SPECIAL EDUCATION PROGRAM EVALUATION
for INSTITUTIONS:
DIAGNOSING MATH LEARNING DISABILITIES /
RECOMMENDED PRACTICES
BY RENEE M. NEWMAN
© MAY 1998 RENEE M. NEWMAN All Rights Reserved
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ii.
CONTENTS
TABLES...............................................................................................................ix.
INTRODUCTION....................................................................................................1.
SCHOLASTIC FACTORS...............................................................................1.
SOCIAL FACTORS...........................................................................................2.
GENETIC FACTORS......................................................................................3.
CONCLUSION and PROGNOSIS................... .......................................3.
PREVENTION.......................................................... ....................................5.
EVALUATING FOR MATH LEARNING DISABILITY...........................................6.
QUALIFICATIONS OF AN EDUCATIONAL EVALUATOR..........................6.
DIAGNOSING DYSCALCULIA........................................................................6.
ISSUES COMPLICATING DIAGNOSIS...........................................................9.
DIAGNOSTIC TEACHING METHODS..................................................................15.
PRESCRIPTIVE & PREVENTATIVE TEACHING...............................................25.
CONCLUSION............................................................................................................26.
REFERENCES............................................................................................................29.
iii.
TABLES
TESTS FOR DIAGNOSING DYSCALCULIA......................................................7.8.
DIAGNOSTIC CLASSIFICATIONS OF DYSCALCULIA..................................10.14.
SEVEN PREREQUISITE MATH SKILLS...........................................................19.
GRAPHIC ILLUSTRATION OF MATH CONCEPTS.............................................21.
MAIN METHODS OF FACILITATING LANGUAGE EXPRESSION..................23.
THE SIX LEVELS OF LEARNING MASTERY....................................................24.
BEST TEACHING METHOD: FROM INDUCTIVE TO DEDUCTIVE..................25.
iv.
INTRODUCTION
America, the richest country in the world, is dedicated to providing a 13year, free and appropriate public education to each of its citizens, beginning at age 5. In Shiawassee County, Michigan, there is one teacher for every 70 people. That amounts to 1 teacher for every 17 students between the ages of 4 and 18, although typical classrooms contain 2630 students. So why are our students loosing the numbers game when the odds are in their favor?
Almost 93% of America's 17 yearolds graduate without proficiency in multistep problem solving and algebra. (NCES 1997, 123124) An alarming 1 of every 4.5 American adults, or 22%, cannot perform simple arithmetic. (NCES 1997, 416) Sharma estimates that 6% of children have true developmental dyscalculia. (CTLM 1989, 86) How can this be explained?
SCHOLASTIC FACTORS: Cambridge College dean, Mahesh Sharma, asserts that math outcomes are terrible for a number of reasons. Our mathematics curriculums are not reflecting what we know about how children learn mathematics. Typical math curriculums are guided by chronological age. Math is presented in a pile up fashion. Each year, more math concepts are added to the pile of previously presented concepts. This is a tragic approach. (Sharma 1989)
Sharma believes that teacher trainers are not bringing all the known aspects of math learning into the teacher preparation curriculum. New math teachers have not been taught the latest developments in learning theory and math conceptualization, and do not know how use technology as a learning tool. (Sharma 1989)
In the end, teachers teach as they were taught. Their teaching style reflects their own learning style. (It is natural to believe that everyone thinks like you do.) Teachers need to realize that if students are experiencing difficulty, they should ask themselves the following questions:
1.
2.
Is my teaching style excluding students with certain learning styles? Are the methods and materials I am using appropriate for and compatible with the student's cognitive level and learning style? Has the student mastered requisite skills and concepts? (Sharma 1989)
Recent studies show that student achievement is strongly influenced by teacher levels of expertise. An expert teacher's students perform 40% better students of an illprepared teacher. Presently, the average K8 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. (USDE 1998)
SOCIAL FACTORS: Cohn (1968) explains that having a disability in math is socially acceptable. He asserts that math ability is more regarded as a specialized function, rather than a general indication of intelligence. As long as one can read and write, the stigma and ramifications of math failure can be diminished and sufficiently hidden.
Sharma, concurs, explaining that in the West, it is common to find people with high IQ's who shamelessly accept incompetency in math. At the same time, they find similar incompetence in spelling, reading, or writing, totally unacceptable. Prevailing social attitudes excuse math failure. Parents routinely communicate to their children that they are "no good at math." (Sharma 1989)
Shelia Tobias, in 1978, realized that because only 8% of girls took 4 years of math in high school, 92% of young women were immediately eliminated from careers and study in science, chemistry, physics, statistics, and economics. Half of university majors were closed to them. Tobias states that women avoid math, not because of inability, but because women are "socialized" away from studying math.
3.
She advocates "math therapy" for both sexes to overcome "math anxiety." (Tobias 1978, 1213)
Sharma asserts that gender differences in math skills are due more to social forces than to genderspecific brain construction and function. He believes that gender differences can be eliminated by equalizing the activities and experiences of both boys and girls at every level of development. The social forces that direct a child's experiences and choice of activities lead to the differences in the neurological sophistication of boys and girls. (Sharma 1989)
For example, most studies show that girls do better than boys in math until the age of 12. Then boys dominate the subject. This difference can be explained by analyzing the genderspecific development of math prerequisite, spatial orientation skills. The main reason for this is the methodology of teaching in preschool and elementary grades, where focus is on finemotor skill development. (Sharma 1989)
Boys and girls are given ample opportunities to play with blocks, Legos, board games, and various materials requiring finemotor coordination. Naturally, girls have better finemotor skills early in their development. (Sharma 1989) And the learning environment, at this point, exercises these skills.
But as the children age, social biases preclude boys and girls from choosing to play with certain things and in certain ways. At this point of divergence, objects and activities acquire a definite gender appropriateness. Blocks, Legos, tree climbing, outdoor activities, and ball sports become "boys' activities." Dolls, playing house, dressing up, talking, cooking, reading, sewing, crafting, and planning social activities become "girls' activities." By avoiding intricmechanical manipulations and "rough and tumble" physical activities, girls loose ground in spatial organization abilities.
4.
Their more sedentary activities offer few exercises in space/motion judgement, symmetry, parttowhole constructions, and development of visualization, muscle memory, and geometric principles. But boys, are gaining ground in all of these areas, and their improving spatial organizational abilities better prepare them for mathematics tasks.
GENETIC FACTORS: As with all abilities, math aptitude can be inherited or an inborn disposition. Studies of identical twins reveal close math scores (Barakat 1951, 154). Research into exceptionally gifted individuals shows high levels of math knowledge in early childhood, unexplained by external influences. Family histories of mathematically "gifted" and "retarded" individuals, revealed common aptitudes in other family members. (CTLM 1986, 53)
Even the most "mathematically gifted" individual can be hindered by inadequate math education. Likewise, a "mathematically retarded" individual will not attain competency in math despite intensive systematic training. (CTLM 1986, 53)
CONCLUSION: A majority of dyscalculia cases, experienced by individuals with average or superior intelligence, are exclusively caused by failure to acquire math fundamentals in school. Worldwide, math has the highest failure rates, and lowest average grade achievements. Almost all students, regardless of school type or grade, cannot perform in math on par with their intellectual abilities. This is not surprising because sequential math instruction requires a perfect command of acquired fundamentals. (CTLM 1986, 52) The slightest misunderstanding makes a shaky mathematical foundation.
5.
PROGNOSIS: Because such a minute fraction of our intellectual potential is utilized, scientists believe that even the worst math performances can be improved considerably. Compensatory strategies and appropriately organized instruction remediate deficiencies. (CTLM 1986, 52)
PREVENTION: When a 5, 6, or 7yearold student is not cognitively ready to learn math concepts, their early introduction will only result in negative experiences and attitudes toward mathematics, and eventually, math anxiety. Parents and teachers must wait until the child is developmentally ready. In the mean time, continually provide plenty of varied informal experiences that teach the desired ideas, but do not eagerly expect mastery of these concepts early on. (Sharma 1989)
EVALUATING FOR MATH LEARNING DISABILITIES
QUALIFICATIONS OF AN EDUCATIONAL EVALUATOR: To achieve highquality educational evaluations, the evaluator must be an expert in the field of inquiry, and should possess the following competencies: 1.) ability to describe the situation being evaluated; 2.) ability to describe the evaluation's context; 3.) ability to conceptualize the purpose and appropriate framework for the evaluation; 4.) ability to select and identify relevant information needs, sources, and questions; 5.) ability to identify, select, and apply effective procedures and techniques for data collection, processing and analysis; 6.) ability to determine the value of objects being evaluated; 7.) ability to communicate plans and results effectively; 8.) ability to successfully manage the evaluation; 9.) ability to maintain ethical standards; 10.) ability to make adjustments in external factors that influence an evaluation; and 11.) ability to critique, revise, utilize, and learn from the evaluation experience. (Worthen, Sanders, and Fitzpatrick 1997, 511513)
DIAGNOSING DYSCALCULIA
(1.) Quantitative dyscalculia is a deficit in the skills of counting and calculating. (2.) Qualitiative dyscalculia is the result of difficulties in comprehension of instructions or the failure to master the skills required for an operation. When a student has not mastered the memorization of number facts, he cannot benefit from this stored "verbalizable information about numbers" that is used with prior associations to solve problems involving addition, subtraction, multiplication, division, and square roots. (3.) Intermediate dyscalculia involves the inability to operate with symbols, or numbers. (CTLM 1989, 7172)
6.
7.
L. Kosc, of Bratislava, advocates in his Slovak "Psychology of Mathematics Abilities" (19711972), the use of a battery of 3 tests which diagnose disorders of math functioning while differentiating from educational deprivation, scholastic deficiencies, organically caused difficulties, and "retardation in school knowledge." (CTLM 1989, 6969)
This battery of tests has been studied extensively and used successfully at the Center for the Teaching and Learning of Mathematics in Framingham, Massachusetts and London, England. The recommended tests are outlined below:
TEST & PROCESS 
MEASURES 
USES 
Kalkulia II, III Subject must determine the exact number of balls arranged in patterns, by manipulating groups using addition, multiplication or the symmetry of arrangements. *(May not be suitable for assessing math abilities in adults.) 
"An accurate perception of quantity, transforming the quantity into numbers, then performing the operations of summation." "Examines the capacity to process quantitative functions (a distinct capacity to group and calculate) in a manner distinct form serial counting, and rote number knowledge expressed in linguistic form." Measures "lateralizable dysfunctions." 
"The added step of quantifying the elements to be counted distinguishes dyscalculia from the verbal information processing difficulties of arithmetic dyslexia or verbal dyslexia." "Test discriminates between mathematically gifted and mathematically untalented disabled children (Kosc 1974), especially when the spatial component in the structure of their mathematical abilities is disturbed." 
The ReyOsterrieth' Complex Figure Test (CFT) An athematic perceptive, visuomotor drawing task like the MPD or Bender Gestalt. Subject is first asked to copy the complex figure made of basic geometric shapes, then to draw it from memory. 
Measures attentive analytical and perceptualorganizational skills, relative position of element to the whole and degree of precision. 
Distinguishes diagnosis of mental retardation and brain damage deficits. Identifies cases of spatial difficulties that interfere with math performance. 
The Number Triangle Test 
Subject is asked to write dictated digits beneath each other, then is asked to add every two, placing the sum in a position between and to the right of the addends. When a sum exceeds 9, only the ones place is recorded. The operation is continued until a column containing only one digit is created. 
Determines basic addition skills, and ability to form an appropriate numerical matrix from given instructions. 
The Number Square Test 


Key Math Diagnostic Test 
Arithmetic information, computation, ability level, and content mastery. 
"Differentiates developmental dyscalculia from dyslexia, brain damage, slow learning, mental retardation and other disabilities." 
Stanford Diagnostic Test 
Arithmetic information, computation, ability level, and content mastery. 
"Differentiates developmental dyscalculia from dyslexia, brain damage, slow learning, mental retardation and other disabilities." 
Wide Range Achievement Test 
Arithmetic information, computation, ability level, and content mastery. 
"Differentiates developmental dyscalculia from dyslexia, brain damage, slow learning, mental retardation and other disabilities." 

Table created with information presented in Focus on Learning Problems in Mathematics (summer & fall).(CTLM 1986, 71119) 

8.
9.
Standardized tests like the Wechsler Intelligence scales and tests of math ability are used to compare individual performance with majority peer group performance. The formula for calculating "Math IQ" is: Math Q= Math Age divided by Chronological Age x 100. A score of 12 standard deviations below the mean (middle) score of thgroup is considered "deficient." A score of 7075 is extremely deficient. (CTLM 1986, 4950)
A dyscalculia diagnosis in preschool age children can be made whea child cannot "perform simple quantitative operations" that should be "routine at his age." (CTLM 1986, 50) 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. (CTLM 1986, 67)
ISSUES COMPLICATING DIAGNOSIS:
The multidisciplinary participants involved in its inquiry complicate the nomenclature of math learning disability. The field of education deals with learning difficulties in math. Psychology is concerned with the disorders and disturbances of math abilities. Neurology and psychiatry deal with the disturbed functions resulting from brain damage. (CTLM 1986, 64) Each profession uses specific terminology to describe math disabilities. The result is the categorical fragmentation of classes and types of dyscalculia, as seen in the table below. At the end of the table, several terms are introduced with definitions of their prefixes.

Class 
Name 
Definition 
Examples: 
1 
Class A 
Developmental Dyscalculia 
Dysfunction in math, in individuals with normal mental functioning, resulting from brain anomalies inherited or occurring during prenatal development. Discrepancy 12 standard deviations below the mean, between mental age and math age. Clear retardation in math development. 
Numerical difficulties with: counting, recognizing numbers, manipulating math symbols mentally and/or in writing, sequential memory for numbers and operations, mixing up numbers in reading, writing, recalling, and auditory processing, memory. Much more effort is required. 
2 
Class B 
PostLesion Dyscalculia 
Math disability that is the result of brain damage/ head injury, cerebral dysfunction, or Minimal Brain Damage. 

3 
Class C 
PseudoDyscalculia, Falsely called dyscalculia. Closely resembling dyscalculia. 
Environmentally Caused Dyscalculia. Math inability caused by: lack of, inconsistent, poor, or inappropriate systematic math instruction; inattention, fear, anxiety, illness, absence, or emotion. 

4 
Class AType 1 
Dyscalculia Subtypes owith Normal Mental Ability 


5 
Class A Type 2 
Secondary Dyscalculia 
Dyscalculia coexisting with oligophrenia, mental retardation, or dementia. 

6 
Class A1a 
Dyscalculia MODERATELY SEVERE: 
Total inability to abstract, or consider concepts, numbers, attributes, or qualities apart from specific, tangible examples. 

7 
Class A1b 
Acalculia 
Complete inability of math functioning. 

8 
Class A1c 
Oligocalculia 
A relative decrease of all facets of math ability. 

9 
Class A2a 
Secondary Dyscalculia 
Dementia with dyscalculia. 

10 
Class A2b 
Secondary Acalculia 
Mental retardation with dyscalculia. 

11 
Class A2c 
Secondary Oligocalculia 
Oligophrenia with dyscalculia. 

12 
Class A2d 
Secondary Paracalculia 
A neurotic aversion to numbers. 

13 
Class C 
Environmentally Caused Dyscalculia 
10. Math inability caused by: lack of, inconsistent, poor, or inappropriate systematic math instruction; inattention, fear, anxiety, illness, absence, or emotion. 

14 
Class CType 1 
Pseudoacalculia 


15 
Class C Type 2 
Pseudodyscalculia 


16 
Class C Type 3 
Pseudooligocalculia 


17 
Class D 
Paracalculia 
MOST SEVERE: Dyscalculia with Learned Math Avoidance. 

18 
Class D1 
Motor Verbal Paracalculia 


19 
Class A1aD 
Dyscalculia + Learned Math Avoidance 


20 
Class A1bD 
Acalculia + Learned Math Avoidance 


21 
Class A1cD 
Oligocalculia + Learned Math Avoidance 


22 
Class C1aD 
Pseudoacalculia + Learned Math Avoidance 


23 
Class C1bD 
Pseudodyscalculia + Learned Math Avoidance 


24 
Class C1cD 
Pseudooligocalculia + Learned Math Avoidance 


25 
Class A1aI 
Verbal Dyscalculia 
Dysnomia for quantitative terms, elements and relations. Capable of performing operations involved. Counting disorders. 
Cannot verbally name amounts of things, numb, terms, symbols, and operations. Cannot associate numerals to amounts of things. Cannot verbally continue counting patterns. May be able to read and write dictated numbers. Capable of performing operations involved 
26 
Class A1aIa 
Motor Verbal Dyscalculia 
11. Cannot read or write dictated numbers. 
May incorrectly write numbers as they are literally pronounced: "Five hundred and 4" as 5004, etc. 
27 
Class B1 
Sensory Verbal Dyscalculia 
Braindamaged. Cannot display a requested number of items physically or pictorially. Cannot read or write numbers, or count items. 

28 
Class A1aII 
Practognostic Dyscalculia 
Impaired ability to manipulate real or pictured items for mathematical purposes. Apraxic (Processing errors that result in inability to perform purposeful motor actions, especially sequences.)Perceptual Dysfunction. 
Cannot manipulate, add, compare, or estimate quantity or magnitude of physical or pictured items. May be able to read, write, and imitate written numbers and operations. Cannot compare, comprehend, or describe partwhole relationships, spatial details, shapes and sizes. 
29 
Class A1aIIa 
Finger Apraxia or Gnosia 
Inability to recognize objects by touching with the fingers. 
Cannot use fingers to assist with math processing, cannot carry numbers or follow computational sequences. Cannot count by heart. 
30 
Class A1aIIb 
Apraxic Dyscalculia 
Inability to purposeful motor acts, especially a sequence of movements. Caused by processing errors. 
Cannot count by heart. Cannot use fingers to assist with math processing, cannot carry numbers or follow computational sequences. 
31 
Class A1aIIc 
Numerical Dyslexia or Literal Dyslexia 
Performs below intellectual, developmental, and academic level. Difficulty with, or inability to read serial numbers, digits, place value, operational signs, math symbols, fractions, squares, roots, decimals, and the language of math. Can be caused by apatic agnosia, or directional confusion. Usually occurs with other types. 
May transpose (mix up) [21 as 12], interchange similar digits[6 and 9], inappropriately insert, or omit digits, words, & signs. May read without acknowledging place value: 5007 as "five hundred seven," or 576 and "five seven six." 
32 
Class A1aIII 
Lexical Dyscalculia or Numerical Dyslexia 
Difficulty with, or inability to read serial numbers, digits, place value, operational signs, math symbols, fractions, squares, roots, decimals, and the language of math. Can be caused by apatic agnosia, or directional confusion. Usually occurs with other types. 
May transpose (mix up) [21 as 12], interchange similar digits[6 and 9], inappropriately insert, or omit digits, words, & signs. May read without acknowledging place value: 5007 as "five hundred seven," or 576 and "five seven six." 
33 
Class A1aIIIa 
Numerical Dysgraphia 
12. Inability to write numbers because of inefficient motor skills or insufficient coordination of visual perception skills and fine motor skills. 

34 
Class A1aIIIb 
Numerical Dyscalculia or 


35 
Class A1aIII(a+b)c 
Numerical Dysmbolia 


36 
Class A1aIV 
Graphical Dyscalculia or Numerical Dysgraphia 
Disability in writing math symbols. Usually occurs with literal dysgraphia and literal dyslexia. 
May be unable to form/write individual digits, or copy them. Cannot encode (write) numbers correctly: [ 5731 as "5000700301" or omits zeros: 5073 as "573"] Writing #s in mixed up order, or opposite direction. May be able to write words for numbers. 
37 
Class A1aIVa 
Numerical Dysgraphia 
Inability to write numbers because of inefficient motor skills or insufficient coordination of visual perception skills and fine motor skills. 

38 
Class A1aIV 
Numerical Dyscalculia 


39 
Class A1aIV(a+b)c 
Numerical Dysmbolia 
Lexical dyscalculia occurring with graphical dyscalculia. OR Numerical Dyslexia occurring with numerical Dysgraphia. 

40 
Class A1aV 
Ideognostic Dyscalculia or Asemantic Aphasia or Dysymbolia TESTS: [a}1007777777.....mentally first/writing 2nd; {b} Series Complet. 
Poor mental comprehension of quantitative concepts. A dysfunction of the cognitive function of forming or assigning numbers & symbols "notions" or meaning. Inability to do mental math. Aphasia is the inability to express meaningful verbal identifications (of math symbols). 
May be unable to calculate the easiest sums mentally, or at an age/ academically appropriate level. May be able to read and write numbers but is oblivious to their meaning. Unable to distinguish colors of objects, or objects from a competing background. Cannot identify a specified number of items. 
41 
Class A1aVa 
Dysymbolia 


42 
Class A1aVb 
Acalculia 
MOST SEVERE: Extreme difficulty in grasping principles and logic of math concepts and reasoning, Gnostic disturbance is noted when can do test mentally but not in writing. 
Is unable to continue the sequence of numbers in the most basic of given series. 
43 
Class A1aVI 
Operational Dyscalculia (Anarithmetie) Tests: Note performance strategy. Have subject verbalize thinking, if possible. 
13. Inability to learn and apply the rules for addition, subtraction, multiplication and division resulting in a disability to successfully perform math operations. 
Frequent errors include: mixing up operations like +/, /¸ , x/¸ , x/+; mistaken or oversimplification of complex operations; insisting on written computation over mental calculation, uses fingers to assist mental or written computation. 
44 
Class A1aVIa 
Sensoryverbal (Cannot count out) 


45 
Class A1aVIb 
MotorVerbal (Cannot name) 


46 
Class A1aVIc 
Sequential Dyscalculia

Poor memory for: counting sequences, operational sequences, math facts, time, direction, schedules. 

47 
Hypocalculia 
Hypo denotes a lack or deficiency in. 


48 
Oligocalculia 
Oligo new, recent 
A serious retardation in developmental math abilities caused by mental retardation. 
Treatment: Appropriately motivate student to learn math, then fill in the knowledge and skill gaps with remedial techniques. 
49 
Calculasthenia (LESS SERIOUS) 

A socially caused deficit in the level of math acquisition, skills, and knowledge. 

50 
Acalculia 
A without, not 


51 
Paracalculia 
Para functionally disordered or diseased condition, or similar to, but not identical to a true condition or form. 

Table created with information found in: 
52 
Learning Disability 
Term from education and educational psychology. 

Focus on Learning Problems in Mathematics. 
53 
Terms containing "calculia" 
Are from the fields of neurology and neuropsychology. 
Numbering and Classification system is the invention of R.M. Newman. 14. 
Summer & Fall edition 1996, Volume 8: # 3 & 4. Framingham, MA: Center for Teaching/ Learning of Mathematics. p. 48116. 
DIAGNOSTIC TEACHING METHODS:
There are 5 critical factors affecting math learning, and they are all essential components of a successful math curriculum. Each of these 5 factors is also a critical diagnostic tool for evaluating learning difficulties in mathematics. (Sharma 1989)
[1.] First, the student's cognitive level of awareness of the given knowledge must be ascertained. There is a range in any class, of low cognition to high levels of cognitive functioning. A teacher must determine each child's prerequisite processing levels, and the strategies he brings to the mathematics task. This information dictates which activities, materials, and pedagogy (teaching theories) are used. (Sharma 1989)
Differences in cognitive ability affect the students' ability, facility, and understanding, and point to the difficulties they will have with specific math concepts. A teacher must not base evaluations of learning mastery, solely on a child's ability to arrive at a correct answer. More important than results, are the level of cognition, and the strategies the student uses to get the answer. The teacher must interview the student, searching out causative factors like a scientist. What is the child thinking? How is the child reasoning through the problem? Does the child have the prerequisite skills? How did the child get a wrong answer? There may be a legitimate reason. (Sharma 1989)
For example, a child with a low level of cognition, is not capable of the higher order thinking required for basic math concepts. When teachers introduce concepts with abstractions and end discussions with abstractions, students are denied the ability to create connections to previous knowledge through the use of concrete modeling. Students will invariably have
15.
16.
difficulties when concepts are presented at a level above their cognitive ability. (Sharma 1989) The child is left with no choice but to memorize the material (if capable) because he has not found a mental hook to hang the new concept on.
If a child has not mastered the concept of number preservation (the idea that 5 represents a set of 5 things), then they are incapable of making the generalizations necessary for performing addition or subtraction. How can you recognize a low functioning child? He is dependent on counting with his fingers or objects. When told that a hand has five fingers, he will have to manually count them when shown a hand and asked how many fingers are showing. (Sharma 1989)
An example of an advanced level of cognition, is a student who uses knowledge of multiplication facts to solve a problem using a least common multiple. At this level of ability, the child is ready for addition and subtraction of fractions. (Sharma 1989)
[2.] Second, the teacher must understand that each student processes math differently. Each person has a unique learning style or "mathematics learning personality." These different styles affect a student's processing, application, and understanding of material. Within every classroom, the student styles are spread across a continuum ranging from purely quantitative to purely qualitative. (Sharma 1989)
Quantitative learners, like to deal exclusively with entities that have determinable magnitudes, like length, size, volume, or number. (Funk and Wagnalls ) Preferring the procedural nature of math, they are methodological, and sequential. They approach math like following a recipe. (Sharma 1989) They break down problems into pieces, solve them, and then assemble the component solutions to successfully resolve the larger problem. They prefer
17.
deductive reasoning, that is, reasoning from the general principal to a particular instance, or reasoning from stated premises to logical conclusions. (Sharma 1990, 22)
Quantitative students learn math best with a highly structured, continuous linear focus. Use handson materials with a counting basis. These include: blocks, unifix cubes, base 10 blocks, and number lines. They prefer one standardized way of problem solving. They experience the introduction of additional ways to solve problems as threatening and uncomfortable a sort of irritating distraction form their pragmatic focus. (Sharma 1989)
Qualitative learners approach math tasks holistically, and intuitively, that is, with a natural understanding that is not the result of conscious attention or reasoning. (Funk and Wagnalls) 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. . (Sharma 1990, 22)
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 procedwith intuitive reasoning. Their work is procedurally sloppy, because they quickly tire of long processes. Their performance is never fluent because they donot practice enough to attain levels of automaticity. Eventually, the qualitative student may show
18.
disinterest in the mastery of mathematics, even though they are able to make connections between math concepts more quickly than the quantitative learner. (Sharma 1990, 22)
Qualitative students learn best with continuous visualspatial materials. They can handle the simultaneous consideration of several problemsolving strategies. A discontinuous style of teaching, stopping for discussion, then resumption of teaching, is agreeable to them. (Sharma 1989) This style may agitate the quantitative learner.
To effectively teach the entire class, the elements of both learning styles must be integrated and accommodated. To teach with one style, exclusively, is to leave out a great many students. If math concepts are not matched to students' cognitive and skill levels, then failure will inevitably result, and the child will be forced into a position of needing remedial services to overcome their academic deficiency in mathematics. (Sharma 1989)
By the age of 12, the academically neglected child has developed anxiety, insecurity, incompetency, and a strong dislike for mathematics because his experiences with it have been hit or miss. At this point, his symptoms become causative factors in the cycle of failure, math avoidance, and limited future educational and occupational opportunities. (Sharma 1989)
[3.] Third, just like with readingreadiness skills, the teacher must assess the existence and extent of mathreadiness skills in each student. Seven prerequisite skills have a profound impact on the ability to learn mathematics. These are nonmathematical in nature, but are extremely important preskills that must be fully mastered before even the most basic math concepts can be successfully learned. (Sharma 1989)
19.
Seven Prerequisite Math Skills
1. 
The ability to follow sequential directions. 
2. 
A keen sense of directionality, of one's position in space, of spatial orientation and space organization. Examples include the ability to tell left from right, north/south/east/west, up/down, forward/backwards, horizontal/vertical/diagonal, etc. 
3. 
Pattern recognition and its extension 
4. 
Visualization: Key for qualitative students. The ability to conger up pictures in one's mind and manipulate them. 
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 principal to a particular instance, or reasoning from stated premise to a logical conclusion. 
7. 
Inductive reasoning: a natural understanding that is not the result of conscious attention or reasoning, easily seeing the patterns in different situations, and the interrelationships between procedures and concepts. (Sharma 1989) 
For instance, if one lacks the ability to follow sequential directions, how can he be presented with the concept of long division without failing miserably? Long division requires retention of several different processes that are performed in a specific sequence. First one estimates, then multiplies, then compares, then subtracts, then brings down a number, then estimates, compares, multiplies, subtracts, brings down a number, and so on.
For the same situation, what if the student has directional confusion? When setting up math problems, he will be chronically unsure of which number goes inside the division platform, or on top of the fraction. The mechanics of moving through the problem will be painful. Consider the directional steps involved. One reads to the right,
20.
then records a number up, then multiplies numbers diagonally, then records the product down below (taking into consideration place value), then brings a number down, then divides diagonally and places the answer up above, then multiplies diagonally, and so on.
If a child has poor perception for things in space, his writing may be disorganized and jumbled. Numbers are not lined up adequately or formed legibly. Operational symbols and notations are often mistaken for numbers in the problem. Geometry may be equally perplexing. Frustration and confusion plague this student.
[4.] Mathematics is a second language and should be taught as such. It is exclusively bound to the symbolic representation of ideas. Most of the difficulties seen in mathematics result from underdevelopment of the language of mathematics. Teaching of the linguistic elements of math language is sorely neglected. The syntax, terminology, and the translation from English to math language, and from math language to English must be directly and deliberately taught! (Sharma 1989)
Historically, mathematicians have operated as if math was an exclusive club, whose members speak a secret language. They taught math in a rigid, complicated manner, and were proud of it. Egotistically satisfying their "fewer the better attitude," they happily weeded out underachievers. Dean Sahrma calls the status quo in mathematics' education, irresponsible and unacceptable, (Sharma 1989) especially in an age where "90% of new jobs require more than a high school level of literacy and math skills." And math educators have failed so miserably, that although 90% of kids want to go to college, paradoxically, 50% of them also want to drop out of math classes as soon as possible.(USDE 1998)
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Every math concept has 3 components. (1.) The first is the linguistic, composed of the words (the specific terminology), arranged in definite ways to convey meaning (the syntax), and the rules of translation from English into math, and from math into English. (2.) The second component is conceptual, or the mathematical idea or mental image that is formed by combining the elements of a class, into the notion of one object or thought. (3.) Third, is the procedural skill component of problem solving, which schools focus on almost exclusively. (Sharma 1989)
Sharma offers examples of poor math language development: Students are frequently taught the concept of "least common multiple", without sufficient linguistic analysis of the words (definitions) and how their order or arrangement (syntax) affects their meaning. This can be demonstrated by asking students to define the terminology. Several incorrect answers will be generated. This proves that students have memorized the term without understanding it linguistically. Teachers do a great disservice to students by treating math as a collection of recipes, procedures, methods, and formulas to be memorized. (Sharma 1989)
BEST PRACTICES: When introducing a mathematical term or concept, a teacher must create a parallel English language equivalent. The new term must be related or made analogous to a familiar situation in the English language. Students must be taught the relationship to the whole, of each word in the term, just as students of English are taught that a "boy" is a noun that denotes a particular "class." An adjective, like "tall" is a descriptive word that restricts or modifies an element (boy) of a particular class (of all boys). Adding another adjective, like "handsome," further restricts, narrows, or defines the boy's place in the class of all boys. This can be graphically illustrated: (Sharma 1989)
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The language of mathematics has a rigid syntax, easily misinterpreted during translation.
For example, "94 take away 7, " might be written correctly, in the exact stated order, as "947." But when the problem is presented as "subtract 7 from 94," the student following the presented order will mistakenly write, "794." Therefore, it is extremely important that students learn to identify and correctly translate math syntax. (Sharma 1989)
Some students are linguistically handicapped by teachers, parents, and textbooks that use "command specific" terminoloto solicit certain actions. For example, most children are told with informal language to "multiply, "add, "subtract," and "divide." They are clueless wthey encounter formal terminology prompting them to find the "product" or "sum" of numbers. (Sharma 1989)
To eliminate this problem, matteroffactly interchange the formal and informal terms in regular discourse. Seek to extend the expressive language set of the student to include as many synonyms as possible. Use at least two terms for every function. (Sharma 1989) 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." Sharma proposes a standard minimum math vocabulary for each stage of mathematics instruction. (Sharma 1989)
The dynaof language translation must also be directly 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 expression. Instead of story problems, Sharma advocates giving the child mathematical expressions to be translated into a story in English. (Sharma 1989)
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For example, present "74=____," saying, "write (or tell me) a number story using these numbers in a way involving subtracting or reducing 7 by 4." The student may respond, "I got 7 candy bars last Halloween and this year I only got 4. How many more candy bars did I get last Halloween than this Halloween?" NOTE: Only 510% of 7 to 9 yearolds use the phrase "how many more than" in their normal speech, and with complete understanding. (Sharma 1989)
Now, to facilitate the child's discovery of extraneous information, ask them to add the dates to each Halloween event in the story. The child may respond, "On October 31, 1998, I got 7 big candy bars. But on October 31, 1999, I only got 4. How many more candy bars did I get in 1998 than I got in 1999?" Then ask them if the answer to the question has changed? Why not? The child will respond, "Because I just added the dates in there. The number of candy bars did not change." (Sharma 1989)

This important linguistic exercise in translation should be added* to become the 4th main method of facilitating language expression. The standard ways are to: 
(1.) 
Show a visual stimulus, or picture, and ask the child to write or talk about it. 
(2.) 
Engage the child in concrete experiences and activities, like field trips or experiments, and ask them to describe their experiences. 
(3.) 
Supply story starters, like "Once upon a time...," and have the child continue and finish the story. 
*(4.) 
Supply a mathematical equation, like 63=___, and have the child construct an example of this. (Sharma 1989) 
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[5.] Before a mathematical concept is learned fully, the student moves through six levels of learning mastery.
1. 
Intuitive Connections: Student connects or relates the new concept with existing knowledge and experiences. 
2. 
Concrete Modeling: Student looks for concrete material with which to construct a model or show a manifestation of the concept. 
3. 
Pictorial or Representational: Student draws to illustrate the concept. In this way he connects the concrete (or vividly imagined) example to the symbolic picture or representation. 
4. 
Abstract or Symbolic: Student translates the concept into mathematical notation, using number symbols, operational signs, formulas, and equations. 
5. 
Application: Student applies the concept successfully to real world situations, story problems, and projects. 
6. 
Communication: Student can teach the concept successfully to others, or can communicate it on a test. Students can be paired up to teach one another the concept. (Sharma 1989) 
WHAT MASTERED IS MASTERY ENOUGH? A student can go quite far on either extreme of the continuum. A quantitative personality can accomplish a lot being strong in mathematical procedures. Qualitative personalities are able to solve a wide range of problems intuitively and holistically. But an excellent mathematician must have command of both learning styles. (Sharma 1989)
PRESCRIPTIVE & PREVENTATIVE TEACHING
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 and12, their ability to acquire math knowledge is disturbed at the point when handson explorations were abandoned in favor of abstractions. This clearly sets them up for mathematical disabilities in the next developmental period of formal propositional operations. (CTLM 1986, 56)
The best teaching methods are diagnostic and prescriptive. They take all of the above mentioned factors into consideration. The teacher recognizes that she is teaching to students who span the continuum of quantitative and qualitative learning styles, and 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 the researched and proven method below.
BEST PRACTICES:
After determining that your students have all of prerequisite skills and levels of cognitive understanding, introduce the new concept in the following sequence:
1. 
Inductive Approach for Qualitative Learners 
a.) Explain the linguistic aspects of the concept. b.) Introduce the general principle, truth, or law that other truths hinge on. c.) Let the students use investigations with concrete materials to discover proofs of these truths. d.) Give many specific examples of these truths using the concrete materials. e.) Have students talk about their discoveries about how the concept works. f.) Then show how these individual experiences can be integrated into a general principle or rule that pertains equally to each example. 
2. 
Deductive Approach for Quantitative Learners 
Next, use the typical deductive classroom approach. g.) Reemphasize the general law, rule, principle, or truth that other mathematical truths hinge on. h.) Then show how several specific examples obey the general rule. i.) Have students state the rule and offer specific examples that obey it. j.) Have students explain the linguistic elements of the concept. (Sharma 1989) 
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CONCLUSION:
Every student with a normal IQ, can learn to communicate mathematically, if taught appropriately. Curricula in the preschool and early elementary years should focus on the development of the 7 prerequisite mathreadiness skills. Teachers and students need to be aware of, and able to accommodate, the different learning styles or "math learning personalities" and the corresponding teaching methods that address each style.
Mathematics must be taught as a mandatory second language. The specific language of mathematics should be deliberately taught each year of the Kindergarten through 12th grade scholastic program, just as reading and English are taught. It must be communicated to parents, teachers, and students, that competency in the language of mathematics, is just as socially and economically essential as excellent reading and writing skills. Proven programs of prevention, systematic evaluation, identification of learning difficulties, early intervention, and remediation in mathematics must be implemented immediately, to reverse dismal achievement statistics, and secure better educational and economic outcomes for America's students.
The United States government has lofty goals for math achievement. The U.S. Department of Education's math priority reads: "All students will master challenging mathematics, including the foundations of algebra and geometry, by the end of 8th grade." And advocates that all K12 students eventually master challenging mathematics, which include "arithmetic, algebra, geometry, probability, statistics, data analysis, trigonometry, and calculus."(USDE 1998)
But compared to other countries, af4th grade, American students fall behind because the curricula continues to emphasize fractions, decimals, and whole number operations,
26.
27.
while international students study advanced concepts including algebra, geometry, and probability. Even with the extra four years of basic study, 21% of 8th graders still cannot add, subtract, multiply, do whole number division, and solve onestep problems. (USDE 1988)
Progress facilitated by professionals will not be realized until the concerns of math teachers and special educators, converge. Typically, math educators are concerned with how best to teach concepts. Special educators are concerned with communicating the abilities and limitations of students. Each is working in isolation on the problem of math learning. A change needs to take place. The teacher needs to focus more on the abilities and learning styles of the child, and the special educator needs to focus more on achieving the content of the mathematics curriculum. (Sharma 1989)
Government statistics report that math education translates into educational and economic opportunities. Taking tough math courses is more predictive of college attendance than is family background or income. Of the students taking algebra I and geometry, 83% go on to college, whereas, only 36% of students who do not take these courses ever go to college. (USDE 1998) Low income students who take these courses are three times more likely to go to college than their peers: 71% attend college, whereas only 27% of low income students, without gateway mathematics courses, go on to college within 2 years of graduation. (Winters 1997)
Clearly, unless we find ways to rescue and rehabilitate the 20% of elementary students, who are unable to grasp mathematics, we will perpetrate the cycle of undereducation, underemployment, and underdevelopment of a significant portion of America's human resources. In January 1998, Vice President, Al Gore, called attention to the shortage of technical workers. Technical careers require high levels of math competence. According to the Department
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of Labor, the demand for engineers, system analysts, system technicians, and computer scientists will double by 2008. (Winters 1998)
Progress is imminent. The government has teamed up with major mathematics organizations to develop voluntary standards and a framework for the careful preparation of math teachers. Many organizations, like the Public Broadcast System, businesseseducation partnerships, and math and science mentoring programs, are finding creative ways to engage student interest in math, and demonstrate real applications of math in daily life. (USDE 1998)
REFERENCES
Blaine, Worthen R., James R. Sanders, and Jody L. Fitzpatrick. 1997. Program Evaluation: Alternative Approaches and Practical Guidelines 2nd ed. White Plains, NY: Longman Publishers USA.
Booth, Wayne, Gregory G. Colomb, and Joseph M. Williams. 1995. The Craft of Research. Chicago: University of Chicago Press.
Center For Teaching/ Learning of Mathematics (CTLM). 1986. III. Progress of Dr. Ladislav Kosc's Work on Dyscalculia. Focus on Learning Problems in Mathematics Volume 8: 3&4. (summer & fall edition).
Funk and Wagnalls. N.D. Standard Desk Dictionary. Harper & Row, Publishers.
NCES. 1997. Snyder, Thomas D., with production manager, Charlene M. Hoffman. Program Analyst, Claire M. Geddes. Digest of Education Statistics 1997, NCES 98015. U.S. Department of Education. National Center for Education Statistics. Washington, DC. {online document] Available at: http://nces.gov/pubs/digest97/98015.html. Internet.
Sharma, Mahesh 1989. How Children Learn Mathematics: Professor Mahesh Sharma, in interview with Bill Domoney. London, England: Oxford Polytechnic, School of Education. 90 min. Educational Methods Unit. Videocassette.
Sharma, Mahesh. 1990. "Dyslexia, Dyscalculia, and Some Remedial Perspectives for Mathematics Learning Problems." Math Notebook: From Theory into Practice 8, no. 7, 8, 9 & 10. (September, October, November, & December).
Tobias, Sheila. 1978. Overcoming Math Anxiety. Boston, MA: Houghton Mifflin Company.
Turabian, Kate 1996. A Manual for Writers of Term Papers, Theses, and Dissertations. 6th ed. Chicago: University of Chicago Press.
United States Department of Education (USDE). 1997. Secretary Riley. Mathematics Equals Opportunity (October 20). [online document] Available at: http://www.ed.gov/pubs/math/. Internet.
United States Department of Education (USDE). 1998. The State of Mathematics Education: Building a Strong Foundation for the 21st Century. (January 9). [online document] Available at: http://www.ed.gov/inits.html#2. Internet.
Winters, Kirk. 1997. Update. ED Initiatives. (19 December). Promising & Exemplary Mathematics Programs Sought. Washington, D.C.: U.S. Department of Education. [journal online] Available at: http://www.ed.gov/pubs/EDInitiatives/. Internet.
Winters, Kirk. 1998. Update. ED Initiatives. (22 January). Washington, D.C.: U.S. Department of Education. [journal online] Available at: http://www.ed.gov/pubs/EDInitiatives/. Internet.
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