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Article Excerpt
Computer-based mathematical cognitive tools (MCTs) are a category of external aids intended to support and enhance learning and cognitive processes of learners. MCTs often contain interactive visual mathematical representations (VMRs), where VMRs are graphical representations that encode properties and relationships of mathematical concepts. In these tools, interaction enables learners to perform epistemic actions on VMRs to explore and learn mathematical concepts. Interactivity of VMRs refers to the feel, form, properties, and quality of this interaction. As such, interactivity of VMRs can influence how and what learners learn. A number of factors affect learners' cognitive processes while interacting with VMRs. Researchers from several disciplines have attempted to characterize interactivity and the multiplicity of factors that affect it. However, as many of these characterizations and factors are inapplicable to VMR-based MCTs, understanding of the factors that affect learning and cognitive processes can help in the analysis of interactive VMRs. This article draws on research from various disciplines to identify and describe the applicability of 12 interactivity factors that affect learning and cognitive processes of learners who use VMR-based MCTs. Collectively, the factors can then serve as a descriptive and conceptual framework to help in the design and evaluation of MCTs and to allow designers to discuss and substantiate their design choices of interactive VMRs.
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The cognitive perspective of learning primarily deals with how the mind interacts with external information and how it processes the information internally (Eysenck & Keane, 1990; Thagard, 1996). In this perspective, learning involves a set of internal cognitive processes such as perceiving, attending, recalling, reasoning, interpreting, evaluating, sense-making, decision-making, and understanding (Ormrod, 1995).
External cognitive aids and artifacts, such as representations and tools, greatly influence and affect learning and cognitive processes (Zhang & Norman, 1994; Glasgow, Narayanan, & Chandrasekaran, 1995; Hutchins, 1995; Scaife & Rogers, 1996; Arcavi & Hadas, 2000; National Council of Teachers of Mathematics [NCTM], 2000; Jonassen, 2003). Computer-based cognitive tools are a category of external aids that can support cognitive, knowledge-based tasks, that is, support performing epistemic actions on information. These tools can support epistemic activities by enhancing, amplifying, transforming, and guiding cognitive processes of learners (Norman, 1993; Pea, 1993; Card, MacKinlay, & Shneiderman, 1999; Lajoie, 2000; Beynon, Nehaniv, & Dautenhahn, 2001; Jonassen, 2003). By sharing and distributing the cognitive processing load and activity of learners, these tools can transform the way cognitive tasks are performed (Kieran, Boileau, & Garancon, 1996; Lajoie, 2000; de Leon, 2002; Yerushalmy, 2004). Cognitive tools can act as partners in cognitive activities of learners (Salomon, Perkins, & Globerson, 1991), and their interface can act as an augmentative prosthetic support for the perceptive and cognitive capabilities of learners (Stojanov & Stojanoski, 2001). To support epistemic activities of learners, many cognitive tools provide structural, logical, and visuospatial formalisms of ideas and concepts in the form of interactive representations (Jonassen & Carr, 2000).
Mathematical cognitive tools (MCTs) are a subset of cognitive tools that allow learners to investigate and explore mathematical information (Sedig, 2004). Similar to other forms of epistemic activities, much of mathematical learning and thinking also involves working with and elaborating on representations in order to understand them or use them for problem solving (NCTM, 2000; Pimm, 1995; Cuoco & Curcio, 2001). Visual mathematical representations (VMRs) constitute an important subset of mathematical representations. VMRs are used extensively to support the investigation and exploration of mathematical ideas--both in static, noninteractive media such as books, as well as in interactive media such as MCTs (English, 1997; Duval, 1999; Whiteley, 2002; Sedig & Sumner, in press). VMRs are graphical representations that encode causal, functional, structural, logical, and semantic properties and relationships of mathematical structures, objects, concepts, problems, patterns, and ideas (Hitt, 2002; Sedig & Sumner, in press). Some examples of VMRs include 2D and 3D visualizations of geometric structures, patterns, graphs, and diagrams.
MCTs incorporating interactive VMRs that can effectively support learning and cognitive processes of learners are difficult to design. To design epistemologically sound MCTs, designers of these tools need to consider several interrelated issues, such as the pedagogical goals of the tool, the kinds of VMRs used, the possible techniques for interacting with VMRs, and the degree and types of interactivity or cognitive support (Sedig, Klawe, & Westrom, 2001; Gadanidis, Sedig, & Liang, 2004; Sedig, 2004; Sedig & Sumner, in press). Designers of MCTs need to make critical decisions about how much cognitive support is appropriate for effective learning. A few studies indicate that there are interactivity factors that affect learners' cognitive processes and overall learning (Jackson, Krajcik, & Soloway, 1998; Sedig et al., 2001; Sedig, Rowhani, Morey, & Liang, 2003; Travaglini, 2003; Chai, 2003). For example, Sedig et al. (2001) conducted a study in which a few interactivity factors of 2D transformation geometry VMRs were operationalized differently. They found that these factors significantly affected children's processing of the information and their degree of learning. (1) Therefore, characterizing the interactivity factors that affect learners' cognitive processes and learning is an educational imperative, as it can help designers of MCTs to design and analyze these tools in a systematic fashion.
Part of the challenge of analysis and design of MCTs is that there is a lack of a proper conceptual framework and a common language describing and characterizing the interactivity of VMRs. Such a framework can help designers clarify their thinking and provide a vocabulary for them to express and substantiate their design decisions. Research in visual reasoning, cognitive technologies, information visualization, and human-computer interaction design has identified a number of factors that affect cognition. Many of these factors apply to interactive VMRs. Unfortunately, they are discussed and reported in different research domains and are scattered across different bodies of literature. Often, the characterizations and descriptions of these factors are bound to the domain in which they are discussed, making it difficult to apply them to the design of interactive VMRs (Sedig & Sumner, in press). Knowing what factors affect learning and cognitive processes during interaction with VMRs and developing a language to describe these factors can be a step in the direction of creating a conceptual framework for the design and analysis of VMR-based MCTs.
This article is part of a larger research plan aimed at developing a systematic framework to provide guidelines for the design and evaluation of MCTs. Earlier research has developed a framework that categorizes and characterizes a set of interactions by which learners can explore and investigate VMRs (Sedig & Sumner, in press). However, this framework looks only at interaction with VMRs, and does not address their interactivity. It does not address how different interactivity factors can affect the quality of interaction, and hence, learning and cognitive processes of learners. This article aims to extend the earlier research. The purpose of this article is to discuss interactivity factors that affect learning and cognitive processes of learners who use VMR-based MCTs. Collectively, the factors can then serve as a framework to help the design and evaluation of such tools. To achieve its purpose, this article draws together research from mathematics learning, visual reasoning, cognitive technologies, information visualization, and human-computer interaction design. Before discussing the factors, conceptual and terminological issues with regard to interaction and interactivity are presented next.
Interaction and Interactivity
Interaction and interactivity are essential features of MCTs. VMRs by themselves are noninteractive. This means that much of their semantic and relational properties are hidden and latent. In order to reason and learn with static, noninteractive VMRs, learners need to analyze, evaluate, process, and elaborate on them (Ormrod, 1995; Peterson, 1996). In the context of MCTs, interaction can allow learners to perform "epistemic actions" on VMRs to adapt the visual information according to their needs (Kirsh & Maglio, 1994; Neth & Payne, 2002; Schwan, 2002). Interaction can act as an epistemic extension of static representations. It extends the communicative power of VMRs by adding a temporal dimension to them, making them dynamic and allowing their latent meanings to become visible. It allows learners to customize the "what" and "how" of the presentation of visual information. Interacting with VMRs allows learners to perform numerous cognitive activities, such as visualizing, analyzing, interpreting, modeling, and organizing. Allowing learners to interact with VMRs leads to some general benefits, such as: supporting dialectic reasoning with and through VMRs; providing opportunistic experimentation and exploration of hypothetical "what if" queries; making mental manipulation of concepts easier; facilitating the acquisition of qualitative insight into and understanding of the nature of VMRs; and coordinating learners' internal mental models with external VMRs (Sedig & Sumner, in press). Sedig and Sumner have identified and characterized 12 different micro-level, task-based interactions for performing cognitive tasks involving VMRs; animating, annotating, chunking, composing, cutting, filtering, fragmenting, probing, rearranging, repicturing, scoping, and searching.
Interaction and interactivity are closely related, yet they are distinct concepts. They have different connotations and meanings in different contexts (Norman & Draper, 1986; Laurillard, 1993; Kirsh, 1997; Sims, 1999, 2000; Yacci, 2000; Otero, Rogers, & du Boulay, 2001; Preece, Rogers, & Sharp, 2002; Roussou, 2004; Shneiderman & Plaisant, 2004). In the context of this article, interaction refers to a learner communicating with one or more VMRs through a human-computer interface (Sedig & Sumner, in press). Interaction with a VMR takes place in the time-space continuum and has two implications: (a) the learner acting upon a VMR and (b) the VMR responding or reacting in some form for the learner to interpret. Interactivity refers to the feel, form, properties, quality, and dimensions of interaction (Svanaes, 1999; Burgoon et al., 2000). Both, interaction and interactivity implicitly suggest how and what a learner learns. As such, a VMR can be interactive, but depending on its interactivity, interaction with it may require different amounts of cognitive effort, engage and support different thought processes, facilitate different degrees and depths of reasoning and learning, and allow different types of interchange between the learner and the content (Steuer, 1992; Laurillard, 1993; Golightly, 1996; Kirsh, 1997; Sims, 1999; Burgoon et al., 2000; Sedig et al., 2001; Preece et al., 2002). For instance, learners may interact with a VMR; however, interacting with the VMR directly or indirectly, through another intermediary representation, can affect the learners' attentive processes and degree of learning (Sedig et al., 2001). Similarly, whether interaction is scaffolded or not can influence learners' decision-making processes (Kirsh, 1997). As such, an understanding of the factors that affect learning and cognitive processes can help in the analysis and evaluation of interactive VMRs. The benefit of interactive VMRs may not be fully harnessed if designers of MCTs do not consider these interactivity factors in their designs.
Researchers from numerous disciplines have attempted to characterize interactivity and the multiplicity of factors that affect it. For instance, human-computer interaction researchers often discuss interactivity in terms of efficiency, affect, helpfulness, learnability, and usability of interactive tools (Teoa, Oha, Liua, & Weib, 2003; Shneiderman & Plaisant, 2004). Media and communication scientists have...
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