Educators often worry that some students just don’t “get” math. In truth, some fundamental difficulties with math may be indicators of mild dyscalculia, or “number blindness.”
It’s hard to imagine an aspect of daily life that is not touched by numerical ideas: telephone numbers, street addresses, radio stations, bus schedules, currency, and time all depend on a rich awareness of numbers and their meanings. The concept of numbers is ubiquitous in the modern world, and it seems at first inspection that anyone with fundamental misconceptions about numbers would be quite apparently handicapped.
This conclusion is so compelling that it remains virtually unquestioned in mathematics education. Students who appear to have fundamental difficulties with mathematics are told to study harder, to pay more attention, and to spend more time working through drills. In many cases, they are tracked into increasingly rudimentary mathematics courses until they are hopelessly behind their peers. In all cases, both teachers and students are frustrated by what seems to be a tremendous chasm separating those who “get” mathematics and those who don’t.
Recent research on how the brain processes math offers some hope for bridging that chasm, suggesting that the fundamental misconceptions students have about numbers may stem from a subtle mathematical impairment. A 1998 report published in the Journal of Pediatrics estimated that approximately five percent of the school age population has some degree of dyscalculia, a sort of “number blindness” that is an impairment of the ability to recognize or manipulate numbers. This [suggests] that for some children, creative teaching techniques and studious discipline are not enough for a productive math education. Mild dyscalculia may easily go unnoticed, leading some students into educational settings that can offer only frustration.
As data about dyscalculia accumulates, it is becoming clear that math education could be improved for all children through a better understanding of how the brain processes math and what can go wrong with that processing.
Developing a Sense of Number
Our ability to understand and manipulate numbers goes through a series of important developmental stages well before we posses any recognizable mathematical talents. Dr. Karen Wynn, a child psychologist at the University of Arizona, has demonstrated that infants as young as 6 months expect that adding one object to another object will produce a set of two objects, and that infants are quite surprised when these arithmetical expectations are violated.
The infantile sense of numbers is restricted to collections of only four or five objects, and the data suggests that infants and adults manipulate such collections using a mental process quite distinct from counting. For small collections, both adults and infants perceive the “numerosity” of the collection directly, somewhat like we perceive shape or color. This direct, intuitive perception of numerosity is called subitization, and it is the first number skill that we develop. When we see three objects, we don’t count “one, two, three,” instead we are simply aware of the group’s “threeness.” Most people can subitize up to seven or eight objects, switching to a variety of counting strategies for larger collections.
Children with Dyscalculia Count in a Unique Way
Subitization is an innate rather than a learned skill, and it forms the basis for the rest of mathematical development. As they get older, children learn to count by comparing the results from a counting procedure to the directly perceived number, a process that fosters an intuitive understanding about how counting and numbers work. That “number sense” guides children later as they learn addition and subtraction, multiplication and division; at each stage of education, new rules about numbers are evaluated in light of the number sense and are incorporated into that sense.
A hallmark of dyscalculia is an impairment of the ability to subitize. Because children with dyscalculia have difficulty directly seeing the “threeness” of three objects, they learn to count in a very different way than other children; most notably, they rely on sequencing and memorization.
At the most basic level, counting is simply putting a list of number words (one, two, three, and so on) into one-to-one correspondence with a group of countable objects. Children with dyscalculia typically have no difficulty remembering the sequence of the number words, and they can put those words into a one-to-one correspondence with a group of objects. To even a careful observer it looks like these children are counting normally, if a tad slowly. But at the end of the sequence, when these children pronounce that there are three objects present, they do not have an intuitive sense of “threeness” – they simply have faith that their counting procedure has led them to the correct answer.
Detecting Number Blindness and Implications for Education
As they progress through the educational system, children with mild dyscalculia may invent increasingly sophisticated coping strategies for learning mathematics. One strategy that is probably quite common is sheer memorization; without a real sense of what a number is, students who are having fundamental difficulties with math will simply memorize a huge volume of rules and patterns. They may never truly understand how numerical concepts generalize, and so may experience difficulty throughout their mathematical education when they have to relate new mathematical rules to each other in an abstract way.
A few cognitive tests can point, albeit crudely, to the possibility of dyscalculia. Foremost amongst these is a reaction time test in which subjects are presented with two numbers and asked which is the larger. For most people, as the distance between numbers increases, the task becomes easier; that is, it’s easier to tell that 9 is bigger than 4 than it is to tell that 5 is bigger than 4. For people with dyscalculia, the response to this test is precisely the opposite. Without the ability to subitize, without a sense of what a number really is, dyscalculics must rely on counting and sequencing. If you’re counting, it takes longer to determine that 9 is bigger than 4 than it does to determine that 5 is bigger than 4.
Causes of Dyscalculia
While the fundamental causes of dyscalculia remain unclear, the evidence suggests that dyscalculia is a developmental problem associated with the a part of the brain’s cortex in the inferior parietal lobe. The root of the problem may lie in the evolutionary history of the human cortex. The part of inferior parietal lobe that seems to be responsible for representing numbers lies adjacent to a part of cortex that is responsible for touch sensation in the fingers. This has led some scientists to suggest that the “number area” in the cortex arose as an evolutionary specialization of finger representation, implying a deep link between number sense and sensation in the fingers. And while many children who have dyscalculia do have impaired sensation in their fingers (a condition known as finger agnosia), there are children who appear to be dyscalculic but do not have any obvious difficulty with finger sensation.
These types of inconclusive results have frustrated research into the underlying factors of dyscalculia, and the neurobiology of the condition remains poorly described. What is clear is that the brain has certain identifiable ways in which it processes mathematical information, and that subtle compromises in those processes can be masked by compensation.
Although the neuroscience of dyscalculia is still in its infancy, an awareness of the condition alone might lead to immediate improvements in the process of mathematical education. Simply knowing that certain types of mathematical difficulties may require special attention is an important step. People with dyscalculia are neither lazy nor stupid, and presenting them with the same information repeatedly is unlikely to improve their mathematical understanding. The root difficulties of number sense must be addressed before further progress can be made. Since math is a highly cumulative discipline, with each new concept building on previously learned concepts, addressing any difficulties with number sense early on is important.
Perhaps more significantly, the existence of the specific difficulties associated with dyscalculia are evidence that math is an integral part of the human brain. For many, mathematics is an alien subject, something highly abstract and difficult to master. A greater appreciation for our sense of numbers, an increased recognition that numbers are a natural part of who we are, may go some distance to remediating our fear of mathematics. Many scientists and philosophers have observed that math is the language of the natural world; as understanding of math and the brain increases, it seems that math is one of the many languages of human nature as well.