Sources of Students’ Difficulties with Proof By Contradiction

The literature on proof by contradiction (PBC) is nearly unanimous in claiming that this proof technique is “more difficult” for students than direct proof, and offers multiple hypotheses as to why this might be the case. To examine this claim and to evaluate some of the hypotheses, we analyzed student work on proof construction problems from homework and examinations in a university “Introduction to Proof” course taught by one of the authors. We also conducted stimulated-recall interviews with student volunteers probing their thought processes while solving these problems, and their views about PBC in general. Our results suggest that the knowledge resources students bring to bear on proof problems, and how these resources are activated, explain more of their “difficulties” than does the logical structure of the proof technique, at least for this population of students.

A Hypothesis Framework for Students’ Difficulties with Proof By Contradiction

In mathematics education, the research on proof by contradiction (PBC) often claims that this activity is more difficult for students than direct proof, or simply difficult in general. Many hypotheses have been offered to support or explain this belief, yet they span a disorientingly wide swath of journal articles, conference papers, dissertations, book chapters, etc. In addition, few attempts have been made to organize these hypotheses or carefully test them. In this paper, we conduct a thorough literature review on PBC, organize existing hypotheses about challenges with PBC into a Hypothesis Framework for (Students’ Difficulty with) Proof By Contradiction (HFPBC), discuss the state of research related to each hypothesis, and offer thoughts on the future study of these hypotheses.

Assessing Proof Reading Comprehension Using Summaries

In this paper, we explore the role of mathematical proof summaries as a tool for capturing students’ reading comprehension of a given proof. We present an interview study based on mathematicians’ pairwise evaluations of student-produced summaries of a proof demonstrating the uncountability of the open unit interval. We present a thematic analysis, exploring features of mathematicians’ pairwise decision-making and their priorities in evaluating summaries. We argue that the students’ proof summaries shared several properties with traditional modes of proof-writing and were frequently evaluated against similar conventions. We consider the consequences for research and practice with proof comprehension and conclude that proof summaries have the potential to form the basis of a new approach to assessment in this area.

The Interplay between Individual and Collective Activity: an Analysis of Classroom Discussions about the Sierpinski Triangle

This article uses a cultural-developmental framework to illuminate the interplay between collective and individual activity in the mathematical reasoning displayed in a university Masters level lesson on fractals. During whole class and small group discussions, eleven students, guided by an instructor, engage in inductive reasoning about the area and perimeter of the Sierpinski triangle, a unique mathematical object with zero area and infinite perimeter. As participants conceptualize and communicate about the Sierpinski problem, they unwittingly generate a linguistic register of action word forms (e.g., fencing, zooming) and object word forms (e.g., area, infinity) to serve Sierpinski-linked mathematical reasoning functions, a register that we document in the first analytic section of the article. In the second analytic section, we report a developmental analysis of microgenetic, ontogenetic, and sociogenetic shifts in the word forms constitutive of the register and their varied functions in participants’ activities. In the third analytic section, we provide a cultural analysis of the classroom’s collective practices, practices that enable and constrain participants’ constructions of form-function relations constituting the register. We examine the ways in which participants work to establish a common ground of talk in their communicative exchanges, exchanges supported by classroom norms for public displays of reasoning and active listening to one another’s ideas. We show that it is as participants work to establish a common ground that the register emerges and is reproduced and altered. We conclude by pointing to ways that the analytic framework can be extended to illuminate learning processes in other classroom settings.