Anna E. Bargagliotti

The impact of a technology-based mathematics after-school program using ALEKS on student's knowledge and behaviors

The effectiveness of using the Assessment and LEarning in Knowledge Spaces (ALEKS) system, an Intelligent Tutoring System for mathematics, as a method of strategic intervention in after-school settings to improve the mathematical skills of struggling students was examined using a randomized experimental design with two groups. As part of a 25-week program, student volunteers were randomly assigned to either a teacher-led classroom or a classroom in which students interacted with ALEKS while teachers were present. Students math performance, conduct, involvement, and assistance was needed to complete tasks were investigated to determine overall impact of the two programs. Students assigned to the ALEKS classrooms performed at the same level as students taught by expert teachers on the Tennessee Comprehensive Assessment Program (TCAP), which is given annually to all Tennessee students. Furthermore, students conduct and involvement remained at the same levels in both conditions. However, students in the ALEKS after-school classrooms required significantly less assistance in mathematics from teachers to complete their daily work.

What Can We Learn About Effective Early Mathematics Teaching? A Framework for Estimating Causal Effects Using Longitudinal Survey Data

Abstract This study investigates the impact of teacher characteristics and instructional strategies on the mathematics achievement of students in kindergarten and first grade and tackles the question of how best to use longitudinal survey data to elicit causal inference in the face of potential threats to validity due to nonrandom assignment to treatment. We develop a step-by-step approach to selecting a modeling and estimation strategy and find that teacher certification and courses in methods of teaching mathematics have a slightly negative effect on student achievement in kindergarten, whereas postgraduate education has a positive effect in first grade. Various teaching modalities, such as working with counting manipulatives, using math worksheets, and completing problems on the chalkboard, have positive effects on achievement in kindergarten, and pedagogical practices relating to explaining problem solving and working on problems from textbooks have positive effects on achievement in first grade. We show that the conclusions drawn depend on the estimation and modeling choices made and that several prior studies of teacher effects using longitudinal survey data likely neglected important features needed to establish causal inference.

Simulation of the Sampling Distribution of the Mean Can Mislead.

Although the use of simulation to teach the sampling distribution of the mean is meant to provide students with sound conceptual understanding, it may lead them astray. We discuss a misunderstanding that can be introduced or reinforced when students who intuitively understand that “bigger samples are better” conduct a simulation to explore the effect of sample size on the properties of the sampling distribution of the mean. From observing the patterns in a typical series of simulated sampling distributions constructed with increasing sample sizes, students reasonably—but incorrectly—conclude that, as the sample size, n, increases, the mean of the (exact) sampling distribution tends to get closer to the population mean and its variance tends to get closer to ś2/n, where ś2 is the population variance. We show that the patterns students observe are a consequence of the fact that both the variability in the mean and the variability in the variance of simulated sampling distributions constructed from the means of N random samples are inversely related, not only to N, but also to the size of each sample, n. Further, asking students to increase the number of repetitions, N, in the simulation does not change the patterns.

Aggregation and decision making using ranked data

Nonparametric procedures are frequently used to rank order alternatives. Often, information from several data sets must be aggregated to derive an overall ranking. When using nonparametric procedures, Simpson-like paradoxes can occur in which the conclusion drawn from the aggregate ranked data set seems contradictory to the conclusions drawn from the individual data sets. Extending previous results found in the literature for the Kruskal-Wallis test, this paper presents a strict condition that ranked data must satisfy in order to avoid this type of inconsistency when using nonparametric pairwise procedures or Bhapkars V procedure to extract an overall ranking. Aggregating ranked data poses further difficulties because there exist numerous ways to combine ranked data sets. This paper illustrates these difficulties and derives an upper bound for the number of possible ways that two ranked data sets can be combined.

How well do the NSF Funded Elementary Mathematics Curricula align with the GAISE report recommendations

Statistics and probability have become an integral part of mathematics education. Therefore it is important to understand whether curricular materials adequately represent statistical ideas. The Guidelines for Assessment and Instruction in Statistics Education (GAISE) report (Franklin, Kader, Mewborn, Moreno, Peck, Perry, & Scheaffer, 2007), endorsed by the American Statistical Association, provides a two-dimensional (process and level) framework for statistical learning. This paper examines whether the statistics content contained in the NSF funded elementary curricula Investigations in Number, Data, and Space, Math Trailblazers, and Everyday Mathematics aligns with the GAISE recommendations. Results indicate that there are differences in the approaches used as well as the GAISE components emphasized among the curricula. In light of the fact that the new Common Core State Standards have placed little emphasis in statistics in the elementary grades, it is important to ensure that the minimal amount of statistics that is presented aligns well with the recommendations put forth by the statistics community. The results in this paper provide insight as to the type of statistical preparation students receive when using the NSF funded elementary curricula. As the Common Core places great emphasis on statistics in the middle grades, these results can be used to inform whether students will be prepared for the middle school Common Core goals.

Statistical Significance of Ranking Paradoxes

When nonparametric statistical tests are used to rank-order a list of alternatives, Simpson-like paradoxes arise, in which the individual parts give rise to a common decision, but the aggregate of those parts gives rise to a different decision. Haunsperger (2003) and Bargagliotti (2009) showed that the Kruskal-Wallis (Kruskal and Wallis, 1952), Mann-Whitney (Mann and Whitney, 1947), and Bhapkars V (Bhapkar, 1961) nonparametric statistical tests are subject to these types of paradoxes. We further investigate these ranking paradoxes by showing that when they occur, the differences in rankings are not statistically significant.

Linear rank tests of uniformity: understanding inconsistent outcomes and the construction of new tests

Several nonparametric tests exist to test for differences among alternatives when using ranked data. Testing for differences among alternatives amounts to testing for uniformity over the set of possible permutations of the alternatives. Well-known tests of uniformity, such as the Friedman test or the Anderson test, are based on the impact of the usual limiting theorems (e.g. central limit theorem) and the results of different summary statistics (e.g. mean ranks, marginals, and pairwise ranks). Inconsistencies can occur among statistical tests’ outcomes – different statistical tests can yield different outcomes when applied to the same ranked data. In this paper, we describe a conceptual framework that naturally decomposes the underlying ranked data space. Using the framework, we explain why test results can differ and how their differences are related. In practice, one may choose a test based on the power or the structure of the ranked data. We discuss the implications of these choices and illustrate that for data meeting certain conditions, no existing test is effective in detecting nonuniformity. Finally, using a real data example, we illustrate how to construct new linear rank tests of uniformity.

Combinatorics and Statistical Issues Related to the Kruskal–Wallis Statistic

We explore criteria that data must meet in order for the Kruskal–Wallis test to reject the null hypothesis by computing the number of unique ranked datasets in the balanced case where each of the m alternatives has n observations. We show that the Kruskal–Wallis test tends to be conservative in rejecting the null hypothesis, and we offer a correction that improves its performance. We then compute the number of possible datasets producing unique rank-sums. The most commonly occurring data lead to an uncommonly small set of possible rank-sums. We extend prior findings about row- and column-ordered data structures.

The Statistical Education of Teachers: Preparing Teachers to Teach Statistics

19 In an increasingly data-driven world, statistical literacy is becoming an essential competency, not only for researchers conducting formal statistical analyses, but also for informed citizens making everyday decisions based on data. As Mark van der Laan recently wrote in Amstat News (see http://magazine. amstat.org/blog/2015/02/01/statscience_feb2015), “We need to take the field of statistics (i.e., the science of learning from data) seriously.” Because statistics and data are everywhere, it is imperative that we educate a population of data-literate people. If not in school, where are individuals expected to learn to be data literate? Teachers need to be able to teach students how to navigate the data world. They are the gatekeepers of the transfer of knowledge to young people. If teachers are the key, then they must be educated in statistics and on how to teach the subject at the different school levels. The Statistical Education of Teachers (SET) report (www.amstat.org/education/ SET/SET.pdf), commissioned by the American Statistical Association (ASA), addresses how teachers should be prepared in statistics. The purpose of this article is to describe the recommendations put forth in the report.

The Impact of a Mathematical Intelligent Tutoring System on Students’ Performance on Standardized High-Stake Tests

Recent research has suggested that standardized high-stakes tests, such as the SAT, have become increasingly important to policy makers, school districts, and society in general. Scoring well on these tests may determine the access to educational opportunities beyond high school. Unfortunately, recent reports have shown that Americans are falling behind their peers in other nations on comparable assessments (Gollub et al., 2002). Additionally, schools in the U.S. must adhere to the demands of the No Child Left Behind Act (NCLB). The policy states that federal district funding is dependent on student overall performance on standardized tests in mathematics, reading and other content areas. To alleviate the problem of U.S. students underachieving on standardized tests, educators must explore areas of pedagogy that have been empirically shown to be effective.