Nearly everyone has heard of developmental dyslexia - a learning disorder characterized by poor reading skills despite otherwise sufficient schooling - but have you heard of developmental dyscalculia? Many people have not. Here is part 3 in a week-long series on this lesser-known learning disorder. (See parts one, and two, and a companion post at The Thoughtful Animal)
Cognitive Explanations for Developmental Dyscalculia: Strategy, Memory, Attention
Experimental studies of developmental dyscalculia and math disability in children have focused primarily on skill development in arithmetic, which can be divided into two sections: counting knowledge, and strategy and memory development.
Counting is governed by five principles:
- the one-to-one correspondence rule, where one word is assigned to each counted object;
- the stable order rule, where the order of counting words must be stable across different sets of counted objects;
- the cardinality rule, which states that final counting word assigned represents the total number of objects in a set;
- the abstraction rule, which states that objects of any kind can be counted;
- the order irrelevance rule, which states that items in a set can be counted in any order.
A mastery of counting is essential to the discovery of the most efficient strategies for basic arithmetic procedures such as addition and subtraction, and later, multiplication and division.
In many models of cognitive development, it is thought that children think or behave a certain way under certain rules for an extended period of time. Then, they undergo a brief and sometimes mysterious transition and begin to act and think in a new way. Robert Siegler, however, prefers to conceptualize developmental change as variable and gradual, with multiple different strategies available to a child as the child’s brain matures. There are, naturally, some problems where there is really only one logical or efficient strategy. Indeed, after some time experimenting with different strategies, both in progressive and regressive directions, most children will focus on the best, most logical strategy, and lock onto it for much of the remainder of their lives.
It makes sense that the normally developing child will use a variety of different strategies when faced with the same or similar problems. For example, there are at least three common strategies that children can use for addition. The most efficient is direct fact retrieval: 3 + 3 always equals 6. Another is the min strategy, where kids count up from the larger number: 9 + 2 = (9 + 1) + 1 = 10 + 1 = 11. A third is decomposition into easily manipulated numbers: 19 + 22 = 19 + 20 + 2 = 39 + 2 = 41. Typically-developing children will ultimately lock into one of these or another strategy when faced with a random addition problem.
Artist's Rendition of Siegler's Strategy Choice Model
Given Siegler's model, we might hypothesize that children who have been diagnosed with dyscalculia may be unable - for any of a number of reasons - to settle on the optimal problem-solving strategy. While this does not preclude children from gaining efficiency over a long period of time, it is likely to leave them lagging behind the rest of their peers, significantly slowed down by the wide variety of problem-solving strategies available to them. Indeed, empirical evidence shows that children with developmental dyscalculia are generally two grade levels below their peers in arithmetic and mathematics.
After the period of exploration in which the child eventually uncovers the best problem-solving strategies, 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. Therefore, even if a child has successfully discovered the most efficient strategy, deficits in memory could lead to disabilities in arithmetic and mathematics.
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.
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). Proficiency 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 slow counting speed, the working memory representation of the augend is more likely to decay before the addend and solution have been fully represented. Under these circumstances, even if the child reaches the correct answer, its association with the rest of the equation in long-term memory will be weak.
Each of these possibilities have been found observationally as well as experimentally:
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. (Geary, 1993)
Brian Butterworth further refines the role of working and long-term memory in the storage of arithmetic facts by presenting 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, non-impaired Italian children age 6-10 took longer to solve a “smaller” x “larger” multiplication problem 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). Both of these findings reflect the limitations of counting speed on arithmetic problem-solving. This seems contradictory to the earlier theory, which suggests 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 finding suggests more complex numerical organization to the storage and representation of arithmetic facts in long-term memory, beyond 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. Examples include breathing and 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 nonimpaired student can complete the task with fairly efficiently – even if the solution isn’t accessed via direct fact retrieval. For a student with math disabilities, however, the process is laborious and takes a significant amount of energy. When a nonimpaired 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 on the assignment, he has already expended a considerable amount of mental energy - significantly more than the first student has - and has likely taken more time to complete each problem. After the first three or four items, his or her 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.
Also associated with information processing theory is inhibition, which is the active suppression of irrelevant sensory input. Related to this is the idea of resistance to interference, or attention, which is the ability of an individual to concentrate on “central” information and ignore “peripheral” information. Normally-achieving students can complete an arithmetic worksheet in a noisy classroom with minimal distraction, and accuracy is usually quite high. Students with developmental dyscalculia, however, may have issues with processing due to a deficit in inhibition.
Peter Rosenberger found evidence that low achievement in math is related to attentional deficits. He tested 102 children in one study of paper-and-pencil tests and questionnaires; those children for whom the math achievement quotient was below 100, the reading achievement quotient above 100, and the difference between the two was 20 points or greater were designated “dyscalculic.” Children who met the opposite criteria were designated “dyslexic.” Seventy-two children qualified as dyscalculic, and thirty qualified as dyslexic. The groups were highly comparable in overall scholastic aptitude; in fact, only the arithmetic score could distinguish the two groups**.
He found that the “freedom from distractibility” quotient of the Weschler scale was lower for the dyscalculics, although this was confounded with the score of the arithmetic subtest. Of four factors calculated from the DSM-III questionnaire that each participant received, only the factor of inattention was statistically different between the groups, and was higher for dyscalculics. Rosenberger thus suggests that specific math underachievement is, in at least some cases, the result of failure of children with attention deficits to automatize number facts in the early grades. If true, he writes,
this finding would suggest that [an attentional deficit] is not merely an additive or aggravating factor in problems with math performance, but in fact interferes with the development of aptitude for this skill [in the first place].
**Which makes me wonder why he chose to discuss dyslexia in the first place.
Get Your Literature On
Gallistel CR, & Gelman R (1992). Preverbal and verbal counting and computation. Cognition, 44 (1-2), 43-74 PMID: 1511586
Siegler, R. (1994). Cognitive Variability: A Key to Understanding Cognitive Development. Current Directions in Psychological Science, 3 (1), 1-5 DOI: 10.1111/1467-8721.ep10769817
Geary, D. (1993). Mathematical disabilities: Cognitive, neuropsychological, and genetic components. Psychological Bulletin, 114 (2), 345-362 DOI: 10.1037/0033-2909.114.2.345
Butterworth, B. (2005). The development of arithmetical abilities Journal of Child Psychology and Psychiatry, 46 (1), 3-18 DOI: 10.1111/j.1469-7610.2004.00374.x
Rosenberg PB (1989). Perceptual-motor and attentional correlates of developmental dyscalculia. Annals of neurology, 26 (2), 216-20 PMID: 2774508