DYSCALCULIA DIAGNOSTIC REPORT FOR JANE DOE
(NO PERMISSION IS GRANTED FOR REPRODUCTION OF THIS SAMPLE REPORT)
PLEASE CONTACT THE AUTHOR WITH ANY REQUESTS: Renee Newman, renee@dyscalculia.org
EVALUATION DATE: December 14, 2000  AGE: 11  GRADE: 7.4 
DIAGNOSIS: Developmental Dyscalculia
EVALUATOR: Renee M. Newman, M.S., Special Education
CONTENTS:
EVALUATION: On December 14, 2000, Jane, in grade 7.4, underwent a 5hour independent evaluation to determine her learning
styles and the presence of dyscalculia and other learning difficulties.
STRENGTHS: Jane has the ability to motormimic what she sees, and readily learns dance routines. Jane follows sequential
directions well, except when she has to deal with "abstract" concepts. She enjoys and excels at artistic expression in
dance, choir, drama, writing, story telling, and community work. She has excellent auditory and visual memory for the vivid
details of emotionally significant experiences, even those of early childhood. She recalls the smallest details and
figuratively paints robust accounts, with recurring descriptions of the smells present during the experiences. Jane is able
to recall the details of stories both read and read to her. She can vividly recall simulations, movie scenes, and personal
encounters.
Jane is usually able to plan ahead and use her time wisely. She usually does not leave behind books and assignments or lose
important belongings. Most important, Jane is a diligent worker. She strives for perfection and is tenacious and ambitious.
She consistently puts an inordinate amount of effort into her math homework, and is dedicated to excellent academic
performance. Socially, Jane is cheerful, popular, and well liked by adults and her peers. Jane speaks fluently, is a great
storyteller, and writes well, especially beautiful poetry.
DIAGNOSIS*: After careful review of her academic record, extracurricular activities, student and parent input, along with
direct observation of her performance on current school math problems, I conclude that Jane has a specific learning
disability in mathematics, properly termed "developmental dyscalculia." This makes her eligible for special education
services under IDEA R 340.1713LD  (f) in mathematics calculation, and (g) mathematics reasoning.
*[According to 34 CFR 300.532 (e), R 340.1721a(1): "The evaluation must be made by a MULTIDISCIPLINARY TEAM. This is a group
of people with different skills, i.e. psychologist, physical therapists, educational diagnosticians, or other professionals
qualified to perform evaluations. The group must include at least one teacher or other specialist with knowledge in the area
of the suspected disability."]
*[According to 34 CFR 300.503 (c), R 340.1723c(4): "If a parent chooses to obtain an independent evaluation, that evaluation
must be considered in decisions made regarding the student's education and may be introduced as evidence at any
administrative hearing."]
Jane was found to be deficient in these prerequisite math skills: Those marked with asterisks denote deficiencies found only
in the area of math reasoning. (1) The ability to follow sequential directions when applied to abstract and math concepts; *
(2) The ability generalize and apply understood classifications; * (3) to order, organize, and sequence quantitative ideas;
(4) to have a command of spatial orientation and organization; * (5) to understand and employ estimation; * (6) to visually
cluster objects; (7) to recognize and extend patterns; * (8) to visualize quantitative ideas; (9) to think deductively; *
and (10) to think inductively easily seeing patterns in situations, and interrelationships between procedures and
concepts.* Without these prerequisite skills, any math learning that takes place is essentially temporary.
PROGNOSIS: Every student with a normal IQ can learn to communicate mathematically, if taught appropriately. Scientists
believe that even the worst math performances are considerably improved when compensatory strategies are coupled with
appropriately organized instruction, proven to remediate skill deficiencies.
Sequential math instruction requires a perfect command of acquired fundamentals. The slightest misunderstanding makes for a
shaky mathematical foundation. Jane can become proficient in mathematics only if she is taught math, from the very
beginning, with a different approach. More of the same will not be successful. Her inconsistent visual and auditory memories
must become secondary pathways for the learning of mathematics. Her strongest memory pathways must become primary. Her good
memory and performance in writing, English, story telling, drama, and kinesthetic movement must be employed for the
acquisition of prerequisite math skills and more advanced concepts.
The Language of mathematics must be deliberately taught and Jane must become proficient in its use and demonstration at
every level. The language of mathematics should be the focus of math instruction, and Jane should practice command of the
language as it relates to the basis of the logic and reason for math concepts and operations.
Discovery methods and lessons dependent on visualspatial aptitude should not be relied upon for the acquisition of obvious
conclusions about mathematical truths. Jane is weak in these areas. Overt teaching is best, with expectations and
opportunities for Jane to immediately prove her understanding with multiple and varied, narrated, handson demonstrations.
These "tests" must be performed daily to move her math knowledge from temporary to permanent status.
Current performance error & reasoning patterns, other disabilities, etc.
Jane avoids and is frustrated by puzzles, and spatial manipulation tasks. Even verbalizing is no guarantee of motor accuracy,
as she says, "I am doing multiplication," when in fact she is performing addition. She experiences leftright confusion,
both verbal and auditory.
In math class, she copies work from transparencies without understanding, and performs her math problems by routine without
reasoning through the operations.
Jane has "outgrown" most of her visual perception difficulties, marked by difficulty organizing the position and shape of
input (u=n, E=3,W, or M.b, d, p, g, q rotation/ reversals.) Sometimes she looses her place when reading or playing the
piano. Although Jane does well in dance, she appears to have difficulty catching balls thrown to her. She does experience
disorientation about position in space and still has problems deciding between left and right.
Jane appears to have no auditory perception or sensory integration disorders. Smell does play a large part in Jane's
emotional memory.
Some integration difficulties were noted. Jane has sequencing difficulties in spelling and math situations. She leaves out
letters in spelling words, but does not seem to realize that the resulting word is not phonetically correct. When working
math problems, she miswrites numbers and signs, and mixes up directional and operational sequences. She has difficulty
finding words in a dictionary; and has some trouble in applying known memory rhymes to the order of operations in
math.
Jane showed some abstraction difficulties, like the inability to make generalizations. She tends to take things literally
and misunderstands jokes, puns, and idioms in conversation, on TV, and in movies. Jane has some organizational difficulties.
She is unable to relate new math facts to previously taught facts and concepts that she, at one time, appeared to have
mastered.
Jane shows some shortterm memory difficulties specific to mathematics. While her memory for other content is normal, her
memory for mathematical content is poor and characteristically inconsistent.
She has difficulty learning and retaining basic addition, subtraction, and multiplication facts. She seems to lack the
"number sense" that is the foundation of all math learning. Even when adding numbers less than 10, Jane counts on her
fingers, in an effort to circumvent her experience of inconsistent recall of studied math facts. Jane suffers from a
frustrating inconsistent visual memory for mathematical facts and concepts like those presented on flash cards, chalkboards,
transparencies, and pencil and paper demonstrations. Even her auditory memory for these experiences is insufficient to
support the recall prerequisite of successful math performances.
Jane has a qualitative learning style. Qualitative learners approach math tasks holistically and intuitively with a natural
understanding that is not the result of conscious attention or reasoning. They define or restrict the role of math elements
by description and characterization of an element's qualities. They are social, talkative learners who reason by verbalizing
through questions, associations, and concrete examples. They draw parallels and associations between familiar situations and
the task at hand.
Most of their math knowledge is gained by seeing interrelationships between procedures and concepts. They focus on
recognizable patterns and the visual/spatial aspects of math information, and do better with math applications. They have
difficulty with sequences and elementary math.
Qualitative learners dislike the procedural aspects of math, and have difficulty following sequential procedures, or
algorithms. Their work is fraught with careless errors, like missing signs, possibly because they avoid showing their work
by inventing shortcuts, eliminating steps, and consolidating procedures with intuitive reasoning. Their work is procedurally
sloppy because
they quickly tire of long processes. Their performance is never fluent because they do not practice enough to attain levels
of automaticity. Eventually, the qualitative student may show disinterest in the mastery of mathematics.
JANE'S HISTORY OF MATH DISABILITY:
In first grade, Jane had no trouble adding sets of pictured objects, but counted on her fingers. Finger counting persisted
into 2nd grade and is still used by Jane in the 7th grade. She recalls struggling in 3rd grade and having no idea of the
purpose of using measuring cups and water. She also had trouble extending patterns, understanding place value, estimating,
and telling time.
Fourth grade was hard and intimidating. She had severe difficulties learning the multiplication tables, which were introduced
in 2nd grade. She also had difficulty with grouping and generalizing. Now in 7th grade, Jane can go through the motions,
repeating a pattern for an entire page of math problems, but cannot explain what she is doing or why. She still has trouble
telling time on a face clock and is thankful for a digital watch.
Upon review of the MET report dated 452000, several conflicting statements were found. On page 7, the team states that
"Jane does not have a severe discrepancy between achievement and intellectual ability." On page 5, it states that "Jane's
Verbal IQ [105Normal or Average / 63%] is considered the best estimate of her ability." It lists her Performance IQ as 89
[23%], which is low average. Jane performed near the low average on tests of usable knowledge, abstract reasoning, visual
attention to details, understanding of social situations, seeing parts in a visual whole, and the psychomotor skill of
coding. On her KTEA, she scored in the 30th percentile on math applications, a grade equivalent of 4.8 in grade 6.8. That
is a performance gap of 2 years.
In corroboration with my findings, the report went on to correctly identify her dyscalculic characteristics: "Jane will
continue to require extra support in math. Pre teaching.... helpful. She will also require reteaching and more repetitions
before skills are solid.... she needs extra time to process the information." p.7
"She sometimes misinterpreted what the question was asking. She also struggled when a twostep process was needed.... she
incorrectly arranged four fractions from least to greatest. .... was slower when "processing" math problems. With time she
was sometimes successful. ...occasionally made computational errors...multiplied 4 digit by 2digit number but made errors
in the process. She occasionally subtracted a few numbers in the final step instead of adding them. ...general math
computational skills are not yet solid. Her occasional computational or procedural errors interfere with her accuracy. She
often completes problems at a slower rate."p.6 Jane demonstrates below average math skills...weak in immediate recall of
basic facts and use of computational skills. She seems to lack number sense and has difficulty determining when answers make
sense. Making inferences also appears to be difficult for Jane." p.3
Now in grade 7.4, all of these conditions are still true, despite the use of "multiple strategies for solving problems and
reaching conclusions,"p.3 diligent work habits, daily professional tutoring before and after school throughout grades 5 [Title 1]
and 6, and daily delivery of an approved curriculum of mathematics instruction by a licensed teacher.
RECOMMENDATIONS:
LOG: All home and school instruction, comprehension, and performance details, as well as strategies, successes and failures
should be logged in a binder that will serve as a record, both diagnostic and prescriptive. Be sure to include the hours of
work and detailed lesson plans. Note the length of time required for certain performances, the attitude and morale of both
teacher and student, and any relevant events that have a bearing on performance. This record should travel daily between
home and school.
As advised, Jane and her parents have committed to working on the acquisition of prerequisite math skills for 30 minutes each
day, following the directions in "A Linguistic Approach to K4 Basic Number Concepts."
Her current placement in 7th grade math should be altered, only if that block of time will be committed to the type and
frequency of instruction here described. If the remedial instruction takes place before or after school, Jane should spend
her math period teaching a weak first or second grader the math concepts that she has recently acquired. This will serve
three purposes. It will reinforce the relevance and permanence of the math concepts for Jane. It will assist a younger child
who is experiencing similar difficulty grasping math concepts.
Instead of the shameful "inferiority" and unwanted attention from friends that Jane would feel if she had to be removed from
regular instruction, Jane's selfesteem and spirit will be buoyed because she is assuming an important role as "teacher of
someone in need."
If these activities cannot be scheduled during her regular math period, Jane should remain in class, but with the following
accommodations. Her teacher must become familiar with the condition and implications of dyscalculia. Jane should be graded
on effort over performance and on a passfail basis so that her grade does not unfairly alter her otherwise respectable
grade point average. Any tests and quizzes should be taken with the class to save face, but should be discarded and replaced
by oneonone private testing sessions with the teacher, where Jane is able to verbalize and employ all of the strategies
mentioned below. The teacher must diligently watch for dyscalculic errors and not penalize for these. Most important, the
teacher must genuinely understand Jane's frustrations and desire to help her succeed. The teacher must be committed to
Jane's success, in spite of the incredible hurdles that stand between Jane and her goal.
It is recommended that Jane be taught math concepts in a holistic method that includes deliberate specific, differentiated
instructional attention to the three components: (1) linguistic [which must include the teaching of symbols, concepts,
vocabulary, syntax, voice, and translation]; (2) conceptual; and (3) procedural or skill. Introduce math ideas with concrete
objects and logical language emphasis that is tied to common quantitative expressions.
The school may wish to confirm the disabilities noted here through testing by the school psychologist and the district
occupational therapist. Disabilities can exist in isolation but are most often found in combination. An individualized program
of treatment should address all areas of difficulty, or "disability." It may take 1015 repetitions over several days for a
learning disabled child to retain what the average child retains in 35 repetitions on one occasion.
It is therefore imperative that remedial training take place daily for at least thirty minutes, but more advantageously for 45
60 minutes. Severe regression will occur when training is inconsistent, boring, review, or "more of the same in nature," and
occurs with less than daily frequency. Spotty or disorganized attempts at remediation will be futile, and will further add
to Jane's frustration with the subject.
Jane's remedial training should be preventative, prescriptive, individualized, multisensory, kinesthetictactile, sequential,
and involve drama and the sense of smell. It must be oneonone with someone knowledgeable about dyscalculia, and trained in
the language and the operations of mathematics. This instructor must be trained to watch for characteristic dyscalculic
mistakes. The instructor must vigilantly look for dyscalculic behaviors, point them out immediately, and offer coping and
compensatory strategies. When a mistake is identified, Jane is to start over and verbalize while demonstrating why and how a
certain thing is being done.
This instruction can be delivered by a specially trained teacher, tutor, mathematically strong highschool student, or
volunteer.
Watch for these common mistakes: Misunderstanding presented problems, the perseveration of ideas and operations where they are
no longer appropriate, midoperation switching of operations, loss of directional sense for operations, loss of sequential
rules of operations, misaligned series of numbers, miswritten numbers and signs, lack of place value awareness, forgotten
purpose of operations, forgotten goal of problem solving, and "blank slate syndrome," where the student suddenly tears up,
having forgotten what they are doing and having realized that all their work, thus far, has been in vain.
Remediation should take place in an area free from distractions. The area must have a white board or chalkboard where Jane
can play teacher, teaching the material back to the instructor. It should have multicolored markers or chalk for the color
coding of operations. Multicolored, erasable pencils or pens should be used on graph paper or columnar pads. Jane must be
trained to place her numbers inside the guide boxes and use the paper rules to line up numbers with proper place value.
Operations or steps should be colorcoded. Jane should describe aloud what she is doing, why, and how, as she works through
math problems. A ruler must be used to view columns of numbers in isolation and to line up digits.
After determining that Jane has all prerequisite skills and levels of cognitive understanding, introduce new concepts in the
following sequence:
[A] Inductive Approach for Qualitative Learners: (1) Explain the linguistic aspects of the concept. (2) Introduce the
general principle, truth, or law that other truths hinge on. (3) Let her use investigations with concrete materials to
discover proofs of these truths. (4) Give many specific examples of these truths using the concrete materials. (5) Have Jane
talk about her discoveries about how the concept works. (6) Then show how these individual experiences can be integrated
into a general principle or rule that pertains equally to each example.
[B] Deductive Approach for Quantitative Learners: Next, use the typical deductive classroom approach. (7) Reemphasize the
general law, rule, principle, or truth that other mathematical truths hinge on. (8) Show how several specific examples obey
the general rule. (9) Have Jane state the rule and offer specific examples that obey it. And finally, (10) have her explain
the linguistic elements of the concept.
Once through a problem is not sufficient experience to cement the routine in permanent memory. The demonstration must be
repeated with Jane demonstrating the concept successfully at least 10 times each session, for a week, from the vantage of
expert or teacher, until complete automaticity and confidence is achieved.
Be sure to move through these six levels of learning mastery before introducing a new concept. (1) Jane intuitively relates
the new concept to existing knowledge and experiences. (2) Then she looks for concrete material to construct a model or show
a manifestation of the idea. (3) Next, she draws to illustrate the concept, connecting the concrete or vividly imagined to
the symbolic representation. (4) Now, Jane translates the concept into mathematical notation, using digits, operational
signs, formulas, and equations. (5) Then, she applies the concept successfully to realworld situations, story problems, and
projects. (6) Finally, Jane must teach the concept successfully to others, and lastly, communicate it on a test. Jane is not
considered to have mastered the material until she "owns" it, which means she has learned it well enough to teach it,
remember it for all time, and use it meaningfully, quickly, easily, appropriately, and profitably.
INSTRUCTOR'S RESPONSIBILITY:
There are five critical factors affecting math learning. Each is an essential component of the successful math curriculum, as
well as being a critical diagnostic tool for evaluating learning difficulties in mathematics. [1] A teacher must first
determine each student's cognitive level of awareness of the knowledge in question, and [2] must ascertain the strategies he
brings to the mathematics task. Low functioning children, like Jane, have not mastered number preservation and are dependent
on fingers and objects for counting. Findings dictate which activities, materials, and pedagogues are used.
[3] The teacher must understand that each student processes math differently, and this unique learning style affects
processing, application, and understanding. Jane is a qualitative learner. [4] The teacher must also assess the existence
and extent of math readiness skills. Nonmathematical in nature, mastery of these seven skills is essential for learning the most
basic math concepts. Jane is deficient in all of them.
The seven prerequisite math skills are: (1) The ability to follow sequential directions; (2) A keen sense of directionality,
of one's position in space, and of spatial orientation and organization; (3) Pattern recognition and extension; (4)
Visualization key for qualitative students is the ability to conjure up and manipulate mental images; (5) Estimation the
ability to form a reasonable educated guess about size, amount, number, and magnitude; (6) Deductive reasoning the ability
to reason from the general principle to a particular instance; and (7) Inductive reasoning natural understanding that is
not the result of conscious attention or reason.
[5]Teachers must teach math as a second language that is exclusively bound to the symbolic representation of ideas. The
syntax, terminology, and translation must be directly and deliberately taught. Two different skills are required: (1)
Students are usually taught to translate English expressions into mathematical expressions. (2) But first they should be
taught to translate mathematical language into English expressions. Instead of story problems, give the child mathematical
expressions to be translated into a story in English.
To eliminate the problem of a limited math vocabulary, matteroffactly interchange the formal and informal terms in regular
discourse. Seek to extend her expressive language set to include as many synonyms as possible. Use at least two terms for
every function. For example, say, "You are to multiply 7 and 3. You are to find the product of 7 and 3. The product of 7
times 3 is 21."
The teacher must ask, "Is my teaching style excluding students with certain learning styles? Are the methods and materials I
am using appropriate for, and compatible with, the students' cognitive levels and learning styles? Have the students
mastered requisite skills and concepts?"
The best teaching methods are diagnostic and prescriptive. They take all mentioned variables into consideration. The
competent teacher supplies plenty of presentation methods and learning activities to stimulate each type of math learning
personality.
Without becoming overwhelmed with the prospect of addressing each child's needs individually, the continuum can be easily
covered by following Dr. Mahesh Sharma's researched and proven method, stated in the RECOMMENDATIONS section of this
report.
RATIONALE:
According to Paiget (1949, 1958), children learn primarily by manipulating objects until the age of 12. If children are not
taught math with handson methods, between years 1 and 12, their ability to acquire math knowledge is disturbed at the point
when hands on explorations were abandoned in favor of abstractions. Early transition to abstract instruction sets children up
for mathematical disabilities in the next developmental period of formal propositional operations.
Brain lateralization, or hemisphere specialization, takes place earlier in girls than in boys. Because of girls' social
development away from spatial tasks, the maximum spatial capability of the right hemisphere is fixed (when it stops growing) in
a premature stage of development.
Research shows that girls are overspecialized in lefthemisphere functioning, and must talk through spatialvisualization
tasks, resulting in slow, unnatural performance. By avoiding intricate mechanical manipulations and rough and tumble
physical activities, girls loose ground in spatial organization skills. Girls' more sedentary activities offer few exercises
in space/motion judgment, symmetry, parttowhole construction, and development of visualization, muscle memory, and
geometric principles. But boys are gaining in all of these areas, and their improving spatial organization skills better
prepare them for mathematics tasks.
Psychologist, Julia Sherman, believes that earlier female verbal and reading development leads females to prefer verbal and
reading teaching and learning approaches to nonverbal righthemisphere problem solving approaches. Other researchers see
spatial visualization as essential to all levels of math learning. These skills exist on a continuum from lowlevel,
requiring no image transformation, to highlevel, involving the visualization and mental manipulation of 3dimensional
figures. Research on athletes suggests that spatial visualization skills can be learned.
Preoperational Stage (Ages 2 to 7): Children can imagine and think before acting, but do not use logical reasoning. They are
egocentric and have difficulty considering others' points of view. They reason by intuition and appearances, not by logic
and implication. They cannot combine parts into wholes, coordinate variables, or consider more than one variable. Space is
restricted to the neighborhood, and time is restricted to seasons, days, and hours. Simple classifications are made but the
child has difficulty arranging objects into a long series and inserting new objects into existing series. The child cannot
conserve size, shape, and volume when objects are rearranged or changed in appearance. The child centers attention on one
property while excluding others. The child cannot reverse his thinking to the point of origin, and does not comprehend that
actions and thoughts are reversible. (In mathematics, Jane is functioning in this preoperational stage.)
Not yet able to reason abstractly, instruction must use object manipulation and concrete situations, in addition to verbal
reasoning. The ability to think conceptually is emerging and proceeds slowly.
The RATIONALE section of this report focuses on the developmental implications of dyscalculia. Callahan, Clark, and Kellough
report that 5% of middle school students operate at this preoperational level of development. These children will have grave
difficulties with mathematical concepts.
Mahesh Sharma (1989), leading dyscalculia expert, estimates that 6% of all students have developmental dyscalculia, marked by
an inability to function beyond the preoperational level. (Keep in mind that coordination of variables, classification,
combining parts, spatial orientation, reversibility, and conservation, are essential premathematics skills.)
Effective mathematics reasoning and conceptualization cannot take place without these foundational attributes. Teachers of
students performing at this stage must successfully build these skills before proceeding with mathematics
instruction.
Concrete Operations Stage (Ages 7 to 11): The child can perform logical operations and with less egocentrism, and can
observe, judge, and evaluate to solve physical problems. Thinking is still concrete, not abstract. Early on, the student is
unable to generalize, weigh possibilities, or
consider hypothetical situations. The child can make multiple classifications, arrange objects and place new items in series,
and make sense of geographical space and historical time.
The concepts of conservation are mastered in the following order: number of objects between ages 6 and 7; matter, length, and
area at age 7; weight between ages 9 and 12; and volume after age 11.
The concept of reversibility of physical and mental processes has emerged. The child can correctly interpret a rearranged
number of objects and changed size and shape of matter. Later on, the child can hypothesize and do higherlevel thinking,
but cannot yet reason abstractly and is just beginning to think conceptually. The most effective teaching/learning methods
are the use of hands on concrete manipulations, in addition to verbal instruction.
Callahan, Clark, and Kellough (1998) report that most middle and junior high school students have not reached the formal
operations stage. Since this stage in prerequisite for advanced mathematical cognition, what teaching methods will
successfully take students from primitive levels of mental processing to the formal and abstract levels required for mastery
of middle school math concepts?
The "threephase learning cycle" is on the right track. It guides students from concrete handson learning through concept
formulation, abstraction, and then to application. Variations on this basic idea are successful because they reach each
student's learning style by using all sensory channels. Through direct experiences, students sense and feel, watch, reflect,
think, question, develop and test theories, evaluate, synthesize, and apply new knowledge to situations, building a
foundation as they go. This emphasis is on doing.
According to Gagne, learning occurs on 8 successive levels, so a teacher should ascertain the child's highest level of mastery before proceeding with diagnostic and prescriptive instruction.
(1) Signal Learning is basic instinctive responses to stimuli.
(2) StimulusResponse Learning is an acquired response to a discriminated stimulus.
(3) Chaining is the linking of physical, nonverbal sequences of simple stimulusresponse events.
Accuracy increases with reinforcement, prior experiences, and practice.
(4) Verbal Association involves object recognition and verbal naming, at the start, and graduates to rote memorization of
numerical and alphabetical sequences.
(5) Multiple Discrimination involves sophisticated naming of individual nouns and the ability to organize and categorize
objects and object concepts.
(6) Concept Learning involves the recognition of objects by their abstract characteristics, as opposed to recognition limited
to purely physical attributes. Abstract properties are conserved and discerned in situations where appearance is
altered.
(7) Principle Learning is the recognition of the relationship between two or more distinct concepts.
(8) Problem Solving involves the application of principles to solve a problem, then a class of problems. This application of
principles to achieve goals creates a higherorder principle: a combination of lowerorder principles (Gagne 1970).
When considering dyscalculia, many students (and adults) do not show mastery of Gagne's late level four. A breakdown appears
in sequencing ability. Although verbal numerical sequencing occurs (counting ability), disruptions are seen in transitions
from 9 to 10, 19 to 20, 29 to 30, etc.
These students also show weakness in recalling the position of sequential items in isolation. If you ask a child what number
comes before 7, they are likely to recall the entire sequence before giving an answer. The items in the sequence do not take
on structural permanence in memory, even after the child has graduated to other levels in other areas of childhood
development.
Later, the child, despite ordinary presentation and drill techniques, is unable to memorize basic addition, subtraction,
multiplication, and division facts. Memory for all mathematical sequences is weak and transient. Soon forgotten, are order
of operations for even the simplest of algorithms, like addition, subtraction, and multiplication.
A pattern of directional confusion emerges, as the frustrated child is unsure of the nature of the number line, placement of
numbers on the clock, and placement of arithmetic components during operations (the nature of place value, numeration, lining
up numbers for successful operations, directions for borrowing, carrying, etc.).
The multiple discrimination that is required of Gagne's level five becomes impossible for the dyscalculic child. Ignorant of
basic mathematical ideas, he is unable to link numbers to concepts of negative and positive, and then to concepts of
reversibility or more advanced concepts.
A diagnosis of dyscalculia can be made as long as the child cannot perform simple quantitative operations that should be
routine at his age. Developmental dyscalculia is present when a marked disproportion exists between the student's
developmental level and his general cognitive ability, on measurements of specific math abilities.
Developmental dyscalculia (Class A) is defined as dysfunction in math in individuals with normal mental functioning.
Mathematical difficulties are the result of brain anomalies inherited or occurring during prenatal development. Deficiency
is defined as a discrepancy of 12 standard deviations below the mean, between mental age and math age (MathQ), marked by a
clear retardation in mathematical development. Characteristics include numerical difficulties with counting; recognizing
numbers; manipulating math symbols mentally and/or in writing; poor sequential memory for numbers and operations; mixing up
numbers in speech, reading, writing, and recalling; and difficulties with auditory processing and memory. Extraordinary
effort is required for math reasoning and function.
All subtypes categorized as Class AType 1 dyscalculia, exist in persons with normal mental ability or IQ. Class A1a,
dyscalculia, is moderately severe. It is marked by a total inability to abstract, extend, or consider concepts, numbers,
attributes, or qualities apart from specific, tangible examples.
Class A developmental dyscalculia can be thought of as several types of difficulty. Quantitative dyscalculia is a deficit in
the skills of counting and calculating. Qualitative dyscalculia is the result of difficulties in comprehension of
instructions or the failure to master the skills required for an operation.
For instance, when a student with qualitative dyscalculia has not mastered memorization of number facts, he cannot benefit
from this "stored verbalizable information about numbers." This essential information is used, with prior associations, to
solve problems involving addition, subtraction, multiplication, division, and square roots.
According to a 1998 report issued by the U.S. Department of Education, student achievement is strongly influenced by teacher
levels of expertise. An expert teacher's students perform 40% better than students of a novice teacher. Presently, the average K
8 teacher has taken only 3 or less math or math education classes in college. Not even 50% of 8th grade math teachers have
taken a single class on math teaching at this level, and 28% of high school math teachers lack a major or minor in
math.
CONCLUSION:
Every student with a normal IQ, like Jane, can learn to communicate mathematically if taught appropriately.
The United States has lofty goals for math achievement. The U.S. Department of Education's math priority reads: "All
students [including Jane] will master challenging mathematics, including the foundations of algebra and geometry by the end
of 8th grade." It hopes that all kindergarten through 12th grade students will eventually master challenging mathematics
that include "arithmetic, algebra, geometry, probability, statistics, data analysis, trigonometry, and calculus (The State
of Mathematics Education Address: Building a Strong Foundation for the 21st Century, 8 January, 1998)."
Armed with the knowledge of developmental theory and best methods, teachers now have the responsibility and the tools to
insure the mathematical success of all students even those, like Jane, who struggle with learning disabilities in
mathematics. 
