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.     

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.     

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.     

Adults' understanding of inversion concepts: how does performance on addition and subtraction inversion problems compare to performance on multiplication and division inversion problems?

Problems of the form a + b - b have been used to assess conceptual understanding of the relationship between addition and subtraction. No study has investigated the same relationship between multiplication and division on problems of the form d x e / e. In both types of inversion problems, no calculation is required if the inverse relationship between the operations is understood. Adult participants solved addition/subtraction and multiplication/division inversion (e.g., 9 x 22 / 22) and standard (e.g., 2 + 27 - 28) problems. Participants started to use the inversion strategy earlier and more frequently on addition/subtraction problems. Participants took longer to solve both types of multiplication/division problems. Overall, conceptual understanding of the relationship between multiplication and division was not as strong as that between addition and subtraction. One explanation for this difference in performance is that the operation of division is more weakly represented and understood than the other operations and that this weakness affects performance on problems of the form d x e / e.     

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.     

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.     

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 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 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.     

Use of the mathematical principle of inversion in young children

An important issue in the development of mathematical cognition is the extent to which children use and understand fundamental mathematical concepts. We examined whether young children successfully use the principle of inversion and, if so, whether they do so based on qualitative identity, length, or quantity. Twenty-four preschool children and 24 children in Grade 1 were presented with three-term inversion problems (e.g., 3+2-2) and standard problems of similar magnitude (e.g., 2+4-3). Problems were presented in three conditions to determine whether children used inversion at all and, if so, whether their decisions were based on quantitative or nonquantitative features of the problems. Both preschool and Grade 1 children showed evidence of using inversion in a fully quantitative manner, indicating that this principle is available in some form prior to extensive formal instruction in arithmetic.     

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.     

Individual differences in children's understanding of inversion and arithmetical skill

Background and aims. In order to develop arithmetic expertise, children must understand arithmetic principles, such as the inverse relationship between addition and subtraction, in addition to learning calculation skills. We report two experiments that investigate children's understanding of the principle of inversion and the relationship between their conceptual understanding and arithmetical skills. Sample. A group of 127 children from primary schools took part in the study. The children were from 2 age groups (6–7 and 8–9 years). Methods. Children's accuracy on inverse and control problems in a variety of presentation formats and in canonical and non‐canonical forms was measured. Tests of general arithmetic ability were also administered. Results. Children consistently performed better on inverse than control problems, which indicates that they could make use of the inverse principle. Presentation format affected performance: picture presentation allowed children to apply their conceptual understanding flexibly regardless of the problem type, while word problems restricted their ability to use their conceptual knowledge. Cluster analyses revealed three subgroups with different profiles of conceptual understanding and arithmetical skill. Children in the ‘high ability’ and ‘low ability’ groups showed conceptual understanding that was in‐line with their arithmetical skill, whilst a 3rd group of children had more advanced conceptual understanding than arithmetical skill. Conclusions. The three subgroups may represent different points along a single developmental path or distinct developmental paths. The discovery of the existence of the three groups has important consequences for education. It demonstrates the importance of considering the pattern of individual children's conceptual understanding and problem‐solving skills.     

Evidence for Use of Mathematical Inversion By Three-Year-Old Children

The principle of inversion—that a + b ? b must equal a—requires a sensitivity to the relation between addition and subtraction that is critical for understanding arithmetic. Use of inversion, albeit inconsistent, has been observed in school-age children, but when use of a computational shortcut based on inversion emerges and how awareness of the inversion principle relates to other mathematical or numerical skills remain unclear. Two possibilities were explored in 3-year-olds by adapting a method used previously with older children involving the addition and subtraction of blocks differing in length. These children were significantly more accurate on inversion than standard problems, this difference was observed even in children who did not count well, and performance did not differ between formats that afforded qualitative or quantitative solutions. Thus, 3-year-olds appear to develop an early sensitivity to quantitative inversion.     

Using addition to solve large subtractions in the number domain up to 20

This study examined 25 university students’ use of addition to solve large single-digit subtractions by contrasting performance in the standard subtraction format (12 − 9 = .) and in the addition format (9 + . = 12). In particular, we investigated the effect of the relative size of the subtrahend on performance in both formats. We found a significant interaction between format, the magnitude of the subtrahend (S) compared to the difference (D) (S > D vs. S < D), and the numerical distance between subtrahend and difference. When the subtrahend was larger than the difference and S and D were far from each other (e.g., 12 − 9 = .), problems were solved faster in the addition than in the subtraction format; when the subtrahend was smaller than the difference and S and D were far from each other (e.g., 12 − 3 = .), problems were solved faster in the subtraction than in the addition format. However, when the subtrahend and the difference were close to each other (e.g., 13 − 7 = .), there were no significant reaction time differences between both formats. These results suggest that adults do not rely exclusively and routinely on addition to solve large single-digit subtractions, but select either addition-based or subtraction-based strategies depending on the relative size of the subtrahend.     

Use of indirect addition in adults' mental subtraction in the number domain up to 1,000

This study examined adults' use of indirect addition and direct subtraction strategies on multi-digit subtractions in the number domain up to 1,000. Seventy students who differed in their level of arithmetic ability solved multi-digit subtractions in one choice and two no-choice conditions. Against the background of recent findings in elementary subtraction, we manipulated the size of the subtrahend compared to the difference and only selected items with large distances between these two integers. Results revealed that adults frequently and efficiently apply indirect addition on multi-digit subtractions, yet adults with higher arithmetic ability performed more efficiently than those with lower arithmetic ability. In both groups, indirect addition was more efficient than direct subtraction both on subtractions with a subtrahend much larger than the difference (e.g., 713 - 695) and on subtractions with a subtrahend much smaller than the difference (e.g., 613 - 67). Unexpectedly, only adults with lower arithmetic ability fitted their strategy choices to their individual strategy performance skills. Results are interpreted in terms of mathematical and cognitive perspectives on strategy efficiency and adaptiveness.     

Children’s strategies in complex arithmetic

Strategies used to solve two-digit addition problems (e.g., 27 + 48, Experiment 1) and two-digit subtraction problems (e.g., 73 – 59, Experiment 2) were investigated in adults and in children from Grades 3, 5, and 7. Participants were tested in choice and no-choice conditions. Results showed that (a) participants used the full decomposition strategy more often than the partial decomposition strategy to solve addition problems but used both strategies equally often to solve subtraction problems; (b) strategy use and execution were influenced by participants’ age, problem features, relative strategy performance, and whether the problems were displayed horizontally or vertically; and (c) age-related changes in complex arithmetic concern relative strategy use and execution as well as the relative influences of problem characteristics, strategy characteristics, and problem presentation on strategy choices and strategy performance. Implications of these findings for understanding age-related changes in strategic aspects of complex arithmetic performance are discussed.     

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.     

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.     

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.     

Preschoolers doing arithmetic: the concepts are willing but the working memory is weak.

The study of early mathematical development provides important insights into young children's emerging academic competencies and, potentially, a basis for adapting instructional methods. We presented nonverbal forms of two- and three-term arithmetic problems to 4-year-olds to determine (a) the extent to which certain information-processing demands make some problems more difficult than others and (b) whether preschoolers use arithmetic concepts spontaneously when solving novel problems. Children's accuracy on simple arithmetic problems (a + b and a - b) was strongly related (r2 = .88) to representational set size, the maximum number of units that need to be held in working memory to solve a given problem. Some children also showed spontaneous use of procedures based on the arithmetic principle of inversion when solving problems of the form a + b - b. These results highlight the importance of identifying information-processing and conceptual characteristics in the early development of mathematical cognition.