One Area of Basic Misunderstanding among Teacher Candidates

Abstract

Understanding the calculation of area is important to mathematics learning and real world applications and contributes to teachers' diagnosis and correction of students' errors. Because our experience led us to suspect that many teachers may lack adequate concepts of measurement of area, we tested 102 undergraduate (sophomore) students (46 men and 56 women) in an introductory education class at a state university. Each subject was given a test paper printed with instructions and a model rectangle marked to indicate it was 5 units wide x 6 units long. Four other similarly marked lines were printed below, each representing the left side of a new rectangle to be drawn by the subject so as to have an area equal to the model rectangle. These Lines, positioned vertically along the left margin to become the left sides of new rectangles, were 2, 3, 4 and 10 marked units long. Subjects were asked to draw the three missing sides of these new rectangles. This task was similar to one developed by Piaget and also used in a recent study of children's judgments of area (Hirstein, et al., 1978), except college-age adults were tested. Responses were correct if the missing sides were drawn to the correct lengths or if the appropriate unit marks were visible, even if actual proportions were distorted. Based on subjects' written explanations and drawings, errors were classified according to the misconceptions found in children's thinking by Hirstein, et 01. (1978). To identify possible influencing factors chi-squared values were calculated for performance on the area test and each of the following: number of high school mathematics courses, college GPA, year in college, sex, and teaching major. Only sex was statistically significant; men performed better than women (p < .01). Of the 102 students tested 57 (35 women and 19 men) or 56% did not draw the correct length of the missing side of one or more of the four new rectangles. Virtually all errors were classifiable as one of the misconceptions identified by Hirstein, et al. as common among children. Four (two men and two women) employed the mOSK primitive of misconceptions (constructing all new rectangles the same length as the original figure even though each had a different width) and 37 confused the concepts of perimeter and area: area is thought to be calculated by counting the units around the edge. Eighteen subjects saw the given units as indivisible rather than subdividable, that is, they were unable to deal successfully with the rectangle which required a fractional side. Only 44% (18 of 56 women and 27 of 46 men) were able to perform this simple task of measurement of area (drawing all four rectangles correctly), while 56% apparently were thinking about area measured in the same naive ways as many young children.