Exploring students' conceptual understanding of the averaging algorithm

School Science and Mathematics, Feb 1998 by Cai, Jinfa

Conceptual understanding of arithmetic average includes both an understanding of the computational algorithm and the statistical aspects of the concept. This study focused on the examination of 250 sixth-- grade students' understanding of the arithmetic average by assessing their understanding of the computational algorithm. The results of the study showed that the majority of the students knew the "addthem-all-up-and-divide" averaging algorithm, but only about half of the students were able to correctly apply the algorithm to solve a contextualized average problem. Students were able to use various solution strategies and representations to solve the average problem. Those who used algebraic and arithmetic representations were better problem solvers than those who used pictorial and verbal representations. This study not only suggests that the average concept is more complex than the simplicity suggested by the computational algorithm, but also indicates the need for teaching the concept of average, both as a statistical idea for describing and making sense of data sets and as a computational algorithm for solving problems.

Arithmetic average is not only an important concept in statistics but also an important concept for informed citizens (National Council of Teachers of Mathematics [NCTM], 1989). Statistical analysis and inferences are conducted almost exclusively through the determination of measures of central tendency, such as the mean or arithmetic average, and measures of dispersions, such as the standard deviation. Data reported and used in daily life, scientific journals, and public media frequently use averages. The arithmetic average is a statistic used to describe and make sense of a data set. It is also a tool, used in conjunction with the standard deviation, for summarizing a data set and comparing data sets (e.g., Gal, Rothschild, & Wagner, 1990; Mokros & Russell, 1995). Computationally, the arithmetic average is defined by adding the values to be averaged and dividing the sum by the number of values that were summed.

Although the average concept seems to be as simple as the averaging algorithm suggests, previous research indicates that precollege and college students do not understand the properties related to the averaging algorithm (e.g., Mevarech, 1983; Pollatsek, Lima, & Well, 1981; Strauss & Bichler, 1988) In addition, students experience many difficulties using the algorithm to solve problems such as finding a weighted mean, which require more than a direct application of the algorithm. In these studies, the misconceptions were not due to students' lack of procedural knowledge in calculating the average. Rather, they were due to students' lack of conceptual understanding of the average, affecting their ability to utilize the algorithm.

The arithmetic average is both a key concept in statistics and a computational algorithm. Therefore, conceptual understanding of average includes both an understanding of the computational algorithm and the statistical nature of the concept. With statistical understanding of the concept, students should be able to use the concept to summarize and make sense of a data set or to compare data sets. With conceptual understanding of the averaging algorithm, students should be able to correctly apply it to solve problems. That is, students not only know the averaging algorithm but also know when and how to correctly apply the algorithm to problem situations. The purpose of this study was to examine students' conceptual understanding of the arithmetic average by focusing on their ability to solve the problem requiring the computational algorithm.

Method

Subjects

A total of 250 sixth-grade students from the Pittsburgh metropolitan area participated in the study. Boys and girls were about evenly distributed in the sample. Subjects were selected from eight private schools and one suburban public school.

Tasks and Administration

An average task (see Appendix A) was administered along with six performance assessment tasks, all taken from a mathematics education reform project (Silver & Lane, 1992). The students had 40 minutes to complete all seven tasks. In the average task, students were asked to provide a numerical answer and to explain how they found their answer. In the other six tasks, students were also asked to construct solutions and provide a written record of their solution processes. These six tasks addressed a variety of important content areas, such as number sense, pattern recognition, number theory, prealgebra, ratio and proportion, and estimation. They were used as a measure of students' mathematical performance.

Data Coding and Analysis

Each response to the average task was coded with respect to four aspects: (a) numerical answer, (b) mathematical error, (c) representation, and (d) solution strategy. The detailed categorization scheme is described in the results section. To ensure a high reliability, about 50 student responses to the average task were randomly selected and were independently coded by two raters. The inter-rater agreements range from 90% to 98%. Each response to the other six tasks was scored according to a holistic scoring scheme, ranging from 0 to 4. The inter-rater agreements based on several selected tasks for the holistic scoring ranged from 84% to 89%. The detailed description of holistic scoring can be found in Lane (1993) and Silver and Lane (1992).