Supporting the Development of College-Level Students’ Conceptions of Statistical Inference

The transition from descriptive to inferential statistics is a known area of difficulty for students taking introductory statistics courses. This chapter shares the experiences from a teaching experiment in a college-level introductory statistics classroom that implemented an informal, data-driven approach to statistical inference using the dynamic statistics software Fathom© as an investigation tool. Findings from the study indicate that the informal inferences on which instruction focused in the first part of the course helped students develop understandings of fundamental aspects of inferential and argumentative reasoning that served as foundations for the formal study of inferential statistics in the latter part of the course. The affordances offered by the tool for delving deeply into the data to make sense of the situation at hand were instrumental in supporting student understanding of both informal and formal inferential statistics.

Coping with Infinity

This chapter outlines the structure of the initial module of a pilot project investigating the benefits of using dynamical mathematical software when teaching students about cross-curricular models of thinking. Focus is on the use of conceptual metaphors when teaching the concept of infinity in mathematics as well as in literature. The use of dynamical mathematics software facilitated constructions of multiple representations in problem solving in a series of carefully selected problems involving infinity. The pilot project is currently being evaluated, but preliminary findings indicate an increased student awareness of the usefulness of the software when learning how to use mathematical inscriptions for problem solving. The pilot project will be expanded in the Fall 2011 to include 6 classes across Denmark to test the applicability of the software, the teaching model and the teaching materials in a variety of classes to see if the initial successes of the project can be replicated.

Vectors and Differential Equations

This chapter is inspired by a session the author gave at the 10th EMAC Conference (Engineering, Mathematics, and Applications) at the University of Technology, Sydney, in December 2011. The audience was university teachers, but the software, Autograph, was designed for use in High Schools. The author was able to show how a simple, pedagogically focused interface could be used to create a highly visual approach to the teaching of two favourite topics: Vectors (in 2D and 3D) and Differential Equations (1st and 2nd Order).

Dynamical Mathematical Software

Understanding mathematical concepts is many-folded. Traditional mathematics mostly emphasizes the algebraic/analytical aspect of a problem with minimal reference to its graphical aspect and/or numerical one. In a modern learning environment, however, multiple representations of concepts are proving to be essential for the teaching of mathematics. The availability of user-friendly dynamical software programs has paved the way for a radical yet smooth way for changing the way mathematical concepts are perceived. This chapter presents some of the author’s attempts for employing innovative methods in teaching topics in calculus, in differential and difference equations. The focus is on the use of dynamical programs that boost the visual component of the topics being investigated, hence contributing to a more complete understanding of these topics.

“Click, Drag, Think!”

The author points out that to fully exploit the heuristic potential of Dynamic Geometry Software (DGS) and to increase the heuristic literacy of students, extant DGS teaching units have to be ameliorated in several ways. Thus the author develops a twofold conceptual framework: heuristic reconstruction and heuristic instrumentation of problems. Its origin is rooted in the literature, its use is demonstrated by various examples, and its value is made plausible by an introductory teaching unit an advanced case study.

Sometimes Less is More

As described in the communities of practice literature (Lave & Wenger, 1991; Wenger, 1998), boundary objects are material things that interface two or more communities of practice. Extending this, Hoyles, Noss, Kent, and Bakker (2010) defined technology-enhanced boundary objects as, “software tools that adapt or extend symbolic artefacts identified from existing work practice, that are intended to act as boundary objects, for the purposes of employees’ learning and enhancing workplace communication” (p. 17). The authors adapt this idea to the undergraduate mathematics classroom and use the phrase “classroom technology-enhanced boundary object” to refer to a piece of software that acts as a boundary object between the classroom community and the mathematical community. They provide three extended examples of these objects as used in a first semester differential equations classroom to illustrate how students’ mathematical activity may advance as they interact with the software. These examples show how the applets operate to provide a way for the classroom community to implicitly encounter the mathematical community through the authentic practices of mathematics (Rasmussen, Zandieh, King, & Teppo, 2005). The first example centers on students beginning experience with a tangent vector field applet. The second example develops as the students learn more about solutions to differential equations and leads to a statement of the uniqueness theorem. In the third example, students use a specially designed applet that creates a numerical approximation and its associated image in 3-space relating to a non-technological visualization task that introduces solutions to systems of differential equations.

String Art and Linear Iterative Systems

String art dates back to the 19th century when it was initially invented to ease the delivery of some mathematical ideas. Since then, the art has evolved and so has its use in mathematics. In this chapter, the authors see how some of the phase portraits for 2 x 2 linear homogeneous iterative systems exhibit some artistic behavior that resembles this form of art. The investigation gives a sufficient condition for the solutions of such systems to form closed cycles. However, in other situations the cycles formed are infinite, producing some fascinating examples of string art.

Interactive Applets in Calculus and Engineering Courses

This chapter reports on the use of specialized computer applets (“Mathlets”) in two different contexts: on-line instruction in calculus through MIT OpenCourseWare and on-campus laboratory exercises on stability of difference schemes in class. Specific applets are described. Three use cases with varying levels of sophistication (elementary, intermediate, and advanced) are outlined.

Dynamical Software and the Derivative Concept

Modern teaching trends impose the need of spending less time on the manipulative approach to differential and integral calculus, putting the accent on the conceptual understanding of the subject. This chapter presents the standard approach and method used to teach the derivative of a function and indicates some critical points in the teaching of the derivative, offering, at the same time, suggestions for overcoming them. As a supplement, the author gives e-resources that can make possible the implementation of a stimulating, visual, dynamic, and broadened method for teaching the derivative of a function.

Nonlinear is Essential, Linearization is Not Enough, Visualization is Absolutely Necessary

Linear differential equations have been well understood for some time and are an important tool for studying the nonlinear systems that most frequently arise in mathematical models of real world systems. Nonlinear systems do not usually have formula solutions, but with graphics, we can see the behaviors of the solutions and thereby “understand” the differential equations. Dynamic and interactive presentations provide students with a streamlined route to understanding these behaviors, resulting in immense power and efficiency that was not previously available at the undergraduate level.