With the awareness that math understanding is actively constructed by each learner, we can intervene in this process to advocate for or provide experience with manipulatives, time for exploration, discussion where the “right” answer is irrelevant, careful and accurate language, access to helpful technologies, and understanding and support.
What constitutes a learning disability in mathematics?
There is no single mathematics disability. In fact, mathematics disabilities are as varied and complex as those associated with reading. Furthermore, there are some arithmetic disabilities which can exist independent of a reading disability and others which do not. One type of learning disability affecting mathematics can stem from an individual’s difficulty processing language, another might be related to visual spatial confusion, while yet another could include trouble retaining math facts and keeping procedures in the proper order. While extremely rare, there are some learners who cannot successfully compare the lengths of two sticks and others who have almost no ability to estimate. Finally, some people experience emotional blocks so overwhelming as to preclude their ability to think responsibly and clearly when attempting math, and these students are disabled, as well.
How is mathematics learning related to mathematics learning disabilities?
Ginsburg (1977) and Baroody (1987) have identified the initial, intuitive stages of mathematics learning as the “informal” stage. A young child learns the language of magnitude (more, less; bigger, smaller) and equivalence (same) at home, long before schooling begins. In much the same way a child learns to chant the alphabet before knowing how to use it, children learn the counting sequence. This sequence is a kind of song, they discover, and it must go in a particular order.
Informal mathematics includes the ability to match one item with another item, as in setting the table. Later, sometime during the first years of formal school, the child comes to realize that five objects, no matter what size, no matter how spread out, no matter what the configuration, are still counted as five. This gradual realization, called “conservation” of number is an exciting transition and cognitive metamorphosis. It heralds the child’s growing ability to use numerals symbolically with real meaning.
A learning disability at this age may revolve around using language, manipulating objects, or judging size at a glance. Those who are visually impaired require experiences touching and judging more/less, bigger/smaller. There is a very small group of children who seem unable to visually compare length and amount.
When children enter school, they will gradually learn the format aspects of number, i.e., adding with exchanging and trading. In the best circumstances, children begin with informal mathematics, usually with manipulatives, and gradually build to the more abstract, less inherently meaningful formal procedures.
Many children do not make this connection and characterize math as a collection of unconnected facts which must be memorized. They don’t look for patterns or meaning and can feel puzzled by classmates who seem to learn with so much less effort. In other cases, adults move in prematurely with children who are eager and excited to memorize, teaching them procedures which they can imitate but not understand. While this informal/formal gap is not, strictly speaking, a learning disability, it probably is a factor in a majority of math learning difficulties.
The pace at which children move from informal to formal arithmetic is far more gradual than most educators or parents realize. Even as adult learners we need a considerable chunk of time with the concrete, “real” aspect of a new piece of learning before we move on to making generalizations and other abstractions.
There are some children who have a language impairment, who do not easily process and understand the words and sentences they hear. Sometimes these children also have difficulty grasping the connection and the organizing hierarchy of”little” ideas and “big” ones. These children are also likely to view math as an ocean full of meaningless facts and procedures to be memorized.
Visual processing difficulties play a different sort of role in reading than they do in mathematics. In math there are fewer symbols to recognize, produce, and decode, and children can “read” math successfully even when they cannot yet read words. Children with visual/spatial perceptual difficulties may exhibit two kinds of problems. In the less severe instance, some will understand math quite clearly but be unable to express this using paper and pencil. More severe is the case where children cannot translate what they see into ideas which make sense to them.
How do you assess a mathematics disability?
One need not be a mathematics expert to evaluate a child’s ability and style of doing math. A one-to-one mathematics interview is the best format for noting details. In the interview one focuses as intently on how the child does mathematics as on what or how correct they do it. It is essential to keep in mind that you are searching for what does work at the same time as you are probing to find out what doesn’t work.
A mathematics interview should include the use of manipulatives, i.e. coins, base ten blocks, geoboards, cuisenaire rods, and tangrams. A calculator is an important tool and can be used to uncover the difference between comprehension and computation difficulties.
The interviewer needs to remember to look at the full range of mathematical areas. In addition to computation, one should explore the child’s ability to make predictions based on understanding patterns, to sort collections of blocks or objects in a logical way, to organize space with flexibility, and to measure.
To aid in making a diagnosis which will result in useful recommendations, look carefully at strengths and weaknesses. Note whether the child talks to herself, whether she draws a picture to help her understand a situation, or whether he asks you to repeat. See if the child has a mathematics “proofreading” capacity by asking him to estimate before he computes. This is an important strength.
How do you help a child who is having difficulty?
The fundamental principle in helping a child with a disability in mathematics is to work with the child to define his or her strengths. As these strengths are acknowledged, one uses them to reconfigure what is difficult.
When learners have lost (or never had) the connection between mathematics and meaning, it is helpful to encourage them to estimate their answers before they begin computing. When children work together in small groups to solve problems, they often ask more questions, get more answers, and do more quality thinking than when they work quietly, alone.
When children have difficulty organizing their written work on a page, they often do better with graph paper. A less expensive solution is to turn lined paper sideways so that the lines serve as vertical columns. This is especially helpful for long division.
The task of learning the facts can be transformed into one requiring Verbal reasoning. Instead of being asked to memorize 7 + 8, one boy was asked, “How do you remember that 7 + 8 = 15?” His strategies, in this case, that 7 + 7 = 14, so 7 + 8 = 15, were practiced and reinforced and he became able to retain his facts. A general principle is that through drill and practice children will get faster at whatever they’re already doing. This technique of focusing on strategies is one which fosters a healthy sense of self reliance and diminishes the need for meaningless memorization.
When children do not have a strong language base, it is even more important for the language of explanations to be absolutely accurate (concrete) and parsimonious. In other words, elaborations confuse rather than help this type of child. Give the instructions or explanation once and give the child time and the materials to think about what has been said so that he or she can formulate a meaningful question, if necessary. Asking these children to process quickly is unrealistic and not helpful.
By contrast, the group of children who use language as a tool to keep themselves on track and to organize their thinking are often extremely quick to respond. Language is their preferred medium, after all. These children often respond well to the use of metaphor in explanations. These children are often impatient and do not understand that good thinking is not instantaneous. They need reassurance and a relaxed structure so that they go beyond the superficial quickness and do some real thinking.
Finally, those who are afraid to even attempt math are often unaware of their very real strengths. This group believes that math = computation, when in fact computation is but a small slice of mathematics. The increasing acceptance of calculators refocuses teachers and students on the real issue at hand: problem solving. Math anxious students often will take risks if their fears are acknowledged and support is provided. Students will gradually feel more powerful as they experience themselves as successful thinkers.
Mathematics learning disabilities do not often occur with clarity and simplicity. Rather, they can be combinations of difficulties which may include language processing problems, visual spatial confusion, memory and sequence difficulties, and/or unusually high anxiety. With the awareness that math understanding is actively constructed by each learner, we can intervene in this process to advocate for or provide experience with manipulatives, time for exploration, discussion where the “right” answer is irrelevant, careful and accurate language, access to helpful technologies, and understanding and support.