# Mathematical Disabilities: What We Know and Don't Know

By: David C. Geary

Over the past several decades important advances have been made in the understanding of the genetic, neural, and cognitive deficits that underlie reading disability (RD), and in the ability to identify and remediate this form of learning disability (LD). Research on learning disabilities in mathematics (MD) has also progressed over the past ten years, but more slowly than the study of RD. One of the difficulties in studying children with MD is the complexity of the field of mathematics. In theory, MD could result from difficulties in the skills that comprise one or many of the domains of mathematics, such as arithmetic, algebra, or geometry. Moreover, each of these domains is very complex, in that each has many subdomains and a learning disability can result from difficulties in understanding or learning basic skills in one or several of these subdomains.

As an example, to master arithmetic, children must understand numbers (e.g., the quantity that each number represents), counting (there are many basic principles of counting that children must come to understand), and the conceptual (e.g., understanding the Base-10 number system) and procedural (e.g., borrowing from one column to the next, as in 43-9) features involved in solving simple and complex arithmetic problems. A learning disability in math can result from difficulties in learning any one, or any combination, of these more basic skills. To complicate matters further, it is possible, and in fact it appears to be the case, that different children with MD have different patterns of strengths and weakness when it comes to understanding and learning these basic skills.

So, how does one approach the study of MD? Researchers in this area have narrowed the search for the deficits associated with MD by taking the theories and techniques used to study normal mathematical development and using them to study children with low achievement scores in mathematics, despite an average or better intelligence (IQ). Nearly all of these studies have focused on elementary-school children and the domains of number, counting, and arithmetic; unfortunately, not enough is know about the normal development of algebraic and geometric skills to provide the foundation necessary to systematically study learning disabilities in these mathematical domains. Because of this research, we now understand some of the deficits that contribute to MD, at least in these basic areas. Before we get to the description of these deficits, the more basic issues of how many children have MD and the diagnosis of MD are addressed. The final section presents brief discussion of future directions.

## How common are math disabilities?

There have only been a few large-scale studies of children with MD and all of these have focused on basic number and arithmetic skills. As a result, very little is known about the frequency of learning disabilities in other areas of mathematics, such as algebra and geometry. In any case, the studies in number and arithmetic are very consistent in their findings: Between 6 and 7% of school-age children show persistent, grade-to-grade, difficulties in learning some aspects of arithmetic or related areas (described below). These and other studies indicate that these learning disabilities are not related to IQ, motivation or other factors that might influence learning.

The finding that about 7% of children have some form of MD is misleading in some respects. This is because most of these children have specific deficits in one or a few subdomains of arithmetic or related areas (e.g., counting) and perform at grade-level or better in other areas of arithmetic and mathematics. The confusion results from the fact that standardized math achievement tests include many different types of items, such as number identification, counting, arithmetic, time telling, geometry, and so fourth. Because performance is averaged over many different types of items, some of which children with MD have difficulty on and some of which they do not, many of these children have standardized achievement test scores above the 7th percentile (though often below the 20th).

In other words, the mixing of many different types of items on math achievement tests makes the identification of the learning disability of many of these children difficult at best, and, at the same time, results in an impression that they are generally poor at mathematics. In fact, many of these children have average or better skills in some areas of mathematics and very poor skills in more specific areas. The averaging of performance over different types of test items gives the false impression of generally poor performance in all areas of mathematics when, in fact, the difficulty may be confined to one or a few specific areas. To complicate matters further, recent studies suggest that children with MD are a heterogeneous group, with different children showing different patterns of knowledge and learning strengths and deficits. For instance, two children with math achievement scores at the 10th percentile, and average IQ scores (i.e., children with MD), may have different forms of MD, that is, different types of deficits.

To more reliably diagnose MD and to better understand the cognitive strengths and weaknesses of these children, tests that provide information on very specific arithmetical and related skills, such as counting knowledge, arithmetic facts, and so fourth are needed. Unfortunately, such tests are not currently available.

## What are the common features of MD?

As stated above, there has been very little research on learning disabilities in mathematical domains outside of the areas of number, counting, and arithmetic. Thus, little can be said, at this time, about whether learning disabilities exist in the areas of geometry and algebra, for instance and, if they do exist, what their form or developmental course is. Some general conclusions can, however, be made about the basic number, counting, and arithmetic skills of children with MD and these are briefly discussed in the respective sections below. The final section presents a discussion of the relation between RD and MD.

### Number

The learning of basic number skills is much more complicated than many adults would assume. Children must learn English number words and their correct sequence (i.e., "one, two, three "), as well as the associated Arabic numbers and sequence (i.e., "1, 2, 3," ). Children must learn the quantities associated with these number words and Arabic numbers (e.g., that "three" and "3" are symbols that represent a collection of any three things) and learn to translate numbers from one form to another, as in translating "thirty seven" into "37." Equally important, children must develop an understanding of the structure of numbers, for instance, that numbers can be decomposed into smaller numbers or combined to create larger numbers. The most difficult feature of the number system is it's base-10 structure, that is, the basic sequence of numbers repeats in series of 10 (e.g., 1, 2, 3, 4, 10 is repeated 10 + 1, 10 + 2, that is, 11, 12). Coming to really understand the base-10 system is difficult for all children, but is essential, as this conceptual knowledge is important for the mastery of other domains (e.g., complex arithmetic).

The base-10 knowledge of children with MD has not been studied, but most of the remaining basic number skills have been assessed, at least for smaller numbers (e.g., 3, 4, 23, 67; but not more complex numbers, such as 1,222,976). Although definitive conclusions cannot be drawn at this point, the available evidence suggests that most children with MD do not have a basic deficit in the ability to understand or learn number concepts. This is not to say that they don't have some difficulties along the way, they do (e.g., learning that teen numbers are composed of 1 "10" and x "1s", e.g., 12 = 10 + 2). Rather, the difficulties these children have with learning number sequences and number concepts do not appear to be any different than the difficulties experienced by children without MD.

### Counting

Learning the basic counting sequence, "one, two, three, four" is not difficult; almost all children learn this sequence. What is important is that children learn the basic concepts or rules that underlie the ability to count effectively. The basic rules are as follows:

One-one correspondence. One and only one word tag (e.g., "one," "two") is assigned to each counted object. Counting the same item twice and tagging the item, "one, two" violates this rule.

Stable order. The order of the word tags must be invariant across counted sets. Many preschool children do not yet know the standard sequence of number words, that is, "one, two, three," but still appear to intuitively understand this rule. For instance, they may count two sets of three objects, "A, B, C."

Cardinality. The value of the final word tag represents the quantity of items in the counted set. One way to test knowledge of this rule is to ask children to count a series of objects and then ask them "how many are there?" If the child understands cardinality then she will just repeat the last number word (e.g., "three" if 3 objects were counted). Children who don't understand cardinality will recount the set.

Abstraction. Objects of any kind, such as rocks, toys, and people, can be collected together and counted.

Order-irrelevance. Items within a given set can be tagged in any sequence, from left to right or right to left, or skipping around.

The principles of one-one correspondence, stable order, and cardinality define the "how to count" rules, which, in turn, provide the skeletal structure for children's emerging counting competencies. While this skeletal knowledge appears to be in-born, it is also known that children make inductions about the basic characteristics of counting by observing standard counting behavior. For instance, because counting typically proceeds from left to right, many young children believe that you must count from left to right; right to left counting would be deemed as wrong. Many children also believe that you must count adjacent items, that skipping around is wrong; in fact, skipping around is OK, as long as each item is counted only once -- this belief is call the adjacency rule. These beliefs suggest that many young children don't fully understand counting concepts.

As noted above, having children count from 1 to 20, for instance, provides virtually no information on their understanding of counting rules, as nearly all children can do this, and is thus of no value in diagnosing MD. However, the use of more subtle techniques that tap children's intuitive understanding of the just described counting rules does provide a fine-grained assessment of their counting knowledge. Several studies have now used these techniques in the study of children with MD. The results indicate that 1st and 2nd grade children with MD understand the concepts of one-one correspondence, stable order, and cardinality just as well as children without MD -- there have been no studies of older MD children's counting knowledge. Many children with MD have difficulties with tasks that assess the order irrelevance principle, or, in other words, believe that only adjacent items can be counted, suggesting that they understand counting as a fixed, mechanical activity.

Although children with MD understand one-one correspondence, they sometimes make mistakes on tasks that assess this concept. In one of these tasks, the child is asked to help a puppet who is just learning how to count. Sometimes the puppet counts correctly and at other times the puppet violates one of the counting rules. The child's task is simply to state whether the puppet's count was "OK and right" or "not OK and wrong." As an example, on some trials, the puppet counts six toys from left to right but double counts, "six, seven," the last toy. Double counting violates the one-one rule and children with MD almost always detect this error. However, when the first toy is double counted, "one, two," many children with MD state that this count is "OK" -- the child has to wait until the puppet has finished the count before deciding if the count was "OK" or "not OK." This pattern, and studies using different techniques, suggests that many children with MD have difficulty keeping information (the error notation in this example) in mind while monitoring the counting process. In other words, they understand most of the counting rules, but often forget numerical information during the act of counting.

### Arithmetic

The basic arithmetic skills of children with MD have been extensively studied in the United States, several European nations, and in Israel. These studies have largely focused on the strategies used to solve simple arithmetic problems, such as finger counting or remembering the answer, and the associated reaction times (i.e., speed of problem solving) and error patterns. These studies have revealed several very consistent patterns with children with MD, patterns that are not related to IQ.

First, many children with MD have difficulties remembering basic arithmetic facts, such as the answers to 5+3 or 3x4. It is not that children with MD do not remember any arithmetic facts, but rather they don't remember as many facts as other children do and appear to forget facts rather quickly. Certain patterns in how quickly they remember facts, when they do remember facts, and in the associated error patterns suggest that this is a fundamental memory problem, that is, not something that these children will "grow out of."

Although a definitive conclusion cannot be drawn at this time, there appear to be two sources of this memory problem -- some children with MD show both types of memory problem, while other children show one form but not the other.

First, it appears that many MD children have difficulties getting basic facts into long-term memory and difficulties remembering, or accessing, the facts that are eventually stored in long-term memory. It appears that these difficulties are very similar to word finding difficulties that are common in some children with RD.

Second, it appears that some children with MD can get facts into and out of long-term memory without too much difficulty but have trouble inhibiting other facts when they try to remember the answers to specific problems, such as 2+3. These children will not only remember 5, they might also have 4 (the number following 2 3 in the counting sequence) and 6 (the answer to 2x3) pop into their heads at the same time. With too many facts being remembered, these children take longer to remember the correct answer -- they may have to consider all of the answers that they remembered and then pick one of these -- and they make more errors.

The other consistent finding is that many children with MD use immature problem-solving procedures to solve simple arithmetic problems, that is they use procedures that are more commonly used by younger children without MD. As an example, the most basic -- least mature -- strategy for solving simple addition problem is called counting-all. To solve 5+3, most younger children will uplift five fingers on one hand, counting "one, two, three, four, five, " and then uplift three fingers on the other hand, counting "one, two, three." They will then recount all of the uplifted fingers starting from one. A quicker and more mature procedure is to simply state the largest number, five in this example, and then count-on a number of times equal to the value of the smaller number, as in "five, six, seven, eight."

As a group, children with MD use less mature counting strategies more frequently and for a longer period of time than do other children; children with MD also tend to make more errors when using counting procedures to solve arithmetic problems. This delay in the adoption of mature counting procedures to solve arithmetic problems appears to be related to the earlier described difficulties in keeping track of information during the counting process and to their rather rigid conceptualization of counting.

Although the results are mixed, it appears that many children with MD catch up to their peers in their ability to effectively -- in terms of the maturity of the procedure and error rate -- use counting procedures to solve arithmetic problems by the middle of the elementary-school years. In other words, for many children with MD this appears to be a developmental delay and not a more fundamental deficit. Nonetheless, there appears to be a subset of children with MD who show difficulties in the use of counting procedures throughout the elementary school years and sometimes later.

Finally, there has been some research on the ability of children with MD to solve more complex arithmetic problems, such as 45+97, but considerably less research than with simple arithmetic. The research to date suggests that some of these children have difficulties sequencing the component steps needed to solve these problems. For 45+97, the first few steps involve adding 5+7, noting the 2 in the appropriate column and then carrying the 10 to the next column. Although many children with MD may understand and be skilled at executing each individual step, "putting them all together in the right order" is often difficult.

## Are RD and MD related?

It appears that many -- perhaps more than 1/2 -- children with MD also have difficulties learning how to read and that many children with RD also have difficulties learning basic arithmetic. In particular, children and adults with RD often have difficulties retrieving basic arithmetic facts from long-term memory. The issue is whether the co-occurrence of RD and difficulties in remembering arithmetic facts are due to a common underlying memory problem. The answer to this question is by no means resolved. Nonetheless, some evidence suggests that the same basic memory deficit that results in common features of RD, such as difficulties making letter-sound correspondences and retrieving words from memory, is also responsible for the fact-retrieval problems of many children with MD. If future research confirms this relationship, then a core memory problem that is independent of IQ, motivation and other factors, may underlie RD and at least one form of MD.

## Where do we go from here?

There is much that needs to be done is this area, in terms of basic research, assessment and diagnosis, and, of course, remediation.

### Basic research

There are more unanswered than answered questions in the MD area. Some examples of issues that need to be addressed: We need to know more about the basic counting and arithmetic skills of children with MD, their developmental course -- which of these deficits are simply delays and which are more fundamental problems -- and the basic memory and other cognitive systems that support these skills. We need to begin studies of potential learning disabilities in other areas of mathematics, such as algebra and geometry, although much of this research must await research on normative development in these areas. A few studies suggest that certain forms of MD, such as the fact-retrieval deficit, may represent an inherited risk, although little is actually known about these risks. In other words, we need to learn more about the genetics of MD and the neurological systems that support mathematical cognition and that might be involved in MD. We need to know more about the co-occurrence of RD and MD.

### Assessment and diagnosis

As described earlier, current standardized achievement tests are too general -- they include too many different types of items -- to provide useful diagnostic information on the source of poor math learning. To be sure, low performance on such tests suggest that a learning problem may exist, but these tests do not provide information on the exact source of the poor achievement. The poor scores could be due to problems remembering arithmetic facts, poor counting knowledge, and so on. A standardized diagnostic test that provides more precise information on the counting knowledge, counting procedures used to solve arithmetic problems, ability to remember facts, and so fourth is needed.

### Remediation

Of course the ultimate goal of learning disabilities research is to develop instructional techniques that remediate, or at least compensate for, the learning difficulty. Perhaps it is needless to say, but, in comparison to remediation studies in the RD area, very little research has been done on remediation in the MD area. Part of the difficulty stems from the fact that, except for the areas described above, little is really known about the nature and course of math disabilities -- it is hard to develop effective remedial techniques for a disorder that is not well understood. Nonetheless, this is an area of great need and an area in which we can probably begin to develop remedial programs, at least for basic counting and arithmetic.

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