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There is a longstanding conversation in the mathematics education literature about proofs that explain versus proofs that only convince. In this essay, we offer a characterization of explanatory proofs with three goals in mind. We first propose a theory of explanatory proofs for mathematics education in terms of representation systems. Then, we illustrate these ideas in terms of combinatorial proofs, focusing on binomial identities. Finally, we leverage our theory to explain audience-dependent and audience-invariant aspects of explanatory proof. Throughout, we use the context of combinatorics to emphasize points and to offer examples of proofs that can be explanatory or only convincing, depending on how one understands the claim being made.

In this paper, we explore the role of syntactic representations in set theory. We highlight a common inferential scheme in set theory, which we call the Syntactic Representation Inferential Scheme, in which the set theorist infers information about a concept based on the way that concept can be represented syntactically. However, the actual syntactic representation is only indicated, not explicitly provided. We consider this phenomenon in relation to the derivation indicator position that asserts that the ordinary proofs given in mathematical discourse indicate syntactic derivations in a formal logical system. In particular, we note that several of the arguments against the derivation indicator position would seem to imply that set theorists could gain no benefits from the syntactic representations of concepts indicated by their definitions, yet set theorists clearly do.

In this paper, we present a comparative case study of two students with different epistemological frames watching the same real analysis lectures. We show that students with different epistemological frames can interpret the same lecture in different ways. These results illustrate how a student’s interpretation of a lecture is not inherently tied to the lecture, but rather depend on the student and her perspective on mathematics. Thus, improving student learning may depend on more than improving the quality of the lectures, but also changing student’s beliefs and orientations about mathematics and mathematics learning.

Prospective secondary mathematics teachers are typically required to take advanced university mathematics courses. However, many prospective teachers see little value in completing these courses. In this paper, we present the instantiation of an innovative model that we have previously developed on how to teach advanced mathematics to prospective teachers in a way that informs their future pedagogy. We illustrate this model with a particular module in real analysis in which theorems about continuity, injectivity, and monotonicity are used to inform teachers’ instruction on inverse trigonometric functions and solving trigonometric equations. We report data from a design research study illustrating how our activities helped prospective teachers develop a more productive understanding of inverse functions. We then present pre-test/post-test data illustrating that the prospective teachers were better able to respond to pedagogical situations around these concepts that they might encounter.

We examined the proof-writing behaviors of six highly successful mathematics majors on novel proving tasks in calculus. We found two approaches that these students used to write proofs, which we termed the targeted strategy and the shotgun strategy. When using a targeted strategy students would develop a strong understanding of the statement they were proving, choose a plan based on this understanding, develop a graphical argument for why the statement is true, and formalize this graphical argument into a proof. When using a shotgun strategy, students would begin trying different proof plans immediately after reading the statement and would abandon a plan at the first sign of difficulty. The identification of these two strategies adds to the literature on proving by informing how elements of existing problem-solving models interrelate.

In previous research, we proposed a set of proof reading strategies that we hypothesized can help students better understand the proofs that they read. The goal of the present paper is to report lessons that we learned from two instructional interventions in which students were taught to apply these strategies. We found suggestive evidence that implementing these strategies helped students understand the proofs that they read, but also found students’ implementation of these strategies to sometimes be problematic. We present instructional modifications, as well as refinements to the strategies themselves, that enabled the students to implement the strategies more effectively.

In this paper, I identify five effective proof reading strategies that mathematics majors can use to comprehend proofs. This paper reports two studies. The first study is a qualitative study in which four successful mathematics majors were videotaped reading six proofs. These students used five proof reading strategies to foster comprehension: (i) trying to prove a theorem before reading its proof, (ii) identifying the proof framework being used in the proof, (iii) breaking the proof into parts or sub-proofs, (iv) illustrating difficult assertions in the proof with an example, and (v) comparing the method used in the proof with one’s own approach. The qualitative analysis of these proof reading strategies illustrates how they may foster comprehension. The second study was a quantitative study in which 83 mathematics professors were surveyed about whether they desired that their students use these strategies. Most mathematicians indicated that they desired that students use them.

In this paper we present a case study of an individual student who consistently used semantic reasoning to construct proofs in calculus but infrequently used semantic reasoning to produce proofs in linear algebra. We hypothesize that the differences in these reasoning styles can be partially attributed to this student’s familiarity with the content, the teaching styles of the professors who taught him, and the amount of time he was given to complete the tasks. We suggest that there are factors that have been ignored in previous research, including domain, instruction, and methodological constraints, that researchers should consider when ascribing to students a proving style.

In this paper, we investigate mathematics majors’ perceptions of the admissibility of inferences based on graphical reasoning for calculus proofs. The main findings from our study is that the majority of mathematics majors did not think that graphical perceptual inferences (i.e., inferences based on the appearance of the graph) were permissible in a proof, but the majority of mathematics majors did believe that graphical deductive inferences (i.e., inferences based on what must necessarily be entailed by a graph having a certain property) were permissible.

The received view of mathematical practice is that mathematicians gain certainty in mathematical assertions by deductive evidence rather than empirical or authoritarian evidence. This assumption has influenced mathematics instruction where students are expected to justify assertions with deductive arguments rather than by checking the assertion with specific examples or appealing to authorities. In this article, we argue that the received view about mathematical practice is too simplistic; some mathematicians sometimes gain high levels of conviction with empirical or authoritarian evidence and sometimes do not gain full conviction from the proofs that they read. We discuss what implications this might have, both for mathematics instruction and theories of epistemic cognition.

This paper concerns proof presentation at the university level. We report on a study in which we observed ten mathematicians constructing or revising proofs for pedagogical purposes. We highlight the factors that they claimed to consider when completing these tasks. We found that intended audience and medium (lecture or textbook) influenced proof presentation. We also found that, although mathematicians generally valued pedagogical proofs featuring diagrams and emphasizing main ideas, these mathematicians did not always incorporate these aspects in the proofs they constructed or revised.

In this article, we report a study in which 109 research-active mathematicians were asked to judge the validity of a purported proof in undergraduate calculus. Significant results from our study were as follows: (a) there was substantial disagreement among mathematicians regarding whether the argument was a valid proof, (b) applied mathematicians were more likely than pure mathematicians to judge the argument valid, (c) participants who judged the argument invalid were more confident in their judgments than those who judged it valid, and (d) participants who judged the argument valid usually did not change their judgment when presented with a reason raised by other mathematicians for why the proof should be judged invalid. These findings suggest that, contrary to some claims in the literature, there is not a single standard of validity among contemporary mathematicians.

Understanding what mathematics teachers know, what they need to know about mathematics, and how that knowledge is learned are important goals in mathematics education. Research on mathematics teacher knowledge can be divided into two categories: (a) what knowledge mathematics teachers have or need to have to teach effectively (e.g., Hill, Rowan, & Ball, 2005; Kahan, Cooper, & Bethea, 2003), and (b) how teachers’ mathematical knowledge for teaching can be developed (e.g., Bell, Wilson, Higgins, & McCoach, 2010; Proulx, 2008). This book describes research of the second type. To date, research in this area has focused primarily on how mathematical knowledge develops in university or researcher-led teacher preparation or professional development programs. This book is novel in that it concerns how and what teachers learn through the process of teaching itself. In his contribution to this book, Ron Tzur (chapter 3) lays out three reasons why this research is essential. First, he argues, teacher preparation programs simply do not contain enough time for teachers to learn all they need to know, so teachers’ learning through teaching is essential. Second, for teachers to develop knowledge of how students think about mathematics and how students receive mathematical lessons, teachers must have classroom experience. Third, the experiences that teachers encounter when teaching have the potential to give rise to meaningful changes in their beliefs and practice. Until recently, learning through teaching has largely been overlooked by researchers. The chapters in this book describe both mathematics and mathematical pedagogy that can be learned through teaching and ways in which this knowledge is acquired. In addition, the text describes the conditions that are needed to foster such learning, conditions that come both from within the teacher and from the influence of the researcher or facilitator in a professional development settings

Although proof comprehension is fundamental in advanced undergraduate mathematics courses, there has been limited research on what it means to understand a mathematical proof at this level and how such understanding can be assessed. In this paper, we address these issues by presenting a multidimensional model for assessing proof comprehension in undergraduate mathematics. Building on Yang and Lin’s (Educational Studies in Mathematics 67:59–76, 2008) model of reading comprehension of proofs in high school geometry, we contend that in undergraduate mathematics a proof is not only understood in terms of the meaning, logical status, and logical chaining of its statements but also in terms of the proof’s high-level ideas, its main components or modules, the methods it employs, and how it relates to specific examples. We illustrate how each of these types of understanding can be assessed in the context of a proof in number theory.

This paper reports a teaching experiment in which two students engaged in tasks that challenged them to describe a final state for a variety of infinite iterative processes. The results from the study indicate that the students used multiple reasoning strategies for addressing these tasks. Refinements in the students’ reasoning occurred as students constructed relationships between the problems they were solving and problems they had solved previously, applying some of the reasoning strategies that they used for one problem to make sense of or solve another problem. We discuss how these findings relate to the existing body of research on infinite iterative processes.

Many mathematics education researchers have suggested that asking learners to generate examples of mathematical concepts is an effective way of learning about novel concepts. To date, however, this suggestion has limited empirical support. We asked undergraduate students to study a novel concept by either tackling example generation tasks or reading worked solutions to these tasks. Contrary to suggestions in the literature, we found no advantage for the example generation group on subsequent proof production tasks. From a second study, we found that undergraduate students overwhelmingly adopt a trial and error approach to example generation and suggest that different example generation strategies may result in different learning gains. We conclude by arguing that the teaching strategy of example generation is not yet understood well enough to be a viable pedagogical recommendation.

In this paper, we report a study in which nine research mathematicians were interviewed with regard to the goals guiding their reading of published proofs and the type of reasoning they use to reach these goals. Using the data from this study as well as data from a separate study (Weber, Journal for Research in Mathematics Education 39:431–459, 2008) and the philosophical literature on mathematical proof, we identify three general strategies that mathematicians employ when reading proofs: appealing to the authority of other mathematicians who read the proof, line-by-line reading, and modular reading. We argue that non-deductive reasoning plays an important role in each of these three strategies.

In this paper, we present data from an exploratory study that aimed to investigate the ways in which, and the extent to which, undergraduates enrolled in a transition-to-proof course considered exa...

Trigonometry is an important subject in the high school mathematics curriculum. As one of the secondary mathematics topics that are taught early and that link algebraic, geometric, and graphical reasoning, trigonometry can serve as an important precursor to calculus as well as collegelevel courses relating to Newtonian physics, architecture, surveying, and engineering. Unfortunately, many high school students are not accustomed to these types of reasoning (Blackett and Tall 1991), and learning about trigonometric functions is initially fraught with difficulty. Trigonometry presents many first-time challenges for students: It requires students to relate diagrams of triangles to numerical relationships and manipulate the symbols involved in such relationships. Further, trigonometric functions are typically among the first functions that students cannot evaluate directly by performing arithmetic operations.

The purpose of this article is to investigate the mathematical practice of proof validation—that is, the act of determining whether an argument constitutes a valid proof. The results of a study with 8 mathematicians are reported. The mathematicians were observed as they read purported mathematical proofs and made judgments about their validity; they were then asked reflective interview questions about their validation processes and their views on proving. The results suggest that mathematicians use several different modes of reasoning in proof validation, including formal reasoning and the construction of rigorous proofs, informal deductive reasoning, and examplebased reasoning. Conceptual knowledge plays an important role in the validation of proofs. The practice of validating a proof depends upon whether a student or mathematician wrote the proof and in what mathematical domain the proof was situated. Pedagogical and epistemological consequences of these results are discussed.

In the mathematics education literature, there is currently a debate about the mechanisms by which group discussion can contribute to mathematical learning and under what conditions this learning is likely to occur. In this paper, we contribute to this debate by illustrating three learning opportunities that group discussions can create. In analyzing a videotaped episode of eight middle school students discussing a statistical problem, we observed that these students frequently challenged the arguments that their colleagues presented. These challenges invited students to be explicit about what mathematical principles, or warrants, they were implicitly using as a basis for their mathematical claims, in some cases recognize the modes of reasoning they were using were invalid and reject these modes of reasoning, and in other cases, attempt to provide deductive support to justify why their modes of reasoning were appropriate. We then describe what social and environmental conditions allowed the discussion analyzed in this paper to occur.

Nagging is a persuasive tactic yet to be fully explored in instructional communication. Nagging involves an exchange in which a student makes persistent requests of an instructor who fails to comply. The purpose of the study was to examine student nagging behavior and, specifically, to examine nagging as a potentially face threatening act as part of Politeness Theory. Students (n=189) described a nagging exchange with an instructor by reporting on one of eight nagging strategies. Nagging is threatening to the positive and negative face of both students and instructors, with the Elicit Sympathy nag the most threatening to the students’ positive face, and the Demonstrate Frustration with the Instructor nag the most threatening to the instructors’ positive face. The Strike a Deal nag was found to be the most threatening to the students’ negative face, and the Flatter Instructor nag the most threatening to the instructors’ negative face. The majority of these face threatening acts are committed off record, or...

According to the NCTM Standards (2000), conjecturing and proving should be central activities throughout students' mathematical education. However, the question of how we can help students generate proofs, especially the formal proofs expected at the high school level, is still a difficult one. In this article, we argue that one cause of students' troubles with proof is that they are not accustomed to making the important, but difficult, connections between their intuitive sensemaking and the formal proofs they are supposed to produce. Thus as teachers, we should provide opportunities for students to make these connections.

In this paper, we distinguish between two ways that an individual can construct a formal proof. We define a syntactic proof production to occur when the prover draws inferences by manipulating symbolic formulae in a logically permissible way. We define a semantic proof production to occur when the prover uses instantiations of mathematical concepts to guide the formal inferences that he or she draws. We present two independent exploratory case studies from group theory and real analysis that illustrate both types of proofs. We conclude by discussing what types of concept understanding are required for each type of proof production and by illustrating the weaknesses of syntactic proof productions.

A localization task required participants to indicate which of 4 locations contained a briefly displayed target. Most displays also contained a distractor that was not equally probable in these locations, affecting performance dramatically. Responses were faster when a display had no distractor and almost as fast when the distractor was in its frequent location. Conversely, responses were slower when targets appeared in frequent-distractor locations, even though targets were equally likely in each location. Negative-priming effects were reliably smaller when targets followed distractors in the frequent-distractor location compared to the rare-distractor location, challenging the episodic-retrieval account. Experiment 2 added a 5th location that rarely displayed distractors and never targets, yet responses slowed most when distractors appeared there. The results confirmed that the attentional system is sensitive to first- and higher-order statistical patterns and can make short- and long-term adjustments in preferences based on prior history of inspecting unsuccessful locations.

Two studies based in the autobiographical memory paradigm were conducted to extend knowledge of recollections about school. In Study 1, undergraduates wrote 1 pleasant memory and 1 unpleasant memory about each of 4 grade levels. Memories from some categories decreased across grades, whereas others increased. More than twice as many memories were social as opposed to academic. Gender effects, causal agents, and event salience were reported. In Study 2, undergraduates wrote 8 recollections, unconstrained by grade or affect. There was a recency effect, and memories most frequently concerned misfortunes, sports, misbehavior, honors, and boy-girl relationships. Only 13% of the reported memories were academic. Future investigation might probe reasons for the dominance of unpleasant memories from the early grades as well as the paucity of academic memories of schooling.

The ability to construct proofs is an important skill for all mathematicians. Despite its importance, students have great difficulty with this task. In this paper, I first demonstrate that undergraduates often are aware of and able to apply the facts required to prove a statement but still fail to prove it. They thus fail to construct a proof because they could not use the syntactic knowledge that they had. By comparing doctoral students and undergraduates constructing proofs in abstract algebra, I have hypothesized four types of `strategic knowledge' – knowledge of how to choose which facts and theorems to apply – which the doctoral students appeared to possess and undergraduates did not. The doctoral students appeared to know the powerful proof techniques in abstract algebra, which theorems are most important, when particular facts and theorems are likely to be useful, and when one should or should not try and prove theorems using symbol manipulation.

You might want to try this brief experiment in one of your mathematics courses.

There is no doubt that Microsoft Word is the leading personal word processing software on the Windows platform. The intuitive user interface provides convenient editing as if it was doing so on a real note pad for publication. Even though users apparently see what they put on the electronic document in Word format, not-apparent information is recorded by the software itself based on the users| behaviour or identification. This information is called |Hidden Information|. In this paper, we discuss what is concealed in the Word file and its related issues.