Childrens understanding and use of inversion in arithmetic

Journal of Experimental Child Psychology, 2009

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 À b (e.g., 3 + 26 À 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 À c (e.g., 3 + 27 À 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.

Cognitive Development, 2002

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.

Cognitive Development, 2009

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.

Cognitive Development, 2008

The present research involved gauging preschoolers' learning potential for a key arithmetic concept, the addition-subtraction inverse principle (e.g., 2 + 1 − 1 = 2). Sixty 4-and 5-year-old Taiwanese children from two public preschools serving low-and middle-income families participated in the training experiment. Half were randomly assigned to an experimental group; half, to a control condition. Participants were tested for an understanding of inversion before and after intervention. One-third of the 5 year olds from both groups performed at the marginally or reliably successful levels before the intervention, and three quarters of them did so in the posttest. Only one of the 4 year olds was marginally successful before the intervention and 4 year olds in the experimental group somewhat benefited from the intervention. Significant social class effect were evident.

1984

The primary objective of this study was to provide an experimental model of children's representations of addition and subtraction concepts viewed as constructed schemes. How children with different counting schemes differ in their addition and subtraction concepts and how the types of problems children solve correlate with the addition and subtraction concepts were specifically explored. The 3-week study was conducted as a teaching experiment, with children's behavior observed and their mental processes probed-in interviews, and in teaching episodes. Eight children in grades 1 and 2 were selected to reflect possible variations in counting, addition, and subtraction schemes. Four representations of addition concepts and six representations of subtraction concepts were found, with one or more specific schemes identified with each representation. The schemes were classified by developmental levels. Children who constructed higher level schemes also solved all kinds of addition and subtraction problems which involved larger numbers. Children's uses of their schemes reflected awareness of the difficulty of a problem and basic understanding of adding and subtracting. (MNS)

British Journal of Developmental Psychology, 2008

Constructing inverse relations 1 Running head: CONSTRUCTING INVERSE RELATIONS Can children construct inverse relations in arithmetic? Evidence for individual differences in the development of conceptual understanding and computational skill.

Journal of Child Psychology and Psychiatry, 1995

Abstract—The development of children's understanding of mathematical relations and of their grasp of the number system is described. It is discussed that children easily recognise one-way part-part relations but that the number system at first causes them difficulty. Children's ...

Mathematical Thinking and Learning, 2009

The British journal of educational psychology, 2016

In the last decades, children's understanding of mathematical principles has become an important research topic. Different from the commutativity and inversion principles, only few studies have focused on children's understanding of the addition/subtraction complement principle (if a - b = c, then c + b = a), mainly relying on verbal techniques. This contribution aimed at deepening our understanding of children's knowledge of the addition/subtraction complement principle, combining verbal and non-verbal techniques. Participants were 67 third and fourth graders (9- to 10-year-olds). Children solved two tasks in which verbal reports as well as accuracy and speed data were collected. These two tasks differed only in the order of the problems and the instructions. In the looking-back task, children were told that sometimes the preceding problem might help to answer the next problem. In the baseline task, no helpful preceding items were offered. The looking-back task included...

Background. Two distinct abilities, mathematical reasoning and arithmetic skill, might make separate and specific contributions to mathematical achievement. However, there is little evidence to inform theory and educational practice on this matter.Aims. The aims of this study were (1) to assess whether mathematical reasoning and arithmetic make independent contributions to the longitudinal prediction of mathematical achievement over 5 years and (2) to test the specificity of this prediction.Sample. Data from Avon Longitudinal Study of Parents and Children (ALSPAC) were available on 2,579 participants for analyses of KS2 achievement and on 1,680 for the analyses of KS3 achievement.Method. Hierarchical regression analyses were used to assess the independence and specificity of the contribution of mathematical reasoning and arithmetic skill to the prediction of achievement in KS2 and KS3 mathematics, science, and English. Age, intelligence, and working memory (WM) were controls in these analyses.Results. Mathematical reasoning and arithmetic did make independent contributions to the prediction of mathematical achievement; mathematical reasoning was by far the stronger predictor of the two. These predictions were specific in so far as these measures were more strongly related to mathematics than to science or English. Intelligence and WM were non-specific predictors; intelligence contributed more to the prediction of science than of maths, and WM predicted maths and English equally well.Conclusions. There is clear justification for making a distinction between mathematical reasoning and arithmetic skills. The implication is that schools must plan explicitly to improve mathematical reasoning as well as arithmetic skills.