PROSPECTIVE TEACHERS' PROCEDURAL AND CONCEPTUAL KNOWLEDGE OF MEAN ABSOLUTE DEVIATION.

Groth, Randall E.

The recommendation to study statistical variation has become prevalent in recent curriculum documents. At the same time, research on teachers' knowledge of variation is in its beginning stages. This study investigated prospective teachers' knowledge in regard to a specific measure of statistical variation that is new to many curriculum documents: the mean absolute deviation (MAD). Seventy-six prospective teachers participated in the study. Participants exhibited various procedural and conceptual characteristics in their thinking about the MAD. The majority were able to successfully select and carry out a procedure for computing the MAD. However, some had difficulty dealing with procedures for absolute deviations, and others confused the procedure for the MAD with the procedure for a different descriptive statistic. Conceptually, participants offered a variety of interpretations of the MAD, with some demonstrating deep understanding of the measure and others demonstrating shallower understanding or misconceptions. Those who demonstrated the strongest conceptual knowledge of the MAD also exhibited sound procedural understanding, suggesting that the two types of knowledge are intertwined in the process of fully understanding the measure. Statistical variation has become widely acknowledged as an important object of study for school curricula because of the role it plays in statistical thinking. Cobb and Moore (1997) argued that the discipline of statistics exists because of the "omnipresence of variability" as seen in situations such as variation among individuals and repeated measurements. Wild and Pfannkuch (1999) expanded on the notion of the omnipresence of variability, stating, "Variation is an observable reality. It is present everywhere and in everything. Variability affects all aspects of life and everything we observe. No two manufactured items are identical, no two organisms are identical or react in identical ways" (p. 235). Garfield and Ben-Zvi (2008) argued that the study of variability should permeate statistics curricula, and that understanding variability is necessary for learning the concepts of distribution and statistical inference. Such observations about variability in relation to statistical thinking are reflected in its central role in various curriculum documents (Aliagea et al., 2005; Franklin et al., 2007; National Governors Association for Best Practices & Council of Chief State School Officers, 2010). An important part of the study of statistical variation is its quantification. Statisticians have developed several measures to quantify the variability of data, such as the range, interquartile range, and standard deviation. In this report, I focus primarily on one particular measure: the mean absolute deviation (MAD). It is the mean distance between the individual data values in a data set and the mean of the data set. Recommendations to include the MAD as an object of study in middle school curricula are relatively new (Franklin et al., 2007, National Governors Association for Best Practices & Council of Chief State School Officers, 2010). A primary rationale for its inclusion in middle school curricula is that the MAD can serve as a precursor to the study of standard deviation at the secondary or college level (Kader & Mamer, 2008). The mathematical structure of the MAD is similar to that of standard deviation, but does not require calculations of squares or square roots. Because of this, the MAD might provide a more intuitive introduction to the idea of measuring the typical distance data values are from the center of a distribution. The recent inclusion of the MAD in middle school curricula raises questions about teachers' knowledge. Sanchez, da Silva, and Coutinho (2011) observed that research on teachers' understanding of variation is scarce. They identified an urgent need for further research in this area. The need to understand teachers' knowledge of the MAD is particularly pressing at this time, as many sixth and seventh-grade teachers in the U.S. are now required to teach it as part of the Common Core State Standards (National Governors Association for Best Practices & Council of Chief State School Officers, 2010). In order to support teachers in implementing instruction related to the MAD, it is optimal for teacher educators and researchers to be able to anticipate reasoning patterns and potential difficulties teachers may have with the idea. The current scarcity of relevant research makes it difficult for them to do so. This study thus aims to fill a gap in the literature by helping to characterize teachers' knowledge of the MAD.

MATHEMATICS education; MATHEMATICS students; MATHEMATICAL ability; MATHEMATICS teachers; CURRICULUM

Investigations in Mathematics Learning, 2014, Vol 6, Issue 3, p51

1947-7503

Academic Journal

10.1080/24727466.2014.11790335