The Contribution of Ernst Mach to Embodied Cognition and Mathematics Education

A study of the interactions between mathematics and cognitive science, carried out within a historical perspective, is important for a better understanding of mathematics education in the present. This is evident when analysing the contribution made by the epistemological theories of Ernst Mach. On the basis of such theories, a didactic method was developed, which was used in the teaching of mathematics in Austria at the beginning of the twentieth century and applied to different subjects ranging from simple operations in arithmetic to calculus. Besides the relevance of this method—also named the “Jacob method” after Josef Jacob who proposed it—to teaching practice, it could also be considered interesting in a wider context with reference to the mind-body problem. In particular, the importance that Jacob gives to “muscular activity” in the process of forming and elaborating mathematical concepts, derived from Mach, resounds in the current debate on embodied cognition, where cognitive processes are understood not as expressions of an abstract and merely computational mind but as based on our physicality as human beings, equipped not just with a brain but also a (whole) body. This model has been applied to mathematics in the “theory of embodied mathematics”, the objective of which is to study, with the methods and apparatus of embodied cognitive science, the cognitive mechanisms used in the human creation and conceptualisation of mathematics. The present article shows that the “Jacob method” may be considered a historical example of didactical application of analogous ideas.     

A Cross-Analysis of the Mathematics Teacher’s Activity. An Example in a French 10th-Grade Class

The purpose of this paper is to contribute to the debate about how to tackle the issue of ‘the teacher in the teaching/learning process’, and to propose a methodology for analysing the teacher’s activity in the classroom, based on concepts used in the fields of the didactics of mathematics as well as in cognitive ergonomics. This methodology studies the mathematical activity the teacher organises for students during classroom sessions and the way he manages¹ the relationship between students and mathematical tasks in two approaches: a didactical one [Robert, A., Recherches en Didactique des Mathématiques 21(1/2), 2001, 7–56] and a psychological one [Rogalski, J., Recherches en Didactique des Mathématiques 23(3), 2003, 343–388]. Articulating the two perspectives permits a twofold analysis of the classroom session dynamics: the “cognitive route” students are engaged in—through teacher’s decisions—and the mediation of the teacher for controlling students’ involvement in the process of acquiring the mathematical concepts being taught. The authors present an example of this cross-analysis of mathematics teachers’ activity, based on the observation of a lesson composed of exercises given to 10th grade students in a French ‘ordinary’ classroom. Each author made an analysis from her viewpoint, the results are confronted and two types of inferences are made: one on potential students’ learning and another on the freedom of action the teacher may have to modify his activity. The paper also places this study in the context of previous contributions made by others in the same field.     

Embodied cognition as grounding for situatedness and context in mathematics education

In this paper we analyze, from the perspective of 'Embodied Cognition', why learning and cognition are situated and context-dependent. We argue that the nature of situated learning and cognition cannot be fully understood by focusing only on social, cultural and contextual factors. These factors are themselves further situated and made comprehensible by the shared biology and fundamental bodily experiences of human beings. Thus cognition itself is embodied, and the bodily-grounded nature of cognition provides a foundation for social situatedness, entails a reconceptualization of cognition and mathematics itself, and has important consequences for mathematics education. After framing some theoretical notions of embodied cognition in the perspective of modern cognitive science, we analyze a case study – continuity of functions. We use conceptual metaphor theory to show how embodied cognition, while providing grounding for situatedness, also gives fruitful results in analyzing the cognitive difficulties underlying the understanding of continuity. This revised version was published online in July 2006 with corrections to the Cover Date.     

数学史视野下小学教师数学素养提升的实践研究

小学教师数学素养提升的实践路径以数学史料中概念本质与教育价值分析为核心,落实数学史融入课堂教学的设计实施与评价。从评价角度分析,小学教师将数学史融入教学的实践促进教师数学素养提升,在三个维度上有所表现:树立发展、联系、辩证、应用的数学观,理解蕴含数学知识中的文化育人价值,塑造人文性数学课堂的教学力。     

Complementarity, sets and numbers

Niels Bohr's term‘complementarity' has been used by several authors to capture the essential aspects of the cognitive and epistemological development of scientific and mathematical concepts. In this paper we will conceive of complementarity in terms of the dual notions of extension and intension of mathematical terms. A complementarist approach is induced by the impossibility to define mathematical reality independently from cognitive activity itself. R. Thom, in his lecture to the Exeter International Congress on Mathematics Education in 1972,stated ‘‘the real problem which confronts mathematics teaching is not that of rigor,but the problem of the development of‘meaning’, of the ‘existence' of mathematical objects'. Student's insistence on absolute ‘meaning questions’, however,becomes highly counter-productive in some cases and leads to the drying up of all creativity. Mathematics is, first of all,an activity, which, since Cantor and Hilbert, has increasingly liberated itself from metaphysical and ontological agendas. Perhaps more than any other practice,mathematical practice requires acomplementarist approach, if its dynamics and meaning are to be properly understood. The paper has four parts. In the first two parts we present some illustrations of the cognitive implications of complementarity. In the third part, drawing on Boutroux' profound analysis, we try to provide an historical explanation of complementarity in mathematics. In the final part we show how this phenomenon interferes with the endeavor to explain the notion of number. This revised version was published online in July 2006 with corrections to the Cover Date.     

Mathematics knowledge for understanding and problem solving

Two important aspects of transfer in mathematics learning are the application of mathematical knowledge to problem solving and the acquisition of more advanced concepts, both in mathematics and in other domains. This paper discusses general assumptions and themes of current cognitive research on mathematics learning, focusing on issues of the understanding thought to facilitate transfer of mathematical knowledge. Two studies illustrating these themes are presented, one concerning students' understanding of numerical relationships involved in basic addition and subtraction combinations, the other dealing with students' understanding of algebraic expressions and transformations. Implications of these cognitive perspectives for instruction are discussed.     

Measuring Mathematical Competences of Engineering Students at the Beginning of Their Studies

This article reports about our efforts to determine engineering students' competence in mathematics. Our research is embedded in a larger project, KoM@ING–Modeling and developing competence: Integrated IRT based and qualitative studies with a focus on mathematics and its usage in engineering studies, within the program Modeling and Measuring Competencies in Higher Education (KoKoHS). KoKoHS provides the umbrella organization of several research projects addressing the modeling and measuring of competences at the college level. KoM@ING aims to model the role of engineering students' mathematical competences for their studies from both a quantitative and a qualitative perspective.

Here, we report the development of a large-scale instrument assessing engineering freshmen's competence in mathematics by applying Rasch analysis to determine measures for item difficulties and student abilities. Several analyses were performed to provide insights into the measures' reliability and validity. In particular, to examine cognitive validity, we scrutinized students' think-aloud protocols when solving the items to investigate their problem solving abilities as a proxy for item difficulty. Overall, we found first evidence that our instrument is suitable to assess engineering freshmen's competence in mathematics. This instrument may be helpful to conduct further research and to inform those concerned with college organization and policy.     

Metaphors and Models in Translation Between College and Workplace Mathematics

We report a study of repairs in communication between workers and visiting outsiders (students, researchers or teachers). We show how cultural models such as metaphors and mathematical models facilitated explanations and repair work in inquiry and pedagogical dialogues. We extend previous theorisations of metaphor by Black; Lakoff and Johnson; Lakoff & Nunes; and Schon, to formulate a perspective on mathematical models and modelling and show how dialogues can manifest (i) application of ‘dead’ models to new contexts, and (ii) generative modelling. In particular, we draw in some depth on one case study of the use of a double number line model of the ‘gas day’ and its mediation of communication within two dialogues, characterised by inquiry and pedagogical discourse genres respectively. In addition to spatial and gestural affordances due to its blend of grounding metaphors, the model translates between workplace objects on the one side and spreadsheet-mathematical symbols on the other. The model is found to afford generative constructions that mediate the emergence of new understandings in the dialogues. Finally we discuss the significance of this metaphorical perspective on modelling for mathematics education.     

Classroom Interpreting Games with an Illustration

Classroom communication has been recognized as a process in which ideas become objects of reflection, discussion, and amendments affording the construction of private mathematical meanings that in the process become public and exposed to justification and validation. This paper describes an explanatory model named “interpreting games”, based on the semiotics of Charles Sanders Peirce, that accounts for the interdependence between thought and communication and the interpretation of signs in which teacher and students engage in mathematics classrooms. Interpreting games account both for the process of transformation (in the mind of the learner) of written marks into mathematical signs that stand for mathematical concepts and for the continuous and converging private construction of mathematical concepts. Teacher–student and student–student collaborative interactions establish a mathematical communication that shapes and is also shaped by the conceptual domains and the domains of intentions and interpretations of the participants. A teaching episode with third graders is analyzed as an example of a classroom interpreting game.     

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.     

A constructive response to `Where mathematics comes from'

Lakoff and Nuñez's bookWhere mathematics comes from: How the embodied mind brings mathematics into being (2000) provided many mathematics education researchers with a novel, and perhaps startling perspective on mathematical thinking. However, as evidenced by reviewers' criticisms (Gold,2001; Goldin, 2001; Madden, 2001), their perspective — though liberating for many,with its humanistic emphases — remains controversial. Nonetheless, we believe this perspective deserves further constructive response. In this paper, we propose that several of the book's flaws can be addressed through a more rigorous establishment of conceptual distinctions as well as a more appropriate set of methodological approaches.This revised version was published online in October 2005 with corrections to the Cover Date.     

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.     

A comparison of Hungarian and English teachers' conceptions of mathematics and its teaching

This paper reports on a statistical study of English and Hungarian teachers' conceptions of mathematics and its teaching. A questionnaire was developed and distributed to teachers of mathematics in 200 English and40 Hungarian schools teaching children in the 11–14 age range. Factor analyses identified four conceptions of mathematics and five of mathematics teaching. These were compared with those yielded by an earlier study involving the same English teachers and found to be consistent indicating the existence of similar conceptions in different educational systems. Differences and similarities in the strengths with which those conceptions are held were suggestive of both global and national conceptual traditions. The significant similarity to emerge concerned teachers from both countries sharing, with similar strengths, a general conception of mathematics teaching incorporating the teaching of mathematical skills, a variety of classroom approaches including investigations and problem-solving, and a recognition that mathematics provides an essential lifetool. Multi-dimensional scaling indicated that English teachers have their perspectives informed by two underlying, and possibly conflicting, traditions– pedagogic relevance and mathematical utility. The Hungarians appeared concerned only with notions of pedagogic relevance – those practices perceived to facilitate effective learning of a subject which is untainted by utilitarian perspectives. This revised version was published online in July 2006 with corrections to the Cover Date.     

MATHEMATICS HOMEWORK: A STUDY OF THREE GRADE EIGHT CLASSROOMS IN SINGAPORE

This paper explores the nature and source of mathematics homework and teachers’ and students’ perspectives about the role of mathematics homework. The subjects of the study are three grade 8 mathematics teachers and 115 of their students. Data from field notes, teacher interviews and student questionnaire are analysed using qualitative methods. The findings show that all 3 teachers gave their students homework for instructional purposes to engage them in consolidating what they were taught in class as well as prepare them for upcoming tests and examinations. The homework only involved paper and pencil, was compulsory, homogenous for the whole class and meant for individual work. The main source of homework assignments was the textbook that the students used for the study of mathematics at school. ‘Practice makes perfect’ appeared to be the underlying belief of all 3 teachers when rationalising why they gave their students homework. From the perspective of the teachers, the role of homework was mainly to hone skills and comprehend concepts, extend their ‘seatwork into out of class time’ and cultivate a sense of responsibility. From the perspectives of the students, homework served 6 functions, namely improving/enhancing understanding of mathematics concepts, revising/practising the topic taught, improving problem-solving skills, preparing for test/examination, assessing understanding/learning from mistakes and extending mathematical knowledge.     

VISUAL IMAGES AS EDUCATIONAL MATERIALS IN MATHEMATICS

In general, students in school learn mathematical concepts by words. Some mathematical concepts, however, are difficult to understand by words. This is especially true of some of the more complicated concepts in mathematics taught in higher education. For students who are studying to become engineers, it is very important to understand mathematics intuitively. Ways must be found for them to learn mathematics that will promote intuitive understanding. We often find that a figure helps us understand mathematical concepts and provides important clues for solving problems. A figure may serve as a concrete expression of an abstract mathematical concept; it is a visual image of the mathematical concept. A visual image is a figure with no words but its title. The aim of this article is to introduce some visual images that are effective in mathematical education.     

Conceptualizing sound as a form of incarnate mathematical consciousness

Much of the evidence provided in support of the argument that mathematical knowing is embodied/enacted is based on the analysis of gestures and bodily configurations, and, to a lesser extent, on certain vocal features (e.g., prosody). However, there are dimensions involved in the emergence of mathematical knowing and the production of mathematical communication that have not yet been investigated. The purpose of this article is to theorize one of these dimensions, which we call incarnate sonorous consciousness. Drawing on microanalyses of two exemplary episodes in which a group of third graders are sorting geometric solids, we show how sound has the potential to mark mathematical similarities and distinctions. These “audible” similarities and distinctions, which may be produced by incarnate dimensions such as beat gestures and prosody, allow children to objectify certain geometrical properties of the objects with which they transact. Moreover, the analysis shows that sonorous production is intertwined with other dimensions of students’ bodily activity. These findings are interpreted according to the “theory of mathematics in the flesh,” an alternative to current embodiment/enactivist theories in mathematics education.     

Conceptualising Resources as a Theme for Teacher Education

In this report, I examine resources and their use in school mathematics. I do so from the perspective of mathematics teacher education and with a view to the practice of school mathematics. I argue that the effectiveness of resources for mathematical learning lies in their use, that is, in the classroom teaching and learning context. The argument pivots on the concepts of school mathematics as a hybrid practice and on the transparency of resources in use. These concepts are elaborated by examples of resource use within an in-service teacher education research project in South Africa. I propose that mathematics teacher education needs to focus more attention on resources, on what they are and how they work as an extension of the teacher in school mathematics practice. In so doing, the report provides a language with which mathematics teacher educators and mathematics teachers can investigate teachers' use of resources to support mathematical learning in particular and diverse contexts. This revised version was published online in September 2006 with corrections to the Cover Date.     

Mathematics before formal schooling

This article is concerned with the learning of mathematics before the commencement of formal schooling. The experiences of two students are described. Four main sets of data were gathered: formal and informal measures of mathematics achievement, the learning environment in the home, and observations during teacher-led sessions concerned with mathematical concepts delivered as part of the regular pre-school program. Particular attention was focused on the nature and quality of the students' interactions with their teacher during these sessions.     

Intuitive vs analytical thinking: four perspectives

This article is an attempt to place mathematical thinking in the context of more general theories of human cognition. We describe and compare four perspectives—mathematics, mathematics education, cognitive psychology, and evolutionary psychology—each offering a different view on mathematical thinking and learning and, in particular, on the source of mathematical errors and on ways of dealing with them in the classroom. The four perspectives represent four levels of explanation, and we see them not as competing but as complementing each other. In the classroom or in research data, all four perspectives may be observed. They may differentially account for the behavior of different students on the same task, the same student in different stages of development, or even the same student in different stages of working on a complex task. We first introduce each of the perspectives by reviewing its basic ideas and research base. We then show each perspective at work, by applying it to the analysis of typical mathematical misconceptions. Our illustrations are based on two tasks: one from statistics (taken from the psychological research literature) and one from abstract algebra (based on our own research).

Beyond motivation: history as a method for learning meta-discursive rules in mathematics

In this paper, we argue that history might have a profound role to play for learning mathematics by providing a self-evident (if not indispensable) strategy for revealing meta-discursive rules in mathematics and turning them into explicit objects of reflection for students. Our argument is based on Sfard’s theory of Thinking as Communicating, combined with ideas from historiography of mathematics regarding a multiple perspective approach to the history of practices of mathematics. We analyse two project reports from a cohort of history of mathematics projects performed by students at Roskilde University. These project reports constitute the experiential and empirical basis for our claims. The project reports are analysed with respect to students’ reflections about meta-discursive rules to illustrate how and in what sense history can be used in mathematics education to facilitate the development of students’ meta-discursive rules of mathematical discourse.