Development of schema and cognitive elements

Abstraction of cognitive elements (i.e., knowledge, skills, and combinations thereof) seems to have a compelling role in preparing students for successful transfer outcomes, but essential to this end is the development of a rich schema for the content area (Helfenstein, 2005; Lovett & Greenhouse, 2000; Perkins & Salomon, 1988). Mastery of a rich schema must precede successful transfer outcomes (Bransford et al., 2000; Cooper & Sweller, 1987; Paas, 1992), yet building and mastering schema is no small task. Schema tends to start small with a few similar problems, and then grows organically as the elements of the schema are repeatedly accessed and strengthened (Cooper & Sweller, 1987; Lovett & Greenhouse, 2000). Helfenstein (2005) argued that the functional development and interconnection of schema and abstraction of cognitive elements are rooted in Gestalt ideas related to insight during the course of problem solving. However,

this should not imply that development and organization of such insight require lightning- strike experiences; schema development can also be nurtured through careful direction of learning activities (Rittle‐Johnson, 2006; Salomon & Perkins, 1989). For example, when content is taught with exposure to multiple contexts, students are more likely to abstract relevant features of the subject matter to more readily draw upon them in the future (Bransford et al., 2000; Salomon & Perkins, 1989). In time, the boundaries of a schema domain swell and overlap with other domains such that the problem solver becomes increasingly equipped to assimilate a new problem into existing schema because of a depth and breadth of associated content mastery (Cooper & Sweller, 1987). In fact, some believe that higher ability students distinguish themselves by virtue of achieving greater abstraction and the facility to call on more distant connections (Goska & Ackerman, 1996).

Bransford et al. (2000) assert that all learning requires transfer based on prior learning. For better or worse, students arrive with an existing network of schema that cannot be overlooked (Broers et al., 2004; Garfield, 1995; Lovett & Greenhouse, 2000). In fact, some have attributed the success of cognitive transfer to the degree of overlap between associated cognitive elements, whether concrete or abstracted (Bransford et al., 2000; Goska & Ackerman, 1996; Helfenstein, 2005; Singley & Anderson, 1989; Sternberg, 1998; E. L. Thorndike & Woodworth, 1901a). Probability topics, for example, notoriously suffer from challenges rooted in poor intuition and contradictory understanding of relevant terms due to inconsistencies with their conversational use (Garfield, 1995; Lovett & Greenhouse, 2000; Singley & Anderson, 1989) resulting in

negative transfer outcomes. Students must reconcile disparities between principles of statistical reasoning and the many fallacies that permeate common culture outside the classroom (Garfield, 1995). Unfortunately, simply providing contradictions to these fallacies is insufficient to depose them and rebuild intuition (Bransford et al., 2000; Garfield, 1995).

While cultivation of a deep and diverse schema is necessary for successful transfer outcomes, it is not sufficient. It should be acknowledged that all learning occurs within some context, and successful transfer outcomes require sufficient abstraction of cognitive elements to transcend the context in which content has been learned (Lovett & Greenhouse, 2000; Perkins & Salomon, 1988). However, students have difficulty recognizing problem structure and generalizing cognitive elements on their own, so the educator must engage strategic methods to facilitate the desired abstraction (Reed et al., 1985). One strategy to encourage successful transfer outcomes involves creating opportunities for students to receive feedback to help them understand when the content is applicable and when it is not applicable; absent this instruction, students tend to use inappropriate mnemonics such as chapter and textbook location in place of appropriate integration to their greater knowledgebase (Bransford et al., 2000).

Further complicating things, schools tend to emphasize abstraction of subject matter discussed, while the scenarios in the real-world that will demand use of that subject matter will almost certainly require contextualized reasoning (Bransford et al., 2000). Lovett and Greenhouse (2000) discuss a strategy implemented using “synthesis labs” in an introductory statistics course in which students are challenged on a regular basis

throughout the semester with tasks that leverage material that integrates cumulative content from the course to date. This creates practice with concepts like tool selection, which comes naturally to an expert statistician but is difficult for the novice to exercise in earnest; the labs give students an opportunity to combine cognitive elements in different ways during problem solving in order to promote synthesis and transfer (Lovett & Greenhouse, 2000). As summarized by Wild and Pfannkuch:

Statistics is itself a collection of abstract models (‘models’ is used in a very broad sense) which permit an efficient implementation of the use of archetypes as a method of problem solution. One abstracts pertinent elements of the problem context that map onto a relevant archetypical problem type, uses what has been worked out about solving such problems, and maps the answer back to context domain. There is a continual shuttling between the two domains and it is in this shuttling or interplay, that statistical thinking takes place. (1999, p. 244)

To be clear, the process of schema development and abstraction of cognitive elements should start with specific examples from which the student is responsible for abstracting understanding. Instruction that is too general may become too vague to be useful for any specific application (Singley & Anderson, 1989). It is important to distinguish that the information should be available in the abstract, but must be usable for a particular situation (Singley & Anderson, 1989). For example, anyone who knows the rules of chess can theoretically execute the perfect game by generating all possible moves and counter- moves and thereby choose the optimal strategy in every scenario; however, this is not a

realistic outcome for a player that has only learned the rules abstractly (Singley & Anderson, 1989).