Following thirteen months behind my only sibling’s academic footsteps was hard. From elementary school on, she was a glowing student; absorbing, digesting, and understanding information seemingly by osmosis. She maintained straight A’s throughout her academic career and earned a college scholarship. I, on the other hand, struggled to maintain a C average and ran away from home at the age of fifteen.

Let’s take a moment and rewind…

I thought I was stupid. Compared to my sister, it certainly appeared that way. However, it wasn’t until many years later I discovered that I learn differently from how I was being taught.

There are three recognized learning styles:

• Auditory learners grasp things by hearing—the worst type of test for them involves reading passages and writing answers. They’re best at writing responses to lectures they’ve heard, and are also good at oral exams.
• Visual learners comprehend through seeing—the worst test type for them includes having to listen and respond. They’re best at diagramming, reading maps, essays (if they’ve studied using an outline), and showing a process.
• Tactile (kinesthetic) learners understand through experiencing/doing—the worst test type for them usually has lengthy tests and essays. They’re best at short definitions, fill in the blanks, and multiple choice.

When I was in school, the general teaching population were auditory teachers. A heavily tactile learner—with a smidgen of visual thrown in for good measure—I was missing the boat!

Fast forward…

Not having done any planning before running away, I didn’t know that if you leave high school before you graduate, you can’t test for a GED—General Education Diploma—until two years after your graduating class.

Biding my time, I went to work for a large, everything-under-one-roof store. Over the next few years I worked my way up to managing the women’s wear department, then added men’s wear, and topped it off with housewares and furniture.

I gained valuable life experience during that window of time. Part of this seat-of-the-pants wisdom was learning to say, “I don’t understand. Can you please explain that differently?” And that’s when I noticed that no matter how many times someone “told” me, it wasn’t until they “showed” me that I got it! When shown, I not only met, but exceeded what was expected of me.

I left high school as a sophomore in 1973. For four long years I waited and prepared to take the GED examination. On a hot day in late June of 1977, with the cut-grass tang of summer in the air, I slipped into a front row seat at the testing center; one of about twenty other people enveloped in the sterile classroom setting.

The all-day test was given in seven parts: Language Arts (writing), Language Arts (reading), Social Studies, Science, Math (calculator allowed), Math (calculator not allowed), and United States Constitution.

Hours later—head held high with a face-splitting grin—I left the facility with every confidence that I’d aced the test. Six weeks later I received my GED certificate, and that was just the beginning. Over time I earned my associate degree, then bachelor’s, followed by a master’s degree. Finally, two weeks before my fiftieth birthday, I sat and defended my Ph.D. thesis.

Hard-wired for buoyancy and tenacious as a terrier, when I set my mind on something I go after it with tremendous resolve. It took a while, but I eventually went from GED to Ph.D. And while I’m Ph.inisheD with brick-and-mortar academia, I’ll never be done learning.

The single biggest problem regarding mathematics and the sciences is motivating younger students to study them.

While the United States excels at welcoming people from all over the world to travel to the U.S. and study science and math, the number of aspiring mathematicians at universities is decreasing. About only 2% of all students in America that pursue a bachelor’s degree are in the fields of mathematics or other physical sciences. On top of that, roughly 48% of students studying a STEM field in a bachelor’s program, and 69% of students pursuing an associate’s degree, changed majors or exited college before earning their degree [1]. One possible explanation for this is the reputation that math has for being boring, uninteresting, and complicated. However, there could hardly be a better time to become a mathematician.

I was watching a documentary about deserts with my 2 year old. At first, she found the documentary to be boring because the desert appeared to be bland and void of life. Then, the screen started showing the lively and colorful aspects of a desert including its oases, various cacti, and small creatures. She was shocked that the arid desert could possess so much color, life, and intrigue. It reminded me of my feelings towards mathematics. For some people, mathematics seems like a dry and dull subject that makes little sense. However, it too becomes full of life and color once one looks in the right places and from the correct perspective.

Ten years ago, the Wall Street Journal ranked jobs based upon a number of parameters such as salary, freedom, the possibility of promotion, and ease of employment. It may surprise you to find out that being a mathematician was ranked as the number one job in the world! [2] Career Cast, a company that started as a spin-off for this kind of ranking in 1988, also ranked mathematician as the number one job in 2014 [3].

In job advertisements, the definition of a mathematician is somebody who applies mathematical theories and formulas to teach or solve problems in a business, educational, or industrial setting. In other words, a mathematician is someone who insists on solving problems, which is what mathematics was invented for. However, a crucial point is missing from this description. There are mathematicians who not only teach or apply mathematics but do math, generate math, and change math.

If a mathematician is one of the best jobs in the world, how many mathematicians are there around the world? Thankfully, it is in the hundreds of thousands: almost 250,000, and it is growing every year, particularly in India and China [4]. These countries encourage their children to participate in STEM fields from a young age and place a very high emphasis upon education that can help foster a love for STEM careers at a young age.

Although mathematics is the science in which theorems last for a long time, it is also a field that is being renewed constantly. There are not many other subjects in which knowledge can last thousands of years and always remain both correct and relevant. For instance, the theorems of Euclid, a Greek mathematician who lived in the third century BC, are still accurate today. If we look at the modern applications of computer science and mathematics we see that people are still applying the contributions of Laplace about the central limit theorem, using the contributions of Shannon in sampling, and applying Fourier’s ideas about signal analysis and signal processing. Paradoxically, mathematics is ancient and contemporary at the same time.

In 2014, Emmanuel Candes of Stanford University gave a short speech about how he had collaborated with medical doctors and mathematicians, and used Fourier’s ideas in analysis and data processing, to drastically reduce the time needed to develop a scan [5]. His machine was a significant evolution for the medical field. In experiments, it was shown that to reconstruct images with the best accuracy, they need to have only 2% of the Fourier transform. Candes managed to reduce this time by a factor of eight by knowing only a very small portion of the Fourier transform. Netflix also uses this technology for reconstructing missing information and predicting the preferences of users for movies.

However, the life of a mathematician is not always full of success. The field is full of uncertainty, intense problem solving that can often take weeks or even months, and waiting. Most of a mathematician’s time as a researcher is spent in failure. That is an objective fact. A common motto is, “Every day is a failure. This is our life.” But when we look back on the amount of time that is spent solving these problems, everything pays off. Every year, the number of new theorems added in mathematics runs into the thousands. Failures are not actually failures, but merely bumps along the path of progress.

Another common problem that befalls mathematicians is that they will be expected to perfectly predict what will happen should their discovery work as intended. This is especially true of work done under the auspices of government agencies, since they will often not grant money without a very clear path towards a desired outcome. Things do not always go as planned, and thus it can be challenging to argue that a desired result will occur 100% of the time.

Despite these drawbacks, one of the most exciting aspects of mathematics is the potential to discover something revolutionary and groundbreaking. History has shown that the ideas and discoveries of one individual can make a tremendous difference in the world. By encouraging and enabling more and more students to study math and science, we thus increase the odds that more discoveries, be they groundbreaking or not, will continue to be made in the future.

Alan Turing was a mathematician who had an outsized impact on human history. He was instrumental in cracking the secret codes that were used by the Nazis during the Second World War. Some have argued that World War II would have lasted at least two more years without Turing’s intervention. In particular, the Normandy operation would have been impossible. What was Turing’s motivation? It was not patriotism, honor, or a strong duty to his country. It was simple: his main motivation was solving riddles and difficult problems. He lived for the thrill of solving his next big challenge.

Paul Erdos, a Hungarian, was the most productive mathematician of the 20th century. Erdos had no home, no car, no bank account, and no salary. He lived with just one suitcase and his ideas. He worked on theorems his entire life. When he got some money from some reward, he would always use part of it to put a reward on another theorem.

Leo Szilard was another revolutionary mathematician. He was the first person to understand the concept of chain reactions between atoms. His inspiration came from a public lecture, in which famed physicist Ernest Rutherford said, “It would be moonshine talking if you are willing to extract energy from the atom.” Rutherford acknowledged that energy existed within atoms but thought it was impossible to do anything with it. Szilard felt that Rutherford was wrong and decided to prove it. He worked and thought for days. And one day, while he was crossing a street in London, he was struck with an idea about the principle of chain reaction and exponential growth of atomic energy. This lead to him eventually meeting with Albert Einstein and the subsequent development of the Manhattan Project.

Steps in developing ideas

But how do scientists find those perfect ideas? What are the steps? Henri Poincare gives us some hints about how the ideas were coming to him. Poincare, who was always considered a genius, experienced many discoveries after working very hard on the problem and he had a lot of failures. He said: “Disgusted from my failure, I went to spend a few days near the sea thinking anything else. And one day, while walking on the cliff, the idea came to me. And as before, it was very brief, sudden, with immediate certainty, that arithmetic transforms of indefinite ternary quadratic forms are identical to those of non-Euclidean geometry.”

So, under which circumstances does a big idea come? In popular culture, we have those myths like Newton saw an apple falling down and changed the world. But in real life, it’s not the way it happens. For Poincare’s example, the cliff has nothing to do with quadratic forms. Poincare was saying that he worked very hard and then he decided to rest a little bit, and finally, he had the enlightening moment. What is important here is if the brain had not been prepared by hard work, the illuminating idea wouldn’t have struck.

So what is the process of discovery?

Of course, a publication will be the last step for the discovery of an idea. Because publication means that the idea is out and going to be seen and read by the world. But before that, we have many steps or ingredients which make our idea stand up on its feet.

1. Fecundation

The first step is fecundation. In other words, conversations and discussions you have with your colleagues. Your interaction has a potential to bring about a new phase, a new idea out of different projects. It may take months or years to decide what you want to prove.

2. Documentation

Documentation is built upon previous research and insights. We may need to document things from several centuries ago or from recent times. Nowadays, all information is stored in computers and the internet, which makes this step easier than before.

3. Motivation

Motivation is the most important ingredient when we pursue a discovery. Psychologists believe that childhood experiences play a big role. For instance, both Szilard and Turing’s lives were strongly influenced by a book they read before they found their ideas. In the case of Turing, the book Natural Wonders Every Child Should Know was his inspiration. When I was a child, I watched “Donald in Mathmagic Land” and thought it was fascinating. It might have played a big role in my choice to become a mathematician.

4. Ecosystem

A discovery or an idea never arrives on its own. A scientist is never alone and there is a whole ecosystem around them. For instance, at one point in history, Persepolis was the most innovative city in the world. Then it was Paris, and then Budapest. Today, we have the well-known Silicon Valley.

If you are working in a lab, you need to have an atmosphere where people can meet, discuss, and be creative. A good idea comes from teamwork.

5. Constraints

The next ingredient that you need for a good idea is constraints. Without constraints, discovering new ideas is not as likely as when there are. Rigorous findings come about mostly with constraints. One of the most famous problems in mathematics is the Riemann hypothesis, which was tested in thousands of experiments. After those experiments, the proof was only a set of logical rules with constraints. Constraints usually help generate authentic results and artistic quality as in rhyming poems.

6. Intuition

Intuition usually comes together with hard work, yet it is not easy to understand its process.

In addition to these six ingredients, whether one has good fortune or not also plays an important role. It is human condition that things may not come out as we like although we might have done everything necessary.

Henri Poincare says, “Thought is only a flash between two long nights, but this flash is everything.” Yet, a big idea might take years of work. Still, even after working so hard, our knowledge is like a tiny island in an ocean of unknown. We know almost nothing, but this is such a precious nothing, because that that knowledge came out of the richness of the thought of many people and their efforts. There are still so many ideas just waiting for us to discover them.

Mathematics, like many scientific fields, is an infinite universe with an infinite amount of problems waiting to be discovered and solved. Their applications within our world are endless, and the possibilities to use these results to create real change in our world are also endless. One tends to ask, is math a stand-alone, abstract concept while its vast horizons of knowledge explains the intricacies of our universe, or does it imply an infinitely bigger wisdom from which this abstract knowledge rises from and that relates to our existence? We must continue to motivate students to study math so that they may fall in love with it and continue to develop humanity.

References

1. United States, Congress, Chen, Xianglei, and Matthew Soldner. “STEM Attrition: College Students’ Paths Into and Out of STEM Fields.” STEM Attrition: College Students’ Paths Into and Out of STEM Fields, National Center for Education Statistics, Nov. 2013. nces.ed.gov/pubs2014/2014001rev.pdf.
2. Needleman, Sarah E. “Doing the Math to Find the Good Jobs.” The Wall Street Journal, Dow Jones & Company, 7 Jan. 2009, www.wsj.com/articles/SB123119236117055127.
3. “Jobs Rated 2014: Ranking 200 Jobs from Best To Worst.” CareerCast.com, CareerCast.com, 9 Mar. 2017, www.careercast.com/jobs-rated/jobs-rated-2014-ranking-200-jobs-best-worst.
4. “Mathematics Genealogy Project.” Welcome! – The Mathematics Genealogy Project, genealogy.math.ndsu.nodak.edu/
5. Emmanuel J. Candes. “Mathematics of sparsity (and a few other things), August 14, 2014, Seoul. ICM2014 VideoSeries PL3. https://www.youtube.com/watch?v=W-b4aDGsbJk

In this issue of The Fountain, Bruce Parker expounds on the “math wars” among parents, students, and educators. Questioning the necessity of math has been going on for many decades and the range of counter arguments against math have varied from “harmful to thinking for ordinary living” and “nonfunctional and nearly valueless”. So, Parker asks, do we really need to have math in middle and high school, as well as college, curriculums? His answer is an emphatic yes.

Supporting Parker in the need for math, Guvenalp writes that “mathematics can make enormous contributions to other disciplines,” and that “an understanding of the universe is intricately linked to mathematics.” Many people think that a scientific viewpoint and a religious viewpoint are incompatible. Mesut Sahin argues that a scientific mind can be deeply complementary to a religious soul – and vice-versa. “Scientific pursuit demands a personality with a strong dedication and faith in a cause. Those are the traits that are well developed in a person with religious experience,” says Sahin, whose essay is a substantive contribution to the age-old debate on whether science and religion can be reconciled.

Who are Sufis? How different from, or similar to, Muslims are they? The Western experience of Islam has historically been more confrontational than peaceful. Sufism, however, has been welcomed rather warmly despite its origins being deeply rooted in Islam. In his “A History of Sufism for Western Readers” Sait Yavuz explains how Sufi orders came into being and systematized their practices over time. Western sympathy towards Sufism seems to arise from the latter’s emphasis on genuine love rather than the prescriptions of legalistic scholars. Sufism, according to Yavuz, is, “setting aside everything in this temporary world to reach purity, and achieve a state of direct experience of God, in which one can even annihilate himself in God’s Divine being.”

Not much different than Sufis, we all seem to have annihilated ourselves in something these days, but this thing is not as holy as the Divine, unfortunately: the internet. Onyeagolu Tochukwu writes that the virtual world in which we have immersed ourselves has created an illusion of greater connection, yet “a great majority of human relationships have been transferred from the world of reality, where they involve personal encounters between humans, to the virtual world of the internet and other online platforms where personal encounters between humans are either ruled out or vastly different.”

Fethullah Gülen, the author of our lead article, was the 2015 recipient of the prestigious Gandhi King Ikeda Award for Peace. The award ceremony took place at the Martin Luther King Int’l Chapel Crown Nave at Morehouse College, Atlanta, Georgia. Mr. Gülen was awarded due to his “interfaith and cross-cultural dialogue about science, democracy and spirituality and opposition to violence and the misuses of religion as political ideology [which] made [him] the humanizing face of Islam.” The keynote speech delivered during the ceremony by Dr. Scott C. Alexander is in print for the first time in this issue of The Fountain. This piece is a must read for those who research both Mr. Gülen’s ideas, and Hizmet, the global movement of education and dialogue inspired from these ideas.

You know education is always a hot topic. Everybody has a share and interest in education: Parents, educators, government officials, elected public representatives, nonprofit organizations, religious groups, and so on. Besides, there are many aspects of education, such as finance, law, administration, etc. However, do you also think that teachers have long been overlooked regarding their authority in their own classes? Haven’t they been overlooked regarding their professional development, even though they are the backbone of the education system and they need some assistance in their efforts more than ever? It is good if you also think like me. It is even better if you don’t. Thus, you can share your recommendations with me once you see this school where things are done differently than many other contemporary schools. These teachers at this school see themselves as a learning society. They criticized the status quo they were in some time ago, and renewed themselves collectively. I want you to see the product. Let’s start our journey, shall we?

Room 114

We enter through the big doors of the school. Physically, it is not the best school you would ever see, but it can definitely be said that vitality and cleanliness of the halls stand out. The atmosphere is very welcoming; something is different, but it is hard to tell what it is. I point to the very first door on the right hand side. With pride, I ask if you would like to see this classroom. The sound coming from room 114 is not different from the sound that bees make when they work. As we approach the door, I understand that this is Ms. Williams and her eighth-grade students studying math. She welcomes us and tells us, “please come on in; there are some empty chairs over there,” pointing to two empty chairs, and we make our way towards to the back of the class and have our seats. This is a math class, but I don’t see any numbers on the board or equations being discussed. Soon, we understand that they finished the chapter on expressing roots of cubic equations last week, and today they are discussing a person named Omar Khayyam (1044 – 1123 C.E.). Ms. Williams knew that Omar Khayyam was a philosopher, mathematician, astronomer, and poet. She had formed her students into four groups, and assigned each of his professions to a group, so that they could do some research and present their findings to the whole class. I understand what you mean from the way you look at me: you’re sure this is a math class? you seem to be saying. You’re right to ask. Don’t they need to finish a chapter and pass to the next one immediately, and stay focused on the possible topics that are asked at the end-of-year tests?

Let’s hold on to that question; we can ask this after the class is dismissed. The first three groups already talked about his contributions to philosophy, astronomy, and poetry. We feel very fortunate not to miss at least the last group, who talk about him as a mathematician. I can see brightness in students’ eyes, and their excitement to share everything they have learned with their classmates. The first student, Omar who coincidentally happens to share a name with Omar Khayyam, begins. Omar Khayyam was a scholar among many others of his time. He is best known today for his poetry, but his contribution to mathematics was great. He showed how to express roots of cubic equations by line segments obtained by intersecting conic sections… Another student, Margaret continues: his work on algebra was known throughout Europe in the Middle Ages, and he also contributed to calendar reform. The algebra (from Arabic al-jabr) of Khayyam is geometrical, solving linear and quadratic equations by methods appearing in Euclid’s Elements… April inserts that Khayyam also gave important results on ratios giving a new definition and extending Euclid’s work to include the multiplication of ratios.

They continue their discussion, asking how these scholars tried to seek knowledge and an answer to who they were; and the great contributions they made towards peace and understanding among people of different origins. However, I am currently very much impressed and I lose myself in the beauty of these students’ discussion that I have found myself holding my breath and forgot about your presence next to me. It was no different than waking up from a dream when the bell was ringing. Sure, I will introduce you to Ms. Williams. I understand that you want to learn more about the uncommon method she adopted for this class: Ms. Williams, do you have a minute, please?

Of course, she says: “I have some time. It’s break time now.” You ask her about the topic and the way she formed her teaching method: Why would you choose a person from old history to discuss? Why do you want your students to talk about poetry or astronomy or philosophy in a math class? Do you think they know enough math to solve the questions on the end-of-year test? Why did you choose a Muslim scholar? Is this an Islamic or a secular school? Would you… and you realize you have bombarded her with your questions. You halt and smile.

She smiles back and says: “Yes, this is a secular state school. We try to practice secularism in a true sense that we can talk about religion, and bring religious materials to school to study, but we never try to impose these ideas into our students’ minds. They actually study very different kinds of perspectives, and they enjoy it very much. We take each student as a whole. Since our topic was roots of cubic equations, and Omer Khayyam was the father of the idea, you happened to see this example.”

“But why would you touch on different areas of study in a math class?” You say without taking a second breath.

Ms. Williams smiles again. “This is very common in our school. I am sure Mr. Wraga next door will also get into some math in his literature class since he knows my course syllabus. We try to keep our teachings interdisciplinary as much as possible. I receive feedback that kids enjoy it greatly. I understand your concern about the tests. However, testing is the very last thing we think of. Most of our students do well at those tests anyway, and those who cannot do so well are seen to be still successful mostly as artists or musicians…”

There is a long pause and silence now, but hurricanes are happening in our minds. How is it possible? What kind of philosophy is this school following? Is it the principal who leads these teachers like a conductor leading an orchestra? As Ms. Williams answered our questions, more questions kept popping up in our minds. The bell rings again, and the break is over. We thank her for letting us sit in the class. We make our way back to the hall again, and I am already elated with the intelligence of these students. You look at my eyes without saying a word, and smile: Room 115?

For many, math is the most difficult class. Is it because of our own laziness, or is much math too abstract to comprehend? Ali Unver says, “educators who see the beauty at the center of mathematics and can make their students see it that way, are more likely to be able to get their students’ attention and teach them more effectively.” He explains in “Mathematics and the Universe” that mathematics is not only for keeping track of our checkbooks.

Mary and Jesus, peace be upon them, are the subject of a considerable amount of Islamic literature. Ahmet Cetinkaya contributes with a thesis on the question of whether Jesus had any brothers and sisters. Drawing on sources of Islamic tradition and the Bible, Cetinkaya lays out interesting information on the topic, which is also a matter of debate between some Christian denominations.

“Romania – Dar-ul Sulk” discusses how historical truths are sometimes forsaken for the sake of political interests. Victor Nitelea presents an exemplary account of the mutually beneficial centuries-long relationship between the Ottomans and Romania and how the positive image of Turks as a tolerant society was replaced by an image of barbarians. This account also explains how the school curriculum can be manipulated according to political trends at the expense of causing intolerance and conflict between East and West.

“Can I Be Your Baggage?” Barney Zwartz, a journalist from Melbourne asks Abdullah Aymaz, who spoke about the emergence of the Gulen Movement in an international conference organized by the Australian Catholic University back in July. Whose baggage did he want to be? Zwartz was referring to the educational movement started in the 1960s among a small circle of young people like Mr. Aymaz who were inspired by Fethullah Gulen. In this issue, Aymaz shares with us his observations of the conference with some flashbacks to his first encounters with Mr. Gulen and how the movement started to take shape.

Americas gifted population

America has approximately 3 million gifted students, a number that represents 6 percent of its entire student population. The Council for Exceptional Children for the U.S. Office of Education reported that 39 states have legislation on gifted and talented education. These states generally define giftedness in terms of intellectual ability, creativity, leadership, and artistic talent (Zettel 1980).

Mitchell (1984) has reported that most states select gifted and talented students through multiple criteria. However, many school districts still use intelligence tests as the key identifier. Intelligence is measured through established IQ tests, and identifying the gifted is based on those scores. The population of gifted individuals is referred to as moderately gifted (IQ range 130-144), highly gifted (IQ range 145-159), and exceptionally gifted (IQ >160) (Clark 2002). However, using test scores alone is not sufficient to identify the great potential of highly creative students, for other factors need to be considered.

Giftedness traits

If a child displays outstanding traits, it is likely that he or she is gifted. In a study by Rogers (1986), the following characteristics distinguished the 38 gifted from the 42 average third and fourth graders: a brisk learning ability, an excellent memory, a long attention span, perfectionism, a fondness for older buddies, a broad vocabulary, a sense of humor, an interest in books, a knack in puzzles and mazes, maturity, inquisitiveness, perseverance, and intense observation.

Other factors that set gifted students apart from their peers are a rich memory, curiosity, reflectivity, and an openness to experiences. Resnick (1997) identifies six types of gifted students. The challenging student is creative and rebellious. The successful gifted student achieves most of the time and prefers to conform to the environment. The underground type is shy and lacks self-confidence. The gifted dropout student is angry and explosive. The autonomous student is intrinsically motivated and passionate. The sixth type consist of the double-labeled students who are both gifted and disabled and sometimes labeled as having ADD (Attention Deficit Disorder) due to their lack of focus.

Modified programs

Gifted students learn faster, understand more complex issues, and have distinctive emotional needs. So, regular school programs must be modified to accommodate their needs. Gifted students should be provided with acceleration options and various forms of enrichment that broaden the conventional school curriculum.

Organizing a comprehensive program to satisfy gifted and talented students is a complex task. Schools should create such opportunities as acceleration, enrichment, self-paced classes, or advanced classes. Staff members should be educated to identify and provide an appropriate curriculum for gifted students. Differentiated Educational Plans (DEPs), which briefly describe the modified content, activities, and assessment planned for such students and his or her group should be filed for each student.

Once the DEPs have been implemented, students should be encouraged to move through content areas at their own pace. If they master a particular unit, they need to be provided with more advanced learning activities, as opposed to more of the same activity. In other words, burdening gifted students with more trigonometry homework or drills, just because they can do it and need to be kept engaged, is not considered to be part of acceleration or enrichment.

Two very helpful strategies in running optional activities toward enrichment and acceleration are compacting and contracts.

Compacting. Students are provided with enrichment and spend less time working on skills they have already mastered in the regular curriculum. For example, a seventh grade student who has a grade equivalence of above 8 in math is usually uninterested in math classes. The teacher can provide him or her with challenging puzzles, competition questions (such as Math Counts or International Math Olympiads), math project samples, and help the students develop advanced and versatile skills separately from their peers.

Contracts. Students and teachers agree on what students will learn, how they will learn it, how long it will take them to learn it, and how they will be evaluated. A science teacher and a student might agree on a cycle of 6-week periods when the student will complete and present a science project or a portfolio. Contracts should include a calendar, written student and teacher roles, a scoring guide (a rubric), and a product format. Students and parents must join the teacher in the decision-making process while outlining this contract (Parke 1989).

Compacting and contracts do not stop students from attending regular classes. Depending on the student’s characteristics, availability on campus, reports from teachers, and the curricula, a student, for example, might stay in regular science and social studies classes, spend half of his or her time working independently in language arts, and not attend regular math classes at all so that he or she can work with the school Math Olympiad team.

Before each campus decides which format will best serve their students’ needs, they should decide what is applicable and feasible. Special magnet schools, pull-out programs, a gifted school within a school, resource rooms, flexible grouping provision based on definite needs, and mainstreaming are some of the available alternatives (Daniel and Cox 1988).

When gifted students are taught in a traditional slow-paced classroom and subjected to an inadequate curriculum, they do not apply themselves, have low stimuli, and even drop out. If schools do not allow gifted and talented students to flourish, their achievement and motivation will fade in a relatively short period of time. Parents of gifted students may choose to enroll their children in alternative programs, such as home schooling or other gifted and talented schools.

Grouping

Rogers (1991) believes that gifted and talented students should spend the greater part of their school day with others of corresponding abilities and interests. Gifted students benefit from learning together and need to be placed with similar students in their areas of strength so that they can understand their learning differences in gifted classes.

Grouping gifted students cannot be considered elitism or a violation of equity in education. It is an erroneous belief that all students are best assisted in heterogeneous learning environments. There are many grouping methods in which average and below-average students may benefit as well as the gifted do. Rogers (1991) suggests many strategies and grouping options. He recommends a cluster grouping of students if schools cannot maintain a full-time gifted program.

In enrichment clusters, advanced learning experiences and higher-level thinking skills are combined in groups of various graded members. Students and teachers learn authentically, work on real problems, and act as practicing professionals to produce a product or effect a change for a real audience. The clusters work according to an extended and flexible schedule (Renzulli 1997).

Accountability

Prosperous people are accountable to the state and are obliged to pay taxes on what they possess. In return, the state has the duty and responsibility to provide services for these people. Gifted and talented people should feel accountable to God, state, and their nation for how they use their potential (for good or bad) and the extent to which they use it (efficiently or wastefully). They should not enjoy the relaxation of saying: This is the best I can do, for their potential is dynamic and subject to improvement. Pure talent must be accompanied by industriousness, perseverance, and conscientiousness. As Thomas Edison stated: Genius is one percent inspiration and ninety-nine percent perspiration. As a result, a genius is often a talented person who has simply done all of his or her homework.

On the other hand, the state, society, and educational components are responsible for buttressing gifted and talented programs and for using these students’ special abilities instead of wasting them. Gifted and talented people should be identified and helped so that they can assist others.

Another critical component of educating the gifted generation is preventing the developing a corrupt, threatening mob of talented people obsessed with selfishness, evil, or lawlessness and deprived of moral values, ethics, and good intentions. Such an outcome is even worse than wasting the gifted generation’s talents. Notorious dictators, criminals, and militants are all known to be geniuses. We are held accountable for developing a fruitful, qualified, and fully activated gifted generation, one that is decent, just, and ethical and can lead and help the world.

References

• Clark, B. Growing up Gifted. Upper Saddle River, NJ: Pearson Education, Inc., 2002.
• Daniel, N. and Cox, J. Flexible Pacing for Able Learners. Reston, VA: The Council for Exceptional Children, 1988.
• Mitchell, B. An Update on Gifted and Talented Education. U.S. Roper Review 6 (1984): 161-63.
• Parke, B. N. Gifted Students in Regular Classrooms. Needham Heights, MA: Allyn & Bacon, 1989.
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While almost everybody who has access to a computer is somehow involved with the Internet for a variety of personal reasons, scientists and the academic community also use it for their own purposes. Examples are sharing data and information, browsing technical papers, searching for related documents, and posting their research activities to colleagues and other interested parties. Recent developments have proven to researchers and academia, both of which publish large amounts of technical literature, that the day of electronic publishing is close at hand.

The easiest way to publish electronically is to post documents on the Internet by generating Web pages. In this case, however, the user remains a passive recipient of information, for the Internet’s interactive communication ability is not being used. This particular capability of the Web allows a scientist or researcher to generate Web pages with which the user can interact. One example of such interaction is performing mathematical operations over the Internet. This is very useful and powerful, for it allows the user to become an active receiver by gathering the needed information from that particular home page.

For instance, if you visit Rice University’s home page for its Department of Mathematics, you will find several interactive math tools designed to help students taking the Ordinary Differential Equations course.3 These tools are activated by using Java or MATLAB programming language. For example, Figure 1 shows a tool called PPLANE, which draws the magnitude and direction of a differential vector in an x-y plane. PPLANE also graphs the linearization about equilibrium points, and displays eigenvalues, eigenvectors, nullclines, and stable and unstable orbits. The user can change the differential equation’s variables and then run the associated MATLAB code over the Internet. A licensed MATLAB copy in the user’s personal computer is not required, for the MATLAB routine is run on the server and displays the output on the user’s browser. This makes it easy for the student to understand how the changes made alter the equation’s features.

This technique is very efficient and powerful for a student who is still in the learning process. It also suggests that the Internet’s interactive feature will affect the education system and the way courses are taught in the future.

Another home page that contains a wide variety of interactive math tools is found at the Web site for Dartmouth College’s mathematics department.4 Professor Richard Williamson has written about 30 interactive math programs for common scientific problems. These vary from differential equations to heat equation solvers, from Newton’s method of calculating the root of an equation to simulating a swing’s motion. All of these programs are activated by the user’s input parameters, and display the answer in the same manner.

Another interesting and very useful Web site is http://www.integrals.com (see Figure 2). This interactive site allows the user to take any symbolic integral over the Internet. It is provided by Wolfram Research, which also produces the well-known and widely used Mathematics tool MATHEMATICA. After the user enters an expression, the integrator automatically runs MATHEMATICA on the server, integrates the expression, and sends the result back to the user’s browser.5 This site is already helping many calculus students with their homework, and is quite handy for researchers who deal with complex integrals in their everyday research.

Another useful interactive math Web site can be found at the Geometry Center Web page of the University of Minnesota, Science and Technology Center.6 This site offers such interactive math tools as hyperbolic triangles, Lorenz simulation, and interactive proofs of popular theorems. There is also a tool for taking numeric integrals. If you get tried of doing mathematics, you can take a break and play some Tetris games at the same site. In fact, interactive games on the Internet are also a particular type of interactive math tool. A better graphical version of Tetris can be found at http://www.geocities.com/SiliconValley/Pines/522 7/tetris.htm.

All of the above Web pages show that cybermath has found its way onto the Internet, thanks to the Web’s interactive communication capability. It is not difficult to imagine that students in other majors will apply this useful and efficient tool to their own field, thus making the Web even more interactive

FOOTNOTES

1. http://www.flifo.com/and http://www.reservations.com/, respectively.
2. http://mallblvd.net/and http://www.internet.net/index.html, respectively.
3. http://math.rice.edu/.
4. http://www.dartmouth.edu/~rewn/index.html.
5. http://www.integrals.com.
6. http://www.geom.umn.edu/.