The concept of number constitutes the basis of the mathematics , being therefore its acquisition the foundation on which the 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 math . This development is due to the fact that children show a biological propensity to acquire basic arithmetic skills and to be stimulated by the environment, as children from an early age find 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 the schooling. Instruction in early childhood education in classification, serialization and number retention produces gains in reasoning ability and academic performance that are sustained over time.
Numeracy 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 ‘mathematical mind’ of the child
The first quantitative knowledge is acquired through three protoquantitative schemes:
- The protoquantitative scheme of the comparison : thanks to this scheme children can have a series of terms that express judgements of quantity without numerical precision, such as bigger, smaller, more or less, etc. This scheme assigns linguistic labels 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 some element is added or removed.
- E he part-to-all prototyping scheme : allows 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. They implicitly 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.
The count is an activity that may seem simple in the eyes of an adult but needs to integrate a number of techniques.
Some consider counting to be a memorized and meaningless learning process, especially of the standard numerical sequence, in order to gradually equip these routines with conceptual content.
Principles and skills needed to improve counting task
Others consider that counting requires the acquisition of a set of principles that govern the skill and allow for a progressive sophistication of the counting:
- The one-to-one matching principle : involves tagging each element in a set only once. It involves the coordination of two processes: participation and tagging, by means of partitioning, control the counted and uncounted elements, while having a series of tags, so that each one corresponds to an object in the counted set, even if they do not follow the correct sequence.
- The established order principle : 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 cardinality principle : establishes that the last label of the numerical sequence represents the cardinal of the set, the quantity 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 elements and with heterogeneous elements.
- The principle of irrelevance : indicates that the order in which the elements are listed 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. Based on the child’s own experiences, he or she will acquire the conventional numerical sequence and will be able to establish how many elements a set has, i.e. to master the counting.
Often, children develop a belief that certain non-essential characteristics of the count are essential, such as standard direction and adjacency. It is also the abstraction and irrelevance of order that serves to ensure and relax 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:
- Repertoire of strategies : different strategies a student uses when performing tasks.
- Frequency of strategies : frequency with which each of the strategies is employed by the child.
- Efficiency of strategies : accuracy and speed with which each strategy is executed.
- Selection of strategies : ability of the child to select the most adaptive strategy in each situation and that allows him/her to be more efficient in the accomplishment of tasks.
Prevalence, explanations and manifestations
The different estimates of the prevalence of learning disabilities in mathematics differ due to the different diagnostic criteria used.
The DSM-IV-TR indicates that the prevalence of calculus disorder has only been estimated in approximately one in five cases of learning disability . It is assumed that about 1% of school-aged children suffer from a calculus 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 are children with learning disabilities like in math?
Many studies have pointed out that basic numerical skills such as number identification or number magnitude comparison are intact in most children with Math Learning Difficulties (hereafter, MAD ), at least for simple numbers.
Many children with GAD have difficulty in understanding some aspects of counting : 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 in tasks involving understanding the irrelevance of order and adjacency.
The greatest difficulty for children with GAD lies in learning and remembering numerical facts and calculating arithmetic operations. They have two major problems: procedural and fact retrieval from the MLP. Knowledge of facts and understanding of procedures and strategies are two dissociable problems.
Procedural problems are likely to improve with experience, their difficulties with recovery do not. This is because procedural problems arise from a lack of conceptual knowledge. Automatic recovery, on the other hand, is a consequence of a dysfunction in semantic memory.
Young children with GAD use the same strategies as their peers, but rely more on immature counting strategies and less on fact retrieval of memory than their peers.
They are less effective in implementing the various counting and fact retrieval strategies. As age and experience increases, those who do not have difficulties execute the recovery more accurately. Those with MAD show no change in accuracy or frequency of use of the strategies. Even after much practice.
When they use memory fact retrieval it is often inaccurate: they make mistakes and take longer than those without AD.
Children with GAD have difficulties in retrieving numerical facts from memory, presenting difficulties in automating this retrieval.
Children with GAD do not make adaptive selection of their strategies. Children with GAD underperform in frequency, efficiency, and adaptive selection of strategies. (referred to as counting)
The deficiencies observed in children with MAD seem to respond more to a pattern 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 semantic memory deficit, and subtype based on visual-spatial skills deficit.
Sub-types of children who have difficulty with mathematics
Research has identified three subtypes of MAD :
- A subtype with difficulties in the execution of arithmetic procedures.
- A subtype with difficulties in the representation and recovery of arithmetic facts from semantic memory.
- A subtype with difficulties in the visuospatial representation of numerical information.
The working memory is an important component process of performance in mathematics. Working memory problems can lead to procedural failures such as fact retrieval.
Students with Language Learning Difficulties + GAD appear to have more severe difficulties in retaining and retrieving mathematical facts and solving problems , whether word, complex or real-life, than students with isolated GAD.
Those with isolated GAD have difficulty with the visuospatial agenda task, which required memorizing information with movement.
Students with GAD also have difficulty interpreting and solving verbal mathematical 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 the learning of mathematics
Problem solving requires understanding the text and analyzing the information presented, developing logical plans for solution and evaluating solutions.
It requires: some cognitive requirements, such as declarative and procedural knowledge of arithmetic and ability to apply this knowledge to word problems , ability to carry out a correct representation of the problem and planning capacity to give solution to the problem; metacognitive requirements, such as awareness of the solution process itself, as well as the strategies to control and supervise its performance; and affective conditions such as favourable attitude towards mathematics, perception of the importance of problem solving or confidence in one’s own ability.
A large number of factors can affect math problem solving. There is increasing evidence that most students with GAD have more difficulty in the processes and strategies associated with constructing a representation of the problem than in executing the operations needed to solve the problem.
They have problems with the knowledge, use and control of problem representation strategies, to capture the super-schemes of different types of problems. They propose a classification differentiating 4 big categories of problems according to the semantic structure: of change, of combination, of 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. Based on this representation, the execution of operations to arrive at the solution of the problem by memory strategies or based on the immediate recovery of long-term memory (LTM) is proposed. Operations are no longer solved in isolation, but in the context of solving a problem.
Bibliographic 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.
- Ministry of Education, Culture and Sports (2000) Difficulties in learning mathematics. Madrid: Aulas de verano. Instituto superior e formación del profesorado.
- Orton, A. (1990) Didactics of Mathematics. Madrid: Ediciones Morata.