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- 1. Fractals: A Conceptual Approach to Teaching and Learning Mathematical Tasks Cynthia A.Yakushev Texas A & M University
- 2. Table of Contents The Concept……………………………………………………………………..……1 Historical Background……………………………………………………………….10 Objectives……………………………………………………………………….……15 NCTM Principles and Standards……………………………………………….…….18 Learning Theory……………………………………………………………………...20 Technology…………………………………………………….……………………..21 Mathematical Modeling……………………………………………………………...22 Problem Solving……………………………………………………………………...24 Activities……………………………………………………………………………..25 Reinforcement………………………………………………………………………..25 Assessment…………………………………………………………………………...26 References……………………………………………………………………………32 ii
- 3. For thirty-years, fractals have been studied. Articles about fractals in various disciplines have been written and published. Technology has developed numerous software applications using fractal knowledge to advance the fields of science, mathematics, and the arts. While educational systems adhere to specific principles and standards which attempt to develop populations of students’ knowledge, skills, and competency for daily life, enough time has passed that educational systems can comfortably introduce the concept of fractals to all students from prekindergarten through grade 12. Revision of policies to educate comes from experience and reason, so the question to teach fractal knowledge must be justified. It will be my attempt to justify simply that fractal knowledge can be used as a conceptual approach to teaching and learning mathematical tasks. The concept of fractals is a relatively new body of knowledge in the discipline of mathematics. Many people have never heard of the word fractal. Those who have heard of this word sometimes cannot explain exactly what it means. Many people have seen the beauty of computer generated fractal art, and yet, do not realize the higher level of mathematics behind their creation. In the natural world, we see fractals everywhere. Much of the time, not realizing that what we look at, in nature, is actually structured the way we think it is structured. Fractals are real concepts that structure everything inside of us, around us, and beyond us. Fractals are ubiquitous; they are everywhere. The fractal concept can teach us about real world mathematical systems, which can then help us to understand how the world operates on a mathematical level. Before now, we may not have clearly understood enough about the fractal concept to recognize that most everything is made of fractals. The word “fractal,” comes from the Latin verb, “frangere,” which means to ‘break into’ -1-
- 4. or to ‘create irregular fragments.’ In 1975, this term was coined by the Yale University professor, Benoit B. Mandelbrot, from the term “fractus,” a Latin adjective, while looking through his son’s Latin dictionary. For Mandelbrot, fractal means “shattered, yet retaining a certain symmetry,”1 He describes fractals as “any image that is infinitely complex and self-similar at different levels.”2 The term self-similar is always associated with the description of fractals. It is an intrinsic property of natural and mathematical fractals. Self-similar is a term to explain a comparative process of reducing or magnifying sections of fractals in two different areas that result in an exact image of the two. A geometric like term would be reflection, or mirror-image. Fractals have always existed; they’ve existed for aeons. In the natural world, the environment, our body, the Universe, and phenomena, fractals have existed unaware to us until recently. These are the “geometric shapes that are very complex and infinitely detailed.” 3 Also, “they are recursively defined and small sections of them are similar to large ones. They are related to chaos because they are complex systems that have definite properties.”4 As a newfound body of knowledge of the discipline of mathematics, fractal research has evolved globally and exponentially since the 1970’s with profound conceptual applications in the fields of mathematics, chemistry, space science, earth science, engineering technology, environmental science, physics, computer science, medicine, and the arts. The “chaos/fractals graph”5 (see Fig. 1) below expresses the exponential growth of fractal research since the relatively new use of computers in the early 1970’s. It shows that by the 1990’s, up to 4,000 articles had been written and published around the globe. -2-
- 5. Fig. 1 A graph showing the number of articles written and published about chaos and fractals between 1974 and 1990. I will show you a graphic selection of fractal examples, from the geometrically simple like the Sierpinski Triangle and Koch Snowflake which can be introduced to elementary-age students, to the mathematically complex like the Mandelbrot Set and Julia Set which can be introduced to secondary-age students. This example is the first system of the Sierpinski Triangle (See Fig. 2). It is named after the Polish mathematician, Waclaw Sierpinski. This is a simple geometric fractal which was introduced in 1916. You will see all of four triangles. One white upside-down triangle bordered by three black right-sided triangles that all fit inside one large triangle perimeter. Ignore the white triangle, for the moment. Allow your focus to be upon the three black triangles. These triangles are where the dimension of fractals will recur and reveal the property of self-similarity -3-
- 6. in reduction or magnification. Fig. 2 1st system = 4 triangles This is the second recursion of the Sierpinski Triangle (See Fig. 3), and explains the mathematical task of addition. Recursion is another word for repetition, or reappearance. Each black triangle now has one white upside-down triangle within it, and the original large one white upside-down triangle from the first system is left in tact. Fig. 3 2nd recursion = 13 triangles The property of self-similarity is revealed in scaling the recursion (or reappearance) of the original triangle, three times, inside the one construct of the triangle’s perimeter. -4-
- 7. Do you see the fractal pattern? The fractal pattern reduces to infinity, another inherent property of a true fractal because the original construct is independent to all changes. This next one is the third recursion of the Sierpinski triangle (See Fig. 4). I reiterate, each black triangle now has one white upside-down triangle within it, and the original large one white upside-down triangle from the first system, and now, the second recursion is left in tact, and the original construct has remained consistent. Fig. 4 3rd recursion = 40 triangles This is the fourth recursion of the Sierpinski Triangle (See Fig. 5). This fractal shows an increase from four triangles to one-hundred-twenty triangles through the conceptual approach of reduction by scaling inside the original construct. -5-
- 8. Fig. 5 4th recursion = 120 triangles Although this fractal example is elementary, “it shows what we call linear self- similarity,”6 and the intellectual leap from four triangles to one-hundred-twenty in just four changes is a profound approach to restructuring student’s ordinary and past way of thinking about mathematical tasks. This example introduces a new way of visualizing real world constructs, and situations through fractals by “describing nonlinear chaotic systems or even totally random processes.”7 This profound conceptual approach alters student’s reality of perception in the present world from their ancestor’s perception of the world. It also enhances a student’s state of consciousness toward themselves, and is a concrete revelation that removes them from ignorance and innocence. Fractals are everywhere in the real world. Fractal knowledge is not only about basic mathematical facts of operation, it is relevant to superior mathematical operations. Our next simple fractal example (See Fig. 6) is the Koch Snowflake, (pronounced coke). It is a geometric fractal using triangles. It is named after Helge von Koch, and introduced in 1906. However, this example places the triangles on the outside of the double and overlapping triangle’s perimeter. It illustrates iteration, another fractal property that means repetition of -6-
- 9. reduction or magnification. Fig. 6 The Koch Snowflake Fractal One Fractal Two Fractal Three Fractal Four 1st system = 6 points 2nd system = 18 points 3rd system = 54 points 4th system = 162 points 3(2) = 6 3(3(2)) = 18 3(3(3(2))) = 54 3(3(3(3(2)))) = 162 Two triangles are placed atop the other, one right-sided, the other upside-down. If I split the first system in half anywhere along the six vertices of symmetry, I will get three points on either side. I have 3(2) = 6 triangle points. This is the first fractal, and explains the elementary operation of multiplication. The next iteration, I add two triangle points to the original six triangle points. Now, I have 3(3(2)) =18 triangle points. This is the second fractal. The third iteration repeats the same process off the second iteration and the first system. I now have 3(3(3(2))) = 54 triangle points. This is the third fractal. Also, two additional points are introduced onto the six vertices. The fourth iteration repeats the same process off the third and second iteration, and the first system. Therefore, I now have 3(3(3(3(2)))) = 162 triangle points. This is the fourth fractal. -7-
- 10. The iteration of reduction goes to infinity. The mathematical formula for these fractals follow this pattern: F(x) F (f(x)) F (f (f(x))) F (f (f (f(x)))) The Mandelbrot Set (See Fig. 7) follows this formula: F(x) = x^ (2 + c) It is a simple nonlinear function, and it is recursively defined by its formula. The “time series graphs”8 below show the plots expressed for c. Three of these plots are normal mathematical functions that are deterministic. However, the last function graphing c as –1.9 reveals chaos. It also shows lines of self-similarity, like a natural coastline; and it is how he gained the insight of applying these nonlinear functions to real world situations like the English coastline, and turbulence. Time Series for c = -1.1 Time Series for c = -1.3 Time Series for c = -1.38 Time Series for c = -1.9 Fig. 7 Four Graphs Plotting the Mandelbrot Set Formula for c.
- 11. -8- The Mandelbrot Set formula and the Julia Set formula are mathematically complex, and can create intricate computer generated images of fractals of great beauty. Here is an example of the “Mandelbrot Set Fractal”9 (See Fig. 8), and the “Julia Set Fractal”10 (See Fig. 9). Traditionally, the Mandelbrot Set image will always have the two circles, small and big, next to each other. The Julia Set, traditionally, will always be swirls like lightning bolts or octopus tentacles. These are the images that can represent geometric power and vastness which can then be applied to imaging in medicine, earth science, space science, physics, mathematics, and the arts. Fig. 8 Mandelbrot Set Fractal -9-
- 12. Fig. 9 Julia Set Fractal Historically, the mathematical formula for the original Mandelbrot Set was firmly established in the 13th century by a German monk, “Udolphus von Aachen” 11. He lived around 1200A.D. to 1270A.D. and was a copyist, poet, and theological essayist. He wrote a poem in Latin called “Fortuna Imperatrix Mundi” or “Luck, Empress of the World.” Seven-hundred-years later, a retired professor of combinatorics, Bob Schipke, took a visit to the Aachen Cathedral. He was astounded by finding an illuminated manuscript of a 13th century carol “O Froehliche Weinacht” that had a Star of Bethlehem that looked strange. At the time, what he accidentally identified was the computer generated icon of the Mandelbrot Set. He investigated further with the aid of a University of Munich historian, Dr. Antje Eberhardt. Schipke got hold of ecclesiastical archives, and found the document of Codex Udolphus in -10-
- 13. Latin, and signed by Udo himself! In Udo’s first chapter, “Astragali,” (Dice) it detailed his research on probability theory through simple rules to add and multiply probabilities. His research was driven by helping other clergy with winning strategies; he had devised strategies for several card and dice games for making money. In another poem, “Fortuna et Orbis” (Luck and a Circle) it shows Udo determining “the value of pi by scattering equal sticks on a ruled surface, and then counting what proportion lie across the lines.”12 In the 18th century, “a French mathematician, Comte de Buffon (1707-1788)”13, had been credited for the same experiment, as it became well known as the Buffon Needle Technique. Udo’s intensive method led him to the approximation of 866/275, which is 3.1418, and it confirms the Biblical value of pi as 3. His contemporary, Leonardo of Pisano, known as Fibonacci, had also quoted 3.1418 as pi. In Udo’s beautiful poem, he visually alludes to his math research through natural associations. He writes in his choral, “O Fortuna,”14 “Luck, Like the moon, Changeable in state, We are cast down, Like straws upon a ploughed field. Our fates measuring, The eternal circle.” -11-
- 14. His allusion to his math discoveries certainly points to the Buffon Needle Technique. Schipke looked into the final chapter, “Salas” (Salvation), and the radical concepts that he read not only expressed the famous fractal formula, the Mandelbrot Set, but clearly gave the reason why Udo was finally driven out of his math research and thus placing fractals on the shelf for the next 700 years. Udo talked about how numbers applied to individuals could determine who goes to heaven and who goes to hell. Naturally, his superiors could see how Udo was using his research to pigeon-hole the Almighty down to a determining value, and believing that he knew God’s ultimate plan for mankind. This information could never be calculated or considered credible to any of the church’s hierarchy. But Udo’s intelligence and ego had tempted him into the illusion that his research doors were open forever. Udo had devised a method considering each human body to have two parts, the profane “profanus,” and the spiritual “animi.” He came to final conclusions by manipulating these parts with pairs of numbers. He had, in reality, created the rules for complex arithmetic, whereby the two parts corresponded to real and imaginary numbers. His “Salas” method was allegorically descriptive of the Mandelbrot Set, and it was completed, and it had been illustrated exactly to the same image made today by modern computers. Unfortunately, the last page of “Salas” sadly reads, “that on pain of excommunication I must lay down my dice and my numbers. I have seen into a realm of heavenly complexity, and my heart is heavy that the door is now closed.”15 Opening the fractal research doors seven-hundred years after is France’s own, Gaston -12-
- 15. Maurice Julia. He developed the theory of the modern dynamic system, which we now call the Julia Set. He was just 25-years-old when he wrote Memoiré sur l’iteration des fonctions rationelles (Report on the iteration of rational functions). He is well remembered for his appearance. A combat injury during World War I left him badly scared for life on his nose, and he constantly wore a black leather patch over it until he died. He continued much of his studying between operations on this injury in the hospital. Nevertheless, he published his book in 1918 which described the function J(f), whereby z is a complex number for which the n-th number of this sequence f^n (z) stays equal, and n grows to infinity. His book won the Grand Prix of l’Academie des Sciences. It was in 1925 that, during seminars, H. Cremer described the visualization of the Julia Set. All the work and studies were quietly shelved, up until the early 1970’s when, the mathematics maverick, Benoit B. Mandelbrot went to work on computer experiments of “nested” images. He is one of the most famous of these fractal explorers. He was born in Warsaw, Poland in November of 1924 to a Jewish family from Lithuania. He lived in France in 1936, and by 1944, he was a student of Gaston Maurice Julia. “He is a pioneer of the chaos theory, and he conceived, developed, and applied fractal geometry, which he used to find order in erratic shapes and processes.”16 In the 1960’s, through experiments with raster-based computer display technology at the Massachusetts Institute of Technology (MIT), he produced “nested” pictures that were considered rare and pathological examples with little or no significance. He identified those nested shapes in many natural systems, and in several branches of mathematics. In 1961, he published a series of studies on the fluctuations on the stock market, the turbulent motion of fluids, the distribution of galaxies in the universe, and on the irregular shorelines of the English -13-
- 16. coast. “In 1975, he developed a theory of fractals that became a serious subject for mathematical study.”17 His emphasis on the importance of fractals in natural systems is now well established in science and mathematics. His book on fractals, Les Objets Fractals (1975)18, paved the way to rapid research on the subject. And he has been awarded the Barnard Medal for Meritorious Service to Science, the Franklin Medal, the Alexander von Humboldt Prize, the Nevada Medal, and the Steinmetz Medal. He currently works at IBM’s Watson Research Center and is a professor at Yale University. Thus far, I have explained to you what a fractal is, who coined the term and who the main players are behind the discovery. Also I have explained where fractals are found in our lives and environment, and when they came into existence and realization. Now, I will explain my goal with this research. Beyond the reasons that I wish to educate myself, and I believe wholeheartedly in self-teaching, I pose this question to justify why the educational systems need to be involved: why should we teach fractal knowledge, and how can we learn of their usefulness to us as students of mathematics? Consider that Mandelbrot was largely self-taught, and Udo von Aachen worked on his research single-handedly because the knowledge is that important. In our world, our daily lives operate consistently on various levels of mathematics, and this new body of knowledge is applicable to the operations of human and environmental systems. If a population on the globe is knowledgeable about fractals, and the rest of the population is not knowledgeable, then this one deficiency of math knowledge within our global educational goals promotes mathematical illiteracy, through ignorance, neglect, incompetence, and apathy toward the usefulness of this knowledge. -14-
- 17. Therefore, my objectives with this research are: 1. Teach fractal knowledge from prekindergarten to grade 12. 2. Teach the fractal patterns from prekindergarten to grade 12, so as to recognize fractals in their various forms in nature and mathematics. 3. Teach the fractal geometry language and specific terminology from prekindergarten to grade 12. 4. Teach about the fractal connections to the world in physical, social, and mathematical systems from prekindergarten to grade 12. 5. Teach about the fractal applications presently used in physical, social, and mathematical models from prekindergarten to grade 12. The standards of how these objectives can be achieved are: 1. Learn the concept of fractals through basic knowledge visually from two-, and three- dimensional shapes, computer generated art, and hands-on experiences with manipulatives and activities. 2. Learn by observing natural objects like leaves, internal body parts, land, rock, atmosphere and water formations, etc. 3. Learn from visual examples, which identify the terms of fractal, iteration, self-similarity, recursion, bifurcation, parameter, chaos, order, linear and non-linear systems. 4. Learn from the sciences outside of mathematics how a panorama of fractal applications are used like imaging in medicine, space and earth science, computer science, and art. 5. Learn from books, Internet resources, hands-on experiences, and guest speakers how we -15-
- 18. can use fractal applications to help us interpret the theory of chaos, and self-organizing criticality in human physical systems, social systems, and abstract mathematical models. A great place to start is within the educational system. The National Council of Teachers of Mathematics (NCTM) is an “international organization of teachers and other professionals committed to excellence in mathematics teaching and learning for all students.” 19 In 1997, they began the Standards 2000 Project which articulated their principles and standards for the world. Their six NCTM “Principles”20 are: 1. The Equity Principle: Excellence in mathematics education requires equity—high expectations and strong support for all students. 2. The Curriculum Principle: A curriculum is more than a collection of activities: it must be coherent, focused on important mathematics, and well articulated across the grades. 3. The Teaching Principle: Effective mathematics teaching requires understanding what students know and need to learn and then challenging and supporting them to learn it well. 4. The Learning Principle: Students must learn mathematics with understanding, actively building new knowledge from experience and prior knowledge. 5. The Assessment Principle: Assessment should support the learning of important mathematics and furnish useful information to both teachers and students. 6. The Technology Principle: Technology is essential in teaching and learning mathematics; it influences the mathematics that is taught and enhances students’ learning. This starting point for a global commitment to excellence in inculcating teaching mathematics well introduces all students in all countries to the basic knowledge, skills, and -16-
- 19. competence from this new body of knowledge that is relevant to promoting an increased understanding of the mathematical world around them. The NCTM “Standards”21 (an overview) are as follows: 1. “Number and Operations: Instructional programs from prekindergarten through grade 12 should enable all students to-- • Understand numbers, ways of representing numbers, relationships among numbers, and number systems; • Understand meanings of operations and how they relate to one another; • Compute fluently and make reasonable estimates. 2. Algebra: Instructional programs from prekindergarten through grade 12 should enable all students to – • Understand patterns, relations, and functions; • Represent and analyze mathematical situations and structures using algebraic symbols; • Use mathematical models to represent and understand quantitative relationships; • Analyze change in various contexts. 3. Geometry: Instructional programs from prekindergarten through grade 12 should enable all students to— • Analyze characteristics and properties of two-and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships; -17-
- 20. • Specify locations and describe spatial relationships using coordinate geometry and other representational systems; • Apply transformations and use symmetry to analyze mathematical situations; • Use visualization, spatial reasoning, and geometric modeling to solve problems. 4. Measurement: Instructional programs from prekindergarten through grade 12 should enable all students to— • Understand measurable attributes of objects and the units, systems, and processes of measurement; • Apply appropriate techniques, tools, and formulas to determine measurements. 5. Data Analysis and Probability: Instructional programs from prekindergarten through grade 12 should enable all students to— • Formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them; • Select and use appropriate statistical methods to analyze data; • Develop and evaluate inferences and predictions that are based on data; • Understand and apply basic concepts of probability. 6. Problem Solving: Instructional programs from prekindergarten through grade 12 should enable all students to— • Build new mathematical knowledge through problem solving; • Solve problems that arise in mathematics and in other contexts; -18-
- 21. • Apply and adapt a variety of appropriate strategies to solve problems; • Monitor and reflect on the process of mathematical problem solving. 7. Reasoning and Proof: Instructional programs from prekindergarten through grade 12 enable all students to— • Recognize reasoning and proof as fundamental aspects of mathematics; • Make and investigate mathematical conjectures; • Develop and evaluate mathematical arguments and proofs; • Select and use various types of reasoning and methods of proof. 8. Communication: Instructional programs from prekindergarten through grade 12 should enable all students to— • Organize and consolidate their mathematical thinking through communication; • Communicate their mathematical thinking coherently and clearly to peers, teachers, and others; • Analyze and evaluate the mathematical thinking and strategies of others; • Use the language of mathematics to express mathematical ideas precisely. 9. Connections: Instructional programs from prekindergarten through grade 12 should enable all students to— • Recognize and use connections among mathematical ideas; • Understand how mathematical ideas interconnect and build on one another to produce a coherent whole; • Recognize and apply mathematics in contexts outside of mathematics. -19-
- 22. 10. Representation: Instructional programs from prekindergarten through grade 12 should enable all students to— • Create and use representations to organize, record, and communicate mathematical ideas; • Select, apply, and translate among mathematical representations to solve problems; • Use representations to model and interpret physical, social, and mathematical phenomena. The NCTM Principles and Standards is a strong basis for enabling educators and other professionals concerned about the issue of meeting the challenge demanded for excellence in teaching mathematics well. It takes time, and it takes constant assessment, reevaluation and changes to stay ahead of the obstacles and problems that naturally occur in all populations. Now, how do we begin in practice? The underlying theory of learning about fractals is imbedded in geometry. The actual math behind the fractal images in algebra is called combinatorics, and uses the Lavaurs’ algorithm. This algorithm is a “method for deciding which pairs of rationales to connect, and the abstract Mandelbrot Set is obtained by drawing arcs between these pairs and collapsing each arc to a point.”22 Since fractal knowledge is mathematically advanced, we can offer students elementary geometric knowledge from which to build up to a deeper understanding of fractal geometry. Elementary students must be taught fractal knowledge in concrete ways, tangibly and visually. -20-
- 23. We can teach the iterated functions systems by introducing the “geometry of plane transformations, sequences, convergence, and even basic ideas as ratio and proportion.”23 From that knowledge, students can advance to fractal dimension by introducing “logarithms, exponents, scaling, and linear regression.”24 For example, by adding block fractals, they can learn “computing lengths, areas, and volumes, and the geometric series.” 25 Later, students can learn the Mandelbrot Set by teaching them complex arithmetic, complex iteration, sequences and convergence.”26 Follow that up with graphical iteration, which will help them learn “composition of functions and their domains and ranges,”27 visually. Eventually, try showing Pascal’s triangle by explaining to them how the triangle operates with “arithmetic in different bases.”28 The students can use the aid of computers to generate fractal graphics or ideas. The technology behind fractals began in the early 1960’s with raster-based computer display technology. From there, in 1987, the English mathematician, Michael F. Barnsley, developed the “Fractal Transform™ which automatically detects fractal codes in real-world images.”29 Presently, there exists a large collection of computer programs specifically for creating fractal images for scientific applications and computer generated art. From the most to least “acquired, tried, and/or in use is as follows: FractInt, UltraFractal, XenoDream, Tiera- Zon, Fractal Explorer, Sterling Ware, QuaSZ, Fractal Zplot, Mind-Boggling Fractals, Apophysics, Cubics FraSZle, Crocus, Kai’s Power Tools, QuatPOV, ChaosPro, Bryce, WinFract, Flarium24, XfrantInt, POV-Ray, DynaMaSZ, Fractal ViZion, Vchira, Xaos, Fractal eXtreme, GrafzViZion, Iterations, Chaoscope, Ktaza, TieraZon2, Xenolab, InkBlot Kaos, The GIMP, Aros Fractals, Quat, Atriatix, Flight From Fractal, Fractal Forge, Quetzal, Dofo-Zon, -21-
- 24. SpangFract, Firefly, Fractal Projector, Hydra, Pod ME, Root IFS, Fractal Vision for Windows, Kaos Rhei, QS Flame, Alien Lab, Dust Fractals, Visions of Chaos, Chaos: The Software, Fractal Domains, Fractal Orbits, Lparser, L-System, Mand2000, and ManPWin.”30 These are just 60 computer programs to create fractals, but there are actually close to 400 programs in total. A great way to get going with programs is to understand how they work, and why they are needed. Let’s look at the relationships to real world math modeling from basics. The geometry exploration software (GES) can give the “length of a segment, the degree measure of an angle, and the area of a polygon.”31 It also allows the user to repeat the construction and distort it, and its advantage is its speed and efficiency to model geometric constructions. There are numerous real world examples to share with students how fractal programs are used a research tools. One example, the Amalgamated Research Incorporated uses fractal technology that is “related to structures found in nature such as tree branching and river distribution systems.”32 For mathematical modeling, let us take another look at the Koch Snowflake. The elementary construction of this diagram shows that the sides of the equilateral triangle continues to divide into thirds. This can show a relationship of self-similarity in natural systems. Let the fractal dimension be “D = log 4/log3”33. The snowflake maintains an infinite -22-
- 25. perimeter while enclosed within a finite area. How is this possible? The answer from dividing log 4 by log 3 is the irrational number 1.261859507. It is an irrational number that goes to infinity and therefore, proving that within a finite area, a construct can be reduced to infinity. We can check this by figuring log (4) = 0.60205 and log (3) = 0.47712. Now, simply divide 0.60205 by 0.47712 and you will get 1.26184. If you raise base 10 by 1.26184, you get 18.27468. The inverse through log (18.2746), returns to 1.26184. Cynthia Lanius, of Rice University, writes that we can study of area of these triangles and put the data on tables to study further. She also adds that students can study terms like “similar, and non-similar.”34 Another elementary fractal geometry model is spatial visualization. For this you can use 1-centimeter grid paper35 to mark out patterns of squares, color a 2 x 2 square section purple. Then, on top of that section, color a 1 x 2 section pink. Now, rotate the grid paper, and on the edges, repeat the process until there is no space left on the paper. Then look at the paper as a whole, and you will have a fractal pattern that shows iteration and self-similarity. Students can develop a representative mathematical language to communicate expressions and equations from these drawings. I made this drawing, at the end of this paper, and students can do the same. Questions can be asked about patterns, or about how many 2 x 3 rectangles can be found in 12 x 12 area? Ask them what is the fraction of purple and pink in a 2 x 3 rectangle? Young students can be taught about plane figures visually through shapes of buildings, trucks, trains, houses, windows, sidewalks, etc. They enjoy identifying the concepts of shapes like a triangle seen in the real world as a “yield sign,” circles seen as traffic lights, and hexagons seen as stop signs. And they learn that a quadrilateral is the shape of a kite. With this knowledge -23-
- 26. as a base, it is not difficult for them to problem solve in distinguishing the differences between circles, triangles, squares, and rectangles. They can prove with reasoning by creating on a dot grid paper the shapes of triangles, rectangles, and squares. Young students can also be taught symmetry with pictures of houses, bugs, geometric shapes, and patterns. Teaching young students about fractals is not a far leap from early geometric knowledge already required of them, and presently taught to them. We need just go a step further to introduce the fractal concept and its approach to learn mathematical operational tasks. Although fractals are everywhere in the world, it does require the knowledge of higher mathematics to utilize it for real-world problem solving in medicine, space science, earth science, chemistry, and mathematics. Applications of fractal math models to make predictions use its “key characteristic with systems at a critical point”36 and this application overflows into the theory of chaos. For example, through research, in the beginning with just a pile of sand, the critical point for an avalanche was discovered. Remarkably, observations from these repetitive experiments helped advance studies with the stock market. Its movements showed the world that economists had been old-fashioned, and looked at market systems all wrong. A renewed attitude change resulted, not one of policy change, but one that would make us prepared for changes and perhaps financial disasters in the future. It had been proven that complex systems would drive themselves to a stable critical point. Per Bak, a physicist, called this “self-organizing criticality.”37 Medical research welcomes the application of fractals in their working domain. When life gets tough and we get extremely scared or extremely sick, their studies have revealed that our -24-
- 27. heart does a very remarkable thing to keep it, and we, going on in life. When we move, all of the muscles in our bodies, give out, get tired, and leave us with muscle pain. However, the heart, with its regular rhythm, is not really regular at all. The electrical activity in its sinoatrial node changes intervals when it contracts from different reactions throughout the day. Therefore, our heart is the precious muscle of great loyalty that never tires. It was through the study of “electrocardiograph analyses that revealed the delay process is in fractal form”38, and in contrast, a sick heart does not reveal fractal patterns. These applications of fractal research in the medical field use higher levels of mathematics in geometry, algebra, and computer science. We can introduce these applications to young students, and teach them how to see fractals everyday. Activities to teach fractals can be learned in fun ways for children, just look for exercises that are basic in knowledge of fractal geometry or lessons to learn about symmetry, self- similarity, iteration and recursion. Any activity to teach fractal knowledge must comply with the intrinsic properties of recursion and iteration which is confined to one construct. The following example, “Pascal’s Triangle and its Relatives,”39 is fun and easy, and gives knowledge in steps that students can build upon to go further in their math skills and competency. A advanced level for learning fractals can be learned by the next example, “Block Fractals.”40 This example explains the increase of blocks in symmetrical form, and explains the numerical representation of its mathematical expression, and the equations. Moreover, the variables defined are relatively easy to understand, like “a” for area, and “v” for volume. The reinforcements for teaching fractals can be as easy as asking questions for comprehension. Basically, asking students simple questions about the concept like: 1. What is a fractal? -25-
- 28. A: It is a geometric shape 2. What does a fractal look like? A: It can look like any shape that has symmetry, and little parts within big parts that look alike. 3. Where do you find fractals? A: They are everywhere, in our bodies, the planet, and the universe. You can use “Bloom’s Taxonomy”41 of questions for knowledge, comprehension, application, analysis, synthesis, and evaluation. These types of questions for assessment continue to reinforce their knowledge about fractals. Other types of reinforcement can be easily achieved by showing them to look at patterns, fractal art, geometric shapes, natural objects, and computer imaging visuals from medical science like magnetic resonance imaging (MRI) of the human brain. Engage them in easy and understandable activities that allow them the hands-on opportunity, like geometry, to make easy fractals with triangles, squares and circles. There is so much to see about fractals that are not only interesting and educational, but downright fascinating! The assessment of fractal knowledge in your students can be accomplished by checking if they can “analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships;”42 “specify locations and describe spatial relationships using coordinate geometry and other representational systems;”43 “apply transformations and use symmetry to analyze mathematical situations;”44 and “use -26-
- 29. visualization, spatial reasoning, and geometric modeling to solve problems.”45 Once they are taught the basic knowledge, the geometric patterns and terms, then they will connect this fractal knowledge to the real world. In fact, they will have new eyes to see their new fractal world, and it will be better, brighter, and clearly a model of the mathematical realm that we want them to learn. In conclusion, my goal to justify that the educational systems need to implement fractal knowledge into the curriculum, and employ the standards of learning this knowledge now has been achieved by explaining the following: 1. Fractal knowledge is valuable to our students because it has been long researched by scientists, and other researchers around the globe, so that there is much information to draw from. 2. Fractal knowledge is used presently around the globe as a technical tool for sciences, medicine, computers, and art. 3. Fractal knowledge is related to geometry, and related to the systems of the real world, and can be connected to natural systems in the environment, the human body, the universe, and in social situations. 4. Fractal knowledge can be taught and learned by concrete models, drawings, technology, and activities. 5. Fractal knowledge can be reinforced and assessed to create cognitive foundations of mathematical knowledge for exploring various mathematical tasks in prekindergarten through grade 12. -27-
- 30. Fractals are everywhere, but until we give this information to students, then they will never be able to know they exist, to see their beauty, and to understand their usefulness to everyone. -28-
- 31. Footnotes 1. Ward, Mark, 2001, Beyond Chaos, New York, New York, pg.81 2. Hidalcorp.com; http://webweevers.com/fractals.htm 3. Mendleson, Jonathan; Blumenthal, Elana, Chaos Theory and Fractals, p.5, http://www.mathmendl.org/chaos/ 4. Mendleson, J. 5. Casti, John L., 1997, Reality Rules: I, New York, New York 6. Casti, John L., 1997, Reality Rules: I, New York New York p. 329 7. Casti, John L., 1997, Reality Rules: I, New York New York 8. Mendleson, J. 9. Mendleson, J. 10. Mendleson, J. 11. Girvan, Ray, “The Mandelbrot Monk,” p.1-4 12. Girvan, Ray, “The Mandelbrot Monk,” p.1-4 13. http://www.teacherlink.org/content/mathrelatedlinks/sketchpad.html 14. Girvan, Ray, “The Mandelbrot Monk,” p.1-4 15. Girvan, Ray, “The Mandelbrot Monk,” p.1-4 16. Hidalcorp.com; http://webweevers.com/fractals.htm 17. Hidalcorp.com; http://webweevers.com/fractals.htm 18. Benoist B. Mandelbrot, http://www.math.yale.edu/mandelbrot/printedbooks.html 19. The National Council of Teachers and Mathematics, Inc., 2000, Principles and Standards for -29-
- 32. School Mathematics, Reston, Virginia, pp. 3-6. 20. The National Council of Teachers and Mathematics, Inc., 2000, Principles and Standards for School Mathematics, Reston, Virginia, pp. 7-9. 21. The National Council of Teachers and Mathematics, Inc., 2000, Principles and Standards for School Mathematics, Reston, Virginia, pp.10-14. 22. “The Mandelbrot Set and Julia Sets,” http://classes.yale.edu/fractals/MandelSet/MandelCombinatorics/LavaursAlgorithm/LA.html, p. 1. 23. “Fractal Geometry Lesson Plan Archive,” http://classes.yale.edu/Fractals/LP/LP.html 24. “Fractal Geometry Lesson Plan Archive,” http://classes.yale.edu/Fractals/LP/LP.html 25. “Fractal Geometry Lesson Plan Archive,” http://classes.yale.edu/Fractals/LP/LP.html 26. “Fractal Geometry Lesson Plan Archive,” http://classes.yale.edu/Fractals/LP/LP.html 27. “Fractal Geometry Lesson Plan Archive,” http://classes.yale.edu/Fractals/LP/LP.html 28. “Fractal Geometry Lesson Plan Archive,” http://classes.yale.edu/Fractals/LP/LP.html 29. Hidalcorp.com; http://webweevers.com/fractals.htm 30. “Fractal Census Statistics,” http://home.att.net/~Paul.N.Lee/Tried-Use_Counts.html, pp.1-2. 31. O’Daffer, Phares. G, Mathematics for Elementary School Teachers, 2nd Ed., 2002, p. 530 32. Amalgamated Research Inc., http://www.arifractal.com/ 33. Casti, John L., 1997, Reality Rules: I, New York, New York, p. 356. 34. Lanius, Cynthia, http://math.rice.edu/~lanius/fractals/iter/html -30-
- 33. 35. Bennett, Albert B., Mathematics for Elementary Teachers, 6th Ed. 2004, inside cover, grid paper. 36. Ward, Mark, 2001, Beyond Chaos, New York, New York, pg. 206-207. 37. Ward, Mark, 2001, Beyond Chaos, New York, New York, pg. 143. 38. Ward, Mark, 2001, Beyond Chaos, New York, New York, pg. 143. 39. “Pascal’s Triangle and Its Relatives,” http://classes.yale.edu/fractals/Labs? PascTriLab/PTAns3pl.html 40. “Block Fractals,” http://classes.yale.edu/fractals/AtMa/BlockComp/B3Perim.html 41. Bloom’s Levels of Taxonomy, http://www.splendoraisd.org/curriculum/Curr/Geometry.pdf 42. National Council Of Teachers and Mathematics, http://standardsnctm.org/document/chapter3/geom.htm , p.1. 43. National Council Of Teachers and Mathematics, http://standardsnctm.org/document/chapter3/geom.htm , p.1. 44. National Council Of Teachers and Mathematics, http://standardsnctm.org/document/chapter3/geom.htm , p.1. 45. National Council Of Teachers and Mathematics, http://standardsnctm.org/document/chapter3/geom.htm , p.1. -31-
- 34. References Amalgamated Research Inc., http://www.arifractal.com/ Bennett, Albert B., Mathematics for Elementary Teachers, 6th Ed. 2004. Benoist B. Mandelbrot, http://www.math.yale.edu/mandelbrot/printedbooks.html “Block Fractals,” http://classes.yale.edu/fractals/AtMa/BlockComp/B3Perim.html Bloom’s Levels of Taxonomy, http://www.splendoraisd.org/curriculum/Curr/Geometry.pdf Casti, John L., 1997, Reality Rules: I, New York, New York “Fractal Census Statistics,” http://home.att.net/~Paul.N.Lee/Tried-Use_Counts.html “Fractal Geometry Lesson Plan Archive,” http://classes.yale.edu/Fractals/LP/LP.html Girvan, Ray, “The Mandelbrot Monk.” Hidalcorp.com; http://webweevers.com/fractals.htm Lanius, Cynthia, http://math.rice.edu/~lanius/fractals/iter/html Mendleson, Jonathan; Blumenthal, Elana, Chaos Theory and Fractals, ,http://www.mathmendl.org/chaos/ O’Daffer, Phares. G, Mathematics for Elementary School Teachers, 2nd Ed., 2002. “Pascal’s Triangle and Its Relatives,” http://classes.yale.edu/fractals/Labs? PascTriLab/PTAns3pl.html Ward, Mark, 2001, Beyond Chaos, New York, New York. http://www.teacherlink.org/content/mathrelatedlinks/sketchpad.html “The Mandelbrot Set and Julia Sets,” http://classes.yale.edu/fractals/MandelSet/MandelCombinatorics/LavaursAlgorithm/LA.html -32-
- 35. References The National Council of Teachers and Mathematics, Inc., 2000, Principles and Standards for School Mathematics, Reston, Virginia -33-
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