# Research Publications

Norton, A., Arnold, R., Kokushkin, V. *et al.* Addressing the Cognitive Gap in Mathematical Induction. *Int. J. Res. Undergrad. Math. Ed.* (2022). https://doi.org/10.1007/s40753-022-00163-2

Abstract:

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. However, a cognitive gap remains in transitioning to formal proof by induction. The gap includes the cognitive demand of quantification and the danger of inadvertently assuming what one is trying to prove. Informed by prior research, we designed instruction that builds from students’ conceptualizations of logical implication, utilizes quasi-induction, and introduces a novel set of tasks designed to bridge the gap that remains. We studied this research-based instructional design within two sections of an Introduction to Proofs course at a large public four-year university in the southeastern United States. Our findings, presented as themes, bring together extant literature and highlight nuances not previously reported, particularly regarding reasons students appear to conflate the assumption of the statement to be proved with the assumption of the inductive hypothesis. For instance, quantification of the inductive implication and the inductive assumption require careful use of language, using terms that may seem ambiguous to students, outside of mathematical convention. We conclude with a discussion of links between the Principle of Universal Generalization and mathematical induction.

Norton, A., Arnold, R., Kokushkin, V. *et al.* Addressing the Cognitive Gap in Mathematical Induction. *Int. J. Res. Undergrad. Math. Ed.* (2022). https://doi.org/10.1007/s40753-022-00163-2

Abstract:

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. However, a cognitive gap remains in transitioning to formal proof by induction. The gap includes the cognitive demand of quantification and the danger of inadvertently assuming what one is trying to prove. Informed by prior research, we designed instruction that builds from students’ conceptualizations of logical implication, utilizes quasi-induction, and introduces a novel set of tasks designed to bridge the gap that remains. We studied this research-based instructional design within two sections of an Introduction to Proofs course at a large public four-year university in the southeastern United States. Our findings, presented as themes, bring together extant literature and highlight nuances not previously reported, particularly regarding reasons students appear to conflate the assumption of the statement to be proved with the assumption of the inductive hypothesis. For instance, quantification of the inductive implication and the inductive assumption require careful use of language, using terms that may seem ambiguous to students, outside of mathematical convention. We conclude with a discussion of links between the Principle of Universal Generalization and mathematical induction.

Abstract:

Mathematical tasks are central to students’ learning since they can influence and structure the ways in which students think about mathematics. Carefully selected tasks have potential to broaden students’ views of a subject matter and facilitate their mathematical growth. However, research identifies that cognitive demands of tasks may change as the tasks are enacted during instruction. For this reason, it is important to understand what instructors can do to maintain the intended cognitive demands. In this paper, we investigate a teacher’s actions for maintaining high-level cognitive demands of tasks in teaching proof by mathematical induction. Our findings suggest that the method of quasi-induction (Harel, 2002) may be considered as an example of a productive scaffolding strategy for assisting students in mastering proof by induction.

Abstract:

Proof by mathematical induction poses persistent challenges for college mathematics students. We use an action-object framework to analyze ways that students might overcome these challenges. We conducted three pairs of interviews with students enrolled in a proofs course. Tasks were designed to elicit student understanding of logical implication and components of proof by induction. We report results from one student, Mike, who had constructed logical implication as an object, and who invented a quasi-inductive proof.

Abstract:

As has been well-documented, the epistemological obstacles associated with teaching and learning mathematical proofs persist despite research-based instruction. We describe the ongoing design process of our NSF-funded project aimed at understanding and addressing those obstacles in introductory proofs courses, using proof by mathematical induction as an anchor. Our process is framed by two cycles of designed-based research. The first cycle corresponds to designing and implementing research-based instruction on mathematical induction, whereas the second cycle broadens the scope of our research to other introductory proofs topics. This paper reports on the outcomes of the first cycle, the transition between the first and second cycles, and the project's end products.

Abstract:

When talking about mathematics, teachers and learners actively use hand gestures to support their speech as well as to describe ideas that are not expressed verbally. In this study, I investigate the gestures that were utilized by an instructor and his students during a teaching episode on proof by mathematical induction. Alibali and Nathan’s (2012) typology of gestures are employed to code the observed gestures. The study reveals that the use of gestures plays an integral role in teaching and learning induction. I show that pointing gestures helped to reduce ambiguity in classroom discussion, representational gestures were useful in describing specific subcomponents of induction, and, finally, metaphoric gestures were independently introduced by a teacher and students to describe the nature of proof by mathematical induction.

Abstract:

Proof by mathematical induction is known to be conceptually difficult for undergraduate students. We present a model that may simulate the impact of logical implication on students mastering proof by induction. We combine Piaget’s action-object theory of mathematical development with a psychological model of working memory and Harel and Sowder’s proof schemes. We analyzed three sets of written assessments from two Introduction to Proofs classes: after students learned about logical implication; before and after instruction on proof by induction. We examine the relationship between proficiency with mathematical induction and treating logical implication as an object within these two classes.