Constructing the Mean as a Mathematical Point of Balance

During the proposed interactive paper session, the presenter will share results of a study designed to examine how middle-grade students come to understand the arithmetic mean as a mathematical balance point.

Theoretical Framework:

Prior investigations of central tendency have characterized and categorized student thinking about the arithmetic mean. For example, Strauss and Bichler (1988) conducted an early study of students' mathematical properties of the mean. These properties included (i) “the average is located between the extreme values [in a data set]”and (ii) “the sum of the deviations from the average is zero.” Building on this work, Mokros and Russell (1995) examined how students make decisions about using the mean framed in realistic contexts and how they make instructional decisions to summarize data. They developed a list of five approaches that students tend to exhibit in solving problems requiring the mean: average as mode, as algorithm, as reasonable, as midpoint, and as a mathematical balance point. The latter is the focus of the current study. Students that use the mean as a mathematical balance point portray the following qualities:

·         view an average as a tool for making sense of the data;

·         look for a point of balance to represent the data;

·         take into account the values of all the data points;

·         use the mean with a beginning understanding of the quantitative relationships among data, total, and average; they are able to work from a given average to data, from a given average to total, from a given total to data.

·         break problems into smaller parts and find "submeans" as a way to solve more difficult averaging problems (Mokros & Russell, 1995, p.26)

Although these studies have provided insight into children's characterizations and conceptions of the mean, their methods only examined students' understandings of the mean at one point in time through the use of single clinical interviews. This study fills a void in the current literature, as it examines the long-term growth of students as they develop conceptions of the mean.

Methodology:

As mentioned above, the primary research question of this study was to determine how middle-grade students come to understand the arithmetic mean as a mathematical balance point. One of the initial hypotheses that I developed was that mathematical properties i and ii mentioned above needed to be constructed by students prior to their understanding of the mean as a mathematical balance point. I sought to understand whether this hypothesis was true for the participating subjects.

To examine the possible schemes and operations that students had available to them, the author engaged in a teaching experiment with the students. During the teaching experiment, the researcher also served as a teacher, posing tasks to students and building models of how the students were thinking about the mean. The primary purpose of any teaching experiment is to experience the reasoning and learning of students (Steffe & Thompson, 2000).

This teaching experiment included six students, ranging from fifth to seventh grade. The students worked in groups of two or three and met with the researcher for eleven meetings over the course of a seven-week period. Throughout the study, the students had access to TinkerPlots, a dynamic data analysis software program.  However, in addition to a number of computer-based tasks, the students also engaged in several paper-based and manipulative tasks.

During each session, students were video recorded as they worked on tasks using three separate cameras (one focusing on the computer screen, one focusing on student faces, and a third camera focusing on their table work). In addition, student work was collected and computer work was saved. The video recordings were reviewed, selected sessions were transcribed, and the transcripts were analyzed using open and axial coding techniques (Strauss & Corbin, 1998).

Data Sources:

Data for this study were drawn from 1) video recordings of the students working as they engaged with the mathematical tasks; 2) transcripts of the video recordings; and 3) students' paper-based and computer-based work.

Results:

Students' ability to reason about the mean as a mathematical balance point takes time for children to construct. However, using targeted tasks and intentional questioning allowed students to make progress toward this goal. At the beginning of the study, most students conceptualized the arithmetic mean as an algorithm. Several misconceptions about the mean were found; for example, the students often overestimated values of the mean, choosing numbers that exceeded the maximum value in the data set. Additionally, many of the students considered the mean as an additive structure. That is, they believed that they could raise the mean by 0.6 if they added a value of 0.6 to the data set. They did not recognize that adding a value to the data set below the initial mean would result in a lower mean.

By engaging students in tasks that addressed properties i and ii, the students developed computational understandings about the mean. By the end of the teaching experiment, most students were operating in ways that suggested they had developed an understanding of the mean as a mathematical balance point.

Educational Significance:

As a result of the teaching experiment, the author has developed a hypothetical learning trajectory (Simon, 1995) that characterizes how students can come to understand the mean as a mathematical balance point. This hypothetical learning trajectory has the potential to influence both educational practice and curriculum development for statistics education. Thus, this session is intended to provide an important link between research and practice.

Further research is needed that continues to unravel how students consider and construct meaning about the arithmetic mean. For example, as a result of this research, a new conjecture emerged that questions whether students' ability to make decisions about which measure of central tendency is most appropriate for given contexts is based on their understanding of the mean as a mathematical balance point.

References:

Mokros, J., & Russell, S. J. (1995). Children's concepts of average and representativeness. Journal for Research in Mathematics Education, 26, 20–39.

Simon, M. A. (1995). Reconstructing mathematics pedagogy from a constructivist perspective. Journal for Research in Mathematics Education, 26, 114-145.

Steffe, L. P., & Thompson, P.W. (2000). Teaching experiment methodology: Underlying principles and essential elements. In A.E. Kelly & R.A. Lesh (Eds.), Handbook of research design in mathematics and science education (pp. 267-306). Mahwah, NJ: Lawrence Erlbaum.

Strauss, A., & Corbin, J. (1998). Basics of qualitative research: Techniques and procedures for developing grounded theory. Thousand Oaks, CA: Sage Publishing.

Strauss, S. & Bichler, E. (1988). The development of children's concepts of the arithmetic average. Journal for Research in Mathematics Education, 19, 64-80.