# Cognitive Development Is a Reconstruction Process that May Follow Different Pathways: The Case of Number

## Abstract

**:**

## 1. Introduction

## 2. Conceptual Framework

#### 2.1. The Notion of Reconstruction

#### 2.2. The Notion of Plurality

#### 2.3. The Notion of Interaction

#### 2.4. The Notion of Substitution1

## 3. Initial State of Numerical Cognition Development

#### 3.1. The Approximate Number System (ANS)

#### 3.2. The Parallel Individuation System (PIS)

## 4. Plurality of Processes Supporting Reconstruction

#### 4.1. Analogical Processing

#### 4.1.1. One-to-One Correspondence

#### 4.1.2. Early Finger Counting

#### 4.1.3. The Sequence of Number Words

#### 4.2. Symbolic Processing

#### 4.2.1. Subitizing

#### 4.2.2. Counting

## 5. Interaction

#### 5.1. Does ANS Have an Effect on the Development of the Symbolic System?

#### 5.2. Does Learning the Symbolic System Have an Effect on ANS Acuity?

## 6. Substitution and Variabilities

#### 6.1. An Example of Substitution Relationships in Typical Development

#### 6.1.1. Two Hypotheses about the Route to Cardinality

#### The ANS-to-Word Pathway

#### The PIS-to-Word Pathway

#### 6.1.2. Experiments Aimed at Choosing between These Two Hypotheses

#### 6.1.3. A Possible Interpretation of the Observed Variabilities

#### 6.2. An Example of Compensation in Atypical Development

#### 6.3. An Example of Substitution in Arithmetic Problem Solving

## 7. Discussion

## Conflicts of Interest

## References

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1 | In his model, Reuchlin [16] used the French word “vicariance” to refer to this type of relationship between processes. For lack of an exact equivalent in English, I use the expression “substitution” to translate this term. |

2 | Weber’s fraction w, which is constant across a range of numerosities, is the difference between the two closest discriminable numerosities, normalized by their size. The same information is given by the coefficient of variation (standard deviation/mean). |

3 | In Piaget’s theory, Equilibration is a general process of cognitive functioning which, whatever the knowledge domain, regulates: (1) the equilibrium between the assimilation of objects into the action schemes of the subject and the accommodation of these schemes to the objects; (2) the equilibrium in reciprocal assimilation and accommodation between schemes; (3) at the higher level, equilibrium between the differentiation of schemes and their integration in a more general structure [31]. |

4 | In the literature, these processes are in fact usually called non-symbolic or non-verbal, which emphasizes what they are not and advantageously avoids having to make a statement about what they are. In my mind, they are analogical, but this is clearly a point of discussion. |

5 | This is not yet a relation of numerical equality because if, say, the identity is modified by replacing an item in one of the two collections by an item that is not identical to it, the relation of numerical equality is disrupted in the child’s eyes [33]. |

6 | The successor function is a rule establishing the existence of a minimal quantity—one—that corresponds to the minimal distance between two consecutive numbers. |

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**MDPI and ACS Style**

Lautrey, J. Cognitive Development Is a Reconstruction Process that May Follow Different Pathways: The Case of Number. *J. Intell.* **2018**, *6*, 15.
https://doi.org/10.3390/jintelligence6010015

**AMA Style**

Lautrey J. Cognitive Development Is a Reconstruction Process that May Follow Different Pathways: The Case of Number. *Journal of Intelligence*. 2018; 6(1):15.
https://doi.org/10.3390/jintelligence6010015

**Chicago/Turabian Style**

Lautrey, Jacques. 2018. "Cognitive Development Is a Reconstruction Process that May Follow Different Pathways: The Case of Number" *Journal of Intelligence* 6, no. 1: 15.
https://doi.org/10.3390/jintelligence6010015