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Article Excerpt
In this paper we focus on interactive visualizations that
support mathematical investigation. In the context of analyzing and redesigning an interactive visualization from the National Council of Teachers of Mathematic's (NCTM's) Illuminations Web site, we consider two interrelated issues: (1) how the pedagogy of mathematical investigation may be manifested in a mathematical interactive visualization, and (2) how interface design principles may interactive visualization's potential for supporting mathematical investigation. Mathematical investigation, as a pedagogical tool, is not a simple undertaking. Facilitating investigation adds significantly to the complexity of instructional design. Good design becomes possible when mathematics education and human-computer interaction design experts work together, rather than in isolation, taking into account pedagogical goals and interface design principles.
Online interactive visualizations (IVs) are the most recent manifestation of technological tools used in mathematics education to support mathematical investigation. Current K-12 mathematics curriculum documents, such as the NCTM Standards and the Ontario mathematics curriculum (NCTM 2000; Ontario Ministry of Education 1997, 1999, 2000), value mathematical investigation based on the pedagogical belief that students learn mathematics best when they have opportunities to be active learners and construct personal understandings of mathematical concepts and relationships. There are different types of online activities. Some activities focus on drill-and-practice sequences and games, brief tutorials, or tightly controlled simulations, where students are more likely to be passive learners. In this paper we focus exclusively on IVs that support mathematical investigation. It must be noted that not all online activities fall into this category.
IVs are part of a larger class of Web-based learning tools, currently referred to as learning objects. A learning object is "any digital entity designed to meet a specific learning outcome that can be reused to support learning" (CLOE 2003). Online IVs are unique in their focus on small interactive units, due in large part to the current nature of Web access where large programs are unwieldy and quickly exhaust download capabilities and user patience. Well designed mathematical IVs may enable students to engage in investigations of mathematical relationships without having to spend a lot of time learning how to use the tool that creates the various representations of these relationships. "The power of the unaided mind is highly overrated" and representations may be seen as "aids that enhance cognitive abilities" (Norman, 1993). It is possible that computer-based representations "can make abstract concepts concrete and manipulative, reveal their properties and constraints, relate them to everyday situations they represent, and connect them to other representations of the same information" (Vosniadou, 1996). "While physical objects become more abstract when modeled on-screen (e.g. science simulations), mathematical objects, already inherently abstract, become more concrete" (Lester, 2000). Such cognitive tools may reduce cognitive load by taking over some of the more mundane elements of a task (Kieran, Boileau & Garancon, 1996; Lajoie, 1993; Surgue, 2000) and thus help support student mathematical investigation.
In this paper we consider two interrelated issues: (1) how the pedagogy of mathematical investigation may be manifested in a mathematical IV, and (2) how interface design principles may be used to enhance an IV's potential for supporting mathematical investigation. We start by considering the Side Length, Volume, and Surface Area of Similar Solids IV (see Figure 1) found on the Illuminations Web site of the National Council of Teachers of Mathematics (NCTM, 2003). We chose this IV for two reasons. First, it provides a concrete example on which to base our analysis of the pedagogical intent of mathematical investigation and its manifestation in an IV, and the mediating role of interface research-based design principles. Second, NCTM offers this IV, along with several others, as models of the mathematics content and pedagogy outlined in its Standards document (NCTM, 2000). Below, we analyze and redesign this IV, drawing attention to its pedagogical intent and design, and its use of interface design principles. Although pedagogy and interface design are inextricably linked, we will initially attend to pedagogy and interface design issues separately, for the sake of discussion and analysis clarity, bringing them together again near the end of the paper.
[FIGURE 1 OMITTED]
Our working assumption is that the pedagogical and interface design of an IV, like the design of any other instructional tool or method, greatly affects how students experience mathematics and consequently what mathematics they learn. IVs are neither neutral nor are they uniform in their effect on student learning. How well a learning outcome is met when an IV is used depends on (1) the pedagogical model embedded in the IV, and (2) the interface design that mediates student engagement. What mathematics students learn when using an IV very much depends on how the IV engages students with mathematics. For example, an IV that allows students to change the coefficients of a linear function and observe changes in its graphical representation would afford very different mathematical engagement than an IV that takes the form of a PowerPoint presentation. Likewise, an IV whose interface allows for direct manipulation of mathematical objects would afford very different mathematical engagement than an IV using textual commands as an intermediary for manipulation.
PEDAGOGICAL DESIGN
In this section we compare and contrast the pedagogical intent and the embedded pedagogical model of the Side Length, Volume, and Surface Area of Similar Solids IV. To identify intent, we rely on the support materials that accompany the IV, that is, the goal statements and learning and teaching notes found on the Web page that contains the IV. These pedagogical goals are then used to analyze the design of the IV.
The Pedagogical Intent of the Interactive Visualization
In discussing pedagogical intent, we consider (1) the mathematical content addressed and (2) the process by which this content is to be taught and learned.
Mathematical content. The mathematical content of the Side Length, Volume, and Surface Area of Similar Solids IV is drawn from NCTM's (2000) Geometry Standard. The focus is on the relationships among the edge length, surface area, and volume of similar rectangular prisms. This is reflected in the type of mathematics questions that students are expected to answer when using the IV (NCTM, 2003):
* How does changing the lengths of the sides of a rectangular prism affect the volume and surface area of the prism?
* How does the scale factor relate to side length, volume, and surface area?
* Why does the graph depicting the relationship between side length and surface area differ from the graph depicting the relationship between side length and volume?
* How do the volume and surface area scale factors differ?
* Why is the relationship between side length and volume cubic whereas the relationship between side length and surface area is quadratic?
Although not explicitly mentioned in the stated goals, the IV also addresses algebra outcomes found in the Standards document. NCTM's view of algebra emphasizes multiple representations of relationships between quantities, and a focus of student attention on the mathematical analysis of change in these relationships (NCTM, 2000). All of the questions listed above address algebra outcomes. In fact, one could argue that algebra is the main curriculum focus of the IV and that geometry serves as a context for investigation.
The image of mathematics portrayed by the above questions focuses on making sense of mathematical connections and relationships. This image stands in contrast to a more traditional view of mathematics where students learn procedures and use them to get the right answers (McGowen & Davis, 2001a; McGowen & Davis, 2001b; Romberg, 1992). More specifically,...
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