Interactivity of visual mathematical representations: Factors affecting learning and cognitive processes

Kamran Sedig & Hai-Ning Liang

The University of Western Ontario

London, Canada

Abstract. 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 in order 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 twelve 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.

Summary

Visual mathematical representations (VMRs) are graphical notations that encode a vast array of mathematical ideas and are an essential component of mathematical cognitive tools (MCTs). Adding interactivity to VMRs allows learners to perform epistemic actions on them and can increase their epistemic utility. These epistemic actions involve cognitive activities and processes such as attending, perceiving, reasoning, decision-making, interpreting, planning, and evaluating. Learning is a high-level cognitive activity that encompasses all such cognitive processes. Interaction then acts as a mediatory mechanism that can support, enhance, and/or transform the cognitive tasks that can be performed on or with static, displayed representations.

Interactivity and interaction are central features of interactive VMRs. They are closely related; yet, they are distinct concepts. Interaction with a VMR refers to acting upon it. A VMR’s interactivity refers to the feel, form, properties, and quality of this interaction. A VMR can be interactive, but depending on its interactivity, epistemic actions may require different amounts of cognitive effort and engage and support different cognitive processes. As such, the quality and effectiveness of the learner-VMR interaction can be in large part determined by its interactivity. The interactions used and the interactivity of a VMR implicitly suggest how and what a learner learns. This article has presented twelve factors for analyzing the interactivity of VMRs. These twelve factors are: affordance, cognitive offloading, constraints, distance, epistemic appropriateness, feedback, flexibility, flow, focus, involvement, scaffolding, and transition.

1. Affordance: Provision of interface cues to advertise possible interactions

2. Cognitive offloading: Provision of interactions that can shoulder the load of some cognitive processes

3. Constraints: Restriction of possible interactions

4. Distance: Degree of difficulty in understanding how to act upon a VMR and interpret its responses

5. Epistemic appropriateness: Suitability and harmony of interactions in supporting learning

6. Feedback: Exchange of information and the direction of communication during interaction between learners and the VMR

7. Flexibility: Range and availability of interactive choices and options available to learners

8. Flow: Duration of interaction with the VMR in time and its effect on learners’ perception of the relationship between cause and effect

9. Focus: Locus of learners’ attention during interaction with the VMR

10. Involvement: Learners’ engagement with and contribution to the information content of a VMR allowed by the available interactions

11. Scaffolding: Provision of interactions to cognitively support and augment learners’ reasoning and understanding of embedded concepts

12. Transition: Communication of visual changes to the VMR

These factors have been developed by bringing together and integrating research findings from a number of research areas such as mathematics learning, visual reasoning, cognitive technologies, information visualization, and human-computer interaction design.