Structural scaffolding via diagrams to teach students syntactic proof production

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. 

Results and learnings

We performed a literature study to assess the best way to introduce students to syntactic proof production. Based on this study we developed an initial assignment set for ‘proof by contrapositive’. This assignment set was trial run in the course ‘Algorithms and Data Structures’. Students made the assignment in groups of four with only minimal intervention. The process in which they tackled the exercise set was recorded and the outcome was used to improve the set further. Based on the perceived problems we further refined the exercise set and once again ran a brief evaluation in a later iteration of the course ‘Algorithms and Data Structures’.

The self-study material was implemented in the course 2IHA10 Algorithms and Data structures. The implementation was framed by a pre-post test where students worked on 8 items, including proving (mostly number theoretical statements), formulating contrapositives for everyday situations and formulating contrapositives in formal proofs. The pre-post research design allows to draw conclusions about the students learning progression resulting from the self-study material.

There was limited response to the post-test, resulting in severe limitation with respect to drawing general conclusions about the efficacy and effects of the self-study material. In this report, the effects of the self-study material on student learning will be investigated qualitatively, based on N=4 students who participated both, in the pre- and post-test. Also, conclusions about factors causing the test results cannot be identified from the test, as students’ thinking behind their answers is hardly evident.

With respect to the students’ proving proficiency (Items 1 and 2 in pre-/post-test), there are slight improvements with respect to the clarity and elegance of the reasoning. However, in attempting a higher formalism in his post-test proof, one student adopted an overgeneralized use of the “ó” sign.

With respect to the students’ proficiency to formulate contrapositives in everyday contexts (Items 3 – 5), two students did not substantially develop, because they already show a high proficiency in the pre-test. The two remaining students improved their proficiency to state the contrapositive. However, interestingly, one of these students failed to formulate a correct contrapositive for the easier items, both in the pre- and post-test.

With respect to formulating the contrapositive for a mathematical statement (Item 7), three out of four students show no improvement of their proficiency, while one student shows the same level of proficiency. In the three cases, students do not reverse the order of implication. Interestingly, in the pre-tests, these students were able to correctly state the contrapositive. We suspect that in the pre-test, students followed a rule-based approach for formulating the contrapositive. The intervention supported students conceptual understanding of the contrapositive, temporarily lowering their proficiency to correctly formulate a contrapositive. Possibly, a longer intervention with more focus on formulating the contrapositive in mathematical statements could have improved this issue. However, a closer analysis of students’ thinking processes is needed to substantiate our suspicion.

Overall, it is likely that the self-study material supports students in formulating the contrapositive in everyday situations. However, more work is needed to understand how to improve the self-study material with respect to extending this proficiency into the realm of mathematics.

Recommendations

For more information or recommendations, please contact a.i.v.goethem@tue.nl and/or a.k.schuelermeyer@tue.nl.

Practical outcomes

We have implemented our exercise set in OnCourse and made it available to the students who participated in 2IHA10 – Algorithms and Data Structures. The exercise set will be made available for use in algorithms courses.