Area and perimeter: Preservice teachers' confusion

School Science and Mathematics, Feb 1997 by Reinke, Kathryn S

Measurement topics, including perimeter and area, are basic to the competency domain of elementary teachers. This study examines solution strategies, used by elementary preservice teachers, to find the perimeter and area of a shaded geometric figure. Many of these preservice teachers incorrectly found the perimeter of the figure by the same method they used to find the area. Examination of incorrect strategies by preservice teachers provides an opportunity to investigate inadequacies in the mathematical backgrounds of these students as well as to make curriculum changes in teacher education programs.

Measurement, in particular the concepts of perimeter and area, are present in the spectrum of the K-12 curriculum. It is part of the important interaction of children with their environment (National Council of Teachers of Mathematics [NCTM], 1989). Very young children walk around patios or sidewalks and they often draw circles in the dirt. This intuitive notion of measurement gradually develops into more formal study and definitions in school. The NCTM specifically addresses the need for inclusion of measurement, perimeter, and area (NCTM, 1989): "In grades K-4, the mathematics curriculum should include measurement so that students can understand the attributes of length, capacity, weight, mass, area, volume, time, temperature, and angle" (p. 51). In addition, it is stated: "In grades 5-8, the mathematics curriculum should include the study of the geometry of one, two, and three dimensions in a variety of situations so that students can... represent and solve problems using geometric models;...develop an appreciation of geometry as a means of describing the physical world" (p. 112).

Since measurement is a major concern of a mathematics program, it is expected that all teachers of grades K-8 as well as high school geometry teachers have a thorough understanding of all concepts in that area. However, with reference to area, Woodward & Byrd (1983), found that prospective elementary teachers were just as naive as eighth graders.

With the current extensive call for reform in elementary mathematics programs, it is imperative that the hurdles that preservice teachers face as part of their mathematical preparation be examined. There is little research that investigates how preservice elementary teachers approach perimeter and area. In order to understand the nature of the obstacles that perimeter and area present to elementary preservice teachers, one must explore and diagnose incorrect strategies.

Method

The subjects for this study were 76 preservice elementary teachers enrolled in the second semester of an elementary mathematics content course of a large university. The group consisted of 4 males and 72 females. These students all had a prerequisite course in college algebra prior to enrolling in this course. Geometry and related measurement were topics encountered during the second half of the course.

The study took place during the middle of the semester just prior to the introduction of geometry and measurement. During the first part of the semester, the students had studied probability and statistics. In a previous course they had studied the number system and rational numbers.

The class was shown a projection of Figure 1. In addition each student was given a sheet of paper on which the same figure had been drawn. Subsequently, they were asked to respond, in writing, to the following:

1. Explain how you would find the perimeter of the shaded part of this figure (do not use formulas). 2. Explain how you would find the area of the shaded part of this figure (do not use formulas).

Questions were answered by the instructor as to what was meant by an explanation without formulas. They were instructed to tell, in sentence form, how they would go about finding the area. It was also stressed that actual answers were not necessary since no dimensions were given. The subjects were given as much time as they needed to complete the task.

Result and Discussion

The subjects expressed concern that they were unsure of their responses. Upon examination of the explanations, it was found that the preservice teachers were justified in their insecurities. Only 9 of the 76, or 11.8%, gave the correct reply for the perimeter problem. A total of 40, or 52.7%, answered correctly for the area problem.

A variety of strategies were used by the subjects for finding the perimeter (Table 1). The incorrect strategy used most often, 19 times, totally ignored the existence of the semicircle and advocated merely finding the perimeter of the rectangle. Approximately one-fourth of the preservice teachers thought that this was the correct way to work the perimeter portion of the problem.

Approximately 22% of the subjects worked the problem exactly as they would an area problem. These 17 responses indicated a confusion that linear measurement is somehow linked to area measurement. That is, the perimeter of the figure could be found by finding the perimeter of the rectangle and subtracting one-half of the perimeter of the circle. For the eight subjects who merely indicated the necessity of measuring, it is difficult to interpret the thought processes. This 10.5% of the subjects were assumed to have given a wrong answer even though they may or may not have known how to correctly find the perimeter. The explanations were not specific enough to make any conclusions.