Browsing by Subject "conceptual understanding"
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Item Growing Certain: Students’ Mechanistic Reasoning about the Empirical Law of Large Numbers(2019-05) Brown, EthanExtensive research has documented students’ difficulty understanding and applying the Empirical Law of Large Numbers, the statistical principle that larger random samples result in more precise estimation. However, existing interventions appear to have had limited success, perhaps because they merely demonstrate the Empirical Law of Large Numbers rather than support students’ conceptual understanding of why this phenomenon occurs. This dissertation developed a sequence of activities, Growing Certain, which provided support for two mechanistic explanations of the Empirical Law of Large Numbers for students in a simulation-based introductory statistics course: swamping, the decreasing influence of extreme values on the mean as sample size increases, and heaping, the increasing concentration of possible sample means around the population mean. Five students participated in over six hours of one-on-one clinical interviews, with analysis focused on one focal participant, “S”. S’s responses were analyzed using a detailed coding of S’s articulation of mechanism components. S already displayed strong inclination towards swamping in the pre-interview questions, and their articulation of swamping became more sophisticated as they progressed in Growing Certain. However, S’s understanding of the connections between population and sample were weak throughout, and S had a lot of difficulty reasoning about multiple sample means simultaneously in a sampling distribution. S’s lack of abstraction of the sample mean appeared to support them in attending to the dynamics of swamping, but hindered them in being able to reason about heaping. Future research could examine representations that bridge swamping and heaping, and to examine individual differences in attention to the mechanistic components of the Empirical Law of Large Numbers.Item Implementing Standards-Based Mathematics: Toward Improving Conceptual Understanding of Key Ideas of Linear Functions for Middle School Immigrant Students(2016-07) Chidthachack, SousadaMathematics education research on immigrant students indicates that schools are often not prepared to support students with cultural and language barriers (Campbell, Adams, & Davis, 2007; Civil 2014). This descriptive case study documents the ways in which middle school immigrant students demonstrated understanding of key ideas of linear functions prior to and after participation in an eight-week Constructivist Teaching Experiment (CTE) (Cobb & Steffe, 2011; Cohen, Manion & Morrison, 2011). The purpose of this study is to show that all students-- regardless of their English language or mathematical proficiency ratings-- can learn mathematics with conceptual understanding (Hansen-Thomas, 2009; Silver & Stein, 1996; Whitney, 2010). Results prior to the CTE indicated the following. First, middle school immigrant students answered a majority of questions correctly without justifying when it was not required on the pre-assessment. And second, pre-assessment questions influenced the ways that middle school immigrant students communicated and used representations. Results after the CTE revealed three major findings. One, all middle school immigrant student participants increased their conceptual understanding on the post-assessment. Two, post-assessment questions with a real-life situation that asked for multiple representations influenced the ways that middle school immigrant students communicated and used representations. Three, all students filled in procedural and conceptual gaps after listening to peer presentations, individually and as a team. In summary, positive outcomes were associated with student-centered and problem-centered mathematics instruction that aligned with equity goals (National Council of Teachers of Mathematics, 2000).