Design and Implementation of a Framework for Defining Integrated Mathematics and Science Education

Integrated mathematics and science teaching and learning is a widely advocated yet largely unexplored phenomenon. This study involves an examination of middle school integrated mathematics and science education from two perspectives—theory and practice. The theoretical component of this research addresses the ill-defined nature of the phrase integrated mathematics and science education. A conceptual framework in the form of a Mathematics/Science Continuum is presented to lend clarity and precision to this phrase. The theoretical framework is then used to guide analysis of tasks students are engaged in during instructional practice in middle school classrooms, where the goal of instruction is full integration of mathematics and science. Barriers to integrating mathematics and science in the school curriculum are also presented.     

Pre-service mathematics teachers become full participants in inquiry investigations

Research is described concerning the effectiveness of inquiry-based laboratory environments created in US mathematics/science education programme courses. Laboratory projects were conducted using a framework that allowed pre-service teachers to explore, analyse, and communicate ‘investigable’ realms of physical phenomena. Goals were for pre-service teachers to experience the value of learning in an inquiry-enhanced environment and to engage in contextualized mathematics so they would utilize this instruction in their future classrooms. It is proposed that inquiry-based laboratories are needed within the mathematics classroom in order to allow students the opportunity to contextualize, to connect to other disciplines, and to experience mathematical concepts. Pre-service teachers were expected to pursue conjectures, collect data, think critically, and communicate findings. This qualitative research shows how the use of inquiry can complement the learning of mathematical content and educational strategies for pre-service teachers. Results provide detailed information for teacher educators regarding instructional design of contextualized mathematical inquiry.     

Mathematics teacher educators’/researchers’ collaboration with teachers as a context for professional learning

This paper describes our joint activity as mathematics teacher educators and academic researchers in collaborating with both experienced and novice teachers in two contexts: an emergent community of inquiry into mathematics teaching and its development; and a research methods course, offered as part of a mathematics education Master’s program, aspiring to initiate participating teachers into research practice through inquiry. Adopting an Activity Theory (AT) perspective, we analyse our activity, identifying its nature and transformations that frame our professional learning. The results indicate that our professional learning is the outcome of a continuous process of becoming aware of our own activity and its transformation in relation to that of the teachers.     

Demokratische Erziehung im Mathematikunterricht —kann es das geben?

As an introduction to this and the next ZDM-issue tackling the problem of mathematics teaching and democratic education, the following outlines the possibility of a systematical view on the problem starting with the German concept of Bildung. Thus at least a positive answer to the question of the headline becomes irrefutable.     

A false belief about fractions – What is its source?

This paper presents the results of interviews with 174 participants solving a problem of elementary mathematics, connected with the part–whole approach to fractions. The motive for the investigation was a specific kind of difficulty observed during a case study conducted to verify the elementary school student's understanding of the concept of fractions. The authors decided to examine the problem in a broader population of mathematics learners at different levels of education: from elementary school to university students and graduates of science majors. Approximately 65% of respondents reported the wrong answer immediately after reading the fraction problem taken from the fourth grade of elementary school. Detailed analysis of the respondents’ performance showed that the source of many wrong answers was a false belief about fractions: The only way to get 1/n of a given whole is to divide this whole into n equal parts, not yet described in educational literature.     

Mathematics teaching development as a human practice: identifying and drawing the threads

The didactic triangle links mathematics, teachers and students in a consideration of teaching?Clearning interactions in mathematics classrooms. This paper focuses on teachers and teaching in the development of fruitful learning experiences for students with mathematics. It recognises primarily that teachers are humans with personal characteristics, subject to a range of influences through the communities of which they are a part, and considers aspects of teachers?? personhood, identity and agency in designing teaching for the benefit of their students. Teaching is seen as a developmental process in which inquiry plays a central role, both in doing mathematics in the classroom and in exploring teaching practice. The teacher-as-inquirer in collaboration with outsider researchers leads to growth of knowledge in teaching through development of identity and agency for both groups. The inclusion of the outsider researcher brings an additional node into the didactic triangle.     

Mathematics Education & Digital Technologies: Facing the Challenge of Networking European Research Teams

This paper introduces the IJCML Special Issue dedicated to digital technologies and mathematics education and, in particular, to the work performed by the European Research Team TELMA (Technology Enhanced Learning in Mathematics). TELMA was one of the initiatives of the Kaleidoscope Network of Excellence established by the European Community (IST-507838—2003–2007) to promote the joint elaboration of concepts and methods for exploring the future of learning with digital technologies. TELMA addressed the problem of fragmentation of theoretical frameworks in the research field of mathematics education with digital technologies and developed a methodology based on the in field cross-experimentation of educational digital environments for maths. Six European research teams engaged in cross-experimentation in classrooms as a means to begin to develop a common language and to analyse the intertwined influence played, both explicitly and implicitly, by different contextual characteristics and theoretical frames assumed as reference by the diverse teams participating in the initiative.     

Mathematics capital in the educational field: Bourdieu and beyond

Mathematics education needs a better appreciation of the dominant power structures in the educational field: Bourdieu's theory of capital provides a good starting point. We argue from Bourdieu's perspective that school mathematics provides capital that is finely tuned to generationally reproduce the social structures that serve to keep the powerful in power, while ensuring that less powerful groups are led to accept their own failure in mathematics. Bourdieu's perspective thereby highlights theoretical inadequacies in much mathematics education research, insofar as it presumes a consensus about a ‘what works agenda’ for improving achievement for all. Drawing on one case where we manufactured awkward facts, we illustrate a Bourdieusian interpretation of mathematics capital as reproductive, and the crucial role of its cultural arbitrary. We then criticise the Bourdieusian concept of ‘mathematical capital’ as the value of mathematical competence in practice and propose to extend his tools to include the contradictory ‘use’ and ‘exchange’ values of mathematics instead: we will show how this conceptualisation goes ‘beyond Bourdieu’ and helps explain how teaching-learning might (ideally) produce ‘cultural use value’ in mathematical competence, while still recognising the contradictions teachers and learners face. Finally, we suggest how critical education research generally can benefit from this theoretical framework: (1) in exposing the interest of the dominant classes; but also (2) in researching critical pedagogic alternatives that challenge orthodoxy in educational policy and practice both in mathematics education and more generally.     

Teachers and didacticians: key stakeholders in the processes of developing mathematics teaching

This paper sets the scene for a special issue of ZDM—The International Journal on Mathematics Education—by tracing key elements of the fields of teacher and didactician/teacher-educator learning related to the development of opportunities for learners of mathematics in classrooms. It starts from the perspective that joint activity of these two groups (teachers and didacticians), in creation of classroom mathematics, leads to learning for both. We trace development through key areas of research, looking at forms of knowledge of teachers and didacticians in mathematics; ways in which teachers or didacticians in mathematics develop their professional knowledge and skill; and the use of theoretical perspectives relating to studying these areas of development. Reflective practice emerges as a principal goal for effective development and is linked to teachers’ and didacticians’ engagement with inquiry and research. While neither reflection nor inquiry are developmental panaceas, we see collaborative critical inquiry between teachers and didacticians emerging as a significant force for teaching development. We include a summary of the papers of the special issue which offer a state of the art perspective on developmental practice.     

Mathematics education and democracy

In this paper, we investigate the relationship between mathematics education and the notions of education for all/democracy. In order to proceed with our analysis, we present Marx’s concept of commodity and Jean Baudrillard’s concept of sign value as a theoretical reference in the discussion of how knowledge has become a universal need in today’s society and ideology. After, we engage in showing mathematics education’s historical and epistemological grip to this ideology. We claim that mathematics education appears in the time period that English becomes an international language and the notion of international seems to be a key constructor in the constitution of that ideology. Here, we draw from Derrida’s famous saying that “there is nothing beyond the text”. We conclude that a critique to modern society and education has been developed from an idealistic concept of democracy.     

Scientific World Building on the Edge of Chaos: High School Students' Beliefs About Mathematics and Science

This study investigated high school students' beliefs about mathematics and science during a four week summer residential mathematics and science program. Beliefs about mathematical and scientific truths, the value and importance of mathematics and science inquiry, gender equity and ability with respect to pursuit of mathematics and science careers, the relationship between mathematics and technology, and the role of science in society were examined. Habermasian ways of knowing were used to categorize student beliefs and determine student world views. Implications of this study include suggested changes in the organizational dynamics of schooling to better prepare our students for surviving in the complexity of the 21st century and reducing dissonance between the “classical” educational viewpoints and the “chaotic” world.     

A research journey through mathematics coaching

Classroom coaching in mathematics is flexible in its definition, complex in its enactment, variable in its outcomes, and dependent on setting and circumstances. Multiple lines of inquiry are required to navigate this subjective terrain: research on coaching encompasses understanding perceptions of coaching held by coaches, teachers, and administrators, measuring the effectiveness of coaching in terms of teachers’ content knowledge and instructional practices, and exploring the nature of coaching within an educational ecosystem. This paper describes a cumulative sequence of research studies that inform current understanding of classroom coaching in mathematics, highlighting methodological decisions made at various crossroads and elaborating on the populations, methods, and instruments used to investigate coaching. A presentation of findings related to the three domains of perception, effectiveness, and nature is followed by reflections on features of coaching that pose particular challenges, questions that remain to be answered, and promising avenues of future inquiry.     

Mathematics education research embracing arts and sciences

As a young field in its own right (unlike the ancient discipline of mathematics), mathematics education research has been eclectic in drawing upon the established knowledge bases and methodologies of other fields. Psychology served as an early model for a paradigm that valorized psychometric research, largely based in the theoretical frameworks of cognitive science. More recently, with the recognition of the need for sociocultural theories, because mathematics is generally learned in social groups, sociology and anthropology have contributed to methodologies that gradually moved away from psychometrics towards qualitative methods that sought a deeper understanding of issues involved. The emergent perspective struck a balance between research on individual learning (including learners’ beliefs and affect) and the dynamics of classroom mathematical practices. Now, as the field matures, the value of both quantitative and qualitative methods is acknowledged, and these are frequently combined in research that uses mixed methods, sometimes taking the form of design experiments or multi-tiered teaching experiments. Creativity and rigor are required in all mathematics education research, thus it is argued in this paper, using examples, that characteristics of both the arts and the sciences are implicated in this work.     

Felix Klein's “Erlanger Antrittsrede”: A Transcription with English Translation and Commentary

One of Felix Klein's leading interests was the role of mathematics education not only in the German universities but in the secondary schools as well. Klein played a leading role in the educational reform movements that flourished during the twenty-year period prior to World War I, and in 1908 he was elected President of the International Mathematics Instruction Commission. The “Erlanger Antrittsrede” of 1872, presented herein, gives a clear expression of Klein's views on mathematics education at the very beginning of his career. While previous writers, including Klein himself, have stressed the continuity between the Antrittsrede and his later views on mathematics education, the following commentary presents an analysis of the text together with external evidence supporting exactly the opposite conclusion.     

Mathematics and Science Teachers' Use of and Confidence in Empirical Reasoning: Implications for STEM Teacher Preparation

The recent trend to unite mathematically related disciplines (science, technology, engineering, and mathematics) under the broader umbrella of STEM education has advantages. In this new educational context of integration, however, STEM teachers need to be able to distinguish between sufficient proof and reasoning across different disciplines, particularly between the status of inductive and deductive modes of reasoning in mathematics. Through a specific set of mathematical conjectures, researchers explored differences between mathematics (n = 24) and science (n = 23) teachers' reasoning schemes, as well as the confidence they had in their justifications. Results from the study indicate differences between the two groups in terms of their levels of mathematical proof, as well as correlational trends that inform their confidence across these levels. Implications particularly for teacher training and preparation within the context of an integrated STEM education model are discussed.     

Embodying mathematics and science: Microworlds as representations

This paper examines the category of open-ended exploratory computer environments that have been labeled “microworlds.” The paper reviews the ways in which the term “microworld” has been used in the mathematics and science education communities, and analyzes examples of specific computer microworlds. Two definitions of microworld are proposed: a structural definition that focuses on design elements shared by the environments, and a functional definition that highlights commonalties in how students learn with microworlds. In the final section of the paper, the notion that computer microworlds can be said to “embody” mathematical or scientific ideas is addressed, within the context of a re-evaluation of the general concept of representation.     

The Road to Reform: A Grounded Theory Study of Parents' and Teachers' Influence on Elementary School Science and Mathematics

Though elementary teacher educators introduce new, reform‐based strategies in science and mathematics methods courses, researchers wondered how novices negotiate reform strategies once they enter the elementary school culture. Given that the extent of parents' and veteran teachers' influence on novice teachers is largely unknown, this grounded theory study explored parents' and teachers' expectations of children's optimal science and mathematics learning in the current era of reform. Data consisted of semi‐structured, open‐ended interviews with novice teachers (n = 20), veteran teachers (n = 9), and parents (n = 28). Researchers followed three stages of coding procedures to develop a logic model connecting participants' discrete designations of the landscape, regulating phenomena, contextual orientation, and desired outcomes. This logic model helped researchers develop propositions for future research on the interactive nature of parents' and teachers' influential role in elementary science and mathematics education. Implications encourage science and mathematics teacher educators—as well as school administrators—to explicitly develop and support novice teachers' ability to initiate and sustain parent/family engagement in order to create a school climate where teachers and parents are synergistically motivated to change.     

Mobilizing histories in mathematics teacher education: memories, social practices, and discursive games

In the first part of this paper, we share and elucidate the way we mobilize histories in some disciplines that are part of the undergraduate courses in mathematics teacher education offered by State University of Campinas and Federal University of Rio Grande do Norte in Brazil. This way of mobilization can be featured as a set of collective indisciplinary problematizations occurring in a series of student investigations. Mobilizing practices of mathematics culture are the object of these investigations. These practices are performed by different communities both constituted by and constituent of different human activities. In the second part of this paper, we will discuss our way of mobilizing histories, contrasting it with the theoretical perspective of expansive learning, just as it has been defended by Yrjö Engeström, in his article Non scolae sed vitae discimus—towards overcoming the encapsulation of school learning. We will also attempt to highlight the role which this researcher has attributed to history in his model of expansive learning, a perspective based on the current research on activity theory.     

Teaching and learning mathematics in the collective

In this paper we analyse and explore teaching and learning in the context of a high school mathematics classroom that was deliberately structured as highly interactive and inquiry-oriented. We frame our discussion within enactivism—a theory of cognition that has helped us to understand classroom processes, particularly at the level of the group. We attempt to show how this classroom of mathematics learners operated as a collective and focus in particular on the role of the teacher in establishing, sustaining, and becoming part of such a collective. Our analysis reveals teaching practices that value, capitalize upon, and promote group cognition, our discussion positions such work as teaching a way of being with mathematics, and we close by offering implications for teaching, educational policy, and further research.