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Grades 6-12

DEVELOPMENTAL PERSPECTIVES ON DYSCALCULIA: IMPLICATIONS FOR TEACHING IN MIDDLE AND SECONDARY SCHOOL

Copyright 1998 Renee M. Newman | All rights reserved. 

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CONTENTS

INTRODUCTION ...1

TODAY'S MIDDLE AND SECONDARY SCHOOLS ...4

A REVIEW OF COGNITIVE DEVELOPMENT THEORY ...7

ANALYSIS OF MATH FAILURE ...13

DIAGNOSTIC TEACHING METHODS ...16

BEST MATHEMATICAL TEACHING PRACTICES FOR A DIVERSE STUDENT BODY ...19

CONCLUSION ...23

INTRODUCTION

This paper will discuss math learning disability, dyscalculia, in light of developmental theory and resulting implication for teaching mathematics in the middle and secondary schools.

According to America's education statisticians, we are failing miserably in our mission to produce students capable of filling the 90% of the new jobs that require more than a high school level of literacy and math skills.

Specifically, U. S. Education Secretary, Richard Riley, detailed essential math courses to 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).

A brief review of U.S. Department of Education statistics (actual vs. desired outcomes) is intended to shed light on inherent implications for middle and secondary school curricula. What are the causes of this dismal performance? What researched and proven teaching methods have the highest success with students of varying learning style developmental levels, and aptitudes required for mastery of mathematics? Will this reality check result in disappointing admissions that the the U.S. is not in a position to compete internationally in mathematics because of failure of majority of our students to advance mathematically beyond the 4th grade?

The Third International Math and Science Study (TIMSS) of exiting 12th grade students tested abilities of 500,000 students in 41 countries during 1995. The results were released by the National Center for Education Statistics (NCES) in February 1998. U.S. students only outperformed two countries on the assessments of mathematics and science general knowledge, and were outperformed by 14 and 11 countries respectively. The U.S. scored the lowest of 26 countries on the assessment of physics and advanced mathematics (NCES 1998).

The U.S. Department of Education found that 93% of high school graduates cannot solve problems involving fractions and percentages. They cannot solve two-step problems involving variables, identify equal algebraic equations, or solve linear equations and inequalities. An astonishing 93% cannot synthesize and learn from varied specialized reading content, and 91% cannot infer relationships or draw conclusions from detailed scientific information (USDE 1991).

Almost 93% of graduating 17-year-olds do not show proficiency in multi-step problem solving and algebra (NCES 1997, 123--124), and one of every 4.5 American adults (22%) cannot perform simple arithmetic (NCES 1997, 416).

Only 56% of exiting 17-year-olds can compute decimals, fractions, and percentages; and over 46% cannot recognize geometric figures, solve simple equations, or use moderately complex math reasoning (USDE 1991).

In 1994, only 51% of high school graduates completed three years of mathematics. But a recent Harris Poll reveals that while 90% of all parents and students desire a college education, 50% of these students quit math education as soon as allowed (The State of Mathematics Education Address: Building a Strong Foundation for the 21st Century, January, 1998).

The TIMSS also found that U.S. 4th graders scored near the top in science achievement, and above average mathematics. "But . . . also revealed . . . that the U.S. was the only country . . . whose students dropped from above average performance in 4th grade mathematics to below average math performance in the 8th grade."

This international math gap is due to the fact that after 4th grade, the U.S. math curriculum continues to focus on basic arithmetic, fractions, decimals and whole number operations, whereas, Japan and Germany graduate to advanced concepts that include algebra, geometry, and probability (The State of Mathematics Education Address Building a Strong Foundation for the 21st Century, 8 January, 1998).

According to National Education Assessment Progress (NEAP) data released in August 1998, 79% of 8th graders could add, subtract, multiply, and divide, whereas 21% could not. Advanced math is not being introduced to read students.

Could this be because the average K--8 teacher has taken 3 or fewer math courses? Not even half of all 8th grade math teachers have taken even one course on teaching mathematics at the middle school level. Some 28% of high school math teachers do not have a major or minor in mathematics, and recent studies show that teacher expertise can account for 40% of the variance in students' mathematics achievement (The State of Mathematics Education Address: Building a Strong Foundation for the 21st Century, 8 January, 1998).

Mean Scholastic Aptitude Test (SAT) scores have dropped " 23 points on the math section between 1962 and 1992 (Sykes 1995, 21--22)."

Given these figures, what can be discovered by an analysis of the modern middle and secondary school? How can the discrepancy between investment and performance be explained and mediated? What are the reasons for poor mathematical outcomes, and how should each aspect of the problem be addressed within the academic institution?

TODAY'S MIDDLE AND SECONDARY SCHOOLS

The concept of schooling has changed radically in the last century. This paper is concerned with middle schools (grades 5 or 6 through 8, ages 10-14) and secondary schools (grades 7 through 12, ages 12-17), which include junior high schools (grades 7-8 or 9, ages 12-14) (Callahan, Clark and Kellough 1998, 4-16).

Middle schools deal with children ages 10-14 undergoing radical pubescent changes in body chemistry that govern physical appearance, intellect, social, emotional and psychological development-- a period termed transcendence (Eichorn 1966, 3).

Psychologically, the middle school student is energetic, adventurous, egocentric, constantly interpreting the environment with persistent misconceptions, is tenacious and naturally curious of mysterious events and agents of change, and seeks the company of the peer group (Callahan, Clark, and Caliph 1998, 38-40).

Transcendence is a period of insecurity and instability, when children desire independence within a secure and supportive environment. Experimentation continues with emerging socio-sexual roles, peer pressure, conformity for security, and novel forms of dress, expression, and inquiry (Callahan, Clark, and Caliph 1998, 36).

Many children exhibit antisocial behavior at this stage. Sociologist, D. Stanley Eitzen, traces this behavior to young people who have lost their dreams. "Without a dream we become apathetic. . . . Without a dream and the hope of attaining it, society becomes our enemy (Eitzen 1992)."

According to Abraham Maslow's hierarchy of needs, a child's disposition will suffer at the level of the unmet need, and will frustrate the filling of subsequent needs. (1) The most basic needs are for food, clothing, and shelter [physiological needs], then for (2) safety [security]. (3) Next the person seeks love and belonging [social]. (4) Then the person seeks to accept himself and others and make full use of his abilities and talents [self-esteem and self-actualization] (Maslow 1970).

By 2020, a majority of public school students will be at risk of not finishing school due to emotional pain, family tragedy, socioeconomic stress, and/or family instability (Tiedt and Tiedt 1995, 37). Learning disabilities, estimated to afflict 20% of students, are also a major risk factor. In addition, over twenty-five different foreign languages are spoken in one of every six American homes-- instead of English-- with Spanish being the most common (U.S. Census 1990).

Special education laws require that students of diverse learning abilities and behavioral norms be integrated to the maximum extent possible into the regular education classroom. The teacher is expected to consider and manage these variables in a humanitarian environment, and to simultaneously address the special needs of each student, deliver the prescribed curriculum and meet the general and special educational objectives of each child.

The new consensus on teaching and learning embraces organizational dynamics for accommodating student differences in culture, development, and learning styles. It is founded on the belief that all students can learn if given proper environment, encouragement, and opportunity. High expectations are held for all students. Individualized learning plans are considered for each student based on abilities, strengths, weaknesses, and interests. Community and parents are invited to be partners in the educational process.

More time is spent on authentic acquisition of basic thinking, reading, writing and math skills. Rote memory is de-emphasized. More specialist teachers are employed and the goal is smaller class sizes. Students are involved in practical exercises of knowledge which include peer-coaching, cross-age coaching, experimentation, real-life hands-on projects, and problem solving. Emphasis is on teacher responsibility for successful outcomes for each student, and on covering the planned curriculum while employing methods that build student self-esteem (Callahan, Clark, and Kellough 1998, 15-18).

Emphasis is away from learning and teaching in isolation. Students and teachers work in collaborative and cooperative groups to achieve common goals and share responsibility for preassessment, evaluation, goal setting, strategic planning, execution, accountability measures and reporting, correction, management, and completion of projects and goals. School schedules are altered to allow more engaged time on task. Mastery of content is demonstrated before the introduction of new material (Callahan, Clark, and Kellough 1998, 17-22).

A competent and effective teacher is defined as one that is adept in the following five areas of decision making: (1) Diagnosis and preassessment of knowledge, needs, desires, and learning styles; (2) Preparation for instruction which includes goal assessment, content, motivational strategies, setting, materials and equipment; (3) Learner guidance  which includes eclectic methods, activities, instruction, resources, dialog, and feedback; (4) Continuous assessment of learning and instruction; and (5) Follow-up (Callahan, Clark, and Kellough 1998, 22) which is remedial, when necessary, and foundational.

A REVIEW OF COGNITIVE DEVELOPMENT THEORY

Jean Piaget, in the 1920s, formulated four successive stages of mental development. These are regarded as unskippable, therefore, educators must attend to the child's developmental stage by multitasking-- which allows each student to work on individual objectives.

When cognitive disequilibrium results from perplexing situations, learners tend to revert to the programmed behavior of earlier developmental stages. This downshifting is known to occur when learning outcomes are specified and rigid, when there is not an adequate foundation on which to rest new information, when correction and affirmation is immediate and external, when schedules are inflexible and restrictive, and when unfamiliar material is presented without assistance (Caine and Caine 1993, 20).

Paiget's developmental stages are summarized below because the teacher must take care to identify the cognitive stage of each child and proceed with an effective approach that will take the child through the present stage, and allow solid progression through subsequent stages, with eventual graduation from the stage required for grade appropriate content mastery.

The following information is taken from figure 2.3 in Callahan, Clark, and Kellough's book, Teaching in the Middle and Secondary Schools (1998). Sensory-motor Stage (Birth to 2 years): The child has direct interaction with seen and felt stimuli and begins to develop mental concepts through association. Later, the child labels objects and demonstrates imagination.

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.

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.

This paper 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. an 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 historic time.

The concepts of conservation are mastered in the following order: (1) number of objects between ages 6 and 7; (2) matter, length, and area at age 7; (3) weight between ages 9 and 12; and (4) volume after age 11.

The concept of reversibility of physical and mental processes has emerged. The child can correctly interpret 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.

Formal Operations Stage (Ages 12 and over): Paiget believed that most adolescents, by age 15, reach this stage, characterized by a quick understanding of abstract ideas, as opposed to dependency on the concrete. The meta-cognitive skills of reflective abstraction are in place-- the ability to plan, monitor, and evaluate one's own behavior.

The person can develop hypotheses, deduce consequences, test hypotheses by controlled experiments with identical variables except the one in question, discriminate between substantiated and unsubstantiated hypotheses using logical analysis, and can reflect upon thought processes used.

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, like Bernice McCarthy's (1990) 4MAT systems 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 on doing, is reflected from Robert Gagne's (1970) General Learning Hierarchy, where he defines learning as the ability to do what one was unable to do before (Callahan, Clark and Kellough 1998, 46-17).

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 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- 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 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 in stage 4, even if it occurs at preschool age, 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 (CTLM 1986, 50, 67).

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; sequential memory for numbers and operations; mixing up numbers in reading, writing, recalling, and auditory processing; and memory. Extraordinary effort is required for math reasoning and function (Newman 1998, 10).

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-1-b, acalculia, is marked by complete inability of math functioning. Class A-1-c, oligocalculia, is a relative decrease of all facets of mathematical ability (Newman 1998, 10).

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.

When a student 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. Intermediate dyscalculia involves the inability to operate with symbols or numbers (CTLM 1989, 71-72).

Can one assume that the child never achieved mastery learning, as defined by Benjamin Bloom (1968), because he was never given sufficient time on task before instruction proceeded? If so, this type of dyscalculia is termed pseudo- dycalculia (Class C). It encompasses math inability caused by environmental factors including lack of, inconsistent, poor, or inappropriate systematic math instruction; inattention; fear; illness; absence; or emotions (Newman 1998, 10).

Is is true that most every child can become competent in every curricular aspect if given sufficient time on task, good instruction and consideration of knowledge and attitudes? How can middle and secondary school teachers remediate students suck in the early stages of Gagne's level four, while meeting the needs of all other children simultaneously?

Surely normal students will express impatience and displeasure at the chronic failure in understanding characteristic of their slower peers. Will these slower students benefit from group work, advocated by Lev Vygotsky (Bruner 1985), or will they become frustrated further and feel less adequate? Will their peer group resent their incompetency and lack of contributions?

How will the teacher manage these classroom dynamics while optimizing learning and preserving self-esteem? Do dyscalculic children benefit from the efforts of discovery learning? Brain research endorses the benefits of this method advocated by Jerome Bruner (1985).

Knowledge gained from hands-on discovery learning is more readily remembered and recalled. It empowers the learner with independent learning and problem solving skills which result in student confidence, motivation, and intrinsic rewards, as opposed to traditional dependence on extrinsic motivation and praise from others (Callahan, Clark and Kellough 1998, 50).

Later we will see how this method, as well as David Ausubel's method of meaningful verbal learning (1963), is embraced by Mahesh Sharma, a leading expert on remedial methods for math learning disabilities. Ausubel recommends hands-on learning in the primary grades but the use of substitutes thereafter. He asserts that learning is transmitted effectively with teacher explanations, demonstrations, diagrams, concept mapping, and other forms of illustration.

Ausubel suggests the teacher plant "advance organizer" ideas before new instruction that will help link new ideas with familiar ideas (Ausubel 1963). Sharma strongly advocates the linguistic planting of requisite ideas and a solid logical verbal component of instruction, but he uses a more comprehensive method to take all students through his six steps to math concept mastery.

ANALYSIS OF MATH FAILURE

Cambridge College dean of education, Mahesh Sharma, offers a number of reasons for terrible math outcomes: Our mathematics curricula are not reflecting what we know about how children learn mathematics. Typical curricula are guided by chronological age and math is presented in pile up fashion. This approach is tragically flawed (Sharma 1989).

Sharma believes that teacher trainers are not bringing all known aspects of math learning to teacher preparation programs. New math teachers have not been taught the latest developments in learning theory and do not know how to employ technology as a learning tool. Teachers end up teaching as they were taught and to their own learning styles (Sharma 1989).

Teachers must ask themselves these questions when their students experience difficulty: Is my teaching style excluding students with certain learning styles? Are my methods and materials appropriate and compatible with cognitive levels and learning styles (Sharma 1989)?

Cohn (1968) explains that failure in math is socially acceptable. Math ability is more regarded as a specialized intellectual function, rather than a general indicator of intelligence. The ramifications and stigma of math difficulty can be sufficiently diminished as long as one can read and write. Sharma, a global educator, concurs, explaining that in the West, it is common to find people with high IQs who shamelessly accept incompetency in math. At the same time, they find similar incompetencies in reading, writing, and spelling, totally unacceptable. Prevailing social attitudes excuse math failure and parents routinely communicate to children that they are not good in math (Sharma 1989).

In 1978, Sheila Tobias realized that because only 8% of girls took 4 years of high school math, 92% of young women were automatically excluded from careers in science, chemistry, physics, statistics, and economics. Half of university majors were closed to them. Tobias, author of Overcoming Math Anxiety, believes that women are socialized away from math study, not incapable of it. She advocates math therapy to overcome math anxiety (Tobias 1978, 12-13).

Sharma agrees, asserting that gender differences in math skills are due more to social forces than to gender-specific brain construction and function. Gender differences can be eliminated by equalizing the activities and experiences of boys and girls at every level of development. The social forces that direct a child's experiences and choice of activities result in gender differences in neurological sophistication (Sharma 1989).

Boys and girls are given ample opportunities to play with constructivist toys like blocks, Legos, board games and other materials requiring fine-motor coordination. Naturally, girls have superior fine-motor skills early in their development and the learning environment, at this point, exercises those skills. But as children age, social biases preclude boys and girls from choosing to play with certain things in certain ways. At this point of divergence, objects and activities acquire a definite gender appropriateness (Sharma 1989).

Blocks, Legos, tree climbing, outdoor activities and ball sports become "boy activities." Dolls, playing house, dressing up, talking, cooking, sewing, reading, crafting, and planning social activities become "girl activities." By avoiding intricate mechanical manipulations and rough and tumble physical activities, girls loose ground in spatial organization skills (Sharma 1989).

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 (Sharma 1989).

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, reveal common aptitudes in other family members (CTLM 1986, 53).

In conclusion, a majority of dyscalculia cases experienced by persons of normal or superior intelligence are exclusively caused by failure to acquire math fundamentals in school. Worldwide, math has the highest failure rates and the 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 for a shaky mathematical foundation. Because such a minute fraction of our intellectual potential is utilized, scientists believe that even the worst math performances can be considerably improved by compensatory strategies and appropriately organized instruction to remediate deficiencies (CTLM 1986, 52).

Sharma warns against pushing children who are not developmentally ready. When a child between 5 and 7 cannot grasp simple math concepts, their early introduction will result in negative experiences, attitudes, and anxieties.

Parents and teachers are cautioned to have patience and to continually provide many varied informal experiences that teach the desired ideas, without eagerly expecting early concept mastery (Sharma 1989).

DIAGNOSTIC TEACHING METHODS

There are five critical factors affecting math learning and each is an essential component of the successful math curriculum as well as being critical diagnostic tools for evaluating learning difficulties in mathematics.

(1) A teacher must first determine each student's cognitive level (low--high) of awareness of the knowledge in question, and the strategies he brings to the mathematics task. Low functioning children have not mastered number preservation and are dependent on fingers and objects for counting. Findings dictate which activities, materials, and pedagogy are used (Sharma 1989).

(2) The teacher must understand that each student processes math differently, and this unique learning style affects processing, application, and understanding. Quantitative learners like to deal exclusively with entities that have determinable magnitudes. They prefer the procedural sequences of math. They methodologically break down problems, solve them and then assemble the component solutions to successfully resolve a larger problem. They prefer to reason deductively, from the general principle to a particular instance (Sharma 1990, 22).

Quantitative students learn best with a highly structured, continuous linear focus, and prefer one standardized way of problem solving. Introductions of new approaches are threatening and uncomfortable- an irritating distraction from their pragmatic focus. Use hands-on materials, where appropriate (Sharma 1989).

Qualitative learners approach math tasks holistically and intuitively, with a natural understanding that is not the result of conscious attention or reasoning. Based on descriptions and characteristics of of an element's qualities they define or restrict the role of math elements. 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.

Qualitative learners focus on recognizable patterns and visual/spatial aspects of information, and do best with applications. They are social, talkative learners who reason by verbalizing through questions, associations, and concrete examples. They have difficulty with sequences and elementary math (Sharma 1990, 22).

Qualitative learners dislike the procedural aspects of math, and have difficulty with sequences and algorithms. Their work is full of careless errors, like missing signs, possibly because they avoid showing their work by inventing shortcuts, eliminating steps, and consolidating procedures with intuitive reasoning.

Because they tire of long processes, their work is procedurally sloppy, and the amount of practice is insufficient for fluency and automaticity. Eventually, the qualitative student may show disinterest in mastery of mathematics even though they make connections between concepts more quickly than the quantitative learner.

Qualitative learners need continuous visual-spatial materials. They can successfully handle the simultaneous consideration of multiple problem solving strategies and a discontinuous teaching style of demonstration and explanation, stopping for discussion, and resumption of teaching (Sharma 1989); whereas this style may agitate the qualitative learner who resents disruptions to linear thought.

(3) For each student, the teacher must 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 (Sharma 1989).

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;

(7) Inductive reasoning- natural understanding that is not the result of conscious attention or reason, easily seeing patterns in situations, and interrelationships between procedures and concepts (Sharma 1989).

(4) Teachers must teach math as a second language that is exclusively bound to the symbolic representation of ideas. The syntax, terminology, and translation from English to math language, and math to English must be directly and deliberately taught.

(5) The teacher must take each student completely through the six levels of learning mastery of a mathematical concept. These are:

(1) Intuitive connection of the new concept to existing knowledge and experiences.
(2) Concrete Modeling wherein the student looks for concrete material to construct a model or show a manifestation of the concept.
(3) Pictorial or Representational, where the student draws to illustrate the concept and so connects the concrete or vividly imagined example to the symbolic picture or representation.
(4) Abstract or symbolic, where the student translates the concept into mathematical notation using number symbols, operational signs, formulas, and equations.
(5) Application, where the student applies the concept successfully to real-world situations, story problems, and projects.
(6) Communication: The 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).

BEST MATHEMATICAL TEACHING PRACTICES FOR A DIVERSE STUDENT BODY

Math teaching is most effective when math concepts are 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 "boy" is a noun that denotes a particular class, while "tall," an adjective, modifies or restricts an element (boy) of a particular class (all boys). Adding another adjective, "handsome," further narrows or defines the boy's place in the class of all boys. At all times, concepts should be graphically illustrated.

The concepts of "least common multiple" and "tall handsome boys" look like this (Sharma 1989):

Figure 1: Illustrating linguistic concepts

All boys Tall boys Handsome tall boys

All multiples Common multiples Least common multiples

The language of mathematics has a rigid syntax that is easily misinterpreted during translation, and is especially problematic for students with directional and sequential confusion. For example, "86 take away 5," may be written correctly in the exact order stated: 86-5.

When the problem is presented as "subtract 5 from 86," the student may follow the presented order and write 5-86. Therefore, it is essential that students are taught to identify and correctly translate math syntax (Sharma 1989). The dynamics of language translation must be deliberately and directly taught.

Two distinct 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 or exemplified by stories in English.

To teach students to identify extraneous information, ask them to retell the story adding dates to the details and then ask them if the basic facts of the story have changed and why. Sharma advises adding this exercise to the list of the three standard methods used to facilitate language expression (Sharma 1989).

Frequently, students are linguistically handicapped by teachers, textbooks, and parents who use "command specific" terminology to solicit certain responses. For example, most children are told with informal language to "add, subtract, multiply, and divide," but the are clueless when they encounter formal terminology prompting them to "find the product or sum" of numbers (Sharma 1989).

The best way to eliminate this problem is to matter-of-factly 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 a 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 instruction.

CONCLUSION

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 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 set children 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 mentioned variables into consideration. The competent teacher recognizes that she is teaching students who span the continuum of quantitative and qualitative learning styles, and she 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 Sharma's researched and proven method. It is outlined below. After determining that students have 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 the students 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 students talk about their 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 students state the rule and offer specific examples that obey it.

(10) Have students explain the linguistic elements of the concept (Sharma 1989)

Every student with a normal IQ can learn to communicate mathematically if taught appropriately. Parents and teachers must communicate that math competency is 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 to secure better educational and economic outcomes for America's students.

The United States 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." 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.

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Renee Hamilton-Newman,
Dec 30, 2010, 9:18 PM