The concept of number constitutes the basis of the math, therefore its acquisition is the foundation on which mathematical knowledge is built. The concept of number has come to be conceived as a complex cognitive activity, in which different processes act in a coordinated manner.
From a very young age, Children develop what is known as a intuitive informal mathematics This development is due to the fact that children show a biological propensity to acquire basic arithmetic skills and to stimulation from the environment, since children from an early age encounter quantities in the physical world, quantities to count in the social world, and ideas. mathematics in the world of history and literature.
The development of number depends on schooling. Instruction in early childhood education in classification, serialization and conservation of number produces gains in reasoning ability and academic performance that are maintained over time.
Enumeration difficulties in young children interfere with the acquisition of mathematical skills in later childhood.
From the age of two, the first quantitative knowledge begins to develop. This development is completed through the acquisition of so-called proto-quantitative schemes and the first numerical skill: counting.
The schemes that enable the child’s ‘mathematical mind’
The first quantitative knowledge is acquired through three proto-quantitative schemes:
The protoquantitative scheme of the comparison: thanks to this, children can have a series of terms that express judgments of quantity without numerical precision, such as larger, smaller, more or less, etc. Using this scheme, linguistic labels are assigned to the size comparison.
The protoquantitative increment-decrement scheme: With this scheme, three-year-old children are able to reason about changes in quantities when an element is added or removed.
ANDThe part-whole protoquantitative scheme: allows preschoolers to accept that any piece can be divided into smaller parts and that if we put them back together they give rise to the original piece. They may reason that when they put two quantities together, they get a larger quantity. Implicitly they begin to know the auditory property of quantities.
These schemes are not sufficient to address quantitative tasks, so they need to use more precise quantification tools, such as counting.
He count It is an activity that may seem simple to the eyes of an adult but needs to integrate a series of techniques.
Some consider that counting is rote learning that lacks meaning, especially the standard numerical sequence, to gradually provide these routines with conceptual content.
Principles and skills needed to improve the counting task
Others consider that counting requires the acquisition of a series of principles that govern the skill and allow a progressive sophistication of the count:
The principle of one-to-one correspondence: involves labeling each element of a set only once. It involves the coordination of two processes: participation and labeling, through partition, they control the counted elements and those that remain to be counted, at the same time that they have a series of labels, so that each one corresponds to an object of the counted set. , even if they do not follow the correct sequence.
The principle of established order: stipulates that to count it is essential to establish a coherent sequence, although this principle can be applied without having to use the conventional numerical sequence.
The principle of cardinality – States that the last label of the number sequence represents the cardinal of the set, the number of elements the set contains.
The principle of abstraction: determines that the previous principles can be applied to any type of set, both with homogeneous elements and with heterogeneous elements.
The principle of irrelevance: indicates that the order in which the elements are started is irrelevant to their cardinal designation. They can be counted from right to left or vice versa, without affecting the result.
These principles establish the procedural rules for how to count a set of objects. Based on their own experiences, the child acquires the conventional numerical sequence and will allow them to establish how many elements a set has, that is, master counting.
Children often develop the belief that certain non-essential features of counting are essential, such as standard direction and adjacency. They are also the abstraction and irrelevance of order, which serve to guarantee and make more flexible the range of application of the previous principles.
The acquisition and development of strategic competence
Four dimensions have been described through which the development of students’ strategic competence is observed:
Repertoire of strategies: different strategies that a student uses when completing tasks.
Frequency of strategies: frequency with which each of the strategies is used by the child.
Strategies efficiency: accuracy and speed with which each strategy is executed.
Strategy selection: the child’s ability to select the most adaptive strategy in each situation and that allows him to be more efficient in carrying out tasks.
Prevalence, explanations and manifestations
The different estimates of the prevalence of difficulties in learning mathematics differ due to the different diagnostic criteria used.
He DSM-IV-TR indicates that The prevalence of calculation disorder has only been estimated at approximately one in five cases of learning disorder It is assumed that about 1% of school-age children suffer from a calculation disorder.
Recent studies affirm that the prevalence is higher. About 3% have comorbid difficulties in reading and mathematics.
Difficulties in mathematics also tend to be persistent over time.
What are children with Mathematics Learning Difficulties like?
Many studies have indicated that basic numerical skills such as number identification or comparison of magnitudes of numbers are intact in the majority of children with Difficulties in Learning Mathematics (onwards, DAM), at least in terms of simple numbers.
Many children with MAD have difficulty understanding some aspects of the count: most understand stable order and cardinality, at least they fail to understand one-to-one correspondence, especially when the first element is counting twice; and they systematically fail at tasks that involve understanding the irrelevance of order and adjacency.
The greatest difficulty of children with MAD lies in learning and remembering numerical facts and calculating arithmetic operations. They have two big problems: procedural and recovery of facts from the MLP. Knowledge of facts and understanding of procedures and strategies are two dissociable problems.
Procedural problems are likely to improve with experience, your difficulties with recovery will not. This is because procedural problems arise from a lack of conceptual knowledge. Automatic retrieval, on the other hand, is a consequence of a semantic memory dysfunction.
Young children with MAD use the same strategies as their peers, but rely more on immature retelling strategies and less on fact retrieval of memory than his companions.
They are less effective in executing the different retelling and fact retrieval strategies. As age and experience increase, those who do not have difficulties execute the recovery with greater accuracy. Those with DAM show no changes in accuracy or frequency of strategy use. Even after a lot of practice.
When they use retrieval of facts from memory, it is usually not very accurate: they make mistakes and take longer than those who do not have LD.
Children with MAD present difficulties in retrieving numerical facts from memory, presenting difficulties in automating this retrieval.
Children with MAD do not make adaptive selection of their strategies. Children with MAD have lower performance in the frequency, efficiency and adaptive selection of strategies. (referring to count)
The deficiencies observed in children with MAD seem to respond more to a model of developmental delay than to one of deficit.
Geary has devised a classification in which three subtypes of MAD are established: procedural subtype, subtype based on deficits in semantic memory, and subtype based on deficits in visuospatial skills.
Subtypes of children who have difficulties in mathematics
The research has made it possible to identify three subtypes of DAM:
The work memory It is an important component process of mathematics performance. Working memory problems can cause procedural failures such as fact retrieval.
Students with Language Learning Difficulties + DAM appear to have difficulty retaining and retrieving mathematical facts and solving problems both verbal, complex or real life, more severe than students with isolated MAD.
Those with isolated MAD have difficulty on the visuospatial scheduling task, which required memorizing information with movement.
Students with MAD also have difficulties in interpreting and solving mathematical word problems. They would have difficulties in detecting relevant and irrelevant information in problems, in constructing a mental representation of the problem, in remembering and executing the steps involved in solving a problem, especially in multi-step problems, in using cognitive and metacognitive strategies.
Some proposals to improve mathematics learning
Problem solving requires understanding the text and analyzing the information presented, developing logical solution plans, and evaluating solutions.
Requires: cognitive requirements, such as declarative and procedural knowledge of arithmetic and the ability to apply this knowledge to word problems, ability to carry out a correct representation of the problem and planning capacity to solve the problem; metacognitive requirements, such as awareness of the solution process itself, as well as strategies to control and supervise its performance; and affective conditions such as a favorable attitude towards mathematics, perception of the importance of problem solving or confidence in one’s own ability.
A large number of factors can affect the solving of mathematical problems. There is increasing evidence that the majority of students with MAD have more difficulty in the processes and strategies associated with constructing a representation of the problem than in executing the operations necessary to solve it.
They have problems with the knowledge, use and control of problem representation strategies, to capture the superschemas of different types of problems. They propose a classification differentiating 4 large categories of problems based on the semantic structure: change, combination, comparison and equalization.
These superschemas would be the knowledge structures that are put into play to understand a problem, to create a correct representation of the problem. From this representation, the execution of operations is proposed to reach the solution of the problem through remembering strategies or from the immediate retrieval of long-term memory (LTM). Operations are no longer resolved in isolation, but in the context of solving a problem.
