Child Development Research

Volume 2015, Article ID 879258, 7 pages

http://dx.doi.org/10.1155/2015/879258

## The Role of Self-Action in 2-Year-Old Children: An Illustration of the Arithmetical Inversion Principle before Formal Schooling

^{1}CNRS, UMR 8240, LaPsyDÉ, 75000 Paris, France^{2}Université Paris Descartes, Sorbonne Paris Cité, UMR 8240, LaPsyDÉ, 75000 Paris, France^{3}Université de Caen Basse-Normandie, Normandie Université, UMR 8240, LaPsyDÉ, 14000 Caen, France^{4}Institut Universitaire de France (IUF), 75000 Paris, France

Received 1 August 2014; Revised 30 January 2015; Accepted 1 February 2015

Academic Editor: Andrew N. Meltzoff

Copyright © 2015 Amélie Lubin et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The importance of self-action and its considerable links with cognitive activity in childhood are known. For instance, in arithmetical cognition, 2-year-olds detected an impossible arithmetical outcome more accurately when they performed the operation themselves (actor mode) than when the experimenter presented it (onlooker mode). A key component in this domain concerns the understanding of the inversion principle between addition and subtraction. Complex operations can be solved without calculation by using an inversion-based shortcut (3-term problems of the form must equal *a*). Some studies have shown that, around the age of 4, children implicitly use the inversion principle. However, little is known before the age of 4. Here, we examined the role of self-action in the development of this principle by preschool children. In the first experiment, 2-year-olds were confronted with inversion ( or 2) and standard ( or 2) arithmetical problems either in actor or onlooker mode. The results revealed that actor mode improved accuracy for the inversion problem, suggesting that self-action helps children use the inversion-based shortcut. These results were strengthened with another inversion problem ( or 2) in a second experiment. Our data provide new support for the importance of considering self-action in early mathematics education.

#### 1. Introduction

A growing body of research is appearing in the field of embodied cognition [1–4]. Embodied cognition suggests that higher cognitive concepts may be based on bodily (i.e., action) experiences [5]. Self-actions modify how we think about the objects we encounter by interconnecting our representation of these objects with the sensory-motor experience associated with acting on objects [6]. The perspective that action and cognition are linked is, of course, not new in developmental psychology [4]. Piaget [7] first noted the influence of action on the development of intelligence. Seminal studies have shown the importance of motor activity and its considerable link with cognitive activity during childhood [2, 6, 8–14]. Embodied cognition is therefore highly relevant for education such as mathematics [1, 15]. Indeed, the beneficial role of children’s action in facilitating learning and thought has been highlighted in this domain [11, 13, 16–23]. In this study, we examined the role of embodied cognition, specifically the role of self-action (i.e., performing an action by oneself), in the development of the arithmetic principle in preschool children.

A key component in the development of arithmetic concerns the understanding of the inversion principle. Arithmetical inversion between addition and subtraction defines the relationship between these two operations: subtraction is the inverse of addition and* vice versa*. Indeed, adding a particular number to an array can be negated by subtracting the same number (i.e., must equal ). With this principle, a complex operation can be solved more efficiently by using a substitution method that is an inversion-based shortcut. The implicit use of this principle can be tested by determining whether children solve the inversion problem (e.g., calculation is not necessary to solve ) more accurately than a standard problem (e.g., calculation is required to solve ) of comparable size.

This topic has drawn considerable attention in recent years (see special issues [24, 25]), with notable discussion on the age at which children have a full grasp of this principle [26]. This major controversy may be explained by the fact that the definition of the term “inversion” is not universally consensual and sometimes refers to different concepts [24]. According to Piaget’s theory, the notion of arithmetical inversion is fundamental to mastering the nature of addition and subtraction. Children can understand this principle around the age of 6 or 7 years and can explain it later, around the age of 10. Before the age of 6, children can add and subtract without necessarily understanding the relationship between these operations [27, 28]. Along these lines, Canobi and collaborators observed that 6- to 9-year-old children often fail to use the inversion principle even after formal instruction [29–31]. However, some studies have shown that before formal schooling, around the age of 4, children implicitly use the inversion principle [32–36]. Some authors have argued that children seem to be sensitive to this principle even before the age of 4 [37, 38]. In Starkey and Gelman’s study [38], the experimenter placed some small objects in his hand and hid them. He added and subtracted other objects and asked the child how many he had in his hand. The children were presented with some inversion problems (, , and ) and a standard problem (). The 3-year-old children performed better on inversion problems than on the standard problem. Sherman and Bisanz [37] used a similar method. The experimenter presented an initial array of blocks () to the children. Then, the experimenter covered the middle of the array and added a number of blocks (). For the inversion problems, blocks were removed. For standard problems, another number of blocks () were removed. The children were asked to tell how many blocks remained in the array. The authors observed that children were better at the inversion problems. However, no significant correlation was obtained with children’s counting performances. The authors argued that very young children could solve inversion problems before they became skillful counters. Although it was not the objective of Izard et al. [39] to study the use of the inversion principle, the transformation proposed in the identity condition of experiment 4 seemed to be an inversion problem. They used a nonverbal manual search task with 2-year-olds. An experimenter put 5 (or 6) puppets in an opaque box, took a puppet out of the box, and, then, after a short delay, returned the puppet to the box (i.e., ). Children were invited to retrieve the puppets from the box. In this identity condition, the children succeeded, suggesting the use of the inversion principle by 2-year-olds. However, no standard problem was proposed here, as is usual in inversion experiments. Thus, it is difficult to draw accurate conclusions. Vilette [36] observed during a violation-of-expectation paradigm ([40], see also [41, 42]) that children grasp the inverse relationship between addition and subtraction only after the age of 4. Two Babar dolls were presented in a little theatre, and the experimenter closed an opaque screen. One Babar doll was added and immediately removed, and the children were presented with either a possible, , or an impossible, , outcome. They were then asked whether each outcome was correct. Although the 4-year-olds succeeded, the younger children failed to detect the impossible outcome. Note that the author did not present a standard problem (in which calculation was necessary) to assess the use of the shortcut strategy, ruling out the possibility of determining whether 4-year-old children used the inversion-based shortcut or simply calculated.

Given the effects of the method on children’s performance, it has been suggested that developmental psychologists must adapt their methods to the age of the children involved in the testing [13, 32, 43, 44]. In the studies cited above [36–39], the experimenter performed the operations, and the children often responded verbally (except [39], which used a manual search). However, from Piaget’s work, we know that, prior to language emergence, children often solve problems through action [45]. At this age, sensory-motor experiences are essential for gaining knowledge and developing cognitive abilities. Lubin and collaborators [13, 23] previously showed that allowing the child to be the actor during the experiment was effective in revealing early arithmetic abilities. They compared a violation-of-expectation method in which the experimenter presented the problem to the children (onlooker mode) with a new method based on action (actor mode) in which the children performed the operations by manipulating the material themselves. For addition problems (, 2, or 3), the results revealed that 2-year-old children detected erroneous results more accurately in actor mode than in onlooker mode [13]. Moreover, the children had better performance in onlooker mode after they solved a problem in an actor mode, suggesting a pedagogical effect of action on arithmetic performance at this age [23]. This methodology, which allows children to perform the set-up of the displays, favors the demonstration of implicit knowledge of arithmetic principles.

According to the aforementioned beneficial effect of self-action in arithmetical skills, we investigated the early roots of the inversion principle using the same methodology [13, 23]. We hypothesized that children might use the inversion principle to resolve 3-term problems more easily if they are the actors of the arithmetical operation. Indeed, embodying the inversion problem (to undo their self-action) should allow children to better integrate the cognitive shortcut based on inversion.

#### 2. Experiment 1

The first study tested our assumption directly by presenting inversion () and standard () 3-term arithmetic problems to 2-year-old children long before they attended formal schooling. We used a standard violation-of-expectation paradigm with onlooker and actor modes (see [13]). If the children used calculation to solve the presented problems, then the inversion and standard problems should be equally difficult. Alternatively, if 2-year-old children are able to implicitly use the inversion principle, then accuracy should be better in the inversion problem than in the standard problem. Moreover, actor mode should allow young children to detect erroneous arithmetic results more easily during the inversion problem (adding and subtracting with self-action) than simply observing an experimenter performing the same inversion problem (onlooker mode). In agreement with our previous results [13], only actor mode should allow children to spontaneously use the shortcut strategy that is cognitively less costly than calculation.

##### 2.1. Method

###### 2.1.1. Participants

Forty 2-year-old children recruited from childcare centers in Caen, France (mean age: 2 years and 8 months, range: 2 years and 5 months to 2 years and 11 months, 20 boys, middle-class homes, French native speakers), participated in this experiment. Written informed consent was obtained from parents.

###### 2.1.2. Design and Procedure

The children were tested individually for approximately 10 minutes in a quiet room. In the “onlooker mode” group, the experimenter presented the arithmetic operations to the children, whereas, in the “actor mode” group, the children executed the same arithmetic operations themselves (see [13] for more details). Half of the participants were randomly assigned to each of these two presentation modes. Two arithmetic problems of similar magnitude were used: an inversion problem “ or 2” (i.e., possible or impossible outcome) and a standard problem “ or 2.” In the inversion problem, one Babar doll was put in a box behind glass (by the experimenter or the child himself, according to the mode). Next, the screen was closed so that the children could no longer see the doll. One doll was then added to the first doll already in the box, and then one doll was removed (). In the standard problem, three dolls were added successively to the box. Next, the screen was closed, and two dolls were removed one at a time (). Each outcome (possible or impossible) was presented twice, and the order of the problem type was counterbalanced. In the two modes, the children were asked to determine which of the final outcomes (i.e., 1 or 2) was the correct result of the operation (Figure 1).