Early Stages in Mathematical Development

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Mathematical Difficulties.

Difficulties with understanding number have been suggested as one reason why people have difficulties learning mathematics. Indeed some researchers have suggested that the developmental disability, dyscalculia, is characterised as a primary impairment of number skills especially when compared to other developmentally normal skills such as language and memory.

This type of difficulty is specific to number skills and is present early on, when the child is first learning about number. However specific problems with number probably represent just one factor in difficulties with mathematics. Many people report having struggled with mathematics at some point in their lives and their difficulties may arise at varying points in the school curriculum, possibly due to different underlying reasons.

There are also cases where mathematical difficulties do not seem to be due to a basic impairment with number skills or associated with any genetic condition. These children may show clear differences in attainment between mathematics and all the other skills at different stages of their mathematical learning. This makes it very important to understand what a child is expected to learn as they progress through the curriculum and pinpoint areas where learning could be affected by other cognitive deficits as difficulties could occur when a specific underlying cognitive skill is required for that stage of learning.

One model1 based on research evidence in mathematical development, has suggested that there may be three independent skills (or “cognitive precursors”) underpinning mathematical learning:

Linguistic abilities, linked to language processing, form a skill pathway with children’s symbolic number system knowledge, i.e., naming and writing numerals.

Quantitative abilities form a skill pathway linked with processing numerical magnitudes. This is a skill that can be used in problems where you need to represent and operate on quantities. For example questions such as which as "Which has more?" or "Which has less?" They do not require knowledge of the number system.

Spatial attention is a complex set of skills that incorporates spatial ability and the allocation of attention. It is assumed that spatial attention will be important in the management of the complex requirements of mathematical tasks.

There may also be other cognitive skills underpinning mathematical development that are not captured by this model. For example, body representations such as those used in finger counting have been linked to the development of number skills. Finger counting may be used as a strategy to understand and keep track of counting and calculation. However another possibility is that fingers allow numerical knowledge to be represented using sensory and motor features during learning. Another skill not explicitly included in the model is working memory, which can be considered as limited capacity system responsible for keeping relevant information in mind and used during the execution of arithmetic procedures. It has also been found to impact on mathematical understanding.

Nevertheless this model, in its current form, makes two predictions. Firstly that the linguistic, quantitative and spatial attention pathways contribute independently to the early learned number skills. And secondly that they have unique and relative contributions to the different mathematical skills learned at different points in the curriculum. These contributions depend on the task demands. Thus a potential deficit with any one of these pathways will lead to different types of mathematical difficulty. It also means that mathematical difficulties may manifest at any point of the mathematical curriculum.

1. LeFevre, J. A., Fast, L., Skwarchuk, S. L., Smith-Chant, B. L., Bisanz, J., Kamawar, D., & Penner-Wilger, M. (2010). Pathways to mathematics: Longitudinal predictors of performance. Child Development, 81(6), 1753-1767. doi: 10.1111/j.1467-8624.2010.01508.x

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