# The difficulties of children in learning mathematics

The concept of **number ** is the basis of the **maths** , being therefore its acquisition the foundation on which the mathematical knowledge is constructed. The concept of number has been conceived as a complex cognitive activity, in which different processes act in a coordinated way.

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 to acquire basic arithmetic skills and stimulation from the environment, since children from an early age find quantities in the physical world, quantities to count in the social world and ideas mathematics in the world of history and literature.

### Learning the concept of number

The development of the number depends on schooling. Instruction in infant education in classification, seriation and conservation of the number **it produces gains in reasoning capacity and academic performance** that are maintained over time.

The difficulties of enumeration in young children interfere with the acquisition of mathematical skills in later childhood.

After two years, the first quantitative knowledge begins to be developed. This development is completed through the acquisition of so-called proto-quantitative schemes and the first numerical skill: count.

### The schemes that enable the 'mathematical mind' of the child

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 quantity judgments without numerical precision, such as larger, smaller, more or less, etc. Through this scheme linguistic labels are assigned to the comparison of sizes.**The proto-quantitative increase-decrease scheme**: with this scheme the children of three years are able to reason about changes in the quantities when an element is added or removed.- AND
**The proto-quantitative scheme part-everything**: allows preschoolers to accept that any piece can be divided into smaller parts and that if they are put together again they give rise to the original piece. They can reason that when they unite two amounts, they get a larger amount. Implicitly they begin to know the auditory property of the quantities.

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

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

Some consider that the count is a rote learning and meaningless, especially of the standard number sequence, to gradually endow these routines of conceptual content.

### Principles and skills that are needed to improve the task of counting

Others consider that the recount requires the acquisition of a series of principles that govern the ability 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, by means of partitioning, they control the elements counted and those that are still to be counted, at the same time that they have a series of labels, so that each corresponds to an object of the counted set , even if they do not follow the correct sequence.**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 using the conventional numerical sequence.**The principle of cardinality**: establishes that the last label of the numerical sequence represents the cardinal of the set, the number of elements that the set contains.**The principle of abstraction**: determines that the above 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 by which the elements are 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 procedural rules on how to count a set of objects. From the own experiences the child is acquiring the conventional numerical sequence and will allow him to establish how many elements a set has, that is to say, to dominate the count.

On many occasions, children develop the belief that certain non-essential features of the count are essential, such as standard direction and adjacency. They are also the abstraction and the 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 competition

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 performing tasks.**Frequency of strategies**: frequency with which each of the strategies is used by the child.**Efficiency of strategies**: accuracy and speed with which each strategy is executed.**Selection of strategies**: ability that the child has 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.

The *DSM-IV-TR* indicates that **the prevalence of stone disorder has only been estimated in approximately one in five cases of learning disorder** . It is assumed that about 1% of children of school age suffer a calculation disorder.

Recent studies claim that the prevalence is higher. About 3% have comorbid difficulties in reading and mathematics.

The difficulties in mathematics also tend to be persistent over time.

### How are children with Difficulties in Learning Mathematics?

Many studies have pointed out that basic numerical competences such as identifying numbers or comparing magnitudes of numbers are intact in most children with **Difficulties in the Learning of Mathematics** (onwards, *DAM*), at least in terms of simple numbers.

Many children with AMD **they have difficulties in understanding some aspects of the counting** : most understand the stable order and the cardinality, at least fail in the understanding of one-to-one correspondence, especially when the first element is counting twice; and systematically fail in tasks that involve understanding the irrelevance of order and adjacency.

The greatest difficulty for children with AMD lies in learning and remembering numerical facts and calculating arithmetic operations. They have two major problems: procedural and recovery of facts of the MLP. The knowledge of facts and the understanding of procedures and strategies are two dissociable problems.

It is likely that procedural problems will improve with experience, their difficulties with recovery will not. This is so because the procedural problems arise from the lack of conceptual knowledge. Automatic recovery, on the other hand, is the result of a dysfunction of semantic memory.

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

They are less effective in the execution of different counting and recovery strategies. As the age and experience increase, those who do not have difficulties execute the recovery with greater accuracy. Those with AMD do not show changes in the accuracy or frequency of use of the strategies. Even after a lot of practice.

When they use memory retrieval, it is usually not very accurate: they make mistakes and take longer than those without DA.

Children with MAD present difficulties in the recovery of numerical facts from memory, presenting difficulties in the automation of this recovery.

Children with AMD do not perform an adaptive selection of their strategies. Children with AMD have a lower performance in frequency, efficiency and adaptive selection of strategies. (referred to the count)

The deficiencies observed in children with AMD seem to respond more to a model of developmental delay than to a deficit.

Geary has devised a classification in which three sub-types of DAM are established: procedural subtype, subtype based on deficit in semantic memory, and subtype based on deficit in visual-spatial skills.

## Subtypes of children who have difficulties in mathematics

The investigation has allowed to identify** three subtypes of DAM** :

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

The *work memory* it is an important component of performance in mathematics. Work memory problems can cause procedural failures as in the recovery of facts.

Students with Difficulties in Language Learning + DAM **they seem to have difficulties in retaining and recovering mathematical facts and solving problems** , of word, complex or real life, more severe than students with isolated MAD.

Those who have isolated DAM have difficulties in the task of visuospatial agenda, which required memorizing information with movement.

Students with MAD also have difficulties in interpreting and solving mathematical word problems. They would have difficulties to detect the relevant and irrelevant information of the problems, to construct a mental representation of the problem, to remember and execute the steps involved in the resolution of a problem, especially in the problems of multiple steps, 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 the solution and evaluating the solutions.

Requires: **cognitive requirements, such as declarative and procedural knowledge of arithmetic and ability to apply said 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 the favorable attitude towards mathematics, perception of the importance of problem solving or confidence in one's ability.

A large number of factors can affect the resolution of mathematical problems. There is increasing evidence that most students with AMD have more difficulty in the processes and strategies associated with the construction of a representation of the problem than in the execution of the operations necessary to solve it.

They have problems with knowledge, use and control of problem representation strategies, to capture the superstores of different types of problems. They propose a classification by differentiating 4 major categories of problems according to the semantic structure: change, combination, comparison and equalization.

These superstores 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 arrive at the solution of the problem by recall strategies or from the immediate recovery of the long-term memory (MLP). The operations are no longer solved in isolation, but in the context of the resolution of a problem.

#### Bibliographic references:

- Cascallana, M. (1998) Mathematical initiation: materials and didactic 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 Síntesis.
- Ministry of Education, Culture and Sports (2000) Difficulties learning mathematics. Madrid: Summer classrooms. Higher Institute and teacher training.
- Orton, A. (1990) Didactics of mathematics. Madrid: Morata Editions.