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Part of the research community that has followed the Early Algebra paradigm is currently delimiting the differences between arithmetic thinking and algebraic thinking. This trend could prevent new research approaches to the problem of learning algebra, hiding the importance of considering an arithmetico-algebraic thinking, a new approach which underpins the construction of a cognitive structure that links both types of thinking. This paper proposes a theoretical and practical framework for a learning approach that supports the construction of a cognitive structure which fosters arithmetico-algebraic thinking at the beginning of secondary school by means of cultural and technological activities relating to polygonal numbers.

This paper focuses on pattern generalisation as a way to introduce young students to early algebra. We build on research on patterning activities that feature, in their work with algebraic thinking, both looking for sameness recursively in a pattern (especially figural patterns, but also numerical ones) and conjecturing about function-based relationships that relate variables. We propose a new approach to pattern generalisation that seeks to help children (grades 2 and 3) work both recursively and functionally, and to see how these two modes are connected through the notion of variable. We argue that a crucial change must occur in order for young learners to develop a flexible algebraic discourse. We draw on Sfard's (2008) communication approach and on Châtelet's (2000) notion of the "virtual" in order to pursue this argument. We also root our analyses within a new materialist perspective that seeks to describe phenomena in terms of "material entanglement," which include, in our classroom research context, not just the children and the teacher, but also words, gestures, physical objects and arrangements, as well as numbers, operations and variables.

Representations are key to mathematical activities and meaning-making processes as they are part of modeling, connecting, communicating, and understanding mathematical ideas and concepts. The current study sought to examine a group of novice algebra learners' interactions with different representations from an onto-semiotic approach. A case study method was employed to understand how different algebraic practices (abstracting, generalizing, justifying, and operating on symbols) and functional thinking types (recursive, covariational, and correspondence) were facilitated through working with multiple representations. Three 6th graders participated in the study by completing 12 algebra tasks and taking part in two interviews. The onto-semiotic approach guided the data analysis process that involved the identification of mathematical objects that emerged in the participating students' mathematical practices. Then, the configuration of objects and semiotic functions established by the students in the functional situations was examined to understand the role of representations in the students' development of algebraic thinking and practices. Findings showed that abstraction is an essential process for generalization. Thinking about far figures facilitated abstraction and generalization through helping students construct non-ostensive concrete/pictorial representations. Verbal representations interacted with all representations and preceded symbolic representations. Working with near figures promoted recursive and covariational thinking while examining the far figures usually resulted in correspondence thinking. Implications for the school curriculum are discussed in the paper.

Little is known about the cognitive effort associated with algebraic activity in the elementary and middle school grades. However, this investigation is significant for sensitizing teachers and researchers to the mental demands of algebra learning. In this paper, we focus on the relationship between algebraic thinking and domain-general cognitive abilities. The sample of the study comprised 591 students from grades 4 to 7. The students' abilities in algebraic thinking were assessed through a test that involved three task categories: generalized arithmetic, functional thinking, and modelling languages. Test batteries were used to assess students' domain-general cognitive abilities in terms of analogical, serial, spatial, and deductive reasoning. The results of structural equation modelling analysis indicated that: (1) analogical reasoning predicts students' abilities only in generalized arithmetic; (2) serial reasoning predicts students' abilities only in generalized arithmetic; (3) spatial reasoning predicts students' abilities in functional thinking and modelling languages; and (4) deductive reasoning predicts students' abilities in all three categories. Differences between students across grades are also discussed.

This paper focuses on the emergence of abstraction through the use of a new kind of motion detector--WiiGraph--with 11-year-old children. In the selected episodes, the children used this motion detector to create three simultaneous graphs of position vs. time: two graphs for the motion of each hand and a third one corresponding to their difference. They explored relationships that can be ascribed to an equation of the type A - B = C. We examine the notion of abstraction on its own, without assuming a dualism abstract-concrete according to which more of one is less of the other. We propose a distinct path for the attainment of abstraction, which involves navigating a surplus of sensible qualities. The work described in this paper belongs to early algebra, we suggest, because it involves the elementary symbolic treatment of unknowns and generals. More broadly, it advances a perspective on the nature of mathematical abstraction.

Self-efficacy in mathematics is related to engagement, persistence, and academic performance. Prior research focused mostly on examining changes to students' self-efficacy across large time intervals (months or years), and paid less attention to changes at the level of lesson sequences. Knowledge of how self-efficacy changes during a sequence of lessons is important as it can help teachers better support students' self-efficacy in their everyday work. In this paper, we expanded previous studies by investigating changes in students' self-efficacy across a sequence of 3-4 lessons when students were learning a new topic in mathematics (n[subscript Students] = 170, n[subscript Time-points] = 596). Nine classes of Norwegian grade 6 (n = 77) and grade 10 students (n = 93) reported their self-efficacy for easy, medium difficulty, and hard tasks. Using multilevel models for change, we found (a) change of students' self-efficacy across lesson sequences, (b) differences in the starting point and change of students' self-efficacy according to perceived task difficulty and grade, (c) more individual variation of self-efficacy starting point and change in association with harder tasks, and (d) students in classes who were taught a new topic in geometry had stronger self-efficacy at the beginning of the first lesson as compared to those who were taught a new topic in algebra (grade 10), and students in classes who were taught a new topic in fractions had steeper growth across the lesson sequence as compared to those who were taught a new topic in measurement (grade 6). Implications for both research and practice on how new mathematics topics are introduced to students are discussed.

This paper describes the difficulties faced by a group of middle school students (13- to 15-year-olds) attempting to translate algebraic statements written in verbal language into symbolic language and vice versa. The data used were drawn from their replies to a written quiz and semi-structured interviews. In the former, students were confronted with a series of algebraic statements and asked to choose the sole translation, of four proposed for each, that was semantically congruent with the original. The results show that most of the errors detected were due to arithmetic issues, especially around the distinction between product and exponent or sum and product in connection with the notions of perimeter and area. As a rule, the error distribution by type varied depending on the type of task involved.

This paper responds to the burgeoning literature on mathematics teacher noticing, arguing that its cognitive orientation misses the cultural and ideological dimensions of what and how teachers notice. The author highlights Goodwin's concept of "professional vision" as a way of bringing analyses of culture and power into studies of teacher noticing. The case of a high school algebra teacher who learned to notice the mathematical strengths of students from marginalized groups is used to illustrate how this might be done. The teacher's noticing involved not only cognitive processes like attending to, interpreting, and deciding how to respond to students' thinking, but also managing dominant ideologies that position students--especially students from non-dominant communities--as mathematically deficient rather than as sense-makers whose ideas should form the basis for further learning. The paper advances the field's capacity for understanding the challenges that teachers face as they attempt to notice in ways that are ambitious as well as equitable.

Recent scholarship around teaching elementary mathematics supports the learning of early algebra with 5- to 12-year olds. However, in spite of the recognition of the affordances of early algebra, issues about how to introduce it remain open. Within this context, Davydov's work is often cited as a source of impressive demonstration of young learners' capacity for algebraic thinking. This work requires further exploration in order to yield a clearer picture of a very particular teaching approach, which focuses on early abstractions and symbolic language. We argue that in order to fully understand how Davydov's work contributes to current conversations and what Davydov was trying to do, we need to shed light on the context- and time-specific discourse of the 1960 Soviet educational reforms that made it possible for Davydov to develop his vision about algebraic thinking and to set in motion appropriate teaching approaches for young learners. In this paper, we look back to the Soviet debates that unfolded in Russia on the integration of early algebra in elementary school word-problem solving. Drawing on these debates and the results of Davydov's school experiments, we lay out the developmental axes of capacity building. This can be done by emphasizing ascent from the abstract to the concrete using a variety of representational modeling tools to support the emergence of algebraic thinking while targeting particular habits of mind within carefully designed learning activities. We conclude with some insights about current arithmetic-algebra debates, and how these could be enriched and deepened by Davydov's work, which yet remains open to future discussion and reflection.

This paper focuses on the mathematical sensemaking by women of color in the USA as part of the global effort of dismantling deficit narratives about historically marginalized groups of students. Following Adiredja's anti-deficit framework for sensemaking, this cognitive study invited a group of women of color to share their understanding of basis from linear algebra to construct a sensemaking counter-story. Extending the framework, this study examines a task that explores the boundaries and nuances of a concept to support the effort of going beyond students' deficits. Eight women extended the concept of basis (and vector spaces) to 22 distinct everyday contexts, drawing from their everyday lives as well as topics from their academic experiences. Their explanations revealed analytical codes describing roles and characteristics of a basis. These codes suggest ways that students can mobilize the concept of basis beyond its logical underpinnings. Contrasting interpretations using a deficit and an anti-deficit perspective construct a counter-story that showcases these women's creativity and flexibility in understanding the concept, and potential resources for the teaching and learning of linear algebra.

This paper introduces a new mode of variational and covariational reasoning, which we call scaling-continuous reasoning. Scaling-continuous reasoning entails (a) imagining a variable taking on all values on the continuum at any scale, (b) understanding that there is no scale at which the continuum becomes discrete, and (c) re-scaling to any arbitrarily small increment for x and coordinating that scaling with associated values for y. Based on the analysis of a 15-h teaching experiment with two 12-year-old pre-algebra students, we present evidence of scaling-continuous reasoning and identify two implications for students' understanding of rates of change: seeing constant rate as an equivalence class of ratios, and viewing instantaneous rate of change as a potential rate. We argue that scaling-continuous reasoning can support a robust understanding of function and rates of change.

Binary operations are one of the fundamental structures underlying our number and algebraic systems. Yet, researchers have often left their role implicit as they model student understanding of abstract structures. In this paper, we directly analyze students' perceptions of the general binary operation via a two-phase study consisting of task-based surveys and interviews. We document what attributes of binary operation group theory students perceive as critical and what types of metaphors students use to convey these attributes. We found that many students treat superficial features as critical (such as element-operator-element formatting) and do not always perceive critical features as essential (such as the binary attribute). Further, these attributes were communicated across three metaphor categories: arithmetic-related, function-related, and organization-related.

The paper focuses on four upper secondary students' collaborative small-group mathematical reasoning (MR) with respect to a sinusoidal function. The students were collaboratively engaged in a process of MR regarding the relationships between mathematical theoretical descriptions of parameters in the algebraic expression of the sinusoidal function and their counterparts when identifying these parameters together with their numerical value in a problem situation. Students' reasoning on the mathematical tool of this sinusoidal function is manifold and multicoloured. Adopting a dialogical approach to the analyses of student interaction, the study reveals that the students' MR, viewed as comprising both process aspects and structural aspects, is characterised by three features. Firstly, the analyses show that the students' MR is characterised by tensions between the mathematical tool and its use in situ, called resistance. Secondly, the students' MR is characterised by their use of semiotic means of objectification such as deictics, gestures, linguistic devices and the text book. Thirdly, the analyses reveal that MR is characterised by the students' efforts to interthink and collaborate in order to achieve the goal of their collective task solving. All four students contributed to the collective MR, however, in various ways and to a varying degree.

Research focusing on algebra from primary to early secondary school level has made several major advances over the past decades. Students' difficulties have been identified and supportive teaching and learning environments have been set up (Cai & Knuth, 2011; Kieran, 2007; Radford, 2008, "Mathematics Education Research Journal," 26, 257-277, 2014). The effectiveness of these environments relies on the teachers' ability to pay careful attention to students' thinking, which then guides their instructional decisions. This inevitably raises the crucial question of the teachers' knowledge for managing these types of situations in the classroom. In this context, this paper focuses on the mathematical knowledge for teaching figural pattern activities of 100 teachers at the primary and early secondary school levels. The results show that many primary teachers lack the essential knowledge for teaching these types of activities: they do not have a clear idea of their goal, they do not consider non-standard algebraic generalisations to be correct and they generally seem unable to help students to improve their arithmetical generalisation. Secondary school teachers also seem unable to give adequate feedback to improve students' arithmetical generalisation. Although they seem to recognise that these activities are intended to improve algebraic thinking, they do not have a clear perception of the role that primary school learning can play in the development of this algebraic thinking.

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 (2009) and the work of Alcock (2010) 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 (1989) 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.

In this paper, we propose an approach to analysing teacher arguments that takes into account field dependence--namely, in Toulmin's sense, the dependence of warrants deployed in an argument on the field of activity to which the argument relates. Freeman, to circumvent issues that emerge when we attempt to determine the field(s) that an argument relates to, proposed a classification of warrants ("a priori, empirical, institutional and evaluative"). Our approach to analysing teacher arguments proposes an adaptation of Freeman's classification that distinguishes between: epistemological and pedagogical a priori warrants, professional and personal empirical warrants, epistemological and curricular institutional warrants, and evaluative warrants. Our proposition emerged from analyses conducted in the course of a written response and interview study that engages secondary mathematics teachers with classroom scenarios from the mathematical areas of analysis and algebra. The scenarios are hypothetical, grounded on seminal learning and teaching issues, and likely to occur in actual practice. To illustrate our proposed approach to analysing teacher arguments here, we draw on the data we collected through the use of one such scenario, the Tangent Task. We demonstrate how teacher arguments, not analysed for their mathematical accuracy only, can be reconsidered, arguably more productively, in the light of other teacher considerations and priorities: pedagogical, curricular, professional and personal.

In this paper I shall discuss data from a study on Colombian mathematics teachers' conceptions of their own teaching practices of beginning algebra, which led to the development of a theoretical model of teachers' thought structures designed as a thinking tool at the initial stage of the study. With a focus on the perspectives of teachers, the study investigated the relationship between the teachers' conceptions of beginning algebra and their conceptions of their own teaching practices, with a view to unravelling their conceptions of change in their practices. Significant findings which threw light on the aforementioned relationship have been presented in Agudelo-Valderrama, Clarke and Bishop (2007), highlighting a direct association between a teacher's conceptions of the nature of beginning algebra, the crucial determinants of her/his teaching practice, and her/his attitude to change. After an overview of the study, this paper focuses on specific evidence which clearly shows that in contrast to the strong relationship between a teacher's conceptions of mathematics and her/his teaching practice, assumed in the theoretical model of teachers' thought structures, the teachers see a strong relationship between their conceptions of social/institutional factors of teaching and what they do in their teaching. Implications of the findings for teacher education in Colombia are identified.

In this paper I consider the problem of designing strategies for teacher education programs that may promote an aware style of teaching. Among the various elements to be considered I focus on the need to address prospective teachers' belief that they must reproduce the style of mathematics teaching seen in their school days. Towards this aim, I argue that the prospective teachers need a context allowing them to look at the topics they will teach in a different manner. This context may be provided by the history of mathematics. In this paper I shall discuss how history affected the construction of teaching sequences on algebra during the activities of the "laboratory of mathematics education" carried out in a 2 year education program for prospective teachers. The conditions of the experiment, notably the fact that our prospective teachers had not had specific preparation in the history of mathematics, allow us to outline opportunities and caveats of the use of history in teacher education.

The integration of history into educational practice can lead to the development of activities through the use of genetic "moments" in the history of mathematics. In the present paper, we utilize Oresme's genetic ideas--developed during the fourteenth century, including ideas on the velocity-time graphical representation as well as geometric transformations and reconfigurations--to develop mathematical models that can be employed for the solution of problems relating to linear motion. The representation of distance covered as the area of the figure between the graph of velocity and the time axis employed in these activities, leads on naturally to the study of problems on motion by means of functions, as well as allowing for the use of tools (concepts and propositions) from Euclidean geometry of relevance to such problems. By employing simple geometric transformations, equivalent real life problems are obtained which lead, in turn, to a simple classification of all linear motion-related problems. When applied to a wider range of motion problems, this approach prepares the way for the introduction of basic Calculus concepts (such as integral, derivative and their interrelation); in fact, we would argue that it could be beneficial to teach the basic concepts and results of Calculus from an early grade by employing natural extensions of the teaching methods considered in this paper.

Some errors in the learning of algebra suggest that students might have difficulties giving meaning to algebraic symbolism. In this paper, we use problem posing to analyze the students' capacity to assign meaning to algebraic symbolism and the difficulties that students encounter in this process, depending on the characteristics of the algebraic statements given. We designed a written questionnaire composed of eight closed algebraic statements expressed symbolically, which was administered to 55 students who had finished their compulsory education and who had some previous experience in problem-posing. In our analysis of the data, we examine both syntactic and semantic structures of the problem posed. We note that in most cases students posed problems with syntactic structures different to those given. They did not include computations within variables, and changed the kinds of relationships connecting variables. Students easily posed problems for statements with additive structures. Other differences in the type of problems posed depend on the characteristics of the given statements.

This paper offers a critical analysis of Dubinsky et al. (On Learning Fundamental Concepts of Group Theory; 1994) and proposes, as an alternative to the four axioms and the standard definitions, that permutation and symmetry may be regarded as the fundamental concepts of group theory. (Author)

As part of a larger study of student understanding of concepts in linear algebra, we interviewed 10 university linear algebra students as to their conceptions of functions from high school algebra and linear transformation from their study of linear algebra. An overarching goal of this study was to examine how linear algebra students see linear transformation as related to high school function. Analysis of these data led to a characterization of student responses into three categories of mathematical structures used to discuss function: properties, computations, and a series of five interrelated clusters of metaphorical expressions. In this paper, we use this analytic framing for exploring the question: to what extent does each of the students in this study have a unified concept image of function across two mathematical contexts, high school algebra and their study of linear algebra? We found that students who expressed a unified notion of function used metaphorical language to bridge any gaps between the notion of function from high school and the notion of linear transformation from linear algebra. We conjecture that the framing of computations, properties, and metaphorical clusters could be extended to discussions of functions in contexts with other mathematical domains. Future research that further examines the extent to which undergraduate students develop a unified concept image of function could then lead to various design research efforts aimed at explicitly fostering such a unified understanding.

In this paper we report the results of an experiment designed to test the hypothesis that when faced with a question involving the inverse direction of a reversible mathematical process, students solve a multiple-choice version by verifying the answers presented to them by the direct method, not by undertaking the actual inverse calculation. Participants responded to an online test containing equivalent multiple-choice and constructed-response items in two reversible algebraic techniques: factor/expand and solve/verify. The findings supported this hypothesis: Overall scores were higher in the multiple-choice condition compared to the constructed-response condition, but this advantage was significantly greater for items concerning the inverse direction of reversible processes compared to those involving direct processes.

To understand relationships between students' quantitative reasoning with fractions and their algebraic reasoning, a clinical interview study was conducted with 18 middle and high school students. The study included six students with each of three different multiplicative concepts, which are based on how students create and coordinate composite units (units of units). Students participated in two 45-min semi-structured interviews and completed a written fraction assessment. This paper reports on how 12 students operating with the second and third multiplicative concepts demonstrated distributive reasoning in equal sharing problems and in taking fractions of unknowns. Students operating with the second multiplicative concept who demonstrated distributive reasoning appeared to lack awareness of the results of their reasoning, while students operating with the third multiplicative concept demonstrated this awareness and the construction of more advanced distributive reasoning when they worked with unknowns. Implications for relationships between students' fractional knowledge and algebraic reasoning are explored.

In this paper, we explore algebra students' mathematical realities around fractions and division, and the ways in which students reinvented mathematical productions involving fractions and division. We find that algebra students' initial realities do not include the fraction-as-quotient sub-construct. This can be problematic because in algebra, quotients are almost always represented as fractions. In a design experiment, students progressively reinvented the fraction-as-quotient sub-construct. Analyzing this experiment, we find that a particular type of mathematical production, which we call preformal productions, played two meditational roles: (1) they mediated mathematical activity, and (2) they mediated the reinvention of more formal mathematical productions. We suggest that preformal productions may emerge even when they are not designed for, and we show how preformal productions embody historic classroom activity and social interaction.

The purpose of this paper is to present an analysis of the difficulties faced by students when working with different representations of vectors, planes and their intersections in R[superscript 3]. Duval's theoretical framework on semiotic representations is used to design a set of evaluating activities, and later to analyze student work. The analysis covers three groups of undergraduate students taking introductory courses in linear algebra. Different types of treatments and conversions are required to solve the activities. One important result shows that, once students choose a register to solve a task, they seldom make transformations between different registers, even though this facilitates solving the task at hand. Identifying these difficulties for particular transformations may help teachers design specific activities to promote students cognitive flexibility between representation registers.

This paper uses a multilevel analysis of mathematical reasoning rooted in Cultural Historical Activity Theory to examine how mathematical discourse and student reasoning about linear functions developed across 3 weeks in a ninth grade bilingual algebra class. Despite the teacher's expertise teaching with a conceptual focus, and her stated intention to focus on slope as a rate of change, the case study students in her class appeared to appropriate a procedural understanding of slope. By examining the nested activity systems of the students' group discussions, the classroom, school, and district, this analysis shows how external assessment pressures shaped the teacher's selection of tasks, her and her students' use of mathematical discourse, and ultimately, her students' opportunity to learn a critical algebraic concept.

Certain mathematical concepts were not introduced to solve a specific open problem but rather to solve different problems with the same tools in an economic formal way or to unify several approaches: such concepts, as some of those of linear algebra, are presumably difficult to introduce to students as they are potentially interwoven with many types of difficulties as formal ones and far away from the actual knowledge of the students. The purpose of this paper is to propose a methodology for studying the introduction of such concepts in linear algebra during tutorial sessions at the beginning of university, the wording of the concepts being yet presented during lectures. For this purpose, we amend a general methodology of Pariès, Robert and Rogalski inside the general framework of Activity Theory. This methodology lets us take into account several specificities of these concepts and studies the mathematical activity the teacher organises for students and the way he manages the relationship between students' actual activities and mathematical tasks. We also present an implementation of this methodology based on a French university course to illustrate our approach and discuss its possibilities.

In this study we investigated the role of a dynamic online game on students' early algebra problem solving. In total 253 students from grades 4, 5, and 6 (10-12 years old) used the game at home to solve a sequence of early algebra problems consisting of contextual problems addressing covarying quantities. Special software monitored the students' online working when solving the problems. Before and after the intervention a paper-and-pencil test on early algebra was administered. The data analysis revealed that the online working contributed to the students' early algebra performance. There was a significant gain in performance across all grades. The highest effect was found in grade 6. Out of the three strategy profile clusters that could be distinguished in the whole sample, the cluster dominated by using extreme values and the cluster characterized by the trial-and-error strategy were most influential on the gain in early algebra performance. The students' level of online working, which was defined as a combination of online involvement and strategy use, appeared to have a marginally significant effect on the gain score for the total sample. Per grade there was no significant effect, yet the levels of online working were significantly related to grade. Free playing was mostly performed in grade 4, looking for answers in grade 5, and exploring relations slightly more in grades 5 and 6. About 17 % of the effect of grade on the gain score was mediated by the level of online working.

Our study concerns the conceptual mathematical knowledge that emerges during the resolution of tasks on the equivalence of polynomial and rational algebraic expressions, by using CAS and paper-and-pencil techniques. The theoretical framework we adopt is the Anthropological Theory of Didactics ("Chevallard" 19:221-266, 1999), in combination with semiotic aspects from the instrumental approach to tool use. The analysis we present is based on interviews carried out with a 10th grade student who participated in our research. Our findings highlight the mathematical knowledge ("technological discourse") constructed in the process of confronting, differentiating, and articulating the several mathematical techniques and theoretical ideas (pertaining to the "numeric perspective" and the "syntactic perspective" on algebraic equivalence) related to the designed equivalence tasks.

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, 1981) 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[superscript k] is a subspace of R[superscript n] for k less than n. We conclude with a discussion of this and how it may be leveraged to inform teaching in a productive, student-centered manner.

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.

This paper explores how mathematicians build meaning through communicative activity involving talk, gesture and diagram. In the course of describing mathematical concepts, mathematicians use these semiotic resources in ways that blur the distinction between the mathematical and physical world. We shall argue that mathematical meaning of eigenvectors depends strongly on both time and motion--hence, on physical interpretations of mathematical abstractions--which are dimensions of thinking that are typically deliberately absent from formal, written definition of the concept. We shall also show how gesture and talk contribute differently and uniquely to mathematical conceptualisation and further elaborate the claim that diagrams provide an essential mediating role between the two.

The introduction of the concept of the variable represents a critical point in the arithmetic-algebraic transition. This concept is complex because it is used with different meanings in different situations. Its management depends on the particular way of using it in problem-solving. The aim of this paper was to analyse whether the notion of "unknown" interferes with the interpretation of the variable "in functional relation" and the kinds of languages used by the students in problem-solving. We also wanted to study the concept of the variable in the process of translation from algebraic language into natural language. We present two experimental studies. In the first one, we administered a questionnaire to 111 students aged 16-19 years. Drawing on the conclusions of this research we carried out the second study with two pairs of students aged 16-17 years.

The goal of this paper is to explore qualities of mathematical imagination in light of a classroom episode. It is based on the analysis of a classroom interaction in a high school Algebra class. We examine a sequence of nine utterances enacted by one of the students whom we call Carlene. Through these utterances Carlene illustrates, in our view, two phenomena: (1) juxtaposing displacements, and (2) articulating necessary cases. The discussion elaborates on the significance of these phenomena and draws relationships with the perspectives of embodied cognition and intersubjectivity.

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.

A common approach used for introducing algebra to young adolescents is an exploration of visual growth patterns and expressing these patterns as functions and algebraic expressions. Past research has indicated that many adolescents experience difficulties with this approach. This paper explores teaching actions and thinking that begins to bridge many of these difficulties at an early age. A teaching experiment was conducted with two classes of students with an average age of eight years and six months. From the results it appears that young students are capable not only of thinking about the relationship between two data sets, but also of expressing this relationship in a very abstract form.

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.

This paper provides a microanalysis of one Algebra I teacher's instruction to explore the advantages that are afforded us by coordinating two perspectives to document and account for the teacher's mathematical understandings. We use constructs associated with Stein, Grover and Henningsen's domain of mathematical didactics and Realistic Mathematics Education's instructional design theory to infer what the teacher might understand to effectively implement her instructional goals and, more importantly, support student learning. By coordinating these perspectives, we developed a working framework for analyzing the teacher's classroom practice retrospectively. For example, we illustrate how the mathematical possibilities related to one student's question might inform the teacher's decisions as she initiates shifts in students' self-generated models. Additionally, we illustrate how the teacher's decision to capitalize on particular students' models contributes in part to the kinds of mathematical ideas that can be explored and the connections students can make among those ideas. More generally, we explore the utility of coordinating these two perspectives to understand the landscape of ideas that teachers might traverse to align their practices with reform recommendations in the United States.

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].

This paper examines the work of students who, when reasoning about real analysis, do so almost exclusively by means of verbal and algebraic reasoning, and tend not to incorporate visual images into their work. It examines the work of students from two parallel courses of introductory real analysis, whose reasoning ranges from those who introduce definitions appropriately and work with them competently, to those who cannot recall definitions and appear to manipulate notation without regard for its reference. It presents a theory that relates the differences to students' expectations regarding their role as learners of mathematics. Throughout, the argument is illustrated with interview data from which the theory was inductively generated.

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.

Little is known about the cognitive effort associated with algebraic activity in the elementary and middle school grades. However, this investigation is significant for sensitizing teachers and researchers to the mental demands of algebra learning. In this paper, we focus on the relationship between algebraic thinking and domain-general cognitive abilities. The sample of the study comprised 591 students from grades 4 to 7. The students' abilities in algebraic thinking were assessed through a test that involved three task categories: generalized arithmetic, functional thinking, and modelling languages. Test batteries were used to assess students' domain-general cognitive abilities in terms of analogical, serial, spatial, and deductive reasoning. The results of structural equation modelling analysis indicated that (a) analogical reasoning predicts students' abilities only in generalized arithmetic; (b) serial reasoning predicts students' abilities only in generalized arithmetic; (c) spatial reasoning predicts students' abilities in functional thinking and modelling languages; and (d) deductive reasoning predicts students' abilities in all three categories. Differences between students across grades are also discussed. [ABSTRACT FROM AUTHOR]

Underlines the common points in F. Uhlig's approach published in an earlier issue of this journal about the question of proof in linear algebra. Describes some of his ideas in a new light and gives perspective for a further didactical development of Uhlig's first experiments. (Author/KHR)

Answers Burn's question related to a previous study on the nature of knowledge about abstract algebra. Claims that a previous paper presents research that attempts to contribute to knowledge of how students' understanding of certain group concepts may develop instead of teaching abstract algebra with the computer software ISETL. (ASK)

Presents a theory of the development of algebraic abstraction that extends Sfard's and Mason's ideas on learners' progress from operational or process-oriented thinking to the abstract or structural perspective. Analyzes secondary school students' approaches to classifying a set of complex equations. (Author/MKR)

Analyzes the nature and growth of students' algebraic knowledge and thinking from an epistemological perspective supported by historical observations. Focuses on two crucial transitions: from the purely operational algebra to the structural algebra of a fixed value of an unknown and then to the functional algebra of a variable. (Contains 58 references.) (MKR)

This paper focuses on the emergence of abstraction through the use of a new kind of motion detector—WiiGraph—with 11-year-old children. In the selected episodes, the children used this motion detector to create three simultaneous graphs of position vs. time: two graphs for the motion of each hand and a third one corresponding to their difference. They explored relationships that can be ascribed to an equation of the type A – B = C. We examine the notion of abstraction on its own, without assuming a dualism abstract-concrete according to which more of one is less of the other. We propose a distinct path for the attainment of abstraction, which involves navigating a surplus of sensible qualities. The work described in this paper belongs to early algebra, we suggest, because it involves the elementary symbolic treatment of unknowns and generals. More broadly, it advances a perspective on the nature of mathematical abstraction. [ABSTRACT FROM AUTHOR]