Re-Inventing Fourth-Grade Mathematics

Subject: Math
Grade: 1-6

"Many young students believe that mathematics is simply arithmetic -- an often bland, boring subject," admits fourth grade teacher Jack J. Lopez of Roseville, California. "My major goal in this project was to dispel that notion." Taking for his objectives those of standard fourth-grade mathematics texts "where many geometric concepts are introduced and, then, dismissed in one or two pages," Lopez overcame "the difficulty of providing practice work for geometric concepts" handled at fourth-grade level. "My project addresses the need for further curricular material in this area in a way that emphasizes active, creative participation in acquiring knowledge," adds the math teacher with nine years experience in grades 4-6. "As a visiting city librarian commented, 'These activities make you use your left and your right brain.'"

Cirby School (K-6) is located in a rapidly growing community of more than 20,000 people. Approximately 500 students are grouped into heterogeneous, mostly self-contained classrooms. Some teachers exchange students for reading and mathematics. "There is a wide variety of ethnic and cultural backgrounds as well as socio-economic levels in our student population," says Lopez, who himself is Spanish-American. "Our school receives Title I funds, and some of our students participate in the free lunch program. My own fourth grade class of 32 students reflects the diversity of the general school population."

In Lopez' project, these students were given the opportunity to actively study and investigate several important geometric concepts. "The activities by which this is accomplished involve looking for and investigating geometric patterns both in natural and man-made objects," Lopez explains. "The students' findings then are recorded and made permanent by means of photographs. Let me note, too, that although these studies are characterized primarily as 'geometric,' they have broader mathematical and scientific importance."

"Throughout this project, I tried to stress the underlying unity and elegance of geometric concepts," adds Lopez. "I also wanted to give the impression that if Euclid were alive today, he'd be out there taking pictures.

"The role of photography was central and crucial to the attainment of my objectives. The act of photographing forced students to decide, to select, to compare, to search, and to reappraise. It compelled them to take an active role in their own learning. The visual beauty of the photographs and of the children's projects emphasized the aesthetic aspect of mathematics, an aspect too often restricted to the minds of professional mathematicians."

What Specifically Did Students Learn Through The Project?

"By participating in a series of four related activities, students attained several important objectives," notes Lopez. "They acquired a firm, active knowledge of some of the basic terminology of Euclidean geometry. This included learning the special vocabulary for describing various geometric shapes and transformations.

"In the first activity, 'Shape-maps,' students began by learning the names of the most common regular polygons. They also learned the terminology for describing circles and arcs: radius, diameter, degree, and chord. They then practiced finding these shapes in their environment. Because the students called my attention to the frequent occurrence of certain shapes I had not discussed, for example, stars and ovals, these also became part of our inventory of shapes. Weeks after this activity was over, students would still comment that they had noticed that the outline of a certain flower was a pentagon or that they had seen a hexagonal stained-glass window. Some were still trying to find the elusive nonagon.

"In the second activity, 'Symmetry,' students learned to identify the line or lines of symmetry of various objects -- letters of the alphabet, pictures in magazines, polygons, etc. They also learned to extend this concept to non-geometric fields such as palindromes, music, and dance. An unexpected outcome for all of us was the discovery of subtle beauty in seemingly plain objects when they are studied intently and closely.

"The third activity, 'Rubber Rascals,' gave students two main opportunities to demonstrate creatively their understanding of linear growth: first, in a practice activity using magazine pictures, and, secondly, by taking photographs which they cut up to demonstrate linear growth in a very dramatic fashion. In attempting to mount their photographs, which were cut into eight to 12 strips as neatly and appealingly as possible, the students further realized the importance of exact measurement. The final appearance and appreciation by all of their mounted photographs certainly strengthened their realization.

"My primary objective in the last activity, 'Recursive Patterns,' was that each student be able to select from several choices the one that exemplified recursion. Secondarily, I wanted them to be able to describe the degree or level of recursion of simple examples. To reinforce this latter skill, I had them explore the notion of recursion further by using a couple of Logo programs on one of our school computers. I wanted the students to appreciate how video artists are beginning to use recursive techniques in their work."

What Equipment And Supplies Were Necessary To Project Success?

"A fully manual 35 mm camera and lots of color film -- both transparency and color film -- were the basic pieces of equipment used in actual picture-taking. We also used a tripod for indoor shots requiring exposures too long to be handheld. We did not use a flash or any filters. Several students took photographs with their own cameras at home. I deliberately kept equipment to a minimum. This assured that every child would learn to use each piece of equipment -- a worthwhile goal in itself."

How Did You Proceed?

"I began our series of activities by instructing the students in the proper use of a camera -- how to hold it, how to focus, how to press the shutter release. A good hint for teachers is to keep your eye on the distance scale when a student is focusing on an object. This way you'll quickly know if the student is even 'in the right ballpark.' I gave them an opportunity to practice these skills with outdated film. I also demonstrated the use of a tripod, which I made available to them at recesses so that they could experiment with the various controls."

How Did You Introduce The First Math Activity?

"I began this activity with a brainstorming session in which the students had five minutes to list all the things -- animate and inanimate -- they could think of that seemed to emphasize linear growth or extension along a particular dimension. I mentioned a giraffe's neck and a skyscraper as examples.

"Next, I showed them how they could simulate growth or extension along a particular dimension of an object by cutting a picture of that object into uniform strips and then gluing them down, leaving small intervals of space between each strip. This created the illusion of 'stretching' the object in the picture. I showed them how this effect was enhanced if they began with a picture of something with one main dimension.

"Each of the children had an opportunity to practice this technique on a picture which he or she selected from a stack of old magazines donated by parents. The children selected a wide variety of objects to 'stretch' -- grandfather clocks, sports cars, and fashion models among others. From their initial practice, the students learned how best to cut and arrange the strips to obtain the desired effect. They were now to put that knowledge to good use.

"Their next activity was to stretch themselves! Working in pairs, they took great pleasure in posing and photographing each other in ways that accentuated height or width. When the photographs came back from the processor a few days later, I had the students work four at a time at a large table at the back of the classroom. I cut their king-size photographs into eight to 12 strips with a paper cutter. (My private effort with scissors had proved unsatisfactory.) I showed the students a carefully prepared example of my own. I explained that my photograph was so neatly centered because I had carefully measured and marked the placement of everything before I began to glue things down with rubber cement.

"After all the students had completed this last phase of the activity -- it took about a week -- I displayed the results on a large bulletin board which I titled 'Rubber Rascals.' This was one of the most popular and talked about bulletin boards I've ever had. Everyone who came in to see it was amused by the illusion of growth which the children had been able to create."

How Did "Recursive Patterns" Proceed?

"Knowing what a difficult subject recursion can be, I began this lesson slowly by going over very simple graphic patterns I had found in a book on Logo. I would, for example, draw the first two or three levels of a recursive pattern of a tree and then ask the students to continue my drawing by filling in the next two levels. They were all able to do this. I also led the class in tracing the first level of a fern frond and a large elm leaf.

"Because I felt many of the students would have difficulty bringing plants or even leaves to class this time of year (winter), I had the children form large groups of five to eight students. After choosing a group leader, members of each group could plan together to find and bring to class several objects or pictures of objects (I added this latter choice to make the assignment easier) that exemplified recursion. The groups were given class time to search in old books and magazines, but they were asked to spend time outside of class looking for appropriate objects.

"Each group was to select the two or three most interesting objects brought in by their members. The objects were to be photographed in a manner that best displayed the recursive pattern the group was studying. Members of each group had to write brief descriptions indicating the recursive nature of whatever object they elected to describe. They could further illustrate their description with simplified drawings if they wanted to. Everyone also was allowed to photograph the samples he or she brought in, whether or not they had been selected by their group for special attention. Each group received space to display its photographs and descriptions.

"In order to give my students additional practice in studying recursive patterns, I took advantage of the availability of an Apple computer running Logo. I adopted a couple of well-known Logo programs that utilize recursion. The students enjoyed experimenting with these. They also received the further opportunity of learning to photograph drawings on a color monitor.

Allowing students to work in pairs for 20 minutes gave everyone a chance at the computer within three days."

How Adaptable Do You Believe Your Project To Be?

"As even a cursory reading will show, the whole of this project -- disallowing the extra computer activity -- is easily transferable to any school setting. Students can study and investigate man's and nature's use of shape, symmetry, and recursion wherever they live. Only a beginner's knowledge of cameras and photography is really necessary for the actual picture-taking phase of the activities. The more you do know and learn about photography, the more you will enjoy taking pictures and the better those pictures will be.

"Variety is not only the spice of life; it's also a psychological technique for ensuring motivation and retention. With that in mind, I use a variety of approaches in implementing the activities in my project. At various times, the students worked individually, in pairs, in small groups, or in large groups. Many times, the entire class would be involved in a discussion. I think it is clear that the size of the groups I used for each particular activity is not fixed but depends on the maturity and ability of the students.

The major cost is the film processing and printing. I asked for king-size prints on glossy paper since I wanted fairly large prints that would be easy to manipulate and cut. One bit of advice I would pass on to beginning photographers: make certain the viewing frame is completely filled by the object you want to photograph. Beginners tend to forget that everything in the frame will be in the final picture."

How Would You Evaluate Your Results?

"This project -- or rather the students' execution of it -- exceeded many of my initial expectations. Although the aesthetic aspect of this program was one that I thought extremely important -- I did, after all, want my students to experience the beauty of mathematics -- I did not feel that this year's class could do well in activities requiring artistic or creative ability. All year I'd had a great deal of difficulty getting them to complete any class art projects.

"Thus, I was greatly surprised and pleased to note the care and perseverance with which the students completed the first two activities. A few slower workers had even pleaded to be allowed to finish their projects at home. Reluctantly, I acceded. The projects came back -- complete and not folded four times to be stored in a back pocket! More gratifying still was the fact that some students who generally did poor or mediocre work in the regular mathematics curriculum were able to excel in these activities. Everyone was given the opportunity to create a work of mathematical and artistic beauty that was uniquely their own.

"My strictly academic objectives also were accomplished. Weeks after we had completed the symmetry activity, students would use the terminology they had learned to describe their own or others' drawings or photographs. I was pleased to note the transference of their knowledge of linear growth to discussions in science and health classes. This was the best indication that their new knowledge had become an intellectual tool with which to study their environment. The success of the last two activities was demonstrated by the children's correct use of the vocabulary of geometry and by their uniformly high scores on unit tests in geometry in our textbook.

"There was one other important outcome to their project," reports the proud math teacher. "The students could not help but notice the large numbers of teachers, aides, and parents who came into our classroom to look at and admire their projects. The students' sense of pride and accomplishment in themselves and in their own classroom was greatly enhanced by all the favorable attention given to their pictures."

| More Lesson Plans | Digital Learning Center - Educators |