Math LD Report




EVALUATION DATE: December 14, 2000   -  AGE: 11 -  GRADE: 7.4

DIAGNOSIS: Developmental Dyscalculia

EVALUATOR: Renee M. Newman, M.S., Special Education


Evaluation | Strengths | Diagnosis | Prognosis | Current performance | History of Math Disability | Recommendations |Teaching Sequence | Sharma's 6 Levels of Learning Mastery | Instructor's Responsibility | Rationale | Gagne's 8 Levels of Learning Mastery | Diagnosis of Dyscalculia | Conclusion

EVALUATION: On December 14, 2000, Jane, in grade 7.4, underwent a 5-hour independent evaluation to determine her learning styles  and the presence of dyscalculia and other learning difficulties. 

STRENGTHS: Jane has the ability to motor-mimic 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  340.1713-LD - (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 visual-spatial 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, hands-on 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 left-right 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 short-term 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.


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 4-5-2000, 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  [105-Normal 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 K-TEA, 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 re-teaching 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 two-step 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 2-digit 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. 



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 K-4 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 self-esteem 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 pass-fail 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 one-on-one  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 10-15 repetitions over several days for a learning  disabled child to retain what the average child retains in 3-5 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, multi-sensory, kinesthetic-tactile, sequential, and  involve drama and the sense of smell. It must be one-on-one 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 high-school student, or volunteer. 

Watch for these common mistakes: Misunderstanding presented problems, the perseveration of ideas and operations where they are no  longer appropriate, mid-operation 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 multi-colored markers or chalk for the color -coding of  operations. Multi-colored, 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 color-coded. 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 real-world 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. 


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. Non-mathematical 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, matter-of-factly 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. 


According to Paiget (1949, 1958), children learn primarily by manipulating objects until the age of 12. If children are not taught  math with hands-on 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 left-hemisphere functioning, and must talk through spatial-visualization 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, part-to-whole 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 non-verbal right-hemisphere problem solving approaches. Other researchers see spatial  visualization as essential to all levels of math learning. These skills exist on a continuum from low-level, requiring no image  transformation, to high-level, involving the visualization and mental manipulation of 3-dimensional 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 pre-mathematics 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 higher-level 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 "three-phase learning cycle" is on the right track. It guides students from concrete hands-on 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) Stimulus-Response Learning is an acquired response to a discriminated stimulus. 

(3) Chaining is the linking of physical, non-verbal sequences of simple stimulus-response 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 higher-order principle: a combination of lower-order 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 1-2 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 A-Type 1 dyscalculia, exist in persons with normal mental ability or IQ. Class A-1-a,  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. 


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.