Children’s difficulties in learning mathematics

Mathematics: a difficult and frustrating subject for many young learners.

The concept of number is the basis of mathematics mathematicsTherefore, 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 an intuitive informal 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, as 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 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. that are maintained over time.

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

Beginning at age 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 schemas that enable the child's 'mathematical mind'.

The first quantitative knowledge is acquired through three proto-quantitative schemas:

These schemes are not sufficient to address quantitative tasks, so they require the use of more precise quantification tools, such as counting.

The counting is an activity that may seem simple to an adult but needs to integrate a series of techniques.

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

Principles and skills needed to improve in 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 counting:

These principles establish the procedural rules on how to count a set of objects. From the child's own experiences, he/she acquires the conventional numerical sequence and will be able to establish how many elements a set has, that is, to master counting.

On many occasions, children develop the belief that certain non-essential characteristics of counting are essential, such as standard direction and adjacency. They are also abstraction and order irrelevance, which serve to ensure and make more flexible the range of application of the above principles.

The acquisition and development of strategic competence

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

Prevalence, explanations and manifestations

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

The DSM-IV-TR indicates that The prevalence of numeracy disorder has only been estimated to be about one in five cases of learning disability.. It is assumed that about 1% of school-age children suffer from a calculation disorder.

Recent studies claim 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 do children with Math Learning Difficulties look like?

Many studies have pointed out that basic numerical skills such as number identification or number magnitude comparison are intact in most children with Mathematics Learning Difficulties (hereafter, MAD), at least for simple numbers.

Many children with MAD have difficulty understanding some aspects of countingMost of them understand stable order and cardinality, at least fail in understanding one-to-one correspondence, especially when the first element is counted twice; and they systematically fail in tasks involving the understanding of irrelevance of order and adjacency.

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

The procedural problems are likely to improve with experience, their difficulties with retrieval are not. This is because procedural problems arise from a lack of conceptual knowledge. Automatic retrieval, on the other hand, is a consequence of semantic memory dysfunction.

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

They are less effective in executing the different counting and fact retrieval strategies. As age and experience increase, those without difficulties execute retrieval more accurately. Those with DAM show no change in accuracy or frequency of strategy use. Even after much practice.

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

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

Boys with DAM do not perform adaptive selection of their strategies.Boys with DAM underperform on frequency, efficiency, and adaptive selection of strategies. (referring to counting)

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

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 present difficulties in mathematics

Research has identified three subtypes of MAD:

The working memory is an important component process of mathematics performance. Working memory problems can lead to procedural failures such as in fact retrieval.

Students with Language Learning Difficulties + DAM appear to have difficulties in retaining and retrieving math facts and solving both word and complex problems.Students with isolated SLD appear to have more severe difficulties in retaining and retrieving math facts and solving word, complex, and real-life problems than students with isolated SLD.

Those with isolated AMD have difficulties on the visuospatial agenda task, which required memorizing information with movement.

Students with AMD also have difficulties in interpreting and solving mathematical verbal 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 plans for the solution, and evaluating the solutions.

It requires: some cognitive requirements, such as declarative and procedural knowledge of arithmetic and ability to apply that knowledge to word problems.These include: the 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 monitor 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 mathematical problem solving. There is increasing evidence that most students with MDD have more difficulty with the processes and strategies associated with constructing a representation of the problem than with performing the operations necessary to solve it.

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

These superschemas would be the knowledge structures that are brought 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 by recall strategies or from immediate retrieval from long-term memory (LTM). The operations are no longer solved in isolation, but in the context of solving a problem.

• The proto-quantitative increment-decrement schemaWith this schema, three-year-olds are able to reason about changes in quantities when an item is added or subtracted.
• Ehe proto-quantitative part-whole schemaallows preschoolers to accept that any piece can be divided into smaller parts and that putting them back together results in the original piece. They can reason that when they put two quantities together, they get a larger quantity. Implicitly they begin to know the auditory property of quantities.
• The one-to-one correspondence principleThe one-to-one correspondence principle: it 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 counted and uncounted elements, 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 established order principleThe established order principle: 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.
• The cardinality principle: states that the last label of the numerical sequence represents the cardinal of the set, the number of elements contained in the set.
• The principle of abstraction: determines that the previous principles can be applied to any type of set, both with homogeneous and heterogeneous elements.
• The principle of irrelevanceindicates 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.
• Repertoire of strategies.The different strategies that a learner uses when performing tasks.
• Frequency of strategiesfrequency with which each of the strategies is used by the child.
• Efficiency of strategiesaccuracy and speed with which each strategy is executed.
• Strategy selectionThe ability of the child to select the most adaptive strategy in each situation and that allows him/her to be more efficient in the performance of tasks.
  • 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 visuospatial representation of numerical information.
Bibliographical references:
• • Cascallana, M. (1998) Iniciación matemática: materiales y recursos didácticos. Madrid: Santillana.
  • Díaz Godino, J, Gómez Alfonso, B, Gutiérrez Rodríguez, A, Rico Romero, L, Sierra Vázquez, M. (1991) Área de conocimiento didáctica de la Matemática. Madrid: Editorial Síntesis.
  • Ministerio de Educación, Cultura y Deportes (2000) Dificultades del aprendizaje de las matemáticas. Madrid: Aulas de verano. Instituto superior e formación del profesorado.
  • Orton, A. (1990) Didáctica de las matemáticas. Madrid: Ediciones Morata.