"What's wrong?" we all ponder.
"We're not doing something right," bemoan parents.
"The system of introducing math to young children is not effective enough," admit teachers.
And so on...
While the U.S. Department of Education (and authorities worldwide) are undertaking more and more research and attempting to introduce improved and innovative ways of approaching and teaching mathematics, one Bank couple decided to do it themselves, inventing a creative method of introducing young children to arithmetic through the use of fractals.
Behrouz Aghevli, assisted by his wife Hedi (Shahine) Aghevli, developed the Fractal Mathematical Kit using triangles as a visual and manipulative symbol for representing numbers. Mr. Aghevli, Systems Analyst, CODIS, and his wife spent almost two years working on their invention. Mr. Aghevli now says, "This project occupied every spare moment outside our working hours." It all started when their daughter, Aryan, was about 3 years old. As a statistician-he holds a Ph.D. in Statistics from Northwestern University-Mr. Aghevli "always loved and enjoyed mathematics. I was very good at it as a child and wanted my daughter to acquire the same appreciation for math." He and his wife bought and encouraged their daughter to play a few games with some math content at a very early age. "At 3-1/2, she was playing Monopoly with us," he relates with pride. "Of course, she didn't really know the value and amounts of the cards but she would learn the corresponding color for the dollar amount and attempt to transact with us. "Later, when we tried to get Aryan more math games, we found there was really not much to choose from, and we became even more interested in developing something for children," Mr. Aghevli continues. It was while he was plowing through such books as Chaos, by James Gleick, and Godel, Escher, Bach: An Eternal Golden Braid by Douglas R. Hofstadter, that an idea started to formulate. "One day I was doodling and started to draw a large triangle. Then I drew another smaller one inside it and yet another smaller one inside the second, and then the concept clicked. Fractals can be used to represent the decimal system. "
What are fractals? For the economist, one can say, fractals are objects that look similar at the macro and micro level. The use of fractals, in different forms, is relatively new but has found its way into many scientific and mathematical disciplines. In his book, the Fractal Geometry of Nature, (1976), B.B. Mendelbot defines fractals as shapes that tend to be scaling, that is to say, "the degree of their... fragmentation is identical at all scales."
Dr. Super-as Mr. Aghevli is called by his students-says the kit "provides a manipulative and visual instruction tool complete with simple semantics for counting, addition, subtraction, multiplication, division ... The kit can also be used for playing a number of simple, yet fascinating, games ... The kit is composed of manipulative components assembled from solitary unilateral triangles of different sizes ... It includes work books with step-by-step instructions on the different uses, along with a complete set of large base triangles and a number of smaller triangles representing the numbers 1, 2 and 3, all made out of hard plastic. Mr. Aghevli has also developed a visual calculator using his fractal numerals in LOGO, which complements the kit.
The philosophy behind the kit is to teach arithmetic at the one digit level based on symmetric and decomposable shapes. The one digit prime numbers, 1, 2, 3, 5 and 7, are formed by assembling small triangles into special symmetric configurations. The numbers 4, 6, 8, and 9 are then represented by an arrangement of multiple copies of the numbers 1, 2, or 3. The number 10 is made up of 10 ones assembled into a large triangle called a "Super." These special representations of the numbers 1 through 10 are called the Fractal Numerals. [see Figure 1 in box]
The mathematical manipulations using the kit to master one digit arithmetic are so limited that kids learn them quite fast. Not only is the method visual and manipulative but it also has a linguistic simplicity. The two basic properties of the fractal numerals are well displayed in Figures I and 2 (see box). In Figure 1, the fractal numerals that add up to 10 are placed on top of each other. Notice these pairs will fit together like pieces of a puzzle to make a Super. For example, looking at the figure from left to right, you can see that 1 and 9 make a Super, 2 and 8 make a Super, 3 and 7 make a Super, 4 and 6 make a Super, and 5 and 5 make a Super.
Figure 2 shows how the numbers 4, 6, and 9 are made up, respectively, of 2 twos, 2 threes (or 3 twos), and 3 threes. Using these relationships for one-digit numbers, he believes the children discover a new world. "The one-digit arithmetic is mastered with ease and the child can progress to the more complex multi-digit arithmetic," he maintains.
Figure 3 shows how counting is done with the fractal numerals. The language is simple, the numbers are not abstract, and the visual and manipulative framework enhance a child's concept of numbers. Notice that one, one Super and one Super-supers... all have the shape of a single triangle, which is getting bigger and bigger. The same would be true for all numerals; for example, 9, nine Supers and nine Super-supers, and so on, all look like a 9 that is getting bigger and bigger,
Mr. Aghevli explains: "The kit is not meant to replace traditional mathematics, but it will certainly add to it since young children need to learn only 1 1 names and shapes to master arithmetic." He adds, "A favorite question in the first grade is "What is infinity?" The answer is Super-super-super... After a child has repeated super for a minute he notices that he will have to say Super forever to get to infinity!" He concludes, "with Super math, children have an easy time understanding the meaning of orders of magnitude and large numbers both visually and linguistically."
While they were in the process of patenting the Fractals Mathematical Kit (Patent No. 5,076,793) in December 1991, Mr. and Mrs. Aghevli led a pilot project during the academic years 1989-1990 and 1991-1992 using the method at Aryan's school in Falls Church, Virginia. The program was called "Super Math Hour." Mr. Aghevli, with special permission and blessing from the Bank, taught one of the sessions for the first and second grades while his wife taught another first grade class once a week. Most of the parents, teachers and students who were part of the pilot found it useful and exciting. Says Ruth B. Haggstroms, second grade teacher in 1990-91, "It was truly a pleasure to see students enthusiastic about mathematics at a time when math scores have been dropping nationwide ... the Super Math Hour kindled a desire for knowledge while eliminating the fear of mathematics." Another teacher, Laverne E. Gable, first grade, 1989-90, said, "The Fractal kit was a valuable aid to my teaching that year. The system is a unique and exciting tool for learning math. It not only works for the beginning math students, but its principles have implications for more advanced work." "It's been an interesting experience to watch the children grow in their interest and understanding of mathematical concepts through the Super Math Project," commented Headmaster Jay Roudebush. Among the students in the pilot class was Ricardo Jesus, son of Bank staff member Susana B. Jesus. Mr. and Mrs. Jesus praised the program saying, "Ricardo has shown a keen interest in math since being introduced to Super and enjoys solving problems and calculations." Adds Janie S. Price, another parent, "We've watched our son Jason's confidence grow as he worked through this project, and I have seen his attitude change from one of apprehension-and even fear to one of wonder and excitement."
The testimonials are many and rewarding, say the Aghevlis. But a new assignment and project at the Bank has occupied all of Mr. Aghevli's time since April 1991. He is currently working on the Query Facilty for Task Manager's Workstation-a PC tool for task managers to retrieve Bank data from various main frame databases. "We have developed a query product to enable all Bank staff to access and retrieve data from all Bank databases through the same user-friendly PC interface," he explains. "This project has occupied and will continue to occupy my energy and concentration for the next several months. The pilot for Task Manager's Workstation is already on nearly 300 desks in the Bank."
"But I'm not letting our Super Math project fade totally into the background," he promises. He continues to conduct classes at The Vision and Conceptual Development Center in Washington, D.C. The Center, founded and directed by Dr. Harry Wachs (former Professor at George Washington University and author of the book Thinking Goes to School), conducts programs for evaluation and growth of visual/conceptual development to encourage intellectual growth from early childhood through adulthood. Among its many services, the Center offers a special mathematics program called "Math Made Magical" designed for gifted preschoolers as well as for children who have difficulty understanding arithmetical concepts. It is here that their Fractals Kit has found a special niche and Mr. Aghevli, who has been an honorary staff member since 1990, teaches a class Saturdays.
As time permits, he continues to give his invention limited exposure and hopes one day to be able to commercialize it as a game product. "My wife has encouraged me all along to lean more toward the creation of games." He has given hands-on workshops at the Math Education Center at the University of Maryland, to the mathematics teachers of the D.C. Public School System, and the Visual and Cognitive Unit of the Reading Center at George Washington University.
What he would ideally like is to find a sponsoring organization so he could refine and distribute his invention to children worldwide. He smiles and adds, "Why not the World Bank? One must always think of the impossible to make it possible."
Editor's note. If you want more information or to see a demonstration of the kit, please contact Behrouz Aghevli, (202) 473-6838.
Source: Bank's World, January/February 1995, Volume 11/12Number 12/1, Pages 13-17