Vygotskian theory and mathematics education: Resolving the conceptual-procedural dichotomy

The aim of this paper is to demonstrate that in spite of some superficial similarities the current mathematics reform in the US based on constructivist principles differs substantially from mathematical education based on Vygotskian cultural-historical theory (V.V. Davydov’s mathematics program), and to illustrate the manner in which Davydov’s program virtually obliterates the conceptual-procedural division that has fueled the current “math wars”. Both constructivism and Davydov’s approach emphasize the active character of students’ acquisition of mathematical concepts. Constructivists, however, begin the instructional process from the children’s preexistent concepts while Vygotskians reorient it toward acquisition of what Vygotsky defined as “scientific” rather than “spontaneous, everyday” concepts. A three-year study of the implementation of Davydov’s elementary mathematics program in a school setting in the US found that the American children overcame the initial challenges of the program, consistently resolved computational errors conceptually, and finally demonstrated the ability to solve high school level mathematics problems. The curriculum appeared to foster the development of theoretical thinking, an explicit goal of Davydov’s program, which constitutes its major value and educational significance.     

Mathematics-for-Teaching: an Ongoing Investigation of the Mathematics that Teachers (Need to) Know

In this article we offer a theoretical discussion of teachers' mathematics-for-teaching, using complexity science as a framework for interpretation. We illustrate the discussion with some teachers' interactions around mathematics that arose in the context of an in-service session. We use the events from that session to illustrate four intertwining aspects of teachers' mathematics-for-teaching. We label these aspects “mathematical objects,” “curriculum structures,” “classroom collectivity,” and “subjective understanding”. Drawing on complexity science, we argue that these phenomena are nested in one another and that they obey similar dynamics, albeit on very different time scales. We conjecture (1) that a particular fluency with these four aspects is important for mathematics teaching and (2) that these aspects might serve as appropriate emphases for courses in mathematics intended for teachers.     

Mathématiques de la vie quotidienne au Burkina Faso: une analyse de la pratique sociale de comptage et de vente de mangues

Mathematics teaching in Burkina Faso is faced with major challenges (high illiteracy rates, students’ difficulties, and high failure rates in mathematics, which is a central topic in the curriculum). As evidenced in many of these studies, mathematics is reputed to be tough, inaccessible, and far from what students live daily. Students here look as though they are living in two seemingly distant worlds, school and everyday life. In order to better understand these difficulties and to contribute in the long run to a more adapted teaching of mathematics, we tried to document and elicit the “mathematical resources” mobilized in various daily life social practices. In this paper, we focus on one of them, the counting and selling of mangoes by unschooled peasants. An ethnographic approach draws on the observation of the situated activity of counting and selling mangoes (during harvesting) and on “eliciting interviews” of the involved actors. The analysis of results highlights a richness of structuring resources mobilized and distributed through this practice, related to what Lave (1988) call “the experienced lived-in-world” and “constitutive order.” The mathematical resources take the form of “knowledge in action” and “theorems in action” (Vergnaud, Rech Didact Math 10(23):133–170, 1990), embedded in the social, economic, and even cultural structures of actors.     

Using culture as a resource in mathematics: the case of four Mexican–American prospective teachers in a bilingual after-school program

This paper explores Mexican–American prospective teachers’ use of culture—defined as social practices and shared experiences—as an instructional resource in mathematics. The setting is an after-school mathematics program for the children of Mexican heritage. Qualitative analysis of the prospective teachers’ and children’s interactions reveals that the nature of the mathematical activities affected how culture was used. When working on the “binder activities,” prospective teachers used culture only in non-mathematical contexts. When working on the “recipes project,” however, culture was used as a resource in mathematical contexts. Implications for the mathematics teacher preparation of Latinas/os are discussed.     

What Is a “Semiotic Perspective”, and What Could It Be? Some Comments on the Contributions to This Special Issue

This comment attempts to identify different “semiotic perspectives” proposed by the authors of this special issue according to the problems they discuss. These problems can be distinguished as problems concerning the representation of mathematical knowledge, the definition and objectivity of meaning, epistemological questions of learning and activity in mathematics, and the social dimension of sign processes. The contributions are discussed so as to make visible further research perspectives with regard to “semiotics in mathematics education”.     

CHILDREN��S GESTURES AND THE EMBODIED KNOWLEDGE OF GEOMETRY

There is mounting research evidence that contests the metaphysical perspective of knowing as mental process detached from the physical world. Yet education, especially in its teaching and learning practices, continues to treat knowledge as something that is necessarily and solely expressed in ideal verbal form. This study is part of a funded project that investigates the role of the body in knowing and learning mathematics. Based on a 3-week (15 1-h lessons) video study of 1-s grade mathematics classroom (N = 24), we identify 4 claims: (a) gestures support children’s thinking and knowing, (b) gestures co-emerge with peers’ gestures in interactive situations, (c) gestures cope with the abstractness of concepts, and (d) children’s bodies exhibit geometrical knowledge. We conclude that children think and learn through their bodies. Our study suggests to educators that conventional images of knowledge as being static and abstract in nature need to be rethought so that it not only takes into account verbal and written languages and text but also recognizes the necessary ways in which children’s knowledge is embodied in and expressed through their bodies.     

The (Embodied) Performance of Physics Concepts in Lectures

Lectures are often thought of in terms of information transfer: students (do not) “get” or “construct meaning of” what physics professors (lecturers) say and the notes they put on the chalkboard (overhead). But this information transfer view does not explain, for example, why students have a clear sense of understanding while they sit in a lecture and their subsequent experiences of failure to understand their own lecture notes or textbooks while preparing for an exam. Based on a decade of studies on the embodied nature of science lectures, the purpose of this article is to articulate and exemplify a different way of understanding physics lectures. We exhibit how there is more to lectures than the talk plus notes. This informational “more” may explain (part of) the gap between students’ participative understanding that exists in the situation where they sit in the lecture on the one hand and the one where they study for an exam from their lecture notes on the other. Our results suggest that in lectures, concepts are heterogeneous performances in which meaning is synonymous with the synergistic and irreducible transactions of many different communicative modes, including gestures, body movements, body positions, prosody, and so forth.     

Analyzing Mathematical Teaching-Learning Situations — the Interplay of Communicational and Epistemological Constraints

This is a commentary paper in the volume on “Teachings situations as object of research: empirical studies within theoretical perspectives”. An essential object of mathematics education research is the analysis of interactive teaching and learning processes in which mathematical knowledge is mediated and communicated. Such a research perspective on processes of mathematical interaction has to take care of the difficult relationship between mathematics education theory and everyday mathematics teaching practice. In this regard, the paper tries to relate the development in mathematics education research within the theory of didactical situations to developments in interaction theory and in the epistemological analysis of mathematical communication.     

The notion of historical “parallelism” revisited: historical evolution and students’ conception of the order relation on the number line

This paper associates the findings of a historical study with those of an empirical one with 16 years-old students (1st year of the Greek Lyceum). It aims at examining critically the much-discussed and controversial relation between the historical evolution of mathematical concepts and the process of their teaching and learning. The paper deals with the order relation on the number line and the algebra of inequalities, trying to elucidate the development and functioning of this knowledge both in the world of scholarly mathematical activity and the world of teaching and learning mathematics in secondary education. This twofold analysis reveals that the old idea of a “parallelism” between history and pedagogy of mathematics has a subtle nature with at least two different aspects (metaphorically named “positive” and “negative”), which are worth further exploration.

Prospective teachers’ reasoning and response to a student’s non-traditional strategy when dividing fractions

Recognizing meaning in students’ mathematical ideas is challenging, especially when such ideas are different from standard mathematics. This study examined, through a teaching-scenario task, the reasoning and responses of prospective elementary and secondary teachers to a student’s non-traditional strategy for dividing fractions. Six categories of reasoning were constructed, making a distinction between deep and surface layers. The connections between the participants’ reasoning, their teaching response, and their beliefs about mathematics teaching were investigated. We found that there were not only differences but also similarities between the prospective elementary and secondary teachers’ reasoning and responses. We also found that those who unpacked the mathematical underpinning of the student’s non-traditional strategy tended to use what we call “teacher-focused” responses, whereas those doing less analysis work tended to construct “student-focused” responses. These results and their implications are discussed in relation to the influential factors the participants themselves identified to explain their approach to the given teaching-scenario task.

Mathematical contributions of Archimedes: Some nuggets

Archimedes is generally regarded as the greatest mathematician of antiquity and alongside Isaac Newton and C F Gauss as the top three of all times. He was also an excellent theoreticiancum-engineer who identified mathematical prob lems in his work on mechanics, got hints on their solution through engineering techniques and then solved those mathematical problems, many a time discovering fundamental results in mathematics, for instance, the concepts oflimits andintegration. In his own words,“… which I first dis covered by means of mechanics and then exhibited by means of geometry”. In this article we briefly describe some of his main contributions to mathematics.     

Preservice Elementary Teachers’ Voices Describe How Their Teacher Motivated Them to do Mathematics

In this study, data in the form of (preservice teacher) student voices taken from mathematical autobiographies, written at the beginning of the semester, and end-of-semester reflections, were analyzed in order to examine why preservice elementary school teachers were highly motivated in a social constructivist mathematics course in which the teacher emphasized mastery goals. The findings suggest that students entered the course with a wide variety of feelings about mathematics and their own mathematical ability. At the end of the semester, students wrote about aspects of the course that “led to their growth as a mathematical thinker and as a mathematics teacher…” Student responses were coded within themes that emerged from the data: Struggle; Construction of meaning [mathematical language; mathematical understanding]; Grouping [working in groups]; Change [self-efficacy; math self-concept]; and the Teacher’s Role. These themes are described using student voices and within a motivation goal theory framework. The role of struggle, in relation to motivation, is discussed.     

Building intellectual infrastructure to expose and understand ever-increasing complexity

This paper comments on the expanded repertoire of techniques, conceptual frameworks, and perspectives developed to study the phenomena of gesture, bodily action and other modalities as related to thinking, learning, acting, and speaking. Certain broad issues are considered, including (1) the distinction between “contextual” generalization of instances across context (of virtually any kind—numeric, situational, etc.) and the generalization of structured actions on symbols, (2) fundamental distinctions between the use of semiotic means to describe specific situations versus semiosis serving the process of generalization, and (3) the challenges of building generalizable research findings at such an early stage in infrastructure building.     

When Intuition Beats Logic: Prospective Teachers’ Awareness of their <Emphasis Type="Italic">Same Sides – Same Angles</Emphasis> Solutions

This paper indicates that prospective teachers’ familiarity with theoretical models of students’ ways of thinking may contribute to their mathematical subject matter knowledge. This study introduces the intuitive rules theory to address the intuitive, same sides-same angles solutions that prospective teachers of secondary school mathematics come up with, and the proficiency they acquired during the course “Psychological aspects of mathematics education”. The paper illustrates how drawing participants’ attention to their own erroneous applications of same sides-same angles ideas to hexagons, challenged and developed their mathematical knowledge.     

Teaching Euclid in a Practical Context: Linear Perspective and Practical Geometry

This article explores the transmission of practical knowledge in the XV and XVI centuries. According to cosmographer Egnatio Danti, optics and other mathematical sciences had “been banished” from the main philosophical schools of his period, and “the little which remains to us is limited to some practical aspects learned from the mechanical artificers”. The “mechanical artificers” were architects, painters and surveyors whose mathematical training constitutes the subject dealt with in this article. The context of Danti’s remark was the letter to the “Accademici del Disegno” of Perugia which introduce his Italian translation of Euclid’s Optics. After the great Medieval season of optical studies, in effect, this science progressed mainly through its practical applications, especially through “that part of perspective which pertains to painting” (Piero della Francesca), and through the spread of methods and instruments for measuring by sight.     

Conceptions for relating the evolution of mathematical concepts to mathematics learning—epistemology, history, and semiotics interacting To the memory of Carl Menger (1902–1985)

There is an over-arching consensus that the use of the history of mathematics should decidedly improve the quality of mathematics teaching. Mathematicians and mathematics educators show here a rare unanimity. One deplores, however, and in a likewise general manner, the scarcity of positive examples of such a use. This paper analyses whether there are shortcomings in the—implicit or explicit—conceptual bases, which might cause the expectations not to be fulfilled. A largely common denominator of various approaches is some connection with the term “genetic.” The author discusses such conceptions from the point of view of a historian of mathematics who is keen to contribute to progress in mathematics education. For this aim, he explores methodological aspects of research into the history of mathematics, based on—as one of the reviewers appreciated—his “life long research.”     

The “System of Chymists” and the “Newtonian dream” in Greek-speaking Communities in the 17th–18th Centuries

The acceptance of new chemical ideas, before the Chemical Revolution of Lavoisier, in Greek-speaking communities in the 17th and 18th centuries did not create a discourse of chemical philosophy, as it did in Europe, but rather a “philosophy” of chemistry as it was formed through the evolution of didactic traditions of Chemistry. This “philosophical” chemistry was not based on the existence of any academic institutions, it was focused on the ontology of principles and forces governing the analysis/synthesis of matter and formulated two didactic traditions. The one, named “the system of chymists”, close to the Boylean/Cartesian tradition, accepted, contrary to Aristotelianism, the five “chymical” principles and also the analytical ideal, but the “chymical” principles were not under a conceptual and experimental investigation, as they were in Europe. Also, a crucial issue for this tradition remained the “mechanical” principles which were under the influence of the metaphysical nature of the Aristotelian principles. The other, close to the Boylean/Newtonian tradition, was the integrated presentation of the Newtonian “dream”, which maintained a discursive attitude with reference to the “chemical attractions”–“chemical affinities” and actualised the mathematical atomism of Boscovich, according to which the elementary texture of matter could be causally explained within this complex architecture of mathematical “punkta”. In this tradition also coexisted, in a discursive synthesis, the “chemical element” of Lavoisier and the arguments of the new theory and its opposition to the phlogiston theory, but the “chemical affinities” were under the realm of the “physical element” as “metaphysical point”.     

Subjectivity and cultural adjustment in mathematics education: a response to Wolff-Michael Roth

In this volume, Wolff-Michael Roth provides a critical but partial reading of Tony Brown’s book Mathematics Education and Subjectivity. The reading contrasts Brown’s approach with Roth’s own conception of subjectivity as derived from the work of Vygotsky, in which Roth aims to “reunite” psychology and sociology. Brown’s book, however, focuses on how discourses in mathematics education shape subjective action within a Lacanian model that circumnavigates both “psychology” and “sociology”. From that platform, this paper responds to Roth through problematising the idea of the individual as a subjective entity in relation to the two perspectives, with some consideration of corporeality and of how the Symbolic encounters the Real. The paper argues for a Lacanian conception of subjectivity for mathematics education comprising a response to a social demand borne of an ever-changing symbolic order that defines our constitution and our space for action. The paper concludes by considering an attitude to the production of research objects in mathematics education research that resists the normalisation of assumptions as to how humans encounter mathematics.