Thomas J. Pfaff

Do Hands-On Activities Increase Student Understanding?: A Case Study.

This article describes the design, implementation, and assessment of four hands-on activities in an introductory college statistics course. In the activities, students investigated the ideas of the central limit theorem, confidence intervals, and hypothesis testing. Five assessments were administered to the students, one at the beginning and end of the course, and three in between the activities. We found that, despite our attempts to engage our students in active reflection, their performance on the assessments generally did not improve. These results raise important issues about the design of pedagogical tools and activities as well as the need to gather data to assess their effectiveness.

Measuring Resource Inequality: The Gini Coefficient

This paper stems from work done by the authors at the Mathematics for Social Justice Workshop held in June of 2007 at Middlebury College. We provide a description of the Gini coefficient and some discussion of how it can be used to promote quantitative literacy skills in mathematics courses. The Gini Coefficient was introduced in 1921 by Italian statistician Corrado Gini as a measure of inequality. It is defined as twice the area between two curves. One, the Lorenz curve for a given population with respect to a given resource, represents the cumulative percentage of the resource as a function of the cumulative percentage of the population that shares that percentage of the resource. The second curve is the line y = x which is the Lorenz curve for a population which shares the resource equally. The Gini coefficient can be interpreted as the percentage of inequality represented in the population with respect to the given resource. We propose that the Gini coefficient can be used to enhance students’ understanding of calculus concepts and provide practice for students in using both calculus and quantitative literacy skills. Our examples are based mainly on distribution of energy resources using publicly available data from the Energy Information Agency of the United States Government. For energy resources within the United States, we find that by household, the Gini coefficient is 0.346, while using the 51 data points represented by the states and Washington D.C., the Gini coefficient is 0.158. When we consider the countries of the world as a population of 210, the Gini coefficient is 0.670. We close with ideas for questions which can be posed to students and discussion of the experiences two other mathematics instructors have had incorporating the Gini coefficient into pre-calculus-level mathematics classes.

Using Informal Inferential Reasoning to Develop Formal Concepts: Analyzing an Activity

Inferential reasoning is a central component of statistics. Researchers have suggested that students should develop an informal understanding of the ideas that underlie inference before learning the concepts formally. This paper presents a hands-on activity that is designed to help students in an introductory statistics course draw informal inferences about a bag of bingo chips and connect these ideas to the formal T-test and confidence interval. This activity is analyzed using a framework and recommendations drawn from the research literature.

Sustainability themed problem solving in data structures and algorithms

During the past two years, we have been creating curricular material centered around complex problems rooted in sustainability. Since multi-disciplinary learning is one of our primary goals, these projects are most meaningful when they connect students from different disciplines working toward a common understanding. However, strong disciplinary components present in their solutions also allow us to frame these projects from strictly disciplinary perspectives. In this paper, we show how they can be used for increased engagement in the context of data structures and algorithms. We review two new ones to explore (i) the structural characteristics of the western part of the U.S. power-grid, and (ii) the effects of over-harvesting on fish stocks.

Use of satellite imagery in multidisciplinary projects

The most important problems of our century require collaborative work. Therefore, as educators, we must be producing individuals who understand how to communicate and work with professionals from other fields. In this paper, we outline our ongoing efforts in addressing this need, without compromising discipline specific qualities or goals. Specifically, we present a number of programming projects that fulfill curricular requirements as well as making students aware of the Science Technology Engineering and Math (STEM) context of their work.

Using sustainability themes and multidisciplinary approaches to enhance STEM education

Purpose – The purpose of this paper is to report on the Multidisciplinary Sustainability Education Project (MSEP) as a framework using sustainability-themed education modules to introduce students to the need for multidisciplinary approaches to solving twenty-first-century problems while retaining traditional course strengths and content. Design/methodology/approach – The MSEP uses sustainability-themed education models and a multidisciplinary approach to link courses across disciplines. Modules are identified by an overarching question with activities designed to address the overarching question from course-specific perspectives, resulting in students writing short technical reports summarizing their results. Students then read and evaluate technical reports from other classes, and complete a summary activity designed to connect perspectives from different disciplines. Findings – The multi-method assessment identified no loss or gain in discipline-specific learning; increased understanding about the char...

Educating about sustainability while enhancing calculus.

Abstract We give an overview of why it is important to include sustainability in mathematics classes and provide specific examples of how to do this for a calculus class. We illustrate that when students use Excel to fit curves to real data, fundamentally important questions about sustainability become calculus questions about those curves.

Go Figure: Calculus Students' Use of Figures and Graphs in Technical Report Writing

Understanding how to read and use graphs to communicate scientific and mathematical in-formation is critical for STEM majors, as well as an important part of quantitative literacy. Ourstudy suggests that first-semester calculus students do not know how to use graphs in a technicalreport without explicit instruction. Although not a surprising result, it leaves us wondering aboutwhen such skills are developed, and if calculus I is a place to start. Our work is now exploringthe potential benefit on students’ use of graphs by having them formally evaluate other students’reports.KEYWORDS: Calculus, Technical Writing, Quantitiative Literacy

A new approach to Macaulay posets

We develop a new approach for establishing the Macaulayness of posets representable as cartesian powers of other posets. This approach is based on a problem of constructing an ideal of maximum rank in a poset. Using the relations between the maximum rank ideal problem and the edge-isoperimetric problem on graphs we demonstrate an application of our approach to specification of all posets with a special Macaulay order. We also present a new general construction for additive Macaulay posets and introduce several new families of Macaulay posets.

Sustainability Education: the What and How for Mathematics

Abstract In this article we provide a simple way to think about the concept of sustainability and provide a number of examples for incorporating sustainability education into commonly taught mathematics courses. Scientific assessments have concluded that ecosystem services (the benefits that humans derive from the functioning of Earths natural systems) are in general decline worldwide. As a result, there is an increasing emphasis on incorporating sustainability principles into traditional STEM education. It is a challenge to offer expected content in our courses while adding this new socially and scientifically relevant information. However, it is important to do so because it is our obligation to help produce an educated citizenry. The mathematics community is strongly positioned to be a foundational part of this effort because of the large percentage of students that take at least one mathematics course, in particular, calculus and statistics. A concise set of five learning objectives for sustainability education can be overlain upon almost any mathematics course while not sacrificing appropriate content. Use of carefully constructed, data-rich, real-life examples allows for increased student engagement and provides for opportunities to pursue mathematics while focusing at least some of our efforts on the areas of greatest societal need.