Each year as I begin to plan a geometry unit for my students, I consider what I am supposed to teach, what I think I ought to teach, and what I currently know how to teach. This year, my planning has taken on a new dimension as a result of a year's coursework in elementary mathematics. The Curriculum and Evaluation Standards for School Mathematics (NCTM 1989), along with a variety of other readings, has expanded my understanding of the importance of providing rich, appropriate opportunities for children to explore all manner of fundamental geometric concepts.
The coursework exposed me to the work of researchers who have conducted investigations into what, when, how, and why various geometry topics should be taught to successfully influence students' learning.
I came away from my studies with a strong sense of responsibility to create a learning environment for my students that will prepare them for the higher-level instruction that they will encounter in middle and high school. The activities that I select, the reasons that I select them, the ways that I present them, and the degree of skill with which I guide my students' understanding lay the foundations for a geometry unit that will build a framework for future learning.
A Framework for Geometry
The focus of my planning is changing from objectives to framework. Just as the larger branches of a tree, its "framework," determine the tree's overall shape, so must I promote a framework for geometric thinking to shape my students' future successes in mathematics.
The curriculum objectives for my grade level include explorations of lines, paths, and polygons. They also include differentiation among horizontal, vertical, parallel, and intersecting lines. While providing a variety of activities for students to investigate these concepts, such as laying wet string on minichalkboards and using geoboards with rubber bands, I also focus attention on developing the specialized geometry language appropriate to these concepts. Standard 9 of the NCTM's
Curriculum and Evaluation Standards for School Mathematics states that "although a facility with the language of geometry is important, it should not be the focus of the geometry program but rather should grow naturally from exploration and experience" (NCTM 1989). The goal, then, is to dovetail the development of concepts and the mathematics language component as seamlessly as possible. The Professional Standards for Teaching Mathematics mentions that the "teacher should monitor students' use of mathematical language to help develop their ability to communicate mathematics" (NCTM 1991). With these points in mind, I try to find ways in which students can meld mathematical concepts and language.
The mathematics activity in this article is actually a lesson in geometry that uses the computer in an artistic exercise to allow students to see the distinct connection between art and mathematics from a personal perspective. The intent is to offer an enjoyable activity that blends concepts and language and avoids the tedium of memorizing definitions (Van de Walle 1994).
Developing a Vocabulary about Lines
In the course of our geometry unit, students are encouraged to share their ideas using language appropriate to the topic. In discussing different types of lines, the terms horizontal, vertical, diagonal, perpendicular, parallel, and intersecting are naturally introduced in the classroom. The expressive language changes from "It has a line on one side and another line just like it on the other side" to the mathematically oriented "It has parallel lines that are vertical." In one paper-and-pencil drawing activity, students eagerly create designs, then describe the figures in geometric terms.
I find that students love to use the language of these special types of lines. The words are big, and the concepts are easy. Remembering which word goes with which particular line, however, proves difficult for the students. They need plenty of meaningful experiences with the lines before the terms become part of their everyday thinking, writing, and speaking vocabularies.
Developing Vocabulary of Geometric Shapes
In addition to the lines, the geometric shapes used in this activity give students an opportunity to further explore and define the various characteristics of rectangles, squares, ovals, and circles. Some students will be able to describe rectangles in terms of their right angles and parallel lines; other students are ready to use the term parallelogram. As students become ready, the description of the rectangle can help them to recognize that the square, by definition, is also a rectangle. Our curriculum introduces the term polygon, and students frequently use it to identify the rectangles and squares. The descriptions of a circle and oval were more difficult for students to capture. "Round" and "curved" were the most common descriptions suggested, thereby giving me the opportunity to introduce the term closed curve.
Fortunately, this year, I discovered an exciting activity that gave my students another experience to explore both the concepts and the language of geometric lines and shapes. This activity linked technology, art, and geometry to create a spectacular product that we called Op Art Geometry. Creating the op-art picture and presenting it in a slideshow format was an exciting, multidisciplinary activity for students to experience. It gave them another opportunity to link concepts and language. Using a drawing and slide-show computer application program, Kid Pix 2, my students created slide shows of op-art designs.
Op art is a twentieth-century school of abstract art that uses straight lines or geometric patterns and brilliant colors to create visual effects. Creating op-art designs involves using geometric shapes, as well as horizontal and vertical lines. A leading practitioner of op art in the United States was Richard Anuszkiewicz, who worked with "combinations of often-dazzling colors in precise geometrical shapes," also known as "hard-edge painting" (Grolier Electronic Publishing 1996).
Any computer drawing and slide-show application program will allow children to create just such hard-edge painting using the dazzling colors of the software application's painting palette. Although I did not emphasize the dynamics of this particular art form as a lesson, one could certainly do so in an interrelated study of art and mathematics.
The students created their op-art designs by first drawing a series of step-by-step panels, or slides, each representing one step in the process of creating the whole design. In writing the dialogue for the audio portion of the slide show, students were instructed to use the language of geometry to describe the shapes and lines of their op-art pictures and the process of creating the pictures. Although the students had already completed some hands-on and paper-and-pencil activities in the geometry unit, they often required clarification of the geometric terms as they worked to put the oral portions of the slide shows together. This observation made me aware of the need for many exposures to the specialized language of geometry.
Op-Art Slide Shows
The slide shows were created by making a series of drawings using the Kid Pix drawing program. Each drawing was saved and later put into the slide-show feature of Kid Pix, where the audio portion was added. More advanced students, who are able to include more extensive mathematical descriptions for the shapes and lines, will require more drawings in their slide shows. It is helpful for students to have a "bank" of geometric terms from which to work as they plan their descriptions. Such terms might include horizontal line, vertical line, congruent shape, perpendicular line, parallel line, similar shape, angle, right angle, polygon, curved line, and so on. We brainstormed and posted these terms on the chalkboard.
A description of the main components of an opart slide show follows. The first and last slides are identical (see figs. 1a and 1j). The first slide presents the finished op-art painting and functions as a preview of what the viewer is going to learn. One student's audio component for the first slide sounded like a commercial: "Op art is very fun and easy to make. You also use geometric shapes and other things, like perpendicular, parallel, horizontal, and vertical lines. So that's all you need to make a perfect op art. It's really fun! Now make your own." An identical slide is presented at the end of the show to demonstrate the finished product.
The Sequence of Drawings
The second slide begins the sequence of op-art drawings. The student starts by drawing and describing a large rectangle, including its name and properties (see fig. 1b). One student's description was "Draw a large rectangle to fill the screen. A rectangle is a geometric shape that has four right angles. It also has all straight lines."
In the third slide, the student places smaller rectangles inside the large rectangle (see fig. 1c). If students make congruent rectangles, then the properties of congruence can be stated in the audio portion, for example, "Two of my rectangles are congruent because they are the same size and the same shape." For art's sake, the rectangles can be placed in any configuration desired. Nesting the rectangles can give a nice effect to the finished product, but the rectangles often become difficult to work with as the painting proceeds.
Some squares are added in the fourth slide (see fig. 1d). I encouraged the students to compare the properties of squares and rectangles. One student stated, in her audio, "Squares are a lot like rectangles because they have four right angles, but they're different because a square's sides are all equal to each other."
The fifth slide depicts the larger rectangle completely filled with geometric shapes; one student said, "You can draw ovals, circles, squares, or rectangles" (see fig. 1e).
Horizontal and vertical lines are demonstrated in the next slides (see figs. 1f-1h), and the final slides show how colors are applied to the drawings (see figs. 1i-1j). The "paint bucket" tool is used to fill each section of the drawing with color. The tool bar in the drawing program can also be used to place other geometric shapes into the painting. The final slide presents the completed op-art painting.
The oral portion of the slide show requires the student to write clear descriptions of what was drawn in each panel, to accept comments on and revise each description, and to practice speaking the part before actually recording it in the slide show. Students discovered that the special geometric language for lines and shapes was useful in describing the process of how their op-art pictures were created. The language also sounded very important to the students, and they seemed pleased to hear themselves speak in such clear, descriptive geometric terms.
I found this oral portion to be the "meat" of the activity because the work required a good deal of clarification of terms. Students paid much attention to being precise in the descriptions, writing and rewriting their texts. The slide show works best with an economy of words, and in this component I thought that students were most meaningfully experiencing the "intersection" of geometry and language.
After finishing the graphic and oral portions of the slide shows, the students completed their presentations by inserting between successive slides a variety of "transition" screens, provided by the software program. When the slide show is run, it takes on an animated quality as it moves from beginning to end.
The op-art activity proved to be both challenging and fun for students. The students acquired skills in working with the software and were pleased with the artistic component of the finished piece. An unexpected learning opportunity arose when some students discovered that their speech was not clear enough to be understood on the slides. A side lesson on oral speaking skills helped remedy the situation.
Most important, the slide show offers a meaningful application for the geometry objectives taught in the early grades. Although manipulatives, paper and pencil, and kinesthetic activities provide the beginning framework for learning to identify the various geometric lines and shapes, developing an op-art slide show gives students the opportunity to apply and reinforce their understanding of these concepts.
The class referred throughout the school year to the geometric concepts and language learned in this unit: "Were there any parallel lines in your opart?" "Think about your op-art. Do you remember if there were any perpendicular lines?" The slide show furnished a frame of reference whenever a geometric term was used and some students needed to revisit the concept. For a teacher, the ability to refer to a commonly shared interest is a powerful way to clarify concepts. I could quickly and easily draw the frame of a finished slide on the chalkboard, and the students could clarify the point of interest. The slide-show activity proved to be a lasting influence throughout the year.
Evelyn Brewer teaches third grade at DuFief Elementary School, Gaithersburg, MD 20878. She recently received a master's degree in elementary mathematics education.
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