Virginia Tech® home

Instructional Modules

BannerImage_1.png

 

Drawing upon past research on proof and proving, we have identified four main topics for organizing modules.

  1. Logical Implication
  2. Quantification 
  3. Mathematical Induction
  4. Function

Each topic is associated with specific epistemological obstacles (EOs), which we define below. Modules specific to each topic can be accessed via the Topics tab above.  

To support student learning, we suggest sociomathematical norms and heuristics for scaffolding student interactions and reasoning (see tab above). These norms and heuristics are essential for setting the stage for eliciting and addressing EOs.

Finally, we include the research-based instructional tasks that were used by Rachel Arnold in her introductory proofs class. These tasks were either pulled directly from existing literature or designed/adapted by members of The Proofs Project based on prior research.

 

What is an epistemological obstacle (EO)?

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. 

Diagram showing proofs project goals

Awareness of EOs should inform the design of instructional approaches to introductory proofs topics. Traditional instructional approaches may be insufficient for addressing these obstacles, necessitating the development of alternative instructional techniques that support student learning. 

Within each of the instructional modules below, we suggest ways that instructors might elicit and address epistemological obstacles (EOs) that are closely associated with each topic.

  • 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.

Topics

Logical implication is a conditional relation between two mathematical statements, a hypothesis and a conclusion. Reasoning through logical implication is central to proof and proving, and it is associated with many EOs. 

Logical Implication Module

Quantification presents additional challenges for students' logical reasoning. Quantifiers like "for all" and "there exists" make critical distinctions in the meanings of mathematical statements. Logical statements may even be multiply quantified or hiddenly quantified. This module investigates prevalent EOs associated closely with quantification. 

Quantification Module

Mathematical induction is a method of proving statements about natural numbers that 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 of using this method, including EOs introduced in the Logical Implication and Quantification modules. In this module, we illustrate how those EOs manifest specifically in the context of proof by mathematical induction, and we suggest ways to address them.

Mathematical Induction Module

Functions are centrally important objects of study in mathematics. Before taking an introductory proofs course, most students have experienced functions only as rules for computing. In proofs-based mathematics, the formal definition of functions as a relation between spaces is introduced. There is a subsequent disconnect between students' past experiences with functions as computational processes and this new, structural definition. 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. 

Function Module

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. The sociomathematical norms module presents examples of such norms, especially norms that set the stage for eliciting and addressing EOs. This module contains examples for establishing and reinforcing those norms. 

Sociomathematical Norms Module

Heuristics are general strategies for solving a variety of mathematical problems across contexts. An example of a commonly used heuristic is "draw a picture." Heuristics are important tools for supporting mathematical reasoning and should be integrated into introductory proof classrooms' mathematical practices. In this module, we suggest heuristics specific to proof and proving.

Heuristics Module
return to project home
About the Project
Our Team
Instructional Modules
Research Publications
Professional Development Training Materials