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## Research on Students’ Thinking and Reasoning about Averages and Measures of Center Brief

The statistical concepts of average, particularly middles or means, are very powerful in statistics, since on the one hand measures of center are often used in a descriptive role to summarize information about a data set. On the other hand, if a data set is a sample that has been appropriately drawn from a parent population, the sample data might be expected to “mirror” the parent population, and thus the mean of that sample provides some information about the (unknown) mean of the entire population from which the sample was drawn. Better yet, collections of samples and their means can furnish a “likely range” within which the actual unknown population mean is probably located. Measures of center thus can play not only a descriptive role but also an inferential role, since we use information from samples to infer information about populations or to make comparisons between populations. How do our students actually tend to think about the concept of “average”?

Mokros and Russell (1995) interviewed students in grades 4, 6, and 8 in “messy data” situations, using contexts like allowance money and food prices that were familiar to students. These students had been taught the procedure for finding the arithmetic average, so they had some familiarity with computing means. However, Mokros and Russell asked students to work backward from a given mean to some possibilities for the data set that could have produced that mean. For example, in one problem students were told that the mean cost of a bag of potato chips was \$1.35, and then they were asked to construct a collection of ten bags whose prices had a mean of \$1.35. In searching for students’ own preferred strategies while they worked on statistical tasks involving averages, Mokros and Russell identified five different conceptions of average among the students they interviewed: average as mode, average as algorithm, average as reasonable, average as midpoint, and average as point of balance.

Some of these conceptions of average prove to be impoverished, whereas others can lead to students’ developing higher levels of thinking about data.  Students who focus primarily on modes in data sets have difficulty working backward from the mean to construct a data distribution that has that mean, especially if they are not allowed to use the mean value itself as a data value. Mokros and Russell concluded that such modal-thinking students don’t see the whole data set as an entity; they can focus only on individual data values. They also found that students who think of average primarily as an algorithm aren’t able to make connections between their computational procedures and the original context of the data. Students who think of average as reasonable tend to believe that it is an approximation, not something that one can compute. Even some of the students who had more powerful conceptions of average, such as the midpoint or as the point of balance (though Mokros and Russell found the latter conception to be rare among their interviewees), had difficulty re-creating a set of bags of potato chips with an average price of \$1.35 if they weren’t allowed to use \$1.35 as a data point. One conclusion we can draw from Mokros and Russell’s work is that computational facility with average does not guarantee that conceptual understanding or contextual connections about average will follow in our students. We have to create opportunities for our students to connect back to the original context and to interpret what their computations mean in light of the context.

Konold and Pollatsek (2002) and Waton and Moritz (2000a, 2000b) provide us with further evidence for the variety of ways that students think about the meaning of measures of center. Watson and Moritz’s findings suggest that students think predominantly of “middle” when they are asked what average means. For example, when asked what it means for a student to be average, or what average means in the context of “the average wage earner can afford to buy the average home,” students most frequently referenced “middles,” and then “most,” with the mean being a distant third. Watson and Moritz offered strong evidence with a large sample of students that there are developmental trajectories for students’ understanding of the concept of average. They suggest that students’ conceptual development of average starts with idiosyncratic stories, proceeds to everyday colloquial ideas, then to “mosts” and “middles,” and finally to the mean as a representative of a data set.

In their work with secondary and postsecondary students, Konold and Pollatsek postulated four conceptual perspectives for the mean: (1) mean as typical value, (2) mean as fair share, (3) mean as a way to reduce data, and (4) mean as a signal amid noise. From a statistical point of view they argue that “signal amid noise” is the most important and most useful way to think about the mean when comparing two or more data sets. In fact, they recommend that the mean should be first introduced to students in the context of comparing two or more data sets. Children’s thinking of average as a typical value arises naturally from their experience, as documented by Watson and Moritz and by Mokros and Russell. Thus, “mean as typical” may be a good starting point for teachers to connect to students’ own informal knowledge, and “mean as fair share” can provide a conceptual platform for connecting to the algorithm for finding the mean. For example, “leveling” stacks of cubes of varying heights to make all the stacks of equal height (fair shares) allows students to uncover the algorithm for computing the mean as well as to generate alternative algorithms (Foreman and Bennett 1995). However, for Konold and Pollatsek, these two conceptions, mean as typical and mean as fair share, are limited and closely tied to the “data analysis” part of statistics, whereas the latter two conceptions, mean as data reducer and mean as signal amid noise, are connected to the “inference and decision making” part of statistics.

In decision-making from data, the process of data reduction is crucial in order to locate an informative signal amid the noise of the variability in data. Thus statistics often reports the mean of a data set as a “‘representative” for an entire data set. Means also furnish a useful signal for making inferences from samples to entire populations and for comparing multiple data sets. Konold and Pollatsek argue that students do not naturally gravitate to using means to compare data sets or to make inferences from samples to populations and that instruction needs to help students grow past their initial informal conceptions of average by concentrating on the “data analysis and decision making” perspective for averages in which the mean is a representative of a data set. They claim that thinking of average as “typical” or as “fair share” does not provide a helpful basis for making group comparisons, whereas the “data reduction” or “signal amid noise” conceptions are more powerful tools in such inferential settings. Konold and Polletsek’s work suggests that students’ initial informal conceptions of average as “typical value” or a “fair share” may impede their conceptual development unless teachers help them to move toward more conceptually rich notions of average, such as average as “representative” or average as “signal.”

The work of Mokros and Russell, Watson and Moritz, and Konold and Pollatsek presents developmental pathways of students’ conceptions of average and suggests conceptual bases to help create teaching and learning trajectories for the concept of the mean. Their work also clearly points out that in fact students do have a rich variety of conceptions of average that we can build on. However, students’ conceptions are in transition, and thus they may not understand the important differences in concepts like mean and median, or when the use of a certain measure of center is most appropriate. The teacher plays a critical role in helping students to parse out the best appropriate uses of measures of center (e.g., Zawojewski and Shaughnessy 2000).

We should add one caveat here, lest we become too caught up with the concept of average separate from the rest of statistics. Averages do not exist independent of the distributions of data that they summarize, and in that summary, averages alone can mask a lot of information, namely, the “noise,” or variability, in a data set. Centers are only one aspect of a distribution; shape and variability are just as important both in describing data and in aiding in inferential decision making. A close look at the school statistics curriculum, particularly in the United States, reveals that far more time is spent in school on notions of average than on variability or on the shape of data distributions. Often the noise in data, the variability, supplies some essential information that can become lost in a summary statistic like an average (see Shaughnessy and Pfannkuch [2002]). Thus, it is also important for teachers and students to work on describing and analyzing the “noise” in data, to explore variability in data, and not to limit themselves to finding measures of center. We recommend that you also read the companion NCTM research brief to this one that discusses the importance of students’ understanding of variability.

References

Foreman, Linda C., and Albert B. Bennett, Jr. Math Alive Course I. Salem, Oreg.: Math Learning Center, 1995.

Konold, Clifford, and Alexander Pollatsek. “Data Analysis as the Search for Signals in Noisy Processes.” Journal for Research in Mathematics Education 33 (July 2002): 259 –89.

Mokros, Jan. and Susan J. Russell. “Children’s Concepts of Average and Representativeness.” Journal for Research in Mathematics Education 26 (January 1995): 20–39.

Shaughnessy, J. Michael., and Maxine Pfannkuch. “How Faithful Is Old Faithful? Statistical Thinking: A Story of Variation and Prediction.” Mathematics Teacher 95 (April 2002): 252–59.

Watson, Jane M., and Jonathan B. Moritz. “Developing Concepts of Sampling.” Journal for Research in Mathematics Education 31 (January 2000a): 44–70.

———. “The Longitudinal Development of Understanding of Average.” Mathematical Thinking and Learning 2 (2000b): 11–50.

Zawojewski, Judith S., and J. Michael Shaughnessy. “Mean and Median: Are They Really So Easy?” Mathematics Teaching in the Middle School 5 (March 2000): 436–40.