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This study compares the relative influence of syntax, semantics, and pragmatics in university students’ interpretation of multiply quantified statements in mathematics, both before and after instruction. Like previous studies, results show that semantics plays a heavy role in student interpretation, especially before instruction. Unlike previous studies, our data suggests that the patterns of student interpretation rely more upon the mathematical context than upon the order of the quantifiers. We operationalize two of Grice’s (1975) pragmatic Maxims to evaluate whether they help explain which interpretations are harder for students to adopt for various statements. Our data support the claim that students find it easier to construct relevant interpretations, but do not support the claim that students find it easier to construct interpretations that render the statement true. Finally, based on our sample from six Transition to Proof classes across the US, we observe that after their experiences in such courses students became more sensitive to syntax in their interpretation of the statements.

This paper offers instructional interventions designed to support undergraduate math students’ understanding of two forms of representations of Calculus concepts, mathematical language and ...

This study investigates undergraduate students’ meanings for quantified variables in mathematical statements involving multiple quantifiers. Clinical interviews with nine undergraduate students were conducted to explore students’ meanings for quantified variables. More specifically, students were asked to interpret and evaluate the Intermediate Value Theorem (IVT) and three other statements with similar logical structure to the IVT. In this paper, we provide our definitions for a quantified variable and student quantification as theoretical constructs from a constructivist viewpoint. Using these theoretical constructs, we interpret our main results, five categories of students’ meanings for quantified variables, which emerged from our qualitative analysis. We also discuss how our findings of students’ various meanings for quantified variables contribute to literature on theories of meaning and address issues in curriculum and instruction.

The purpose of this study is to explore how an introductory real analysis (IRA) course can be designed to bridge a gap between students’ intuition and mathematical rigor. In particular, we focus on a task, called the e-strip activity, designed for the convergence of a sequence. Data were collected from a larger study conducted as a classroom teaching experiment for a semester-long IRA course. Fischbein’s notion of secondary intuition was employed to elucidate the development of student intuition throughout the e-strip activity. We discuss how the e-strip activity played a role in developing students’ intuition and how it impacted students’ learning of the subsequent topics in the course, including definitions and theorems about convergence.

Students’ apprenticeship into proof-oriented mathematical practice requires that they become aware of a range of conventions and assumptions about mathematical language, reference, and inference that rarely appear as explicit components of the content of undergraduate courses. Because these matters of interpretation are often pre-conscious for students and taken for granted by instructors, students may fail to apprehend the problems at hand and thus misunderstand the mathematical community’s solutions. So, we argue that proof-oriented instruction must simultaneously help students understand problems of ambiguity and reference – which requires them to engage in what we call metalinguistic and metamathematical reasoning – as well as provide some informal language for discussing possible solutions to these problems. We provide three illustrative episodes from real analysis classrooms to highlight our common way of framing this class of learning challenge and the pedagogical difficulties they entail. In each case, we highlight the use of analogies as a method of helping students apprehend a problem of mathematical language, reference, or inference and discuss possible solutions. This paper’s contributions are thus both practical and theoretical: the instructional method is intended to serve as one possible approach to training students in mathematical conventions and assumptions, and it helps demonstrate our theoretical perspective on this class of teaching and learning phenomena.

The purpose of this paper is to provide issues related to student understanding of logical components that arise when solving word problems. We designed a logic problem called the King and Prisoner Puzzle - a linguistically simple, yet logically challenging problem. In this paper, we describe various student solutions to the puzzle and discuss the issues with students’ logic. In particular, it is thought-provoking that invalid arguments in students’ solutions to the puzzle are based on a lack of precise understanding of some basic logical components. This emphasizes the necessity of additional teaching to form mutual understanding of the meaning of the logical components.

Conceptual blending describes how humans condense information, combining it in novel ways. The blending process may create global insight or new detailed connections, but it may also result in a loss of information, causing confusion. In this paper, we describe the proof writing process of a group of four students in a university geometry course proving a statement of the form conditional implies conditional, i.e., ( p → q ) ⇒ ( r → s ). We use blending theory to provide insight into three diverse questions relevant for proof writing: (1) Where do key ideas for proofs come from?, (2) How do students structure their proofs and combine those structures with their more intuitive ideas?, and (3) How are students reasoning when they fail to keep track of the implication structure of the statements that they are using? We also use blending theory to describe the evolution of the students’ proof writing process through four episodes each described by a primary blend.

The original version of this article unfortunately contained two instances where the task names are mislabeled by switching the abbreviations “EA” and “AE.” Under “Rates of Normative Construal” in the “Results” section, second paragraph, fourth sentence, the sentence should be written.

The purpose of this study is to examine the characteristics of students’ thinking about aspects of graphs in the context of evaluating statements about real-valued functions from Calculus. We conducted clinical interviews in which undergraduate students evaluated mathematical statements using graphs to explain their reasoning. From our data analysis, we found two ways students think about graphs, value-thinking and location-thinking. These two ways of thinking were rooted in students’ attention to different attributes of points on graphs we provided: either the values represented by the points or the locations of the points in space. In this paper, we report our classification of students’ thinking about aspects of graphs in terms of value-thinking and location-thinking. Our findings indicate that students’ thinking about aspects of graphs accounts for key differences in their understandings of mathematical statements. We discuss some implications of our findings for instruction and curriculum development in Calculus and beyond.

In this article, we suggest an instructional intervention to help students understand statements involving multiple quantifiers in logical contexts. We analyze students' misinterpretations of multiple quantifiers related to the ϵ-N definition of convergence and point out that they result from a lack of understanding of the significance of the order of the quantifiers in the definition. We introduce the Mayan activity which is designed to cause and then to help resolve students' cognitive dissonance. In particular, the Mayan stonecutter story in the activity is presented in an understandable and colloquial form so that students can recognize the independence of ϵ from N in the ϵ-N definition. Consequently, the Mayan activity can be regarded as a useful instructional intervention to study statements related to the ϵ-N definition of convergence.

This article suggests an activity, called the ϵ–strip activity, as an instructional method for conceptualization of the rigorous definition of the limit of a sequence via visualization. The article also describes the learning objectives of each instructional step of the activity, and then provides detailed instructional methods to guide students to reach the learning objectives.

This study explored students’ understanding of a logical structure in defining the limit of a sequence, focusing on the relationship between e and N. The subjects of this study were college students who had already encountered the concept of limit but did not have any experience with rigorous proofs using the e–N definition. This study suggested two statements, each of which is written by using a relationship between e and N, similar to the e–N definition. By analyzing the students’ responses to the validity of the statements as definitions of the limit of a sequence, students’ understanding of such a relationship was classified into five major categories. This paper discusses some essential components that students must conceptualize in order properly to understand the relationship between e and N in defining the limit of a sequence.

There are many studies on the role of images in understanding the concept of limit. However, relatively few studies have been conducted on how students’ understanding of the rigorous definition of limit is influenced by the images of limit that the students have constructed through their previous learning. This study explored how calculus students’ images of the limit of a sequence influence their understanding of definitions of the limit of a sequence. In a series of task-based interviews, students evaluated the propriety of statements describing the convergence of sequences through a specially designed hands-on activity, called the ɛ–strip activity. This paper illustrates how these students’ understanding of definitions of the limit of a sequence was influenced by their images of limits as asymptotes, cluster points, or true limit points. The implications of this study for teaching and learning the concept of limit, as well as on research in mathematics education, are also discussed.

We present a mathematical theory of the two-dimensional offset curves from the viewpoint of medial axis transform. We explore the local geometry of the offset curve in relation with the medial axis transform, culminating in the classification of points on the offset curve. We then study the domain decomposition from the viewpoint of offsets, and in particular introduce the concept of monotonic fundamental domain as a device for detecting the correct topology of offsets as well as for stable numerical computation. The monotonic fundamental domains are joined by peaks or valleys of the medial axis transform, or by what we call the critical horizonal section whose algebro-geometric properties are rigorously treated as well.