Difficulties of children in learning mathematics

The concept of number forms the basis of math, being therefore its acquisition the foundation on which the mathematical knowledge. The concept of number has come to be conceived as a complex cognitive activity, in which different processes act in a coordinated manner.

From very small, Children develop what is known as a intuitive informal math. This development is due to the fact that children show a biological propensity for the acquisition of basic arithmetic skills and stimulation from the environment, since that children from an early age encounter quantities in the physical world, quantities to count in the social world, and mathematical ideas in the world of history and literature.

Learning the concept of number

The development of the number depends on schooling. Instruction in early childhood education in classification, seriation 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 be developed. This development is completed through the acquisition of schemes called proto-quantitative and the first numerical skill: counting.

The schemes that enable the child's 'mathematical mind'

The first quantitative knowledge is acquired through three protoquantitative schemes:

1. The protoquantitative scheme of the comparison: Thanks to this, children can have a series of terms that express quantity judgments without numerical precision, such as bigger, smaller, more or less, etc. Using this scheme, linguistic labels are assigned to the size comparison.
2. The protoquantitative increase-decrease scheme: With this scheme, three-year-olds are able to reason about changes in quantities when an element is added or removed.
3. 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 numbers together, they get a larger number. Implicitly they begin to know the auditory property of quantities.

These schemes are not enough to address quantitative tasks, so they need to use more precise quantification tools, such as counting.

He count It is an activity that in the eyes of an adult may seem simple but it needs to integrate a series of techniques.

Some consider counting to be rote learning and meaningless, especially the standard numerical sequence, to gradually provide these routines with content conceptual.

Principles and skills that are needed to improve in the counting task

Others consider that the count requires the acquisition of a series of principles that govern the skill and allow a progressive sophistication of the count:

1. The one-to-one correspondence principle: involves labeling each element of an array only once. It involves the coordination of two processes: participation and labeling, through the partition, they control the counted elements and those that are missing by count, 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 sequence correct.
2. The principle of established order: stipulates that in order to count it is essential to establish a coherent sequence, although this principle can be applied without the need to use the conventional numerical sequence.
3. The cardinality principle: sets that the last label in the number sequence represents the cardinal of the array, the number of elements that the array contains.
4. 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.
5. The principle of irrelevance: Indicates that the order in which the elements begin to be enumerated 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 process rules for how to count a set of objects. From their own experiences, the child gradually acquires the conventional numerical sequence and will allow him to establish how many elements a set has, that is, master counting.

Children often develop the belief that certain non-essential features of the count are essential, such as standard address and adjacency. They are also the abstraction and irrelevance of the order, which serve to guarantee and make the range of application of the above principles more flexible.

The acquisition and development of strategic competence

Four dimensions have been described through which the development of students' strategic competence is observed:

1. repertoire of strategies: different strategies that a student uses when carrying out the tasks.
2. Frequency of strategies: frequency with which each of the strategies is used by the child.
3. Strategy Efficiency: accuracy and speed with which each strategy is executed.
4. Selection of strategies: ability of the child 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

Different estimates of the prevalence of mathematics learning difficulties differ due to the different diagnostic criteria used.

He DSM-IV-TR indicates that the prevalence of calculation disorder has only been estimated at about 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.

How are children with Learning Difficulties in Mathematics?

Many studies have indicated that basic numerical skills such as identifying numbers or the comparison of magnitudes of numbers are intact in most of the Children with Difficulties in Learning Mathematics (onwards, DAM), at least for simple numbers.

Many children with MAD have difficulty understanding some aspects of the count: most understand stable ordering and cardinality, at least they fail to understand one-to-one correspondence, especially when the first element is counted twice; and they consistently fail at tasks that involve understanding the irrelevance of order and adjacency.

The greatest difficulty for 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 recovery difficulties are not. This is so because procedural problems arise from a lack of conceptual knowledge. Automatic recovery, on the other hand, is the consequence of a semantic memory dysfunction.

Young boys with DAM use the same strategies as their peers, but rely more on immature counting strategies and less on fact retrieval from memory than his peers.

They are less effective in executing the different fact counting and retrieval strategies. As age and experience increase, those without difficulties perform the recovery more accurately. Those with MAD do not show changes in the accuracy or frequency of use of the strategies. Even after a lot of practice.

When they use fact retrieval from memory it is often inaccurate: they make mistakes and take longer than those without DA.

Children with MAD present difficulties in retrieving numerical facts from memory, presenting difficulties in automating this retrieval.

Children with DAM do not make adaptive selection of their strategies. Children with DAM have lower performance in frequency, efficiency, and adaptive selection of strategies. (referring to the count)

The deficits 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 that establishes three subtypes of DAM: procedural subtype, subtype based on deficits in semantic memory, and subtype based on deficits in skills visuo-spatial.

Subtypes of children with difficulties in mathematics

The investigation has made it possible to identify three subtypes of MAD:

• A subtype with difficulties in the execution of arithmetic procedures.
• A subtype with difficulties in the representation and retrieval of arithmetic facts from semantic memory.
• A subtype with difficulties in the visual-spatial representation of numerical information.

The work memory it is an important component process of achievement in mathematics. Working memory problems can cause procedural failures such as in fact retrieval.

Students with Language Learning Difficulties + DAM seem to have difficulty retaining and retrieving mathematical facts and solving problems, both word, complex or real life, more severe than students with isolated MAD.

Those with isolated MAD have difficulties in the visuospatial diary task, which required memorizing information with movement.

Students with MAD also have difficulty interpreting and solving mathematical word problems. They would have difficulties to detect the relevant and irrelevant information of the problems, to build a mental representation of the problem, to remember and Execute the steps involved in solving a problem, especially multi-step problems, to use cognitive and metacognitive strategies.

Some proposals to improve the learning of mathematics

Problem solving requires understanding the text and analyzing the information presented, developing logical plans for solution, and evaluating solutions.

Requires: cognitive requirements, such as declarative and procedural knowledge of arithmetic and the ability to apply that knowledge to word problems, ability to carry out a correct representation of the problem and planning ability to solve the problem; metacognitive requirements, such as awareness of the solution process itself, as well as strategies to control and monitor its performance; and affective conditions such as a favorable attitude towards mathematics, perception of the importance of solving problems 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 with processes and strategies. associated with the construction of a representation of the problem than in the execution of the operations necessary to work it out.

They have problems with the knowledge, use and control of problem representation strategies, to grasp the superschemes of the 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 super-schemes 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 the operations is proposed to reach the solution of the problem. problem by recall strategies or from immediate retrieval of long-term memory (MLP). Operations are no longer solved in isolation, but in the context of solving a problem.

Bibliographic references:

• Cascallana, M. (1998) Mathematics initiation: didactic materials and resources. Madrid: Santillana.
• Díaz Godino, J, Gómez Alfonso, B, Gutiérrez Rodríguez, A, Rico Romero, L, Sierra Vázquez, M. (1991) Area of ​​didactic knowledge of Mathematics. Madrid: Editorial Synthesis.
• Ministry of Education, Culture and Sports (2000) Difficulties in learning mathematics. Madrid: Summer classrooms. Higher institute for teacher training.
  
• Orton, a. (1990) Didactics of mathematics. Madrid: Morata Editions.