©1998 RENEE M. NEWMAN All Rights Reserved ii. SPECIAL EDUCATION PROGRAM EVALUATION for INSTITUTIONS: DIAGNOSING MATH LEARNING DISABILITIES: RECOMMENDED PRACTICES 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.
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 13-year, 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 26-30 students. So why are our students losing the numbers game when the odds are in their favor? Almost 93% of America's 17 year-olds graduate without proficiency in multi-step problem solving and algebra. (NCES 1997, 123-124) 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 curricula are not reflecting what we know about how children learn mathematics. Typical math curricula 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 ill-prepared teacher. Presently, the average K-8 teacher has taken only 3 or less math or math education classes in college. Not even 50% of 8th grade math teachers have taken a single class on math teaching at this level, and 28% of high school math teachers lack a major or minor in math. (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, 12-13) Sharma asserts that gender differences in math skills are due more to social forces than to gender-specific 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 gender-specific development of math prerequisite, spatial orientation skills. The main reason for this is the methodology of teaching in pre-school and elementary grades, where focus is on fine-motor skill development. (Sharma 1989) Boys and girls are given ample opportunities to play with blocks, Legos, board games, and various materials requiring fine-motor coordination. Naturally, girls have better fine-motor 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 intricate mechanical manipulations and "rough and tumble" physical activities, girls lose ground in spatial organization abilities. 4. Their more sedentary activities offer few exercises in space/motion judgement, symmetry, part-to-whole 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 7-year-old 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 high-quality 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, 511-513) 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, 71-72) 6. 7. L. Kosc, of Bratislava, advocates in his Slovak "Psychology of Mathematics Abilities" (1971-1972), 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, 69-69) 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:
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 1-2 standard deviations below the mean (middle) score of the group is considered "deficient." A score of 70-75 is extremely deficient. (CTLM 1986, 49-50) A dyscalculia diagnosis in pre-school age children can be made when a child cannot "perform simple quantitative operations" that should be "routine at his age." (CTLM 1986, 50) Developmental dyscalculiais 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.
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 hands-on 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 procedures with intuitive reasoning. Their work is procedurally sloppy, because they quickly tire of long processes. Their performance is never fluent because they do not practice enough to attain levels of automaticity. Eventually, the qualitative student may show 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 visual-spatial materials. They can handle the simultaneous consideration of several problem-solving 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 reading-readiness skills, the teacher must assess the existence and extent of math-readiness skills in each student. Seven prerequisite skills have a profound impact on the ability to learn mathematics. These are non-mathematical in nature, but are extremely important pre-skills that must be fully mastered before even the most basic math concepts can be successfully learned. (Sharma 1989) 19. Seven Prerequisite Math Skills
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) 21. 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)
22. 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 "94-7." But when the problem is presented as "subtract 7 from 94," the student following the presented order will mistakenly write, "7-94." 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" terminology to solicit certain actions. For example, most children are told with informal language to "multiply, "add, "subtract," and "divide." They are clueless when they encounter formal terminology prompting them to find the "product" or "sum" of numbers. (Sharma 1989) To eliminate this problem, 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 at least two terms for every function. (Sharma 1989) For example say, "You are to The dynamics of 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)
23. For example, present "7-4=____," 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?" 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)
24. [5.] Before a mathematical concept is learned fully, the student moves through six levels of learning mastery.
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 hands-on methods, between years 1 and12, their ability to acquire math knowledge is disturbed at the point when hands-on 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 pre-requisite skills and levels of cognitive understanding, introduce the new concept in the following sequence:
25. CONCLUSION: Every student with a normal IQ, can learn to communicate mathematically, if taught appropriately. Curricula in the pre-school and early elementary years should focus on the development of the 7 prerequisite math-readiness 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 K-12 students eventually master challenging mathematics, which include "arithmetic, algebra, geometry, probability, statistics, data analysis, trigonometry, and calculus."(USDE 1998) But compared to other countries, after 4th 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 one-step 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 under-education, 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 28. 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, businesses-education 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. Center For Teaching/ Learning of Mathematics (CTLM). 1986. III. Progress of Dr. Ladislav Kosc's Work on Dyscalculia Funk and Wagnalls. NCES. 1997. Snyder, Thomas D., with production manager, Charlene M. Hoffman. Program Analyst, Claire M. Geddes Sharma, Mahesh 1989. Sharma, Mahesh. 1990. "Dyslexia, Dyscalculia, and Some Remedial Perspectives for Mathematics Learning Problems." Tobias, Sheila. 1978. Turabian, Kate 1996. United States Department of Education (USDE). 1997. Secretary Riley. United States Department of Education (USDE). 1998. Winters, Kirk. 1997. Update. Winters, Kirk. 1998. Update. |