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.
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Visualization offers students a window into the world of mathematics. Technology used in mathematics at the university level plays many roles in facilitating visualization. For example, in beginning calculus many classes use software such as Mathematica, Matlab, or Maple to help students to visualize graphs of functions, derivatives, integrals, and series. Animations are often available for use with these programs, and instructors or programmers may set up demonstrations to show students a visual illustration to help them understand the mathematics. Additionally, the development of Java applets to illustrate a specific idea in calculus, differential equations, non-Euclidean geometry, and other courses can provide an opportunity for students to visualize the intended mathematics. Over the past several years, we have been engaged in designing and studying innovative learning environments in differential equations that provide students with significant opportunities to visualize concepts as they reinvent key mathematical ideas (Rasmussen & Kwon, 2007). The use of specially designed technological tools has been a central component of these efforts. The purpose of this chapter is to illustrate how these technological tools function as boundary objects between the classroom community and the broader mathematical community.
As illustrated in the communities of practice literature (Lave & Wenger, 1991; Wenger, 1998), boundary objects are material things that interface two or more communities of practice and satisfy the informational requirements of all of them (Bowker & Star, 1999; Lee, 2007). A community of practice is a collective construct in which the joint enterprise of achieving particular goals, such as doing mathematics, evolves within the social connections of that particular group (Lave & Wenger, 1991; Wenger, 1998). Boundary objects can be mathematical symbols, technology, documents, software, or other items that allow people to cross between different communities and work together. Hoyles, Noss, Kent, Bakker (2010) define Technology-Enhanced Boundary Objects (TEBOs) as, “software tools that adapt or extend symbolic artifacts 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). We adapt this idea, which was actually developed to study learning in the workplace, to the undergraduate mathematics classroom and hence use the phrase “classroom TEBO” to refer to a piece of software that acts as a boundary object between the classroom community and the mathematical community. Classroom TEBOs serve the purpose of promoting student learning and enculturation into the discipline of mathematics. It is important to note that a classroom TEBO is only a boundary object as the students’ interact with it, other students, and the teacher, who in this case is a representative of the mathematics community.
As mathematics instructors may have noticed, most off-the-shelf software tools illustrate the mathematics in more or less the form used by experts. For example, when a solution to a differential equation is given, expert software typically allows students to chose an initial condition and then the graph (or a close approximate) is shown. Analytic solutions, when they exist, might also be provided. There is no need for students to consider the form or context of the differential equation as the complete answer is there in front of them. Moreover, students tend to be passive observers. This might be fine for experts, as they have likely solved the problem for themselves and have deep understanding as the basis for the answer. When they see the answer, there is already a firm background that supports the answer. After students use this type of software, however, they may not have anything more than the answer! Typical examples of such traditional use of technology in education include those from Perge (2007), who shows how to use Maple to plot solutions to a second order differential equation that models a forced pendulum, and those of Maat and Zakaria (2010), who use Maple to symbolically solve differential equations.