Learning Disabilities in Mathematics
by: C. Christina Wright, Ph.D, National Center for Learning Disabilities
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 and Baroody 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.
Summary
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.
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