ABSTRACT

There are many ways to represent information in educational settings: textual descriptions, formulae, photographs, drawings and so on. A good match between the type of representation and learning demands can greatly support learning and contribute to enhanced levels of performance and understanding (Ainsworth, 2006; Greeno & Hall, 1997). Often, more than one type of representation appears to qualify for being used in a learning situation. An informed choice for one type of representation or another can be made on several grounds (Ainsworth, 2006; Scaife & Rogers, 1996). For example, a representation can be used because it causes less cognitive load compared with other representations. Representations can also be selected on the basis of the extent to which they promote clarity or reduce ambiguity (Stenning & Oberlander, 1995). Furthermore, combining two or more representational formats is assumed to have some additional effects on knowledge construction processes (Ainsworth, 1999, 2006). Different formats can complement each other or constrain the interpretation of the other (Ainsworth, 1999, 2006; van der Meij & de Jong, 2006). External representations can be presented to students, but students can

also construct representations themselves. Cox (1999) argues that the process of constructing a representation helps students to improve their knowledge, because the interaction between their internal representation and the external representation they construct, can make them aware of gaps in their internal representations they had not noticed before. Examples of activities in which students construct an external representation are: writing a summary (Hidi & Anderson, 1986), creating a drawing (Van Meter & Garner, 2005), or constructing a concept map (Gijlers & de Jong, submitted; Nesbit & Adesope, 2006; Novak, 1990). Whether it be external representations presented to students or represen-

tations constructed by students, in either way a clear-cut recipe for which representational format to use when does not exist. Moreover, some researchers argue that the effects of representations found in one domain cannot readily be

generalised to other domains (Cheng, Lowe, & Scaife, 2001). In this chapter a concise overview is presented of a series of studies that focused on the effects of representations on knowledge construction in the domain of combinatorics and probability theory. The effects of different representational formats were investigated and compared with each other in the context of simulation-based inquiry learning. In Study I, the effects of different formats used to represent the subject matter in computer simulations were investigated. In Study II, the focus was on the effects of format on learning outcomes when learners (individually or collaboratively) construct domain representations themselves. (For more details about the studies, see e.g., Kolloffel, 2008; Kolloffel, Eysink & de Jong, 2010; Kolloffel et al., 2009.)

There are several ways of representing information in the domain of combinatorics and probability theory. Some of the most commonly used formats will be discussed on the basis of the following problem, which is typical for the domain:

Your bank distributes a random four-digit code as a personal identification number (PIN) for its credit card. What is the probability that a thief finding the card and trying to get money with it will guess the correct code in one go, and will be able to plunder your account?