1. DYSCALCULIA TESTING INSTRUMENTS....40
2. MATH DISABILITY CLASSIFICATIONS....42
3. SEVEN PREREQUISITE MATH SKILLS....89
4. SIX LEVELS OF LEARNING MASTERY....91
5. RECOMMENDED SEQUENCE FOR MATHEMATICS INSTRUCTION....99

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Dyscalculia is a term meaning "specific learning disability in mathematics." People who suffer with a poor memory for all things mathematical have many other symptoms and characteristics. Taken as a whole, these coexisting conditions comprise what this author terms "the dyscalculia syndrome."

Originating with the author's personal experiences with mathematics, a list of relevant characteristics was published on the Internet in February 1997. Since then, some 4,895 people from around the globe have responded via e-mail, phone, and post to corroborate, share similar experiences, and get advice in coping with the disorder.

Respondents range from high school students to doctors and university administrators. Most are looking for definitions, causes, and protocols for diagnosis and treatment. School administrators seek procedural advice from a legal standpoint. Parents search for advice on school issues, tutoring, testing, and college. Students want survival skills, relief from troubling math failure, and concessions from instructors and institutions. Many adults, even after achieving success in other areas of their lives, seek remedial and coping strategies to overcome this baffling and frustrating condition. Almost all dyscalculics seek vindication of their intelligence, and illumination and understanding of their secret disability.

This paper aims to answer all these questions and achieve all of these ends. It will take the reader from darkness to enlightenment. It will leave readers empowered with a full understanding of the complete scope of issues surrounding dyscalculia, and adequately armed with a repertoire of resources for combating the effects of the dyscalculia syndrome.

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1. Inconsistent computation results in addition, subtraction, multiplication and division.
2. Poor mental math ability. May have fear of money and cash transactions. May be unable to mentally figure change due back, the amounts for tips, taxes, surcharges, and discounts.
3. Poor with money and credit. Checkbooks are unbalanced and disordered, overdrafts may be common. Fails to see how small amounts add up, and how interest compounds. May be unable to grasp the concepts of compounding interest, yield, credit, debit, discounting, and other terminology from the financial world.
4. Short term, not long term, financial thinking. Does not succeed with financial planning or budgeting. Fails to see big financial picture. Prone to credit over-extension, poor financial decision making, and debt.
5. When writing, reading and recalling numbers, these common mistakes are made: number additions, substitutions, transpositions, omissions, and reversals. Is almost always unaware of these mistakes. Similar mistakes with letters are rare.
6. Poor math memory. Inability to grasp and remember math concepts, rules, formulas, sequences (order of operations), and basic addition, subtraction, multiplication and division facts.
7. Poor long-term memory (retention & retrieval) of math concept mastery. May be able to perform math operations one day, but draw a blank the next! May be able to do book work but fails all tests and quizzes.

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8. May have difficulty grasping concepts of formal music education. Difficulty sight-reading music, learning fingering to play an instrument, and so on.
9. May be unable to comprehend or "picture" mechanical processes. Lacks "big picture/ whole picture" thinking. Poor ability to "visualize or picture" the location of the numbers on the face of a clock, the geographical locations of states, countries, oceans, streets, etc.
10. Poor memory for the "layout" of things. Gets lost or disoriented easily. May have a poor sense of direction, lose things often, and seem absent minded. (Remember the absent minded professor?) May experience anxiety when forced to navigate under time pressures, like when changing classes, during swim meets, playing football, basketball, or baseball.
11. Experiences directional confusion. Has difficulty discriminating left from right, and north, south, east, and west. Has poor memory for remembering learned navigational concepts: starboard and port, longitude and latitude, horizontal and vertical, and so on.
12. Despite good muscle tone and strength, may have only good to fair athletic coordination. Has difficulty keeping up with rapidly changing physical directions like in aerobic, dance, and exercise classes. Difficulty remembering the physical sequences required for routines, karate moves, dance steps, and "sports plays." Has difficulty remembering the rules for playing sporting games, remembering the order of play, and understanding technicalities. Is quickly "lost" when observing fast action games, like football, baseball, and basketball. As a result, may avoid physical activities and physical games.



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13. Difficulty keeping score during games, or difficulty remembering how to keep score in games, like bowling, etc. Often loses track of whose turn it is during games, like cards and board games. Limited strategic planning ability for games, like chess.
14. Normal or accelerated language acquisition: verbal, reading, and writing. Poetic ability. Good visual memory for the printed word. Good in the areas of science (until a level requiring higher math skills is reached), geometry (figures with logic not formulas), and creative arts.
15. Difficulty with time management. Inability to recall schedules, and sequences of past or future events. Unable to keep track of time. May be chronically late. May be unable to memorize sequences of historical facts and dates. Historical timelines are vague.
16. Mistaken recollection of names. Poor name/face retrieval. Substitution of names beginning with the same letter (Newman 1985a).
17. Tendency to personalize statistics, odds and probabilities due to a lack of appreciation or true understanding of common large numbers relative to a situation (Paulos 1988, 7-8).
18. Tendency to drastically underestimate the frequency of coincidences. Tendency to attribute "great significance to correspondences of all sorts, while attributing too little significance to quite conclusive but less flashy statistical evidence (Paulos 1988, 26)".

1. On tests, please allow me scrap paper with lines and ample room for uncluttered figuring.
2. I need instant answers and a chance to do the problem over once, if I get it wrong the first time. Often my mistakes are the result of "seeing" the problem wrong. To AVOID this, you would have to watch as I went through each problem, correcting any mistakes in recording as they happened.
3. Problems written too closely together on the page cause me mental confusion and distress.
4. Please make the test problems pure, testing only the required skills. They must be free of large numbers and unnecessary distracting calculations. These sidetrack me into frenzy!
5. Please allow me more than the standard time to complete problems and please check to see that I am free of panic (tears in my eyes, mind frozen).
6. If possible, please allow me to take the exams on a one-to-one basis, in your presence.
7. Most importantly, never forget that I WANT to learn this and retain it! But realize that math is very DIFFERENT than other subjects for me. It is traumatic! The slightest misunderstanding or break in logic overwhelms me with tears and panic. Please understand that I have attempted math and failed many times. Math is emotionally charged for me. Pity will not help, but your patience and individual attention will.
8. I do not know why this is so hard for me. It is as if my math memory bank keeps getting accidentally erased. And I cannot figure out how to correct the system errors!

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9. I ask that we work together after class on the material just presented. Or, if that is impossible, sometime that day for at least an hour.
10. I ask that extra problems be given to me for practice and maybe a special TA (teaching assistant) be assigned to me.
11. I know that working with me may be just as frustrating for you. There are no logical patterns to my mistakes. A lot of them are in recording or in "seeing" one part of a problem in another. Sometimes I read 6x(x+3) as 6(x+3). Sometimes I read 9 as 4 or y as 4 and 3 as 8. After you work with me a couple of times, I am sure you will realize how important it is to keep problems as pure and simple as possible because my brain creates enough of its own frustrating diversions.
12. It is typical for me to work with my teacher until I know the material well, and then get every problem wrong on the test! Then 5 minutes later, I can perform the test with just the teacher, on the chalkboard, and get all the problems correct. So, please, do be patient with me, and please do not give up on me!
13. When presenting new material, I must be able to WRITE each step down and TALK it through until I understand it well enough to teach it back to you.
14. Maybe you could go over the upcoming lesson with me. Then the lecture would be more of a review and I would not be sitting through class in tears.
15. Lastly, I am sure you know by now that I am not trying to "get out of" doing what is required of the rest of the class. I am not making excuses for not "pulling my load." I am willing to put WAY more into this class than is required of the average or better student. I am not lazy, and I feel really smart in everything but math.

In summary, there are a great number of students who have serious difficulties in learning mathematics, but find the rest of academic subjects easy. These students have high IQs, are excellent readers and creative writers, and learn quickly. They are frustrated by a paradoxical condition. Superior performance is easily demonstrated in thinking, verbal, reading and writing skills, and in every subject where these skills are the predominant modes of learning and assessment.

But when it comes to any subject that requires understanding and application of the language of mathematics, they fail miserably, to everyone's surprise. These students may become ill, disruptive, easily frustrated, and may use their creative abilities to avoid tasks (Baum 1990, 2) involving mathematics.

Most gifted children teach themselves to read before they are 6, some even reading between the ages of 2 and 4. Gallagher contends that once basic reading skill is attained, the child is able to advance his intellectual breadth of knowledge on his own. He will usually excel in verbally dominated areas like social studies, English, and science (Baskin and Harris 1980, 38).

Mathematics presents a different case because basic skills are dependent upon rigid sequential mastery. It is difficult to advance independently in arithmetic because much guidance is required, whereas skills in logical math reasoning allow for autonomous progress (Baskin and Harris 1980, 38). Learning disabilities in gifted children are frequently not discovered until adulthood (Baum 1990, 2).

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Silverman contends that this discrepancy between reading and mathematical ability is due to advanced visual-spatial ability with underdeveloped sequencing skills. This results in difficulty learning math and foreign languages the way they are typically taught (Delisle and Berger 1990, 3). Many gifted students never achieve their potential because they have never worked at complex tasks and are unprepared for challenging subjects (Winebrenner and Berger 1994, 1).

This paper will discuss the implications for giftedness in language function, co-morbid with authentic disability in mathematics. Each educational area requires extensive study and appropriate educational programming. The gifted-disabled student is at-risk on both fronts, as she will eventually lose interest and respect for schooling that is unchallenging in most areas.

She will matriculate without developing mathematically, suffering an emotional fury of frustration, failure, and avoidance. And again, her lop-sided academic achievement will disqualify her from pursuing half of all careers, especially those in the lucrative technological field.

The actual stories of some exemplary gifted/math-disabled people will follow, highlighting "the dyscalculia syndrome," giving it voices, description, and life.

In a typical e-mail dated 26, November 1998, Leslie writes:

In an e-mail dated 24, November, 1998, MaryJo from Michigan tells about her daughter: "Michelle can grasp all other concepts of school except Math. She has a hard time telling you what number comes before another number when we get into the teens. . . . "

In an e-mail dated 22, November 1998, Tanya, 25, of Arizona, describes childhood giftedness and a head injury that resulted in dyscalculia symptoms. She writes:

In an e-mail dated 23, November 1998, Tanya elaborates on her condition further:

In an e-mail dated 12, November 1998, Eva, age 50, writes from Denmark:

In a letter dated 8, November 1998, Cathy, age 50, of Alabama humorously writes of her dyscalculia experiences:

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In an e-mail dated 12, November 1998, Cathy continues to share her humorous experiences:

On 3, November 1998, Susan writes:

On 12, November 1998, Kathy in Michigan writes seeking help for her daughter:

On 8, November, 1998, Lou, an education professional in Texas writes for personal support and information for his students with dyscalculia:

In an e-mail dated, 20, May, 1998, Tony writes, in desperation, from a college in Sunderland, United Kingdom:

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Here is an interesting letter sent by an accomplished novelist, and senior citizen, who expressed relief and validation upon uncovering her dyscalculia syndrome. It is dated 17, October 1998. (She wishes to remain anonymous.)

In an e-mail dated 7, November 1998, G. Michael Callahan, J.D., assistant Attorney General, recounts his experiences as a mathematics teacher:

In an e-mail to the author dated 24, June 1998, Marge writes:

In a 1997 e-mail to the author, Aris writes: "I was struggling with high school algebra while enrolled in English honors courses, and the disparity was ridiculous. I couldn't keep up with the teacher. I couldn't finish my tests in the time allowed. I did fine on homework but failed all my in-class quizzes, etc. It was almost cripplingly reminiscent of all my math classes as a kid. Big flashbacks to major failure and self-hatred over fractions. I had to do something. So I decided to be tested for a math disability. This testing occurred in college, and showed average IQ scores in mathematics and superior scores in all other areas."

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An anonymous student writes the author on 20, May 1998:

Rose, a British elementary student in the 1950s, beautifully describes her dyscalculia syndrome in an e-mail to the author. She titles her letter, Math Trauma (or why the heck did I never get past Long Division?. Her excellent letter dated 25, May 1998, follows:

On 14, October 1998, a sportswriter in Dallas, writes:

The story of Barbara, a precocious child in the early 1970s, concludes the illustrations of dyscalculia syndrome and sheds light on the workings of a gifted, but troubled, young mind. Barbara recalls reading Reader's Digest when she was 5, 6 and 7, and her mother's college textbooks when she was 12. She remembers finding her father's old high school Algebra text on a bookshelf when she was 7. This was Barbara's first encounter with reading content that she could not comprehend with deliberate effort. The unconquerable book frustrated her to tears and a fit of anger.

When Barbara was in grade school she remembers being bored and impatient with the repetition encountered in each grade. Waiting for slower students to respond, and the instructional repetition they required, provoked an adrenaline response and agitation.

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To fill the downtime during class, she kept a paperback hidden and secretly read it at every opportunity. Teachers were aware of this, and sometimes deliberately asked questions to see if she was paying attention. But she always answered correctly, so they ignored the behavior. (Barbara was extremely good at multi-tasking.)

Sometimes Barbara's attempts to keep herself entertained led to protest from other students who tattled on her. Then the teacher was compelled to ask her to put the book away. When she could not read, Barbara's mind filled with poetry and she filled notebook after notebook with thoughts, poems, and a chronicle of her childhood.

Eventually, Barbara determined to be "mentally absent" from her unstimulating environment. She tuned out the classroom and her peers. Barbara was extra sensitive of the feelings of others, and could never pick on or make fun, like the other kids. She remembers crying because her friends were making fun of a girl's velvet dress in 1st grade. Their conversations over each other, TV shows, and Hollywood heartthrobs seemed ridiculous and trivial to her. She read her books while walking down the hall, walking home, riding the bus, waiting in line, and every chance she could get.

When she got home, she'd watch the news and the educational shows on PBS. While the rest of the girls talked of Soap Operas, Barbara disdained them. (She did try to like them, to be included in the gossip, but could not even seduce herself to enjoy them.) She felt their second class acting, melodrama, and wild forays insulted her intelligence.

Barbara fanaticized about college, and disdained attending elementary and high school all together. The whole school culture didn't make sense anyway. Nobody liked a smart kid, a nerd. The stupider one acted, the cooler they were. It was cool to have an "I don't

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want to be here, I don't care about learning attitude." Life seemed to be all about having fun, playing games, gossiping, hanging out with the cool people, looking pretty, joining the latest fads, accumulating the most stuff, and wearing the right labels. None of that seemed related to anything that Barbara thought was important. She could not wait to be old enough to work.

She decided it was "stupid" to act like a kid. Besides, no one took them seriously, and they were incapable of contributing anything to the world. Kids were just consumers, just prisoners. They couldn't be anything important. They had no power. And Barbara felt especially powerless about her parents' divorce. She tried to intervene, but it was futile. She tried to help her mom by taking on more responsibility, but her mom encouraged her not to worry about grown up problems, and to just enjoy being a child.

Barbara was uncomfortably trapped in her childish body, and did her best to conceal and embellish it. She related more with her babysitters and her mother's friends, than her peers. Several adults took Barbara under their wing and schooled her in grown up things, like politics and literature.

Here's how Barbara's mom introduced her firstborn: "This is Barbara, 4, going on 24.... 12 going on 30." Barbara played "mother hen" for her younger siblings, and they openly resented it with hostility.

Barbara was annoyed when adults laughed at her attempts to be grown up, and shooed her away from their conversations. Barbara avoided playing outdoors with groups of kids. She wanted to stay inside and participate in adult conversations, but of course, her mother told her to go play. Barbara always felt she had important things to say, relevant tidbits to contribute, but adults were never interested. So Barbara joined the adult world by reading adult books.

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In first grade, she was reading the Laura Ingells Wilder series, then Nancy Drew mysteries, then Judy Bloom, then Alice and other books with teenage and adult themes. By 12, she was reading mom's college textbooks, and at 14, began attending classes at a community college under her mom's name. (She got As.) She began hanging around with older kids, doing the things that they did (to appear grown up), like smoke cigarettes, drink alcohol, and experiment with drugs. She did not enjoy or agree with these activities, but they seemed the only way to separate herself from her chronological age.

Barbara's mother kept her involved in all of the regular childhood activities: museum field trips, zoos, libraries, lakes, cider mills, art shows, political rallies, girl scouts, camping, family outings and events, church, religious education, choir, dance, gymnastics, art, and music lessons. And Barbara participated in all of these, appearing quite normal and busy, but below the surface, Barbara, using her honed ability to multi-task, used even these events as opportunities for diversion.

Although Barbara seemed quite mature and responsible for her age, she really needed constant adult supervision when outside the home. Of course, given Barbara's state of mind, she would have felt suffocated by that, and would have rebelled, possibly running away. Barbara developed a penchant for being alone. As much as she liked talking to strangers, she liked being alone, too.

The aforementioned problems stemmed from Barbara's precociousness or giftedness- her early and advanced abilities to speak, reason, read and write. Next we will look at aspects of Barbara's personality that were shaped by the more negative characteristics of her dyscalculia syndrome. Later in the paper, while discussing various perils of giftedness, we again draw attention to Barbara in a section titled The Twisting of a Gifted Child.

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Although Barbara had much going for her- precocious facility in spoken and written language, and success in school- a shameful list of idiosyncrasies precariously undermined her attempts at independence and the rounding-out of her personality.

Barbara was not good at everything. When it came to sports, for some reason, she could not keep track of the play, and could not remember the intricacies and rules that guided each sport. She was always behind the action, wondering what just happened and why.

Not grossly uncoordinated, Barbara was flexible, physically strong, and had good endurance. She even tried to maintain a positive attitude about her athletic ability. She made the cheerleading squad in 8th grade. She was good at gymnastic feats, making up cheers, and enthusiasm, but it was very difficult for her to master physical routines. And she was pretty confused about which cheers to start when, as she did not understand what was going on in the game and how the cheers applied to the circumstances. Sometimes this resulted in public embarrassment.

She was a good hitter in volleyball, but could never remember the set up of players or the order of rotation. In baseball, she was a fair hitter, a worse catcher, a decent pitcher, but got easily confused about where to throw the ball when fielding. She also could not remember the placement of outfielders.

In inactive sports like bowling, Barbara was a fair bowler, but never could remember how to keep score. She was on the swim team for three years in high school, and although a strong and fast swimmer, she never got the fine points of each stroke perfected, and panicked when she had to flip at the end of the lane, even though she could flip. She also had difficulty swimming straight in her lane.

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She had very poor ability to navigate without full sight. Eventually, she developed excuses to preclude her from participating in sports. In college, she joined crew and was a strong and fast rower, but she was ever confused about how to tell starboard from port, north from south, east from west! Eventually she gave up the sport.

All through her childhood, in music classes and choir groups, Barbara tried to get away with mouthing the words because she was unable to sing on key, especially when other voices competed. Apart from learning the mechanics of musical scales, EGBDF and ABCD, Barbara never learned to sight-read music, despite 8 years of regular music instruction and 3 years in band. She could play better by ear, but her ability was marginal. It was difficult to remember the fingering sequences required for flute, piano, and guitar notes, and even for typing without looking at the keys.

In the early days of computers, Barbara was pitifully frustrated by the keystroke sequences required to perform simple operations. Modern computers are not a problem for her. She has improved considerably in her ability to execute sequences of tasks required for computer operations, by practice with real life applications.

Sequences were especially a problem, as was Barbara's memory for the lay out of things. In her small high school, Barbara was easily turned around. When a bell rang, she would often forget where to go next. Time and schedules were hard to keep track of. A lot of the time, Barbara walked aimlessly in a daze trying to remember where she was headed or what class she was due in. When she arrived at her locker, she frequently could not remember her combination.

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When she learned to drive, she was always getting lost, or could follow directions downtown, but was unable to reverse directions to get home. She lacked good "muscle memory." It was very difficult for her to reproduce demonstrated physical sequences or operations. She preferred to write directions down and then follow them.

Eventually, Barbara compensated by affixing a class schedule and combination, to the top of every book, notebook and folder. She made lists for everything, and crossed off items as they were done. If she forgot to look at her calendar or list, trouble resulted, because Barbara had a distinct ability to focus completely on the task at hand and totally forget about past and upcoming events. She was even dangerously able to screen out her surroundings when paying attention to something interesting, and she was chronically late.

On her annual scholastic achievement tests, Barbara consistently scored in the 98^th percentile and above in all areas, except math. She never memorized addition and subtraction facts. In 4^th grade she could not memorize the multiplication tables, or remember the sequences required for division and the manipulation of fractions. In high school she attempted and failed Algebra six times, but got an A in Honors Geometry. She did well in all Honors classes except for Economics, where the concept of stock market tracking eluded her!

In sharp contrast, another gifted adult, Edward (who avoids reading, writing, and school) says to Barbara:

Unfortunately, for Barbara and others with dyscalculia syndrome, learning is not so easy. They actually prefer the permanence of information in written form, because their memories are unreliable for viewed sequences of mechanical operations and processes. They excel in all subjects where thinking, verbal, reading, and writing skills are the predominant modes of acquiring and demonstrating knowledge. Some famous gifted children have had similar experiences. Their trials are briefly described in the next section.

Some of the most famous gifted children suffered strange childhood incongruencies in development. Einstein did not speak a word until he was four and had early difficulties with arithmetic. Thomas Edison did not learn to read until he was 9, and was considered a delinquent. Wernher Von Braun, the father of rocketry, flunked 9th grade Algebra (Moore 1981, 2-3).

William James, an avid reader and prolific writer, wrote, "I am myself a very poor visualizer, and find that I can seldom call to mind even a single letter of the alphabet in purely retinal terms. I must retrace the letter by running my mental eye over its contour in order that the image of it shall have any distinctness at all (Vail 1979, 30)."

Benjamin Franklin never remembered a time when he could not read. At the Boston Grammar School, Franklin, 8-years-old, studied the classics so he could become a minister. He learned quickly and became the top student. By the middle of his first year of school, he was promoted to the next level and was expected to be promoted again before the year's end.

Franklin's father decided to remove him and put him into a private school to better prepare him for a more professional, worldly career. By the end of his second and last year of formal schooling, Franklin performed exceptionally in every subject, but arithmetic, which he failed. His father then removed him from school and put him to work. From then on, Franklin educated himself, worked, apprenticed at his brother's print shop at 12, and eventually became a great scientist, statesman, writer, and publisher (Kelly 1996, 19-26).

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Thomas Jefferson included special provisions for the highly able when he established the first American public university in Virginia. But the first systematic attempt at public education for the gifted appeared in St. Louis in 1868, allowing the gifted rapid advancement through the grade levels (Baskin and Harris 1980, 14).

In 1975, the National Science Foundation estimated that 125,000 of the top 10% of bright children drop out of school (Moore 1981, 3). Terman postulates that too many of our gifted, between 7% and 47%, are underachievers and "languish in idleness (Strang 1960, 7)." Jacques Barzun said, "their discovery of themselves and by others, is not inevitable (Barzun 1959, 139)."

Academically talented children comprise 15-20% of the school population. These children "have the ability to study effectively and rewardingly, advanced mathematics, foreign languages, and tough courses in chemistry and physics (Conant 1958, 16)." They have the ability, interest and industry necessary to succeed in academic programs (Strang 1960, 18).

Only 2% of the nation's smartest students ever earn a Ph.D. Of the brightest students, 25% never finish college. Only 1% of the nation's population has superior intelligence. Children with IQs of 140, waste 50% of their time in the classroom, and those with IQs of 160 waste 100% of their classroom time (Abraham 1958, 5-6).

Without special training in the recognition of gifted children, teachers will identify only 40% of them (Strang 1960, 143). In 1983, the U.S. Commission on Excellence in Education estimated that 50% of all gifted children are underachieving, as defined as a discrepancy between ability and performance (Ford and Thomas 1997, 1-2).

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Canadian, Dr. William E. Blatz, reported "no appreciable difference in the level of achievement of children with IQs of 140 or over who went to a special school (Strang 1960, 203)." The most gains are seen when intervention focuses on the gifts instead of the disability (Whitmore and Maker 1985).

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 all children have true developmental dyscalculia (CTLM 1989, 86).

Dyscalculia has no less predictive merit than illiteracy, a characteristic shared by 80% of the prison population (Weger 1989, 36). The foreigner to math is lucky that there are no debtor's prisons. For all 103 million U.S. households, bankruptcy rates increased 13% in the decade between 1985 and 1995 (Francese 1997), and then increased an astounding 29% in the single year between 1995 and 1996! In 1980, 97% of all bankruptcies, or 288,000, were by non-businesses. By January 1998, personal bankruptcies soared to 1,152,000- a rate that quadrupled in just 18 years (USA Today 1997). Given the dismal math understanding of 90% of recent high school graduates, personal bankruptcies will only continue to increase.

Employment Futures:

Between 1994 and 2005, demand for system analysts will jump by 92%. The demanded number of computer engineers and scientists will increase 90%.

Demand for computer programmers will grow by 12% (USDC 1997).

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The average hourly compensation in 1996 for an intermediate customer support technician was $40.80, a software development architect earned $77.70, an operating systems software architect/consultant earned $85.60, and an operating systems/software programming analyst manager earned $92.20 per hour (USDC 1997)!

America has an acute shortage of information technology (IT) workers. U.S. computer science graduates declined 40% from 1986-1994, and 50% of all U.S. IT students are foreign students. Turnover rates range between 35-45% in these areas: client/server architecture, data modeling, packaged software applications, and distributed databases. Annual trade growth in the software market is 12%, and in computer services, 11%. Both markets had a combined growth of 50% in the 4 years between 1994 and 1998 (USDC 1997).

American youths are leaving high school ill prepared for the advanced study required for these lucrative jobs. Only 56% of exiting 17-year olds can compute decimals, fractions, and percentages. Over 46% cannot recognize geometric figures, solve simple equations, or use moderately complex math reasoning. An astonishing 93% of high school graduates cannot solve problems involving fractions or percentages. They cannot solve 2-step problems involving variables, or identify equal algebraic equations, or solve linear equations and inequalities. An alarming 93% cannot synthesize and learn from varied specialized reading content. An amazing 91% cannot infer relationships and draw conclusions using detailed scientific information (USDE 1991).

If over 60% of all high school graduates now go directly to college, (25% of freshman have taken advanced courses) (Riley 1998, 1), that means that over 90% of entering freshman will need remedial math courses.

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Heavy TV viewing directly corresponds to low achievement scores on NEAP tests (National Assessment of Educational Progress). In 1990, 62% of 9-year-olds watched TV over 3 hours per day, and 23% watched TV more than 6 hours per day. One in every five 9-year-olds could not add and subtract 2-digit numbers, or recognize relationships among coins (USDE 1991).

In 1978, Sheila Tobias realized that only 8% of girls took 4 years of high school mathematics, thus 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).

Standardized achievement tests measure how much a student has learned about a school subject. Standardized aptitude tests measure a student's ability to learn a school subject and are used to predict future school performance. Under the Family Education Rights and Privacy Act of 1974, students and parents have the right to examine their academic records, including test scores (Bagin and Rudner 1998, 1, 3).

Before entrance to first grade, a child should have a reading-readiness and an IQ test. Then IQ tests should be given every 3-4 years until age 18. Annual standardized tests should measure academic achievement. In secondary school, by the 7th grade, aptitude tests (which measure one's capacity to learn certain material), interest inventories, and preference tests, should be given with careful consideration for students who may skip a grade or enter college early, as they will need advance academic planning (Cutts and Moseley 1953, 115-116, 184, 190, 204).

Terman suggests a vocational-interest test to delineate the area of occupational choice. When many options are present, the parents should help the child work through these questions: What types of work are best left to bright individuals? What personal abilities and interests make one area of work preferable to another? What are the job prospects for each suitable vocation (Cutts and Moseley 1953, 115-116, 184, 190, 204)?

When one of several areas of interest and aptitude are discovered, the child should actively investigate the scholastic and vocational requirements of these and plan ahead.

A college preparatory course will include early foreign language and advanced mathematics and science study in high school. A good rule of thumb is 4 years of high school science, math, English, history, and literature, coupled with extra-curricular activities that develop the talents, interests, and social skills of the individual.

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Tests given before a child is 2.5 years of age are unreliable. Test scores become more reliable at age 5. An 8-year-old scoring exceptionally high will continue to do so at ages 15 and 16. A test given at 8th grade is as predictive of college success as a test given in the senior year of high school (Strang1960, 15).

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)."

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 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).

Quantitative dyscalculia is a deficit in the skills of counting and calculating. 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. Intermediate dyscalculia involves the inability to operate with symbols, or numbers (CTLM 1989, 71-72).

39.

40.

41.

The table lists the battery of tests studied extensively and used successfully to diagnose dyscalculia at the Centers for Teaching and Learning Mathematics in Framingham and Wesley, Massachusetts, and London, England.

The varied disciplines involved in the diagnosis of dyscalculia 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).

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Each profession uses specific terminology to describe math disabilities. The result is the fragmentation seen in the following table. At the end of the table, several terms are introduced with definitions of their prefixes.

TABLE 2: MATH DISABILITY CLASSIFICATIONS

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Normal intelligent quotients range between 90 and 110, with 100 being the average. Scores above 110 are superior, and scores above 140 are very superior. The Educational Policies Commission estimates that 10% of the population has IQs of 120-136, while only 1% have IQ's137 or above (Cutts and Moseley 1953, 17). In 1937, Terman and Merrill published the following IQ classifications: 30-69, Mentally Defective; 70-79, Borderline Defective; 80-89 Low Average; 90-109, Normal or Average; 120-139 Superior; 140-169 Very Superior (Moore 1981, 41).

In 1971, the U.S. Dept. of Education concluded that 50% of gifted children are not identified with the use of group tests. Other experts agree that standardized tests, which measure only 8 mental operations, discriminate against the gifted by not evaluating completely the 120 mental functions identified in the Guilford Structure of the Intellect Model (Moore 1981, 52-53).

Use discretion when basing important decisions solely on IQ scores, which can vary over time and across testing instruments. Fluctuations of 10 points have been seen in more than 3/4 of all students: 1/3 of student's scores fluctuate by 20 or more points. Scores of 1/10 of students vary by 30 points, and a few have scores that change by as many as 45 points (Strang 1960, 16).

50.

Since much of life is gauged to accommodate the average (child, size, age, intellectual level, activity level, etc.), to the above average person, the fit is all wrong. When kept in unstimulating circumstances, typical curricula, books, peer conversations and activities bore and frustrate the bright child, causing him to mentally "check out," as Barbara did. Trouble rears its head as apparent "laziness, undermotivation," or an unbalanced devotion to intellectual pursuits at the expense social and physical development (Cutts and Moseley 1953, 3-4).

Even children who excel in every subject may be acquiring inefficient lazy habits because things come too easily for them, and they are not challenged to learn and perform to their potential. Upon college entrance, where competition requires sustained efficient study habits, they may not know how to study because they have never had to! (Barbara found this to be the case.) Hollingsworth reported that bright children waste 50%-100% of class time fooling around, talking, or not paying attention (Cutts and Moseley 1953, 74, 99).

Terman's research shows that accelerated students do better work in school, college, and adult life than their peers who progress normally through grade levels. Of the 200 students allowed to skip their senior year of high school and matriculate at Yale, Chicago, Columbia and Wisconsin before age 16.5, all but 3 completed their freshman year successfully; 51% were in extracurricular activities, and 51% made the dean's list. "In general, the skipper who expects to go to college should have an IQ of 130 or more (Cutts and Moseley 1953, 82-83)."

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52.

In a study of 50,000 children, of the 23 "geniuses" found, most of the parents and teachers of these children did not suspect intellectual prowess, and several actually considered the children stupid. Other studies have pointed to the life-long underachievement of many bright children. These children find trouble along the way, drop out of school, and spend their lives working routine jobs (Cutts and Moseley 1953, 8).

As regards language development, the ability to generalize is a characteristic of superior mental ability. Generalization is the ability to see common elements in situations and the ability to deduce general principles from these isolated events. Personality traits, such as early foresight, originality, self-confidence, curiosity, and eagerness to please and help, can predict brightness (Cutts and Moseley 1953, 11, 13).

Some children, fearing ridicule from their peers, dubiously hide their intellectual abilities. These children often end up leading the group into mischief. Sometimes the child, conscious of his intellectual superiority, will feel entitled to recognition, but will do nothing to earn it. This child may seem conceited and obnoxious (Cutts and Moseley 1953, 34).

Gil Caudhill, a gifted education consultant, believes that gifted children have no inclination to conform; yet they are continually intimidated into traditional molds (Moore 1981, 11).

Terman studied 1,500 gifted individuals and found this occupational breakdown: 46% became professional doctors, teachers, lawyers, engineers, writers, etc.; 26% became business executives, accountants, and military officers; 20% became clerks, salesmen, skilled trades and craftsmen, and insurance agents, etc. Very few became farmers, semi-skilled or day laborers (Terman and Oden 1947, 172-174).

Terman found that 85% of gifted children come from homes with favorable conditions. The parents were in the prime of life and the mother was in good health when the child was born. The birth was without problems, a significant percentage was breast fed, and the children experienced excellent health during the first year (Terman and Oden 1947).

Martin found these home situations conducive to superior child development: Parents expect the best of their children, give of their thoughts and time, more so than material things; do not force their ambitions upon them, and do not guilt trip them. They do plenty of things with the child, as well as for him; give him real chores to do to instill responsibility and importance; treat older children with respect, and constructively work through squabbles. Both parents who enjoy being together head the best homes (Martin 1943, 596-608).

In Terman's 1926 findings, gifted children had better nutrition, were breast fed longer, and had significantly less ear infections and hearing disturbances (Moore 1981, 15).

In Catherine Cox's study of 300 genius children, their most common tell-tale signs were: "quick understanding, great curiosity, retentive memory, early speech, unusual vocabulary, and extensive information (Cox-Miles 1954, 1002)."

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Dorothy Sisk, former director of the U.S. Department of Education's Office of Gifted and Talented, lists the following characteristics of giftedness:

1. Early use of advanced vocabulary.

2. Keen observation and curiosity.

3. Retention of a variety of information.

4. Periods of intense concentration.

5. Ability to understand complex concepts, perceive relationships, and think abstractly.

6. A broad and changing spectrum of interests.

7. Strong critical thinking and self-criticism.

8. Early demonstration of talents in music, art, athletics, and/or the performing arts. (Moore 1981, 20)

9. Unusual alertness in infancy (Berger 1998b, 1).

Terman found that gifted children learned to read earlier than their peers did and they became voracious readers of a wide variety of subjects. Books supply a readily available source of information to satisfy their insatiable curiosity about everything around them (Miller and Price 1981, 112).

The gifted child reads early, easily, and far more than average. He reads more non-fiction, and a wider variety of topics than other children do. He gets totally absorbed, and has the ability to shut out the rest of the world while reading. Terman's studies showed that gifted 7-year-olds read over 20 hours a week, and read an average of 10 books in a 2-month period (Strang 1960, 116-117).

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Dr. Terman described the characteristics of 1,000 children with IQ's higher than 140. He found that, in general, the group had superior physical appearance, health, social adjustment, and performance on tests of character, and scores on school achievement tests. Two-thirds of their parents noted signs of intellectual superiority before 3.5 years of age in girls, and slightly later in boys. Musical ability was recognized around 5 years of age and other special abilities around 6 years of age (Terman 1954, 221-230).

All of Terman's 14-year-olds had reached puberty, and 48% of the 13-year-old girls had begun menstruation. The gifted adolescents in Terman's studies had accelerated physical maturity, and superior health (Strang 1960, 154).

Most gifted children are not lacking in the practical abilities required of shopwork or handicrafts. They usually do quite well in these areas, as long as they are persistent enough to develop these talents (Strang 1960, 122).

Genius comes from the Latin, genere meaning, "to produce." True geniuses produce great original works that stand the test of time. The rare genius has an abundance of good ideas, shows unusual creativity and curiosity, solves abstract problems, and discerns the significant. He finds new relationships among ideas, is spontaneous and focused, has sudden flashes of insight, looks at things from unique intuitive perspectives, and is extremely perceptive and persistent- all at an early age. Care should be taken to provide encouragement, approval, and expert instruction in his areas of interest (Strang 1960, 12-13).

Aside from the ability to retain everything they see and hear, and an intensive desire to execute their curiosity, gifted children can cope with unfamiliar situations and solve problems without help. They ponder adult ideas and concerns (Strang 1960, 23-25).

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They have an abundance of mental energy, and are interested in a plethora of things. They are self-starters and self-taught in areas of interest. Their school performance is often 2-4 years ahead of their grade placement. They are likely to be weakest in handwriting because it requires manual dexterity and coordination. They tend to be "old for their age," comfortably joining conversations and projects with adults and older children, and enjoy verbal facility. As adults, they maintain their superior character and achievement (Strang 1960, 23-25).

Giftedness in creative writing can be identified by the "imaginative imagery of preschool children, and in the vivid speech and spontaneous free writing of older children when they are dealing with something that is exciting to them." Writing is judged for the literary qualities of felicity of expression and originality (Strang 1960, 19).

Socially gifted children can be observed at an early age for their marked social sensitivity, "ability to sense the feelings and responses of others, and ability to handle social situations (Strang 1960, 20-21) ." They suffer along with others (QAGTC 1994, 2).

They are natural leaders that find ways to resolve social difficulties by satisfactorily involving everyone. They keenly help the group achieve its goals while carefully meeting the needs of each individual member. They are popular, kind to everyone, champion the weak, suggest interesting activities, take charge, and are good in games. Some may possess the high moral and spiritual qualities that lead to outstanding religious leaders, philosophers, and thinkers. They have a great capacity for understanding human relationships and have spiritual insights, and translate these into exemplary conduct (Strang 1960, 20-21).

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Abraham describes signposts of giftedness, particularly common in the student with dyscalculia. The student shows a mature ability to express himself with creative writing and picturesque speech. His superior listening skills result in quick wit, and genuine mental and emotional participation. He learns quickly with fewer repetitions. He is usually resentful of "meaningless busy work," and feels the time could more profitably be used. He may be impatient and rebellious at the passive attitudes of those around him. He is adept in objective self-analysis of his abilities and limitations. His acute self-awareness may serve to alienate himself from his peers, and he may react by gravitating towards adults or introspection. He has a long attention span (Abraham 1958, 25-27) and tenacious ability to focus.

RED FLAGS---GIFTED AND AT RISK

Logically, bright children should like and do well in school but when they do not, red flags should go up. An educational psychologist should be consulted and every effort made by parents and educators to find and remedy the problems. Typical difficulties stem from lack of interest, special disabilities in reading or math, poor work and study habits, truancy, disciplinary problems (Cutts and Moseley 1953, 114), and low morale.

Some children do not understand that the teacher cannot listen to and talk to them as much as their mothers because she must accomplish instructional goals, and spread herself among many children. When this is the case, children become impatient with the teacher and acquire a dislike for school (Strang 1960, 180).

They may dominate discussions with their verbal proficiency. They may frequently disrupt class routine, and tend to challenge and question indiscreetly because they are extremely persistent in having their questions answered (QAGTC 1996,1-2). They enjoy arguing points logically and clearly, and do not perceive this as disrespectful (Scott n.d., 1).

Some very common problems befall gifted children for specific reasons. Many dislike basic routine and are impatient with others because they acquire and retain information so quickly. Because they are intrinsically motivated, they are strong-willed and resist direction. They question everything, including teaching procedures, and resist routine practice because they enjoy problem solving and can easily conceptualize, abstract, and synthesize. They frequently resist traditions, expectations, roles, rituals and other seemingly "illogical" activities that are not rooted in cause and effect relationships (Webb 1994, 1).

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Often duties and people are neglected during periods of intense concentration and persistent focus. This can result in overt dissatisfaction with interruptions. Gifted children are frustrated by inactivity because they are highly alert and energetic. They appear to be non-conformist, and rebellious, but only because they prefer self-reliance and working on their own (Webb 1994, 2). They feel they require little direction and supervision and may resent hovering by adults (Scott n.d., 2). Frequently time pressures exasperate them. They appear disorganized and scattered, but only because they try to maximize their vast and diverse interests and abilities (Webb 1994, 2).

Sometimes overt aggression, and/or emotional outbursts result from an inability to construct, draw or write exactly as they see things in their "mind's eye." Especially in young gifted children, sometimes gross motor and usually fine motor skill development lags considerably behind cognitive and conceptual abilities. These children (15-20%) give up when their motor performance fails to meet their standards of perfection. Some even become depressed by weak performance in an isolated area, like math or handwriting (Webb 1994, 2-3).

Gifted children experience stress more than their non-gifted peers because they have a heightened sensitivity and emotional response to personal thoughts, external events, relationships, expectations, and environment. Narrowing career and study paths is stressful, also, because the act of choosing means eliminating many attractive alternatives (Kaplan 1990, 1-2).

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Because they enjoy the challenge of multi-tasking, they assume responsibility for physically and emotionally demanding course loads, extra jobs, and activities. When they are not over-extended they feel nervous and "out of control." Sometimes their stress results in forgetfulness, indecision, poor concentration, impulsiveness, self-destructive behavior, and rash decisions (Kaplan 1990, 1-2).

Watch for these signs of burn out: Lost interest in school; lost personal happiness; lost positive outlook; lost excitement for people and activities; resentment of people, school or work; lost motivation, ambition, and effort; boredom, sleeplessness, emotional volatility, fatigue, personal dissatisfaction, nervous habits, frequent illness or health complaints; dependent and attention-getting behaviors; aggression, despondency, indecision, lost sense of humor and perspective; and physical, mental and emotional exhaustion (Kaplan 1990, 4). Immediate attention should be provided when any of these symptoms occur. Seek the services of a physician, psychologist, and counselor.

To temper natural stress, gifted students need to learn healthy coping skills and life styles, and gain awareness of the ways they are different and like their peers. They