Investigating children’s understanding of inversion using the missing number paradigm

The development of conceptual understanding in arithmetic is a gradual process and children may make use of a concept in some situations before others. Previous research has demonstrated that when children are given arithmetic problems with an inverse relationship they can infer that the initial and final quantities are the same (e.g. 15 + 8–8 = □ ). However, we do not know whether children can perform the complementary inference that if the initial and final quantities are the same there must be an inverse relationship (i.e. 15 + □ −8=15 or 15+8−□ = 15). This paper reports two experiments that presented inverse problems in a missing number paradigm to test whether children (aged 8–9) could perform both these types of inferences. Children were more accurate on standard inverse problems (a + b−b = a) than on control problems (a + b−c = d), and their performance was best of all on rearranged inverse problems (b−b + a = a). The children’s performance on inverse problems was affected by the position of the missing number and also by the order of elements within the problem. This may be due to the different types of inferences that children must make to solve these kinds of inverse problems

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.Psycholog

Sampling from the mental number line: how are approximate number system representations formed?

Nonsymbolic comparison tasks are commonly used to index the acuity of an individual’s Approximate Number System (ANS), a cognitive mechanism believed to be involved in the development of number skills. Here we asked whether the time that an individual spends observing numerical stimuli influences the precision of the resultant ANS representations. Contrary to standard computational models of the ANS, we found that the longer the stimulus was displayed, the more precise was the resultant representation. We propose an adaptation of the standard model, and suggest that this finding has significant methodological implications for numerical cognition research

Process- and object-based thinking in arithmetic

Many influential theorists have proposed that learners construct mathematical objects via the encapsulation (or reification) of processes into objects. These processto- object theories posit that object-based thinking comes later in the developmental path than process-based thinking. In this paper we directly test this hypothesis in the field of early arithmetic. An experiment is reported which studied 8 and 9 year-old children’s use of the inverse relationship between addition and subtraction. We demonstrate that a subset of children were unable to solve arithmetic problems using process-based thinking, but that, nevertheless, they were able to use the inverse relationship between addition and subtraction to solve problems where appropriate. The implications of these findings for process-to-object theories are discussed

Can children construct inverse relations in arithmetic? Evidence for individual differences in the development of conceptual understanding and computational skill

Understanding conceptual relationships is an important aspect of learning arithmetic. Most studies of arithmetic, however, do not distinguish between children's understanding of a concept and their ability to identify situations in which it might be relevant. We compared 8- to 9-year-old children's use of a computational shortcut based on the inverse relationship between addition and subtraction, in problems where it was transparently applicable (e.g. 17+11−11=□) and where it was not (e.g. 15+11−8−3=□). Most children were able to construct inverse transformations and apply the shortcut in at least some situations, although they used the shortcut more for problems where it was transparently applicable. There were individual differences in the relationship between children's understanding of the inverse relationship and computational skill that have implications for theories of mathematical development

Teachers' understanding of the role of executive functions in mathematics learning

Cognitive psychology research has suggested an important role for executive functions, the set of skills that monitor and control thought and action, in learning mathematics. However, there is currently little evidence about whether teachers are aware of the importance of these skills and, if so, how they come by this information. We conducted an online survey of teachers' views on the importance of a range of skills for mathematics learning. Teachers rated executive function skills, and in particular inhibition and shifting, to be important for mathematics. The value placed on executive function skills increased with increasing teaching experience. Most teachers reported that they were aware of these skills, although few knew the term “executive functions.” This awareness had come about through their teaching experience rather than from formal instruction. Researchers and teacher educators could do more to highlight the importance of these skills to trainee or new teachers

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

Children’s mapping between symbolic and nonsymbolic representations of number

When children learn to count and acquire a symbolic system for representing numbers, they map these symbols onto a preexisting system involving approximate nonsymbolic representations of quantity. Little is known about this mapping process, how it develops, and its role in the performance of formal mathematics. Using a novel task to assess children’s mapping ability, we show that children can map in both directions between symbolic and nonsymbolic numerical representations and that this ability develops between 6 and 8 years of age. Moreover, we reveal that children’s mapping ability is related to their achievement on tests of school mathematics over and above the variance accounted for by standard symbolic and nonsymbolic numerical tasks. These findings support the proposal that underlying nonsymbolic representations play a role in children’s mathematical development

Inhibition in dot comparison tasks

Dot comparison tasks are commonly used to index an individual’s Approximate Number System (ANS) acuity, but the cognitive processes involved in completing these tasks are poorly understood. Here we investigated how factors including numerosity ratio, set size and visual cues influence task performance. Forty-­four children aged 7-­9 years completed a dot comparison task with a range of to-­be-­compared numerosities. We found that as the size of the numerosities increased, with ratios held constant, accuracy decreased due to the heightened salience of incongruent visual information. Furthermore, in trials with larger numerosities participants’ accuracies were influenced more by the convex hull of the array than the average dot size. The numerosity ratio between the arrays in each trial was an important predictor for all set sizes. We argue that these findings are consistent with a ‘competing processes’ inhibition-­based account, where accuracy scores are influenced by individual differences in both ANS acuity and inhibitory control skills