A Bridge Anchored on Both Sides: Formal Deduction in Introductory CS, and Code Proofs in Discrete Math

TL;DR

This paper explores integrating formal program reasoning into introductory CS and discrete math courses to bridge the gap between programming and mathematics, supported by educational tools like Orca.

Contribution

It presents curriculum designs that incorporate code reasoning early in CS education and introduces Orca, a proof assistant, to enhance feedback and understanding.

Findings

Code reasoning helps connect programming and math concepts.

Early introduction of formal proofs improves student understanding.

Orca shows promise as an educational proof tool.

Abstract

There is a sharp disconnect between the programming and mathematical portions of the standard undergraduate computer science curriculum, leading to student misunderstanding about how the two are related. We propose connecting the subjects early in the curriculum---specifically, in CS1 and the introductory discrete mathematics course---by using formal reasoning about programs as a bridge between them. This article reports on Haverford and Grinnell College's experience in constructing the end points of this bridge between CS1 and discrete mathematics. Haverford's long-standing "3-2-1" curriculum introduces code reasoning in conjunction with introductory programming concepts, and Grinnell's discrete mathematics introduces code reasoning as a motivation for logic and formal deduction. Both courses present code reasoning in a style based on symbolic code execution techniques from the programming language community, but tuned to address the particulars of each course. These courses rely primarily on traditional means of proof authoring with pen-and-paper. This is unsatisfactory for students who receive no feedback until grading on their work and instructors who must shoulder the burden of interpreting students' proofs and giving useful feedback. To this end, we also describe the current state of Orca, an in-development proof assistant for undergraduate education that we are developing to address these issues in our courses. Finally, in teaching our courses, we have discovered a number of educational research questions about the effectiveness of code reasoning in bridging the gap between programming and mathematics, and the ability of tools like \orca to support this pedagogy. We pose these research questions as next steps to formalize our initial experiences in our courses with the hope of eventually generalizing our approaches for wider adoption.