Teaching and Learning Proof Across the Grades

A K-16 Perspective

Despina A. Stylianou, Maria L. Blanton, Eric J. Knuth

• Publisher: Routledge
• Year: 2010
• Print ISBN: 9780415887311
• eBook ISBN: 9781135856748
• Topic: Education
• Subtopic: Education General

A Co-Publication of Routledge for the National Council of Teachers of Mathematics (NCTM)

In recent years there has been increased interest in the nature and role of proof in mathematics education; with many mathematics educators advocating that proof should be a central part of the mathematics education of students at all grade levels. This important new collection provides that much-needed forum for mathematics educators to articulate a connected K-16 "story" of proof. Such a story includes understanding how the forms of proof, including the nature of argumentation and justification as well as what counts as proof, evolve chronologically and cognitively and how curricula and instruction can support the development of students' understanding of proof. Collectively these essays inform educators and researchers at different grade levels about the teaching and learning of proof at each level and, thus, help advance the design of further empirical and theoretical work in this area. By building and extending on existing research and by allowing a variety of voices from the field to be heard, Teaching and Learning Proof Across the Grades not only highlights the main ideas that have recently emerged on proof research, but also defines an agenda for future study.

IV
Teaching and Learning of Proof in College

Arguments that the “essence of mathematics lies in proofs” (Ross, 1998, p. 2) and that “proof is not a thing separable from mathematics … [but] is an essential component of doing, communicating, and recording mathematics” (Schoenfeld, 1994, p. 76) reinforce the centrality of proof in mathematical thinking. Indeed, the central role of proof in mathematics becomes more obvious at the college level where the level of mathematical content and students’ maturity facilitates the introduction to complex forms of proof. The chapters in Section IV conclude the “proof story across the grades” by shifting our attention to questions and issues specific to teaching and learning proof at the post-secondary level and connecting these questions and issues to those raised in the previous sections.

In Sections II and III, the authors helped conceptualize the role and meaning of proof for young children and high school students respectively. Their questions included: what constitutes an acceptable argument at those levels? What role can proof have in the entire mathematics curriculum? (see Introductions to Sections II and III). They explored students’ intuitive notions of proof and how school experiences might nurture and challenge these natural ways of understanding mathematical argumentation towards developing what constitutes a logically valid argument.

At the college level, the questions of what constitutes proof and what role it might have in the curriculum become peripheral. The Committee on the Undergraduate Program in Mathematics (MAA, 2000) maintains that proof should be central to the teaching and learning of mathematics at the post-secondary level. And as Harel and Sowder (Chapter 16) suggest, college students, in particular those who major in mathematics-related areas, are expected to graduate from college with some degree of facility in reading and writing proofs. The questions that guide the teaching and learning of proof at this level concern how to make the type of reasoning that underlies proof central to the entire mathematics curriculum.

Furthermore, at lower levels, proof is often given the broader definition of the act of arguing to convince one’s community of the truth of an assertion (Balacheff, 1988). As students advance to college courses, an informal argument or explanation is not sufficient (Harel & Sowder, Weber & Alcock in this volume). Students who major in mathematics-centered topics are expected to write mathematical proofs, which have a distinct form and well-defined conventions. As Harel and Sowder (2007) argue elsewhere and Weber and Alcock argue in this volume (Chapter 19), in post-secondary mathematics instruction a sharp shift takes place, and the validity of the proof is dependent not only on its content but also on its form. This section reflects the attempt to understand this shift and, ultimately, its implications for the teaching and learning of proof in the college classroom. In completing the story of proof across the grades, this section highlights the distinct differences in the character of proof and the common goal to teach students the notion of what constitutes a grade-appropriate logically valid argument.

Developing College Students’ Understanding of Proof

Much of the literature on the learning of proof, especially at the college level, has focused on identifying difficulties that students face when reading and writing proofs. This work provides an important starting point in the research base on teaching and learning proof. Indeed, understanding how to support student learning often begins with understanding where students’ difficulties lie. Building from this research, this volume uses perspectives that move towards models and frameworks for supporting students in their growth. The authors in this section use new lenses to identify ways in which students’ learning processes can be viewed not as an obstacle in instruction but as a basis to inform instruction. Some of the chapters challenge our views by combining perspectives commonly used in mathematics education with perspectives used in other fields as a way of opening a dialogue among researchers and practitioners.

Using the DNR framework developed in their earlier work, Harel and Sowder (Chapter 16) explore the critical issue of how faculty might facilitate the development of student learning of proof and what they know about how students learn proof. They focus on the development of student “proof schemes”—a label for the means that one might use in convincing oneself and others about a matter. In this perspective, each student holds some level of understanding and appreciation of proof, so, this examination of students’ proof schemes “highlights the student as a learner.” Moreover, it focuses instructors’ attention in two areas: (1) current student understanding, rather than students’ final proof productions and hence their possible shortcomings; (2) the instructional practices that may help in gradually refining student understanding towards the desirable deductive proof scheme, that is, the proof scheme practiced by contemporary mathematicians.

Blanton, Stylianou, and David (Chapter 17) and Smith, Nichols, Yoo, and Oehler (Chapter 18) examine mathematics learning that takes place in classrooms for which social negotiation of mathematical meaning is commonplace (Cobb & Yackel, 1996). Although such classroom environments are rare at the undergraduate level, the perspective used in these chapters partly as a tool to understand students’ learning of proof is innovative and, more importantly, their effects on the learning of proof appear to be fruitful.

Blanton, Stylianou, and David (Chapter 17) explore possible ways in which students begin to internalize the language of proof and argumentation as they actively participate in whole class discourse. They maintain that the transfer of responsibility for proving a statement from the teacher to the students supports students’ proof learning. They then conjecture that teachers’ prompts that encourage student engagement in the discussion towards proof and the development of a critical stance towards statements and claims made publicly can be crucial in students’ development of proof construction. They further suggest that students internalize public argumentation in ways that facilitate private proof construction.

Smith, Nichols, Yoo, and Oehler, like Blanton and her colleagues, view proving as an activity that develops within a community of learners and that can be appropriated by assisted participation. They discuss the ways in which a college instructor’s choices to assume and relinquish the role of leader in the classroom affect students’ participation in the learning community and facilitate their engagement in mathematical discussions about proofs presented during the course of instruction.

Weber and Alcock (Chapter 19) and Selden and Selden (Chapter 20) shift our attention to individual students’ learning of proof. They write from the perspective that student learning of proof cannot be separated from the mathematical content in which it occurs. These two chapters consider the character of mathematical proof and its implications for student proof construction.

Weber and Alcock (Chapter 19) frame their discussion around the argument that learning to prove in college classrooms requires students to work within a new representation system. Because this process can be simultaneously limiting and empowering, it is important to understand both the challenges and affordances that students face within the (new) representation system of mathematical proof.

Connecting their argument to their earlier work on syntactic and semantic reasoning in proof construction, Weber and Alcock claim that constructing a proof syntactically may allow an undergraduate to produce a valid argument when they could not otherwise do so. However, when students limit themselves to syntactic proof productions, they may not make use of links between the representation system of proof and other informal representations of mathematical ideas, and so may find the proof non-convincing and non-explanatory. By the same token, when students attempt to link the formal representation system in which proofs are produced with their concept images, that is, use semantic reasoning, they may have more choices as to what line of reasoning to pursue. A potential problem with semantic reasoning, however, is that an individual may be able to develop a mathematically correct understanding of why a proposition is true, but be unable to frame their argument within the representation system of proof.

Similar to Weber and Alcock, Selden and Selden shift our attention to mathematical proof as it might be viewed in advanced college courses by proposing a framework for working within proof to facilitate student growth. In particular, they introduce the notion of looking at proofs themselves, because differing types of proof and components of these proofs can correspond to various abilities needed to produce them. They suggest that an analysis of kinds and aspects of proofs should facilitate teaching by coordinating the theorems assigned with student abilities. It should also facilitate the assessment of student abilities and, hence, the study of how students learn about proof.

Teaching Practices that Support Proving

Throughout most of this volume, teaching and learning proof are naturally interwoven. As in the previous sections, the authors in this section draw on their work on student learning of proof to suggest principles for instruction, or use their study of instruction to examine the growth of student learning.

Blanton, Stylianou, and David (Chapter 17) examine teacher practice in an undergraduate mathematics course that embedded the development of proof in the social activity of the classroom. They used selected classroom episodes from this course to design a framework for characterizing whole class discourse on proof, specifically, teacher and student utterances. They then used this framework to analyze classroom discourse, specifically, to study how these utterances impact one another and scaffold student learning of proof.

Smith, Nichols, Yoo, and Oehler (Chapter 18) continue this emphasis on teacher practice. However, while Blanton and her colleagues conduct a fine grain analysis of the teacher’s actions and utterances during selected teaching episodes, Smith et al. focus on teacher actions over time. In particular, they look at how one instructor’s choices about when to interrupt and direct class discussions of students’ work influenced the development of a classroom community of inquiry and encouraged a view of mathematics as a human social activity by engaging students in discourse about mathematics and proof.

These studies examine classrooms that are not common at the college level in that the public negotiation of ideas and collective development of proofs are part of the sociomathematical norms of the classroom. Harel and Sowder (Chapter 16) complement this work by using a wider lens on instructional practices in the teaching of proof and bringing to the forefront the views of a larger group of college instructors who approach the teaching of proof from a variety of perspectives. They study the views of university mathematics faculty of upper division courses, using interviews about students’ success and difficulty with proof in the typical university mathematics curriculum. According to Harel and Sowder, the question of critical importance is “what instructional interventions can bring students to see an intellectual need to refine and alter their current proof schemes into deductive proof schemes?” (Chapter 16, p. 279).

After discussing the relation of the structure of the proof to the learning of proof, Selden and Selden make specific suggestions for the teaching of proof, taking into account these features. They describe an approach to teaching in which instruction is integrated into students’ construction of proofs, and in which features of proofs are more important than the topics of theorems.

Overall Issues and Perspectives on the Teaching and Learning of Proof at the College Level

This section attempts to bring to the forefront issues of student learning and instructional practice regarding proof at the post-secondary level, while taking into consideration the nature of proof itself, the representation system in which it resides and the norms and culture of mathematics. The studies that are described in this section are rooted in different methodological orientations ranging from the notion that the building of proof should be a part of the sociomathematical norms of the classroom (Blanton et al., Smith et al.) to one based on understanding the instructors’ notions of proof (Harel and Sowder), and to orientations rooted in the nature of mathematical nature of proof ...

Table of contents

• Cover Page
• Title Page
• Copyright Page
• Series Editor’s Foreword: The Soul of Mathematics
• Preface
• List of Contributors
• Introduction
• Section I Theoretical Considerations on the Teaching and Learning of Proof
• Section II Teaching and Learning of Proof in the Elementary Grades
• Section III Teaching and Learning of Proof in Middle Grades and High School
• Section IV Teaching and Learning of Proof in College
• References