliably developing part of core cognitive architecture.     

Children's use of addition to solve two‐digit subtraction problems

Subtraction problems of the type M ? S = ? can be solved with various mental calculation strategies. We investigated fourth‐ to sixth‐graders' use of the subtraction by addition strategy, first by fitting regression models to the reaction times of 32 two‐digit subtractions. These models represented three different strategy use patterns: the use of direct subtraction, subtraction by addition, and switching between the two strategies based on the magnitude of the subtrahend. Additionally, we compared performance on problems presented in two presentation formats, i.e., a subtraction format (81 ? 37 = .) and an addition format (37 + . = 81). Both methods converged to the conclusion that children of all three grades switched between direct subtraction and subtraction by addition based on the combination of two features of the subtrahend: If the subtrahend was smaller than the difference, direct subtraction was the dominant strategy; if the subtrahend was larger than the difference, subtraction by addition was mainly used. However, this performance pattern was only observed when the numerical distance between subtrahend and difference was large. These findings indicate that theoretical models of children's strategy choices in subtraction should include the nature of the subtrahend as an important factor in strategy selection.