Experiences with Novices: The Importance of Graphical Representations in Supporting Mental Models

Recursion is an important problem solving technique used in programming. It is also a highly unfamiliar mental activity and many computing novices have difficult understanding recursion and applying recursive techniques in problem solving. Research studies have concluded that novices and experts differ in their mental models of recursion. Novices seem to possess various inadequate models of recursion especially the iterative or loop model. This paper examines whether novices who are aided in acquiring an expert’s mental model of recursion (the copies model) can effectively use this model in evaluating recursive algorithms. Results of a study indicated that a large percentage of novices who had previously demonstrated an understanding of the copies model (using explicit diagrammatic traces) failed do so when not using diagrammatic traces. In fact, they appeared to demonstrate evidence for the incorrect iterative or loop model when trying to mentally evaluate recursive programs. The results provide evidence that mental models are unstable and that graphical representations are a very necessary aid to retrieval of novices’ mental models. This suggests that the teaching of recursion may be best facilitated by teaching students how to simulate the execution of a recursive algorithm using diagrammatic traces. Introduction Cognitive research studies in computer programming and other disciplines have demonstrated the importance of adequate mental representations or models in understanding processes, complex tasks, or systems and have also revealed that learning is facilitated by aiding learners in forming appropriate mental models or by modifying and elaborating learners' incomplete models (e.g. DuBoulay, O'Shea & Monk 1980; Mayer 1981). While novices may acquire an understanding of the syntax of a programming language, without appropriate mental models they are unable to successfully understand complex concepts or engage in more difficult aspects of problem solving. Effective mental models are thus of critical importance in understanding programming concepts. This paper focuses on recursion, a programming concept which many students find difficult to understand and to apply in their problem solving activities (e.g. Anderson et al 1988; Anzai & Uesato, 1982a, 1982b; Baird 1986; Elenbogen & O'kennon 1988; Ford, 1984; Ginat & Shifroni 1999; Henderson & Romero 1989; Kahney 1983; Kessler & Anderson 1986; Kurland & Pea 1983; Pirolli, 1986; Roberts 1986; Wiedenbeck, 1988). Recursion is a misunderstood or barely understood programming concept among programmers (Ford 1982). It is difficult because it is a highly unfamiliar mental activity (Anderson et al 1988) and may be counter-intuitive to beginning computer science students (Elenbogen & O’Kennon 1988). Kahney (1983) concluded that novices and experts differ in their mental models of recursion and further categorised various mental models that novices were seen to possess. Kahney’s conclusions have been supported by many other subsequent studies (cited above). They indicate that most novices, unlike experts, are unable to conceptualise separate and unique invocations of subprograms in recursion (the copies model) and more importantly the flow of control in an executing recursive program. Novices easily adopt the more familiar notion of iteration when attempting to understand recursion (Kessler & Anderson 1986; Kurland & Pea 1983; Wiedenbeck, 1989). In Kahney’s study, students were not explicitly taught how to simulate the flow of execution control and visualise separate invocations of subprograms. They were only required to comment on whether the solution programs given would work and did not seemed to have used any diagrammatic drawings to help them in their problem-solving. This paper will investigate whether novices who are aided in acquiring an expert’s mental model of recursion (the copies model) can effectively use this model in evaluating recursive algorithms. Particular focus will be placed on the role of graphical /diagrammatic execution traces during problem solving. Mental Models The history of the concept of mental models in human cognition may be traced to Craik (1943), who suggested that humans make use of internal models of external reality, which enable them to better understand and react to situations in their environment. These internal models are formed through interaction with external