Identifying Developmental Levels and Learning Trajectories in Statistics for Grade 6-12 Students
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Since the early 80s, reform movements have recommended increasing the content and rigor of statistics in school mathematics curriculum. Two important curriculum documents, the Guidelines for Assessment and Instruction in Statistics Education (GAISE) Report: A Pre-K-12 Curriculum Framework and the Common Core State Standards in Mathematics (CCSS-M) provide detailed descriptions of what students should know and should be able to do in statistics (Franklin et al., 2007; National Governors Association Center for Best Practices, and Council of Chief State School Officers, 2010). These descriptions are based on the hypotheses of learning trajectories of statistical concepts. There is a need to empirically understand the learning development and growth of statistical concepts, particularly those that are related to the investigation cycle in statistics (formulating questions, collecting data, analyzing data, and interpreting results). Understanding the learning trajectories of statistics is important for instruction and assessment of statistical concepts. This study aims to understand how students learn statistics by describing the developmental growth of students’ understanding of statistical concepts and learning trajectories of several concepts in statistics. To reach this goal, an instrument that measures students’ developmental levels in learning statistics and learning trajectories of several statistical concepts was developed. The instrument was administered to 797 high school and middle school students in Central Texas. Three methods of data analyses were conducted: (1) a basic psychometric evaluation of the instrument using classical test theory (CTT), (2) explanatory and confirmatory factor analysis and latent regression analysis by applying structural equation modeling (SEM) approaches, and (3) ordinal regression analysis.
Five structural equation models were developed following the Pre-K-12 GAISE Framework and the CCSS-M that aligned each item in the instrument to the appropriate GAISE developmental level (Level A, Level B, or Level C). The results demonstrate an acceptable fit of the model to the empirical data. The results indicate that the Pre-K-12 GAISE Framework’s suggestions that students develop understanding of statistics through three hierarchical levels were supported by the data.
The descriptive analysis results also demonstrate that students who participated in this study performed well on items measuring the statistical process component of formulating questions and collecting data. Students showed lower performance on items measuring the process component of analyzing data and understanding the nature of variability. The results also indicate that students, who have developed into Level B in the areas of formulating questions, collecting data, analyzing data, and interpreting results, might not necessarily have developed into Level B in understanding variability. For all process components excepting the nature of variability process component, the patterns tend to be similar, where items that measure lower GAISE levels have higher percentage of correct answers (lower difficulty indices) than items that measure higher GAISE levels.
The results of ordinal regression analysis reveal that the more advanced the grade levels and the latest mathematics courses taken, the higher the students’ GAISE level. This indicates that students who have better preparation in mathematics tend to have higher GAISE levels. The items were split into two forms - FORM 1 and FORM 2; the ordinal regression result also reveals that FORM 1 is more sensitive in identifying Level A students than Level B and Level C students compared to FORM 2. The results, however, showed that there is no clear relation between students’ GAISE Levels and their ages.