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Mathematics: What's the Big Idea?

Workshop #8

The Future of Mathematics: Ferns and Galaxies

Content Guide - Andee Rubin

Supplies Needed for Workshop #8:
1 die, 1 ruler per pair of participants, worksheets 1-4,
pencils, paper, scissors, rulers, calculators, tape, a variety of colored markers

About the Workshop

What is the theme of the workshop?
The advent of new technologies in the mathematics classroom is rapidly changing how, what, and when we teach. Not only do higher-level mathematics become accessible to students at a younger age, but new and fascinating worlds of mathematics can open up to them - in their own classrooms - thanks to the power of computers. We will examine two new mathematical tools, recursion and iteration, and introduce a new geometry, Fractal Geometry, which complements the much older and more familiar Euclidean Geometry.

Whom do we see? What happens in the videoclips?
We will see many amazing and beautiful things in this workshop. First we will see three different approaches to the creation of a famous fractal, the Sierpinski Triangle. Second, we will see multiple representations of fractals: fractals on paper, on a computer monitor, in nature, and built with manipulatives. Finally, we will touch on the connection between fractals and chaos, and see how beautifully patterned shapes can result from "chaotic" systems.

What issues does this workshop address?
This workshop will stress the importance of familiarizing both teachers and students with new, burgeoning concepts in mathematics if we are to produce young citizens who will be competent players on the cutting edge of mathematics and science in the twenty-first century. Mathematics scholar and internationally renowned author, Ian Stewart, warns that the scientist/mathematician of the future "will need to combine, in a single integrated world view, aspects of traditional mathematics, modern mathematics, experimentation, and computation (i.e., computer "know-how").

What teaching strategy does this workshop offer?
A variety of teaching and learning strategies will be modeled and discussed. Of note is the use of a variety of media - paper and pencil activities, concrete models, transparencies, graphing calculators, video clips, and computer software - to introduce a completely new topic.

To which NCTM Standards does this workshop relate?
This workshop will stress number sense and spatial sense (Standards 6 and 9 in the K-4 content standards). The activities we will explore involve finding patterns and exploring relationships (Standard 13 in the K4 Standards, Standard 8 in the 5-8 standards), and measurement (Standard 10 in the K4 Standards, Standard 13 in the 5-8 standards). The four process standards (Mathematics as Problem Solving, Communication, Reasoning, and Making Connections) will also figure into this workshop.

Suggested Classroom Activities and Strategies

Exploring Recursion and Iteration in the Primary Grades

A) Think of counting as an iterative process. Ask students to articulate what rule is repeated over and over again as we count from one whole number to the next.

B) Have students find nursery rhymes and folk songs that are recursive in nature, stories that contain the telling of stories within them, movies that show movies inside movies, or plays that illustrate a play within a play.

C) Have students find recursive pictures or labels, such as the well-known Morton Salt box.

D) Have students bring in recursively-built toys or puzzles, such as Russian dolls or the Tower of Hanoi.

E) For those students who use word processing applications at home or in school, explain to them that a file inside a subdirectory, inside a directory, inside a drive, inside the hardware is yet another example of netting and embedding, similar to a small Russian doll inside a medium Russian doll, inside a large Russian doll, inside an extra large Russian doll, etc.

Suggested Strategies
In each example, make sure to point out the element of "permanence" and the element of "change." In other words, pose the questions "What changes?" and "What stays the same?" In example (A), the method of "adding 1" is permanent, and the outcome (i.e., the new and bigger number) after each iteration changes. In example (D), the size of the dolls changes (i.e., becomes smaller as we open them up one by one), but the shape remains the same. After explaining a few examples, have students write a sentence for what changes and what remains the same in each new example. For homework, have them find additional illustrations of recursion in everyday life.

Other Visual Patterns in Pascal's Triangle for Middle School See Activity Sheet 1
In this workshop we created an amazing visual pattern when coloring in the even numbers in Pascal's Triangle. Another phrase for even numbers is "multiples of two." Divide your class into seven groups to further explore the visual patterns created by coloring in multiples of 3, 4, 5, . . ., 9. Assign a whole number n, from 3 to 9, to each group. Have each group color in the multiples of n only. When the groups are done, they can share their visual patterns with the whole class. Have them verbalize and then record at least three observations. Challenge: Are any of these new patterns fractals? How can you justify your answer?

Suggested Strategies
This activity can be assigned to pairs of students within each group. One student uses a calculator and divides each number by the assigned whole number n to see if the remainder is 0. If it is, his or her partner proceeds to color in the hexagon that contains that number. The students take turns using the calculator and coloring the hexagons. This method helps students see the connection between "multiple" and "divisor," a connection which is not always clear to students. Obviously, when the number is small, no calculator is necessary.

Exploring Other Geometric Fractals in Grades 4-8 Activity Sheet 2

  1. Show the Box Fractal to your students. Ask them to verbalize the iterative process from one stage to the next. (You will find that they will provide a variety of explanations.)

  2. Ask students to suppose that the side length of the initial square is 1 linear unit. Have them write a sequence for the common side length of the squares at stages 0, 1, 2, 3, . . . , n.

  3. Then ask students to suppose that the area of the initial square is 1 square unit. Have them write a sequence for the total area of the Box Fractal at stages 0, 1, 2, 3, . . . , n.
  4. Have students try to find other fractals at various Web sites on the Internet. In particular, have students find the (Von) Koch Snowflake or the (Von) Koch Curve. Have them formulate investigative questions.

Suggested Strategies
This activity could be done in a variety of ways individually, in pairs, in small groups, or with the class as a whole. After students find the two sequences in parts (2) and (3), have them articulate additional questions on this - the box fractal - or other fractals. For instance, "Write a sequence for the total perimeter of the Box Fractal at stages 0, 1, 2, 3, 4,..."

Finding Fractals in Nature
Have students collect natural fractal objects or bring in pictures of fractal objects in nature. Stress the similarities and differences between geometric and natural fractals. Ask students to create their own fractals, either with paper and pencil or with the aid of a computer.

Post-Workshop Questions

  1. Did you have any idea that there were other geometries besides Euclidean geometry, the type we learn in grades K-12? If not, how do you think you might incorporate aspects of this new geometry (fractal geometry) into your math lesson plans?

  2. Graphing calculators are slowly but surely trickling down from high school to middle school, and from middle school to primary school. In general, today's children are raised in a much more technological world than many of us were, and therefore are much less technophobic than we are. How do you feel about allowing students to use state-of-the-art technologies in your math class? What are your thoughts about how these technologies may help or hinder their conceptual development?

  3. Rare is the student who is not captivated by the beauty and surprise of fractal geometry. Students will want to take the subject further, and will probably ask questions that neither they nor you can answer. What will you do if such questions arise?

Suggested Resources

Briggs, John. Fractals: The Patterns of Chaos. New York: Simon and Schuster, 1992

Briggs, J. and F. D. Peat. Turbulent Mirror. New York: Harper Row, 1989.

Gardner, Martin. Fractal Music, Hypercards and More.... New York: W.H. Freeman and Co., 1992.

Gleick, James. Chaos: Making a New Science. New York: Penguine Books, 1987.

McGuire, Michael. An Eye for Fractals. Reading, MA: Addison Wesley, 1991.

Peitgen, H.O. and P.H. Richeter. The Beauty of Fractals. New York: Springer-Verlag, 1986.

Wahl, Bernt. Exploring Fractals on the Macintosh. Reading, MA: Addison Wesley, 1995. (Book and Software).

Mathematics: What's the Big Idea?

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