Wednesday, February 6, 2008

Developmental Dyscalculia, Part 4

Part 4 in the series:
Cognitive Domains: Memory

At the end of a child’s exploration of various strategies available, the mastery of elementary arithmetic is achieved when all basic facts can be retrieved from long-term memory without error. Mastery of basic arithmetic is crucial to later competence in more complex mathematical operations such as long division, fractions, geometry, calculus, and so on. Given the importance of memory to basic arithmetic competence, even if a child has successfully reached the most efficient strategy, deficits in memory could lead to mathematics disabilities.

When a computation is executed, the probability of direct retrieval increases for each subsequent solution to the same problem. However, in order for the execution of a computational strategy to lead to the construction of a long-term memory representation between a problem and its solution, both the equation’s augend (i.e. the first number) and addend (i.e. the second number), as well as the answer, must all be simultaneously active in working memory. Thus, arithmetic and mathematical ability is directly related to the function (or dysfunction) of the working and long-term memory stores (Geary, 1993).

In order to create a long-term memory for an arithmetic fact, such as 13 + 7 = 20, an individual must be both proficient (i.e. accurate) and efficient (i.e. speedy). Accuracy is important because if the child commits many computational errors, then the child is more likely to retrieve incorrect answers from long-term memory when later presented with the same problem. Efficiency is likewise important because with a slow counting speed, the working memory representation of the augend is more likely to decay before the addend and solution have been fully represented in working memory. In this circumstance, even if the child reaches the correct answer, it will be less strongly associated with the problem in long-term memory.

These cognitive models are evidenced empirically, as indicated by Geary (1993): “Cognitive studies indicate that when solving arithmetic problems, in relation to their normal peers, [mathematically disabled] children tend to use immature problem-solving strategies, have rather long solution times, and frequently commit computational and memory-retrieval errors.”

Butterworth (2005) further refines the role of working and long-term memory in the storage of arithmetic facts. He presents evidence that retrieval times show a very strong problem-size effect for single-digit problems: the larger the sum or product, the longer it takes to solve. Further, adults without any mathematical disability are quicker to solve an equation in the form of “larger addend” + “smaller addend” than they are to solve the same equation where the addends are reversed. Similarly, normal Italian children 6-10 years old took longer to solve a “smaller” x “larger” equation than a “larger” x “smaller” equation, despite the fact that the Italian education system teaches “smaller” x “larger” first (e.g. the 2x multiplication table is learned before the 6x multiplication table). This seems contradictory to the earlier theory, which offers that equations with which you have more experiences are more strongly stored in long-term memory – since the 2x arithmetic facts were presumably encoded into long-term memory well before the 6x arithmetic facts. This evidence suggests a more complex numerical organization to the storage and representation of arithmetic facts in long-term memory, not just rote association.

Information processing theory offers yet another model for the role that the function or dysfunction of working and long-term memory has in the pathology of developmental dyscalculia and other mathematical impairments. Central to information processing theory is the idea of limited capacity: the human mind has only a finite capacity for information processing at any one time. A fundamental assumption to this theory is that each type of mental process takes up some amount of the “space” or “energy”. At one extreme are automatic processes, which require virtually no space or energy. These processes work without intention or conscious awareness, don’t interfere with other processes, don’t improve with practice, and are not influenced by intelligence, education, motivation, or anything else, such as breathing or sweating. On the other end of the continuum are effortful processes, which use up the resources available in working memory, and have the opposite properties of automatic processes.

When confronted with an arithmetic task, a normal student can complete the task with minimal problems and fairly efficiently – even if the solution isn’t accessed via fact retrieval. For a student with math disabilities, however, the process is likely laborious and takes up significant amounts of energy. Perhaps this has something to do with the continuum of automatic and effortful processing. When a normally-achieving student is confronted with a straightforward arithmetic problem such as 5 + 11 + 37, the student can quickly identify the steps needed to solve the equation and move on to the next item on the worksheet. When a mathematically disabled student is confronted with the same problem, even after having learned and understood the fundamentals of counting and addition, each of the steps necessary to compute the answer takes up significantly more effort to complete. By the time the student moves on to the next item, he has already expended considerably more energy than the first student has, and has likely taken more time to complete each problem. After the first three or four equations, his energy store is perhaps depleted, and the rest of the worksheet is riddled with errors because the student has no mental energy left to tackle the subsequent calculations.

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