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The paper presents a characterisation about argumentation and proof in mathematics. On the basis of contemporary linguistic theories, the hypothesis that proof is a special case of argumentation is put forward and Toulmin’s model is proposed as a methodological tool to compare them. This model can be used to detect and analyse the structure of an argumentation supporting a conjecture (abduction, induction, etc.) and the structure of its proof. The aim of the paper is to highlight the importance of structural analysis between argumentation and proof. This analysis shows that although there are clear cases of continuity between argumentation supporting a conjecture and its proof, there is often a structural distance between the two (from an abductive argumentation to a deductive proof, from an inductive argumentation to a mathematical inductive proof). [ABSTRACT FROM AUTHOR]

This paper is a case study of the teaching of an undergraduate abstract algebra course, in particular the way the instructor presented proofs. It describes a framework for proof writing based on Selden and Selden () and the work of Alcock () on modes of thought that support proof writing. The paper offers a case study of the teaching of a traditionally-taught abstract algebra course, including showing the range of practice as larger than previously described in research literature. This study describes the aspects of proof writing and modes of thought the instructor modeled for the students. The study finds that she frequently modeled the aspects of hierarchical structure and formal-rhetorical skills, and structural, critical, and instantiation modes of thought. This study also examines the instructor's attempts to involve the students in the proof writing process during class by asking questions and expecting responses. Finally, the study describes how those questions and responses were part of her proof presentation. The funneling pattern of Steinbring () describes most of the question and answer discussions enacted in the class with most questions requiring a factual response. Yet, the instructional sequence can be also understood as modeling the way an expert in the discipline thinks and, as such, offering a different type of opportunity for student learning. [ABSTRACT FROM AUTHOR]

This paper focuses on teaching and learning the set of real numbers R and its completeness property at the university level. It studies, in particular, the opportunities for understanding this property that students are offered in four undergraduate correlative courses in Calculus and Analysis. The theoretical framework used in the study draws on concepts developed in the Anthropological Theory of Didactics, especially the notions of praxeology and mathematical organization. The paper shows different expectations concerning the same notion ( R and its completeness) through different school levels, and intends to bring up some reflections about the transition from Calculus to Analysis. [ABSTRACT FROM AUTHOR]

This paper provides an instrumental account of precalculus students’ graphical process for solving polynomial inequalities. It is carried out in terms of the students’ instrumental schemes as mediated by handheld graphing calculators and in cooperation with their classmates in a classroom setting. The ethnographic narrative relays an instrumental sociogenetic account of mathematical knowledge construction and foregrounds a progressive evolution of mathematical knowledge from the concrete to the abstract phase. [ABSTRACT FROM AUTHOR]

Burn (2005) proposes a genetic approach to teaching limits of numerical sequences. The article includes an explanation of the Method of Exhaustion,1 a generalization of this method, and a description of how this method was used for obtaining areas and lengths in the seventeenth century. The author uses these historical and mathematical analyses as a basis for proposing an alternative definition of the limit of a sequence. The paper focuses on the fine mathematical and historical detail of the notion of limit. Reading it made me reflect on the explicitly or implicitly involved didactical aspects, which I would like to share with ESM readers. [ABSTRACT FROM AUTHOR]

Learning is better than knowing, generalization is more illuminating than abstract generality or universality because we perceive and thus become conscious of change or development only. Signs and representations establish the dialectic of fixation on the one hand and transformation on the other, which is so essential to learning and cognition. Mathematical epistemology from a semiotic point of view therefore is above all a genetical epistemology. All real mathematical activity is concerned with representations of mathematical entities rather than with things in themselves and with the processes of continuous transformation of a given representation into others. This paper tries to give an overview of the essential relationships between activity theory, epistemology and mathematical education, using the semiotics of Charles S. Peirce as a unifying reference. It is certainly beyond the scope of such a paper to spell out all the questions involved in every detail. Much of what is said in the four short sections to follow is calling for further concretization and research. [ABSTRACT FROM AUTHOR]

This paper reports on a study investigating students' ways of conceptualizing key ideas in linear algebra, with the particular results presented here focusing on student interactions with the notion of subspace. In interviews conducted with eight undergraduates, we found students' initial descriptions of subspace often varied substantially from the language of the concept's formal definition, which is very algebraic in nature. This is consistent with literature in other mathematical content domains that indicates that a learner's primary understanding of a concept is not necessarily informed by that concept's formal definition. We used the analytical tools of concept image and concept definition of Tall and Vinner (Educational Studies in Mathematics, 12(2):151-169, ) in order to highlight this distinction in student responses. Through grounded analysis, we identified recurring concept imagery that students provided for subspace, namely, geometric object, part of whole, and algebraic object. We also present results regarding the coordination between students' concept image and how they interpret the formal definition, situations in which students recognized a need for the formal definition, and qualities of subspace that students noted were consequences of the formal definition. Furthermore, we found that all students interviewed expressed, to some extent, the technically inaccurate 'nested subspace' conception that R is a subspace of R for k < n. We conclude with a discussion of this and how it may be leveraged to inform teaching in a productive, student-centered manner. [ABSTRACT FROM AUTHOR]

The objective of this paper is to study students' difficulties when they have to ascribe the same meaning to different representations of the same mathematical object. We address two theoretical tools that are at the core of Radford's cultural semiotic and Godino's onto-semiotic approaches: objectification and the semiotic function. The analysis of a teaching experiment involving high school students working on the tangent, shows how students' difficulties in ascribing sense to different representations of a common mathematical object can be traced back to the kind of objectification processes and semiotic functions they are able to establish. [ABSTRACT FROM AUTHOR]

In this paper, a focus is made on defining processes at stake in an unfamiliar situation coming from discrete mathematics which brings surprising mathematical results. The epistemological framework of Lakatos is questioned and used for the design and the analysis of the situation. The cognitive background of Vergnaud's approach enriches the study of freshmen's processes at university. The mathematical analysis and the results specifically underscore the in-action definitions and the zero-definitions and highlight the need of similar mathematical experiences in education, particularly focused on defining processes and the exploration of a research problem. [ABSTRACT FROM AUTHOR]

This paper (1) presents a conceptual framework for analyzing the mathematics addressed in probability lessons and (2) uses the framework to compare the mathematics that two teachers with contrasting teaching approaches addressed in class when teaching the topic of probability. One teaching approach aimed to develop understanding; the other emphasized mechanistic answer finding. Class work on 193 problems was analyzed qualitatively and quantitatively, showing some similarities and some differences in the mathematics that the two teachers offered to students. The differences found seemed to be linked to the teachers’ teaching approaches. The findings suggest that teachers who adopt different teaching approaches, to some extent, make available to learn different mathematics even when they use the same textbooks. [ABSTRACT FROM AUTHOR]

This paper investigates the role of tools in the formation of mathematical practices and the construction of mathematical meanings in the setting of a telecommunication organization through the actions undertaken by a group of technicians in their working activity. The theoretical and analytical framework is guided by the first-generation activity theory model and Leont’ev’s work on the three-tiered explanation of activity. Having conducted a 1-year ethnographic research study, we identified, classified, and correlated the tools that mediated the technicians’ activity, and we studied the mathematical meanings that emerged. A systemic network was generated, presenting the categories of tools such as mathematical (communicative, processes, and concepts) and non-mathematical (physical and written texts). This classification was grounded on data from three central actions of the technicians’ activity, while the constant interrelation and association of these tools during the working process addressed the mathematical practices and supported the construction of mathematical meanings that this group developed from the researchers’ perspective. Technicians’ emerging mathematical meanings referred to place value, spatial, and algebraic relations and were expressed through personal algorithms and metaphorical and metonymic reasoning. Finally, the educational implications of the findings are discussed. [ABSTRACT FROM AUTHOR]

This paper presents a study of instructors’ and students’ perceptions of the knowledge to be learned about limits of functions in a college level Calculus course, taught in a North American college institution. I modeled these perceptions using a theoretical framework that combines elements of the Anthropological Theory of the Didactic, developed in mathematics education, with a framework for the study of institutions developed in political science. While a model of the instructors’ perceptions could be formulated mostly in mathematical terms, a model of the students’ perceptions included an eclectic mixture of mathematical, social, cognitive, and didactic norms. I describe the models and illustrate them with examples from the empirical data on which they have been built. [ABSTRACT FROM AUTHOR]

The study reported here is the third in a series of research articles (Harkness, S. S., D’Ambrosio, B., & Morrone, A. S.,in Educational Studies in Mathematics 65:235–254, ; Morrone, A. S., Harkness, S. S., D’Ambrosio, B., & Caulfield, R. in Educational Studies in Mathematics 56:19–38, ) about the teaching practices of the same university professor and the mathematics course, Problem Solving, she taught for preservice elementary teachers. The preservice teachers in Problem Solving reported that they were motivated and that Sheila made learning goals salient. For the present study, additional data were collected and analyzed within a qualitative methodology and emergent conceptual framework, not within a motivation goal theory framework as in the two previous studies. This paper explores how Sheila’s “trying to believe,” rather than a focus on “doubting” (Elbow, P., Embracing contraries, Oxford University Press, New York, ), played out in her practice and the implications it had for both classroom conversations about mathematics and her own mathematical thinking. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In his 1976 book, Proofs and Refutations, Lakatos presents a collection of case studies to illustrate methods of mathematical discovery in the history of mathematics. In this paper, we reframe these methods in ways that we have found make them more amenable for use as a framework for research on learning and teaching mathematics. We present an episode from an undergraduate abstract algebra classroom to illustrate the guided reinvention of mathematics through processes that strongly parallel those described by Lakatos. Our analysis suggests that the constructs described by Lakatos can provide a useful framework for making sense of the mathematical activity in classrooms where students are actively engaged in the development of mathematical ideas and provide design heuristics for instructional approaches that support the learning of mathematics through the process of guided reinvention. [ABSTRACT FROM AUTHOR]

Before the advent of symbolism, i.e. before the end of the 16th Century, algebraic calculations were made using natural language. Through a kind of metaphorical process, a few terms from everyday life (e.g. thing, root) acquired a technical mathematical status and constituted the specialized language of algebra. The introduction of letters and other symbols (e.g. “+”, “=”) made it possible to achieve what is considered one of the greatest cultural accomplishments in human history, namely, the constitution of a symbolic algebraic language and the concomitant rise of symbolic thinking. Because of their profound historical ties with natural language, the emerging syntax and meanings of symbolic algebraic language were marked in a definite way by the syntax and meanings of the former. However, at a certain point, these ties were loosened and algebraic symbolism became a language in its own right. Without alluding to the theory of recapitulation, in this paper, we travel back and forth from history to the present to explore key passages in the constitution of the syntax and meanings of symbolic algebraic language. A contextual semiotic analysis of the use of algebraic terms in 9th century Arabic as well as in contemporary students' mathematical activity, sheds some light on the conceptual challenges posed by the learning of algebra. [ABSTRACT FROM AUTHOR]

This paper examines the change in young children’s understanding of ‘equal’, ‘more’, ‘less’, and ‘between’, words commonly used in equivalent and non-equivalent situations, over a 3-year period. Seventy-six children participated in the longitudinal study. Each year they were asked to share their understanding of these four words. Past research has indicated that many children have limited understanding of ‘equal’ as quantitative sameness. The results of this research suggested that many children also have limited understanding of ‘more’ and ‘less’ and that these understandings did not significantly change over the 3-year period. [ABSTRACT FROM AUTHOR]

This study reports on how students can be led to make meaningful connections between such structures on a set as a partition, the set of equivalence classes determined by an equivalence relation and the fiber structure of a function on that set (i.e., the set of preimages of all sets {b} for b in the range of the function). In this paper, I first present an initial genetic decomposition, in the sense of APOS theory, for the concepts of equivalence relation and function in the context of the structures that they determine on a set. This genetic decomposition is primarily based on my own mathematical knowledge as well as on my observations of students’ learning processes. Based on this analysis, I then suggest instructional procedures that motivate the mental activities described in the genetic decomposition. I finally present empirical data from informal interviews with students at different stages of learning. My goal was to guide students to become aware of the close conceptual correspondence and connections among the aforementioned structures. One theorem that captures such connections is the following: a relation R on a set A is an equivalence relation if and only if there exists a function f defined on A such that elements related via R (and only those) have the same image under f. [ABSTRACT FROM AUTHOR]

Abstract algebra courses tend to take one of two pedagogical routes: from examples of mathematics structures through definitions to general theorems, or directly from definitions to general theorems. The former route seems to be based on the implicit pedagogical intention that students will use their understanding of particular examples of an algebraic structure to get a sense of those properties which form the basis of the fundamental definitions. We will explain the transition from examples to abstract algebra as a series of shifts of attention and in this paper we will use a case study to examine the initial shift, which we will call apprehending a structure, and examine how one student came to apprehend the structure of the commutative ring Z 99. [ABSTRACT FROM AUTHOR]

To understand the difficulties that many students have with comprehension of mathematics, we must determine the cognitive functioning underlying the diversity of mathematical processes. What are the cognitive systems that are required to give access to mathematical objects? Are these systems common to all processes of knowledge or, on the contrary, some of them are specific to mathematical activity? Starting from the paramount importance of semiotic representation for any mathematical activity, we put forward a classification of the various registers of semiotic representations that are mobilized in mathematical processes. Thus, we can reveal two types of transformation of semiotic representations: treatment and conversion. These two types correspond to quite different cognitive processes. They are two separate sources of incomprehension in the learning of mathematics. If treatment is the more important from a mathematical point of view, conversion is basically the deciding factor for learning. Supporting empirical data, at any level of curriculum and for any area of mathematics, can be widely and methodologically gathered: some empirical evidence is presented in this paper. [ABSTRACT FROM AUTHOR]

It is widely attested that university students face considerable difficulties with reasoning in analysis, especially when dealing with statements involving two different quantifiers. We focus in this paper on a specific mistake which appears in proofs where one applies twice or more a statement of the kind "for all X, there exists Y such that R(X, Y)", and forgets that in that case, a priori, "Y depends on X". We analyse this mistake from both a logical and mathematical point of view, and study it through two inquiries, an historical one and a didactic one. We show that mathematics teachers emphasise the importance of the dependence rule in order to avoid this kind of mistake, while natural deduction in predicate calculus provides a logical framework to analyse and control the use of quantifiers. We show that the relevance of this dependence rule depends heavily on the context: nearly without interest in geometry, but fundamental in analysis or linear algebra. As a consequence, mathematical knowledge is a key to correct reasoning, so that there is a large distance between beginners' and experts' abilities regarding control of validity, that, to be shortened, probably requires more than a syntactic rule or informal advice. [ABSTRACT FROM AUTHOR]

The paper is based on extensive research carried out on students of mathematics who had completed a university course of calculus. The basic purpose of the research was to determine the students' images of the concept of limit, that is to find out their associations, conceptions and intuitions connected with limits and to determine the degree of their efficiency and the sources of their formation. To achieve the objectives an expanded set of selected problems – simple but not quite standard – and various other research instruments were used. Several classes of images of the limit concept have been identified and described, according to their main foci: neighborhoods, graph approaching, values approaching, being defined at x0, limit of f at x0 equals f (x0), and algorithms. [ABSTRACT FROM AUTHOR]

This paper reports research that focuses on student teachers' reflections on their learning process in a collaborative problem-solving context. One group of students with limited mathematical backgrounds worked on two problems in geometry without teacher intervention. We focus on two episodes from the group dialogues. In the first episode (section 5) the students basically reflect on two key issues. The first reflection is related to the concern of making problem-solving tasks too difficult in general while the second reflection has to do with the concern of participation in the solution process. The students discuss how they can give hints or introduce particular ideas before presenting a solution in order to stimulate colleague participation, thus promoting the understanding of the solution process. The second episode (section 6) illustrates the reflection of students on their preparation as future teachers of mathematics. They emphasise that the experience of getting stuck with a problem may help them to better understand the frustration pupils experience while working on unfamiliar problems in classroom. Based on the experience of getting stuck, the students reflect on how they could motivate themselves as well as pupils to work on mathematical problems. They suggest that a good strategy is to start working on an easier problem. If they succeed in coming up with a solution to that problem, they think that it is then more stimulating to proceed to a difficult one. [ABSTRACT FROM AUTHOR]

This case study uses a sociocultural perspective and the concept of appropriation (Newman, Griffin and Cole, 1989; Rogoff, 1990) to describe how a student learned to work with linear functions. The analysis describes in detail the impact that interaction with a tutor had on a learner, how the learner appropriated goals, actions, perspectives, and meanings that are part of mathematical practices, and how the learner was active in transforming several of the goals that she appropriated. The paper describes how a learner appropriated two aspects of mathematical practices that are crucial for working with functions (Breidenbach, Dubinsky, Nichols and Hawks, 1992; Even, 1990; Moschkovich, Schoenfeld and Arcavi, 1993; Schwarz and Yerushalmy, 1992; Sfard, 1992): a perspective treating lines as objects and the action of connecting a line to its corresponding equation in the form y = mx + b . I use examples from the analysis of two tutoring sessions to illustrate how the tutor introduced three tasks (estimating y-intercepts, evaluating slopes, and exploring parameters) that reflect these two aspects of mathematical practices in this domain and describe how the student appropriated goals, actions, meanings, and perspectives for carrying out these tasks. I describe how appropriation functioned in terms of the focus of attention, the meaning for utterances, and the goals for these three tasks. I also examine how the learner did not merely repeat the goals the tutor introduced but actively transformed some of these goals. [ABSTRACT FROM AUTHOR]

This paper begins by considering the cognitive mechanisms available to individuals which enable them to operate successfully in different parts of the mathematics curriculum. We base our theoretical development on fundamental cognitive activities, namely, perception of the world, action upon it and reflection on both perception and action. We see an emphasis on one or more of these activities leading not only to different kinds of mathematics, but also to a spectrum of success and failure depending on the nature of the focus in the individual activity. For instance, geometry builds from the fundamental perception of figures and their shape, supported by action and reflection to move from practical measurement to theoretical deduction and euclidean proof. Arithmetic, on the other hand, initially focuses on the action of counting and later changes focus to the use of symbols for both the process of counting and the concept of number. The evidence that we draw together from a number of studies on children‘s arithmetic shows a divergence in performance. The less successful seem to focus more on perceptions of their physical activities than on the flexible use of symbol as process and concept appropriate for a conceptual development in arithmetic and algebra. Advanced mathematical thinking introduces a new feature in which concept definitions are formulated and formal concepts are constructed by deduction. We show how students cope with the transition to advanced mathematical thinking in different ways leading once more to a diverging spectrum of success. [ABSTRACT FROM AUTHOR]

Through our work in mathematics and science education we have observed that students react similarly to a wide variety of conceptually unrelated situations. Our work suggests that many responses which the literature describes as alternative conceptions could be interpreted as evolving from common, intuitive rules. This paper describes and discusses one such rule, manifested when two systems are equal with respect to a certain quantity A but differ in another quantity B. We found that in such situations, students often argue that ’Same amount of A implies same amount of B‘. Our claim is that such responses are specific instances of the intuitive rule ’Same A–same B‘. This approach explains common sources for students‘ conceptions and has strong predictive power. [ABSTRACT FROM AUTHOR]

The transition from arithmetic to algebra in general, and the use of symbolic generalizations in particular, are a major challenge for beginning algebra students. In this article, we describe and analyze students’ learning in a “computer intensive environment” designed ad hoc and implemented in two seventh grade classrooms throughout two consecutive school years. In particular, this article focuses on the description and analysis of how students initial generalizations (which relied on computerized tools that enabled different students’ to work with different strategies) shifted to recursive and explicit symbolic generalizations. [ABSTRACT FROM AUTHOR]

We examine a commonly suggested proof construction strategy from the mathematics education literature-that students first produce a graphical argument and then work to construct a verbal-symbolic proof based on that graphical argument. The work of students who produce such graphical arguments when solving proof construction tasks was analyzed to distill three activities that contribute to students' successful translation of graphical arguments into verbal-symbolic proofs. These activities are called elaborating, syntactifying, and rewarranting. We analyze how engaging in these activities relates to students' success in proof construction tasks. Additionally, we discuss how each individual activity contributes to the translation of a graphical argument into a verbal-symbolic proof. [ABSTRACT FROM AUTHOR]

Existing research on limits documents many difficulties students encounter when learning about the concept. There is also some research on teaching of limits but it is not yet as extensive as the research on student learning about limits. This study explores the discourse on limits in a beginning-level undergraduate calculus classroom by focusing on one instructor's and his students' discourses through a communicational approach to cognition. The findings indicate that some of the limit-related contexts in which students struggled coincided with those in which the instructor shifted his elements of discourse on limits. The instructor did not attend to the shifts in his discourse, making them implicit for the students. The study highlights that the discrepancies among participants' discourses signal communicational breakages and suggests that future studies should examine whether teachers' explicit attention to the elements of their discourse can enhance communication in the classrooms. [ABSTRACT FROM AUTHOR]

In this study we analyze students' understanding of two-variable function; in particular we consider their understanding of domain, possible arbitrary nature of function assignment, uniqueness of function image, and range. We use APOS theory and semiotic representation theory as a theoretical framework to analyze data obtained from interviews with thirteen students who had taken a multivariable calculus course. Results show that few students were able to construct an object conception of function of two variables. Most students showed difficulties finding domains of functions, in particular, when they were restricted to a specific region in the xy plane. They also showed that they had not fully coordinated their R, set, and function of one variable schemata. We conclude from the analysis that many of the interviewed students' notion of function can be considered as pre-Bourbaki. [ABSTRACT FROM AUTHOR]

nalogical reasoning is believed to be an efficient means of problem solving and construction of knowledge during the search for and the analysis of new mathematical objects. However, there is growing concern that despite everyday usage, learners are unable to transfer analogical reasoning to learning situations. This study aims at facilitating analogy use for conjecturing in discourse-rich mathematics classrooms. We reconceptualized one of the traditional perspectives on analogical reasoning, called classical analogy, as a more dynamic one by providing learners with the opportunity to choose a target object and its property. While shifting attention to particular aspects of mathematical activity, we observed and analyzed how students became aware of hidden relational similarities and utilized them while weakening others to make conjectures. The detailed analysis of the constructs and processes of several similarity-making and conjecturing activities supports the significance of reconceived classical analogy use in mathematics classrooms. [ABSTRACT FROM AUTHOR]

The aim of this article is to develop analytical tools for studying mathematical communication in collaborative activities. The theoretical construct of contextualization is elaborated methodologically in order to study diversity in individual thinking in relation to effective communication. The construct of contextualization highlights issues of diversity in collaborative activities as it emphasizes how students may struggle differently with a learning activity. The interaction of students (12 to 13 years old), playing a specifically designed dice game, is used as an example for illustration. The article shows how accounting for the focal events of the interlocutors, and the contexts in which they contextualize these events, help in organizing our thinking about mathematically effective communication in collaborative activities. [ABSTRACT FROM AUTHOR]

The view of infinity as a metaphor, a basic premise of modern cognitive theory of embodied knowledge, suggests in particular that there may be alternative ways in which one could formalize mathematical ideas about infinity. We discuss the key ideas about infinitesimals via a proceptual analysis of the meaning of the ellipsis “...” in the real formula $\hbox{.999\ldots = 1}$. Infinitesimal-enriched number systems accommodate quantities in the half-open interval [0,1) whose extended decimal expansion starts with an unlimited number of repeated digits 9. Do such quantities pose a challenge to the unital evaluation of the symbol “.999...”? We present some non-standard thoughts on the ambiguity of the ellipsis in the context of the cognitive concept of generic limit of B. Cornu and D. Tall. We analyze the vigorous debates among mathematicians concerning the idea of infinitesimals. [ABSTRACT FROM AUTHOR]

In this research article, I present evidence of the existence of visual templates in pattern generalization activity. Such templates initially emerged from a 3-week design-driven classroom teaching experiment on pattern generalization involving linear figural patterns and were assessed for existence in a clinical interview that was conducted four and a half months after the teaching experiment using three tasks (one ambiguous, two well defined). Drawing on the clinical interviews conducted with 11 seventh- and eighth-grade students, I discuss how their visual templates have spawned at least six types of algebraic generalizations. A visual template model is also presented that illustrates the distributed and a dynamically embedded nature of pattern generalization involving the following factors: pattern goodness effect; knowledge/action effects; and the triad of stage-driven grouping, structural unit, and analogy. [ABSTRACT FROM AUTHOR]

This study is part of a project concerned with the analysis of how students work with two-variable functions. This is of fundamental importance given the role of multivariable functions in mathematics and its applications. The portion of the project we report here concentrates on investigating the relationship between students’ notion of subsets of Cartesian three-dimensional space and the understanding of graphs of two-variable functions. APOS theory and Duval’s theory of semiotic representations are used as theoretical framework. Nine students, who had taken a multivariable calculus course, were interviewed. Results show that students’ understanding can be related to the structure of their schema for R3 and to their flexibility in the use of different representations. [ABSTRACT FROM AUTHOR]

Our work is inspired by the book Imagining Numbers (particularly the square root of minus fifteen), by Harvard University mathematics professor Barry Mazur ( Imagining numbers (particularly the square root of minus fifteen), Farrar, Straus and Giroux, New York, ). The work of Mazur led us to question whether the features and steps of Mazur’s re-enactment of the imaginative work of mathematicians could be appropriated pedagogically in a middle-school setting. Our research objectives were to develop the framework of teaching mathematics as a way of imagining and to explore the pedagogical implications of the framework by engaging in an application of it in middle school setting. Findings from our application of the model suggest that the framework presents a novel and important approach to developing mathematical understanding. The model demonstrates in particular the importance of shared visualizations and problem-posing in learning mathematics, as well as imagination as a cognitive space for learning. [ABSTRACT FROM AUTHOR]

It is widely recognized that purely cognitive behavior is extremely rare in performing mathematical activity: other factors, such as the affective ones, play a crucial role. In light of this observation, we present a reflection on the presence of affective and cognitive factors in the process of proving. Proof is considered as a special case of problem solving and the proving process is studied adopting a perspective according to which both affective and cognitive factors influence it. To carry out our study, we set up a framework where theoretical tools coming from research on problem solving, proof and affect are present. The study is performed within a university course in mathematics education, where students were given a statement in elementary number theory to be proved and were asked to write down their proving process and the thoughts that accompanied this process. We scrutinize the written protocols of two unsuccessful students, with the aim of disentangling the intertwining between affect and cognition. In particular, we seize the moments in which beliefs about self and beliefs about mathematical activity shape the performance of our students. [ABSTRACT FROM AUTHOR]

I investigate the contributions of three theoretical frameworks to a research process and the complementary role played by each. First, I describe the essence of each theory and then follow the analysis of their specific influence on the research process. The research process is on the conceptualization of the notion of limit by means of the discrete continuous interplay. I investigate the influence of the different perspectives on the research process and realize that the different theoretical approaches intertwine. Moreover, I realize that the research study demanded the contribution of more than one theoretical approach to the research process and that the differences between the frameworks could serve as a basis for complementarities. [ABSTRACT FROM AUTHOR]

How do students think about algebra? Here we consider a theoretical framework which builds from natural human functioning in terms of embodiment – perceiving the world, acting on it and reflecting on the effect of the actions – to shift to the use of symbolism to solve linear equations. In the main, the students involved in this study do not encapsulate algebraic expressions from process to object, they do not solve ‘evaluation equations’ such as $$ 3x + 2 = 8 $$ by ‘undoing’ the operations on the left, they do not find such equations easier to solve than $$ 3x - 1 = 3 + x $$ , and they do not use general principles of ‘do the same thing to both sides.’ Instead they build their own ways of working based on the embodied actions they perform on the symbols, mentally picking them up and moving them around, with the added ‘magic’ of rules such as ‘change sides, change signs.’ We consider the need for a theoretical framework that includes both embodiment and process-object encapsulation of symbolism and the need for communication of theoretical insights to address the practical problems of teachers and students. [ABSTRACT FROM AUTHOR]

Simplicity arguments are to be found in most geometrical works, from those of Proclus in his Commentaries on the First Book of Euclid’s Elements, up to those of contemporary manuals. Our goal is to read these arguments in their historical contexts to analyze agreements, disagreements and the multiplicity of points of view. For a better apprehension and a better understanding of the different conceptions, we will focus on the notion of angles and their measurements. We will study the notion of ≪ simplicity ≫ in various Elements of Geometry, in particular those of Euclid, Peletier du Mans (), Arnauld (), Lacroix () and Hoüel (). From there, we will examine French schoolbooks of geometry, beginning from the 1960s up to the 1990s, including those of the so-called period of ≪la réforme des mathématiques modernes≫ in France. [ABSTRACT FROM AUTHOR]

In this article, we take a rapid journey through the history of algebra, noting the important developments and reflecting on the importance of this history in the teaching of algebra in secondary school or university. Frequently, algebra is considered to have three stages in its historical development: the rhetorical stage, the syncopated stage, and the symbolic stage. But besides these three stages of expressing algebraic ideas, there are four more conceptual stages which have happened along side of these changes in expressions. These stages are the geometric stage, where most of the concepts of algebra are geometric ones; the static equation-solving stage, where the goal is to find numbers satisfying certain relationships; the dynamic function stage, where motion seems to be an underlying idea, and finally, the abstract stage, where mathematical structure plays the central role. The stages of algebra are, of course not entirely disjoint from one another; there is always some overlap. We discuss here high points of the development of these stages and reflect on the use of these historical stages in the teaching of algebra. [ABSTRACT FROM AUTHOR]

The study focused on the interaction between two secondary school students while they were working on computerized mathematical investigation tasks related to probability theory. The aim was to establish how such interaction helped the students to learn from one another, and how it may have hindered their learning process. The assumption was that interaction is beneficial for students if they can perform certain key activities, namely showing, explaining, justifying, and reconstructing their work. Both students attained mathematical level raising. However, the student who explained frequently and criticized himself attained more mathematical level raising than the student who did not explain her work frequently or criticize herself. [ABSTRACT FROM AUTHOR]

In the context of a three-year professional development program in mathematics, practicing elementary teachers persistently engaged in collaborative inquiry and reflection to build connected meanings for slope. One teacher invented a compelling representation for slope as a process of repeated addition, using Cuisenaire rods, based on teachers' shared experiences developing recursively defined linear equations. The presence of and tension between different representations of slope, brought forth by the teachers, catalyzed productive cycles of choice and inquiry for the entire class. Personal agency, purposeful choice, and performance provide a valuable lens for fine-grained analysis of mathematical learning. [ABSTRACT FROM AUTHOR]

A semiotic perspective on mathematical activity provides a way of conceptualizing the teaching and learning of mathematics that transcends and encompasses both psychological perspectives focussing exclusively on mental structures and functions, and performance-focussed perspectives concerned only with student' behaviours. Instead it considers the personal appropriation of signs and the underlying meaning structures embodying relationships between signs. It is concerned with patterns of sign use and sign production, including individual creativity in sign use, and the underlying social rules and contexts of sign use. It is based on the concept of a semiotic system, comprising signs, rules of sign production, and an underpinning meaning structure. This theorisation is applied to the learning of number, from counting to calculation. Historical, foundational and developmental (i.e., learning) perspectives are explored and contrasted. It is argued that in each of these domains, the dominant significant activity concerns the production of sequences of signs. [ABSTRACT FROM AUTHOR]

Mathematical signs and symbols have a decisive role for coding, constructing and communicating mathematical knowledge. Nevertheless these mathematical signs do not already contain mathematical meaning and conceptual ideas themselves. The contribution will present basic elements of an epistemology of mathematical knowledge and then apply these theoretical ideas for analyzing case studies of two teaching episodes from elementary mathematics teaching. In this way different roles of mathematical signs as means of communication (oral function), of indicating (deictic function) and of writing (symbolic function) will be elaborated. [ABSTRACT FROM AUTHOR]

The purpose of the present article is to give an account of the emergence of knowledge pertaining to the transition from arithmetic to algebra in the course of a debate in a grade 7 classroom. This debate follows two other instances of work: (1) the adidactic interaction between each student and a given problem, (2) the adidactic interaction of each student with the procedures generated by other students during stage 1. The two kinds of processes – adaptation to a milieu and social interactions – play a critical role in the change of “rationality” required for the move from arithmetic to algebra. Both the design of the initial mathematical problem given to the students and the organization of the interactions leading to the debate under study in this article are based on this hypothesis. The research presented in this article is set in a broader work of didactic engineering that aims at studying didactic conditions for making a connection between arithmetic practices and algebraic practices. [ABSTRACT FROM AUTHOR]

Four groups (three engineering students in each group) were videotaped while constructing concept maps in Linear Algebra. There are two aims of this study. The first is to characterize the discourse in the groups by addressing the following research questions: Do the students communicate in an effective way? Do the students' communications contain the elements typical for a mathematically productive discourse? The analysis indicates that the communication among the students is effective and contains the elements that are characteristic for a mathematically productive interaction. The two types of methods used to analyze the data were focal and preoccupational analysis. The mathematical content and the coherence of the conversations were examined through focal analysis. The participants' engagement in the discourse was examined by preoccupational analysis, carried out by means of interactive flowchart. The second aim of this study is to evaluate the newly developed methodological framework used to characterize the discourses. The study shows that several aspects of the methodological framework need to be developed. [ABSTRACT FROM AUTHOR]

Exercises and problems about functions found in 24 middle-school textbooks from 15 countries were analysed using an adaptation of Balacheff's theory of conceptions and Biehler's notion of the prototypical domain of application of concepts in order to describe the practices associated with the notion of function. The analysis yielded five different practices – symbolic rule, ordered pair, social data, physical phenomena, and controlling image – that were present both across and within the textbooks analysed. The existence of different practices might help to explain the compartmentalised and sometimes contradictory notions that students and teachers have about functions and might shed light on the processes of designing curriculum and instruction on functions. [ABSTRACT FROM AUTHOR]

The purpose of this exploratory study was to investigate the impact of writing during mathematical problem solving. The study involved an analysis of ninth grade algebra students' written and verbal descriptions of their mathematical problem solving processes. Through this comparison, a better understanding of the connection between problem solving and writing is realized. The written and verbal data show a relationship between the number of problem solving strategies tried by students and their success. The majority of problem solving behaviors involve execution actions such as carrying out goals and performing calculations. Students who construct global plans are more successful problem solvers. Students engage in verification behaviors at various stages of problem solving though the majority of students do not verify their final answers. While both oral and written descriptions serve as a tool for understanding students' thinking processes, a comparison of the two modes of reporting, using a metacognitive framework as the lens of analysis, reveals some important variations. Students who wrote descriptions of their thinking were significantly more successful in the problem solving tasks (p< 0.05) than students who verbalized their thinking. Differences in metacognitive behaviors also support the premise that writing can be an effective tool in supporting metacognitive behaviors. [ABSTRACT FROM AUTHOR]