Project introduction and background information
One of the main difficulties for students starting a computer science (or mathematics) degree is the ability to argue in a rigorous manner. Beginning students often strongly rely on intuition and have difficulty embracing the formal mindset where only facts and logical derivation are allowed. Proving is one of the most demanding activities in the transition from school mathematics to tertiary mathematics. It encompasses the manipulation of logical statements and requires students to use a syntactic proof production in contrast to the semantic proof production they are used to. In a syntactic proof production, a student is producing a proof “solely by manipulating correctly stated definitions and other relevant facts in a logically permissible way” (Weber and Alcock, 2004, p. 210). In a semantic proof production, on the other hand, the student uses informal diagrams, representations, examples, and even intuitive means to make inferences while proving (ibid.).
The change in focus to a syntactic proof production is often left implicit. We consider the notion of structural scaffolding for facilitating syntactic proof production. Research in education has shown that students need to be facilitated in their Zone of Proximal Development (ZPD). In the ZPD, students can achieve something with support of a knowledgeable expert, which would be out of reach for them if they would learn on their own, without help (van de Pol et al.,2010). Structural scaffolding refers to the principle of materializing structural features, for example of mathematical argumentation, so that students can explicitly work on these structural features (Hein and Prediger, 2017) and, hence, be guided towards achieving their learning goals in the ZPD.
To achieve structural scaffolding we aim to supply students with a visual diagram-style representation of the underlying logic in the argumentation of a textual proof. This diagram is offered in the context of a series of exercises that explicitly relate the structure of the diagram both with the structure of the purely logic-based argumentation and the underlying structure of a textually written proof. Through a series of exercises, students learn to recognize the structure in the logical proof and match this with the less abstract diagram representation, subsequently the diagram can be used to recognize the same logical structure in the text-based proofs. Conversions to and from logic and textual proofs to the diagram help make explicit the identical underlying structure in all representations. The exercise sets will be directly embedded in the base courses within the computer science curriculum.
Objective and expected outcomes
With the project focus on developing said exercise sets, we ask the research question:
What are the effects of structural scaffolding on students' learning of syntactic proof production?
The project aims to develop exercise material that realizes diagram-based scaffolding to facilitate students in proof production by making logic and proof structures explicit. Accordingly, the project activities are located in the framework of educational design research, where the developed exercise material will be trialed, evaluated and improved in iterative cycles.