Instructional Modules
Drawing upon past research on proof and proving, we have identified five main topics for organizing modules. With the exception of sociomathematical norms, each topic is asccoiated with specfic epistemological obstacles.
We frame EOs as cognitive challenges that persist even in response to research-based instruction. So, EOs can be experienced both by students and teachers during instructional interactions. When instructors experience EOs, there is a tension between the desire to circumvent them and the need to provide students with opportunities to develop logical structures that are fundamental for proving. However, addressing an obstacle head on is essential for overcoming it because “it will resist being rejected and, as it must, it will try to adapt itself locally, to modify itself at the least cost, to optimize itself in a reduced field, following a well-known process of accommodation” (Brousseau, 2002, p. 85). Because EOs are persistent, students must experience an intellectual need to motivate, persevere, and successfully overcome the obstacles.
Awareness of EOs should inform the design of instructional approaches to introductory proofs topics. Traditional instructional approaches may be insufficient in addressing these obstacles, necessitating the development of alternative instructional techniques that support student learning. Within each of the following modules we suggest ways that instructors might elicit and address EOs.
Eliciting refers to the ways instructors might bring forth students' experiences of EOs and generate an intellectual need to address them. Addessing refers to ways that instructors might support students as they productively struggle through an EO, over time.
Sociomathematical norms describe normative ways for students and instructors to interact in the classroom. Unlike other classroom social norms, such as respecting one's peers and asking questions in class, sociomathematical norms are specific to mathematics. For example, we might expect our students to challenge assumptions made in proofs presented in class.
We frame student reasoning about logical implications using an action-object perspective (Piaget, 1970; Dubinsky, 1991) . As an action, a logical implication involves three components: a predicate, P; a conclusion, Q; and a transformation between them. Students who treat implications as actions can reason by modus ponens: the truth of P transfers, by implication, to the truth of Q. However, in treating an implications as actions across three components, rather than as single objects, students might experience persistent challenges in transforming and quantifying them. In particular, they might not reason by modus tollens: Q is false implies P must also be false. This module focuses on related EOs.
Quantification presents another challenge, particularly when quantifiers are hidden (Shipman, 2016). Overlooking hidden quantifiers can result in logical fallacies. Logical statements containing multiple quantifiers (i.e., multiply quantified statements) exacerbate students’ struggles with quantification. For example, Dawkins and Roh (2020) found that students tend to read multiply quantified statements semantically, applying their contextual knowledge to make sense of the formal statements, thereby neglecting the syntax of the statement and the order of the quantifiers in particular (Piatek-Jimenez, 2010). In our framing, such EOs should be deliberately evoked and carefully addressed over time.
Proof by mathematical induction poses a persistent challenge for students enrolled in proofs-based mathematics courses. Prior research indicates a number of related factors that contribute to the challenge, and suggests fruitful instructional approaches to support students in meeting that challenge. In particular, researchers have suggested quasi-induction as an intuitive approach to understanding the role of the inductive implication (Avital & Libeskind, 1978; Harel, 2001). However, a cognitive gap remains in transitioning to formal proof by induction. The gap includes the danger of inadvertently assuming what one is trying to prove. Quantification of the inductive implication and the inductive assumption require careful use of language, using terms that may seem ambiguous to students. This module includes tasks designed to build upon students’ conceptualizations of logical implication, utilize quasi-induction, and bridge the gap that remains.
Functions are centrally important objects of study in mathematics, and yet, before taking an Introduction to Proofs course, most students have experienced functions only as rules for computing. In our Intro to Proofs courses, we introduce functions as objects of study, often by defining them formally as sets of ordered pairs, (x, y). So, there is a disconnect between students' past experiences with functions as computational processes and their structural definition in proofs courses (Sfard, 1991). This disconnect can lead instructors and students to experience related EOs in the classroom, especially as students are expected to consider properties of functions, their inverses, and their compositions.