Joye Thaller

CD 145

Final Paper

May 2, 2005

The Effect of Geometer’s Sketchpad on Seventh Graders’ Understanding of Area and Perimeter

Abstract

Area and perimeter are concepts that are usually hard for seventh graders to fully grasp. While they may memorize formulas, they often show difficulty applying the concepts. Past research has shown that students who construct their own knowledge tend to understand it better and retain it longer. Geometer’s Sketchpad is a dynamic geometry software that may aid in helping students understand more fully the concepts of area, perimeter, and circumference. Students can make their own conjectures and create powerful ideas using this software. This design study proposes that a group of students taught by formal classroom instruction be compared to a group of students learning the same material using Geometer’s Sketchpad. The students’ concept knowledge will be evaluated with pre- and post-tests, including a post-test six months later to determine long term effects. Student attitudes towards the material covered and to the software will be evaluated with student surveys. The results of this study may indicate that math teachers should use Geometer’s Sketchpad more in their classrooms in lieu of more formal teaching methods.

Introduction

The concepts of perimeter and area are usually taught to children in middle school. In the past, an instructivist approach has usually been used, where formulas and definitions are simply given to children, who are then expected to memorize these definitions and regurgitate them on tests. Area and perimeter can be difficult concepts and many children have trouble understanding the relationship between the two. In this paper, I include circumference under the broader heading of perimeter.

I believe computers may aid in students’ understanding of math. In particular, the dynamic geometry software,The Geometer’s Sketchpad, has been used successfully by many teachers for teaching geometry concepts in both the middle grades and in high school geometry classes. However, few research studies have been done to determine what the effect is of Geometer’s Sketchpad versus regular classroom instruction. I believe this is an area in need of more in-depth research and so I propose the following research questions for this study:

1. What are children’s pre-conceptions of area and perimeter/circumference before instruction?

2. Does Geometer’s Sketchpad aid in children’s understanding of area and perimeter/circumference compared to that of children taught by formal classroom instruction?

3. What are the long term effects of using Geometer’s Sketchpad compared to learning by formal classroom instruction?

Literature Review

Mindtools are “computer applications that require students to think in meaningful ways in order to use the application to represent what they know.” (Jonassen, 2000, p. 4) In other words, the computer should not be thinking for the child or telling the child what to do. Vygotsky’s theoriessay that children usetools (or artifacts) to mediate between the user and the environment.The computer is a material artifact, like a pencil, a calculator, or a textbook, that acts as a mediator in the education process. It is not the educator.A child working with a computer program should be able to go at his or her own rate, choose the way he or she approaches the project, and see his or her own work in the computer’s output. It is important that the type of software being used act as a Mindtool.

Dynamic geometry environments are a type of Mindtool. The term “dynamic geometry” implies that parts of a geometric shape can be moved. Its use in a computer environment offers the user an opportunity to develop intellectual activities based on geometrical knowledge (Kuntz, 2002). Key features of a dynamic geometry environment include: 1) a mathematically solid base for the implemented geometry, 2) a user-friendly interface to allow the user to quickly learn the key facets of the software, and 3) fast feedback to let the user have control of the objects he or she has constructed. One of the benefits of using dynamic geometry software is that the user can freely analyze problems in an active and dynamic way without the difficulties of trying to draw and manipulate shapes by hand.

Papert (1980) talks about computer software as a “powerful tool” when it is used by a child in an active and self-directed way and when the knowledge is acquired for a recognizable personal purpose. Dynamic geometry software can act as a powerful tool to identify and examine geometric properties (Santos-Trigo & Espinosa-Perez, 2002). Students can construct, visualize, revise and change their own figures or constructions instantaneously. Thus, students have the opportunities to make their own conjectures. The software becomes a powerful tool in that it allows students to make and verify their own conjectures about geometry. The conjectures may be personally meaningful to the students, particularly if the conjecture is then used for some higher purpose (e.g. coming up with a conjecture about how to maximize area for a garden). Sometimes, these conjectures are existing geometric proofs. Othertimes, students come up with conjectures that have never been thought of or proved before. Both can be seen as powerful ideas and are equally important in students’ learning about geometry.

Schwartz (1989) furthers this point in his paper about computer programs as intellectual mirrors.“Using such environments as ‘intellectual mirrors,’ it is possible for users to probe their own understanding of a domain, as well as to devise new relationships among the objects of a domain (Schwartz, 1989, p. 58). The software he describes in his paper is called the Geometric Supposer. The three main features of the Geometric Supposer that allow it to be called an intellectual mirrorare: 1) it asks no questions of the user and makes no inferences about the user’s intentions; 2) its primitive functions in its tool menu are reasonable close to the elementary operations of the formal system of plane geometry; and 3) it provides an environment for allowing students to understand how the particularity of their efforts fits into a larger mathematical generality. These characteristics are important in helping students construct their own knowledge. There are other types of geometry programs very similar to the Geometric Supposer, such as Cabri-Geometry and Geometer’s Sketchpad. Because it is slightly newer and is already in use in many classrooms around the country, Geometer’s Sketchpad will be the focus of this research study. This proposal does not imply that one piece of software is better than the other. In fact, further research studies could compare different types of dynamic geometry software. The decision to use Geometer’s Sketchpad was mainly because it was easier for the researcher to access.

There have been few detailed studies of the effect of Geometer’s Sketchpad on learning. One study focused on the amount of structure the teacher gave the students when doing a Geometer’s Sketchpad activity (MacGregor & Thomas, 2002). Past research indicated that increased structure, in the method of step-by-step instruction, can increase learners’ perception of guidance and provide uniformity of learning outcomes with specific outcome expectations. Less structure, however, promotes more learner control and self-regulation, more creativity, and better meta-cognitive skills. The study by MacGregor & Thomas was designed to determine if the instructional approach influences students’ conceptual understanding of geometry, and also identify the students’ perceptions of the support provided by the instructor, challenges, frustrations, and successes while working on the project-based activities. The participants were 82 tenth-graders working on a geometry transformation problem with Geometer’s Sketchpad. In the first two groups, students were provided with specific problems to solve and given a set of guidelines to follow. In the second two groups, students were provided with the same problem-based learning activity, but were responsible for identifying the information and strategies they needed to solve the problem. The method of data collection was to have students write in an electronic diary and to write reflections in response to questions from the teacher. The results of the study indicated that the students in the teacher-directed group seemed to have a better understanding of the geometrical concepts. The students in the self-directed groups expressed higher levels of frustration in doing the activity. However, many students in the self-directed groups expressed a sense of self-confidence and pleasure with their accomplishments. An important lesson to take from this study is that the teacher has to be careful not to let students get to the point of frustration with the activity. While it is good to let students construct their own knowledge, the students should still be guided and provided with enough structure to make progress and not get frustrated. The teacher selected for this proposed design study should have a good grasp of how to provide enough structure to guide the students without instructing them step-by-step.

Anotherstudy focused on teacher-student interaction in an activity using two different types of geometry software (Hannafin et al., 2001). Since it is has been established that teachers can provide instructional scaffolding in the form of questioning and prompting, the researchers wanted to determine how teachers would provide scaffolding in an open-ended learning environment. The researchers created an instructivist learning environment using two computer based tools: a dynamic geometry program (Geometer’s Sketchpad) and a computer-assisted tutorial program (Introduction to Geometry). The overall goal of the study was to investigate the effects of the two different types of geometry programs. Students worked in pairs on activities and also kept journals. The students had access to both programs on their computers. The teacher’s role was to circulate through the room and encourage the students to use the tutorial when needed or to ask appropriate questions. Because there were so many variables in this study, it is difficult to say whether the Geometer’s Sketchpad provided a better learning environment for the students or not. The study was mostly qualitative and cited quotes from the students such as, “I think I learned better using the Sketchpad than a normal class because it was hands-on.” While these quotes provided some positive evidence, the small sample size of the study (12 students) was not enough to provide conclusive results. In addition, the focus of this study was more on teacher and student attitudes than on the learning of a particular math topic. The research proposed in this study will focus more on the use of the software to learn a particular topic in geometry.

The geometry topics that will be studied using Geometer’s Sketchpad are area and perimeter of rectangles and area and circumference of circles. There are many reasons why area and perimeter deserve attention in a research study. The Massachusetts State Frameworks for Measurement in Grades 7-8 say students should:

Demonstrate an understanding of the concepts and apply formulas and procedures for determining measures, including those of area and perimeter/circumference of parallelograms, trapezoids, and circles. Given the formulas, determine the surface area and volume of rectangular prisms, cylinders, and spheres. Use technology as appropriate.(MassachusettsState Frameworks, 2001)

The NCTM Principles and Standards for Mathematics for Grades 6-8 say that students should “develop and use formulas to determine the circumference of circles and the area of triangles, parallelograms, trapezoids and circles and develop strategies to find the area of more-complex shapes.” (NCTM, 2004, p. 240) In particular, NCTM says that:

“Whenever possible, students should develop formulas and procedures meaningfully through investigation rather than memorize them. Even formulas that are difficult to justify rigorously in the middle grades, such as that for the area of a circle, should be treated in ways that help students develop an intuitive sense of their reasonableness.” (NCTM, 2004, p. 243)

Area and perimeter are not always easy topics for students, which is why NCTM advocates investigation rather than rote memorization. Boston and Smith (2003) say that “situations in which one attribute of a geometric figure is allowed to vary while another attribute remains constant can provide rich-problem solving opportunities.” (p. 208) In one vignette, Boston and Smith (2003) cite a case where three seventh-graders argued vociferously that a fixed amount of rabbit fencing would always form the same area no matter which shape you put it in. After the teacher asked them to try it on their calculator instead of just guessing, they were convinced that the calculator results were wrong. It was only after going into the hallway, roping off a perimeter, and counting the square tiles on the floor, did they come to the realization that their original theory was wrong and that the same perimeter could result in different areas. Clearly, this was a case of Piagetian disequilibrium. Fortunately, the teacher realized this and kept asking them questions until they saw for themselves that their original conjecture was incorrect.

In a study by Woodward and Byrd (1983), 129 eighth-graders were given a task to find the garden with the largest area out of six differently-shaped rectangles, all with a perimeter of 60 meters. Only 23% answered the question correctly (the square garden is the largest), while 59% of the students said all the gardens were the same size. Students from a mathematics course for prospective elementary school teachers were asked the same question. Almost two-thirds of the teachers stated that the gardens were all the same size.Although the students and teachers had learned the formula for calculating area and the dimensions of each garden were given, somehow the connection was not made that gardens with the same perimeter were not necessarily the same size. Clearly, something was lacking in the way in which area was taught to these people. A formula-based approach using only computational techniques does not work. Formulas should instead be developed inductively.

In my own experience in a seventh-grade classroom, I have seen many children (working on a similar problem) state that the volume of different cylinders of various heights will be the same if the surface area is the same (it is not, the maximum volume is the one whose radius equals its height). However, it is not until the children actually fill these containers with beans and measure the volume for themselves that they see that the volume varies.

Geometer’s Sketchpad allows students to “see for themselves” in a virtual environment. Since they control what the computer is doing, they will know it is not lying when it shows two rectangles of the same perimeter having different areas.They can also develop formulas inductively and test them with the software. Figure 1 shows an example of the way in which Geometer’s Sketchpad can be used to explore rectangles, while Figure 2 shows an exploration of circles. In this proposed design study, students will use Geometer’s Sketchpad to explore area and perimeter of rectangles and area and circumference of circles.

Figure 1. Example of Geometer’s Sketchpad used to explorearea and perimeter of rectangles.

Figure 2. Example of Geometer’s Sketchpad used to explore circumference of circles.

Methodology

The research proposed for this study will be both quantitative and qualitative. The pre-test and post-tests will be analyzed quantitatively, while the videotapes and surveys will provide qualitative data. Portions of the survey will also be assessed quantitatively.

Sample

The sample will consist of two classes of seventh graders (20-25 students in each class) taught by the same teacher, at a school to be determined. The lesson will span two days: one day on rectangles and one day on circles. It is important that the students not have formal instruction in the geometry of rectangles or circles yet, so the study will be done at the beginning of the school year. The teacher will teach one class with instructivist methods without using computers. The lesson can be taken out of a textbook or other source of information that the teacher would like to use. The other class will use computers with Geometer’s Sketchpad to do the activity. The students will work in pairs on the activity. The teacher will guide the students through the activity, providing help when needed and asking provoking questions to stimulate critical and complex thinking. The teacher should not specifically teach the students in the computer group, but should help when needed so that students do not get too frustrated. Each worksheet is broken into two smaller exercises. For the first exercise, one student should use the computer while the other student records the information. For the second exercise, the students will switch roles. On the second day, the students will follow the same routine.