Potential scenarios for Internet use in the mathematics classroom

Research on the influence of multiple representations in mathematics education gained new momentum when personal computers and software started to become available in the mid-1980s. It became much easier for students who were not fond of algebraic representations to work with concepts such as function using graphs or tables. Research on how students use such software showed that they shaped the tools to their own needs, resulting in an intershaping relationship in which tools shape the way students know at the same time the students shape the tools and influence the design of the next generation of tools. This kind of research led to the theoretical perspective presented in this paper: knowledge is constructed by collectives of humans-with-media. In this paper, I will discuss how media have shaped the notions of problem and knowledge, and a parallel will be developed between the way that software has brought new possibilities to mathematics education and the changes that the Internet may bring to mathematics education. This paper is, therefore, a discussion about the future of mathematics education. Potential scenarios for the future of mathematics education, if the Internet becomes accepted in the classroom, will be discussed.     

关于理工科高等数学研究型教学与大学生创新意识培养研究的构想

由于高等数学是高校理工科一门重要的基础课,因此进行高等数学研究型教学与大学生创新意识培养研究是当今高校理工科专业高等数学教学中进行素质教育的必然要求.本文论述了其研究的目的、理论依据和意义,同时阐述了研究型教学与创新意识的培养问题研究的现状和趋势.最后提出了理工科高等数学研究型教学与大学生创新意识培养研究的实施方案和计划、实施细则、效益分析和研究环境.     

Computational Literacy and “The Big Picture” Concerning Computers in Mathematics Education

This article develops some ideas concerning the “big picture” of how using computers might fundamentally change learning, with an emphasis on mathematics (and, more generally, STEM education). I develop the big-picture model of computation as a new literacy in some detail and with concrete examples of sixth grade students learning the mathematics of motion. The principles that define computational literacy also serve as an analytical framework to examine competitive big pictures, and I use them to consider the plausibility, power, and limitations of other important contemporary trends in computationally centered education, notably computational thinking and coding as a social movement. While both of these trends have much to recommend them, my analysis uncovers some implausible assumptions and counterproductive elements of those trends. I close my essay with some more practical and action-oriented advice to mathematics educators on how best to orient to the long-term trajectory (big picture) of improving mathematics education with computation.     

E-learning in secondary–tertiary transition in mathematics: for what purpose?

Mathematics is a challenging subject for many science freshmen, and failing mathematics exams is a major factor in university dropout rates. Many factors, including metacognitive, affective and linguistic ones, play a role in students?? difficulties in mathematics. Starting from this perspective, we conducted a theoretical exploration of the potential of online environments in helping students counteract the mathematical difficulties they face in the transition from secondary school to university. Although this secondary?Ctertiary transition and the use of technology are both widely researched issues in mathematics education, the potential of technology in helping students in the ??rite of passage?? to tertiary education has not yet been researched. This paper reports on the developed theoretical framework and on the preliminary findings from the implementation of an e-learning course that we designed with the aim of supporting students in the critical phase of transition from secondary school to university.     

A Sino-German semi-virtual seminar in mathematics education

In summer 2006 the University of Education in Weingarten, Germany, and East China Normal University, Shanghai, performed a semi-virtual seminar with mathematics students on “Mathematics and Architecture”. The goal was the joint development of teaching materials for German or Chinese school, based on different buildings such as “Nanpu Bridge”, or the “Eiffel Tower”. The purpose of the seminar was to provide a learning environment for students supported by using information and communication technology (ICT) to understand how the hidden mathematics in buildings should be related to school mathematics; to experience the multicultural potential of the international language “Mathematics”; to develop “media competence” while communicating with others and using technologies in mathematics education; and to recognize the differences in teaching mathematics between the two cultures. In this paper we will present our ideas, experiences and results from the seminar.     

Mathematical paradoxes as pathways into beliefs and polymathy: an experimental inquiry

This paper addresses the role of mathematical paradoxes in fostering polymathy among pre-service elementary teachers. The results of a 3-year study with 120 students are reported with implications for mathematics pre-service education as well as interdisciplinary education. A hermeneutic-phenomenological approach is used to recreate the emotions, voices and struggles of students as they tried to unravel Russell’s paradox presented in its linguistic form. Based on the gathered evidence some arguments are made for the benefits and dangers in the use of paradoxes in mathematics pre-service education to foster polymathy, change beliefs, discover structures and open new avenues for interdisciplinary pedagogy.     

The history of mathematics as a coupling link between secondary and university teaching

During the years they spend in university, many mathematics students develop a very poor conception of mathematics and its teaching. This fact is bad in all cases, but even more in the case of those students who will be mathematics teachers in school. In this paper it is argued that the history of mathematics may be an efficient element to provide students with flexibility, open-mindedness and motivation towards mathematics. The theoretical background of this work relies both on recent research in mathematics education and on papers written by mathematicians of the past. Opinions are supported with examples. One example concerns a historical presentation of ‘definition’; it was developed with mathematics students who will become mathematics teachers. For students oriented to research or to applied mathematics, an example is presented to address the problem of the secondary-tertiary transition.     

Language and communication in mathematics education: an overview of research in the field

Within the field of mathematics education, the central role language plays in the learning, teaching, and doing of mathematics is increasingly recognised, but there is not agreement about what this role (or these roles) might be or even about what the term ‘language’ itself encompasses. In this issue of ZDM, we have compiled a collection of scholarship on language in mathematics education research, representing a range of approaches to the topic. In this survey paper, we outline a categorisation of ways of conceiving of language and its relevance to mathematics education, the theoretical resources drawn upon to systematise these conceptions, and the methodological approaches employed by researchers. We identify four broad areas of concern in mathematics education that are addressed by language-oriented research: analysis of the development of students’ mathematical knowledge; understanding the shaping of mathematical activity; understanding processes of teaching and learning in relation to other social interactions; and multilingual contexts. A further area of concern that has not yet received substantial attention within mathematics education research is the development of the linguistic competencies and knowledge required for participation in mathematical practices. We also discuss methodological issues raised by the dominance of English within the international research community and suggest some implications for researchers, editors and publishers.     

Changed views on mathematical knowledge in the course of didactical theory development—independent corpus of scientific knowledge or result of social constructions?

The study tries to show one line of how the German didactical tradition has evolved in response to new theoretical ideas and new—empirical—research approaches in mathematics education. First, the classical mathematical didactics, notably ‘stoffdidaktik’ as one (besides other) specific German tradition are described. The critiques raised against ‘stoffdidaktik’ concepts [for example, forms of ‘progressive mathematisation’, ‘actively discovering learning processes’ and ‘guided reinvention’ (cf. Freudenthal, Wittmann)] changed the basic views on the roles that ‘mathematical knowledge’, ‘teacher’ and ‘student’ have to play in teaching–learning processes; this conceptual change was supported by empirical studies on the professional knowledge and activities of mathematics teachers [for example, empirical studies of teacher thinking (cf. Bromme)] and of students’ conceptions and misconceptions (for example, psychological research on students’ mathematical thinking). With the interpretative empirical research on everyday mathematical teaching–learning situations (for example, the work of the research group around Bauersfeld) a new research paradigm for mathematics education was constituted: the cultural system of mathematical interaction (for instance, in the classroom) between teacher and students.     

Practitioner's Research: A Study in Changing Preservice Teachers’ Conceptions About Mathematics and Mathematics Teaching and Learning

Educators involved in the current reform movement in mathematics education recommend that students be actively involved in constructing their own knowledge and developing powerful mathematical concepts. This study suggests how ideas based on constructivist learning theory can be put into practice in a preservice mathematics education class.     

Moving toward shared realities through empathy in mathematical modeling: An ecological systems theory approach

The mathematics education community has routinely called for mathematics tasks to be connected to the real world. However, accomplishing this in ways that are relevant to students’ lived experiences can be challenging. Meanwhile, mathematical modeling has gained traction as a way for students to learn mathematics through real-world connections. In an open problem to the mathematics education community, this paper explores connections between the mathematical modeling and the nature of what is considered relevant to students. The role of empathy is discussed as a proposed component for consideration within mathematical modeling so that students can further relate to real-world contexts as examined through the lens of Ecological Systems Theory. This is contextualized through a classroom-tested example entitled “Tiny Homes as a Solution to Homelessness” followed by implications and conclusions as they relate to mathematics education.     

Game and decision theory in mathematics education: epistemological, cognitive and didactical perspectives

In the 1950s, game and decision theoretic modeling emerged—based on applications in the social sciences—both as a domain of mathematics and interdisciplinary fields. Mathematics educators, such as Hans Georg Steiner, utilized game theoretical modeling to demonstrate processes of mathematization of real world situations that required only elementary intuitive understanding of sets and operations. When dealing with n-person games or voting bodies, even students of the 11th and 12th grade became involved in what Steiner called the evolution of mathematics from situations, building of mathematical models of given realities, mathematization, local organization and axiomatization. Thus, the students could participate in processes of epistemological evolutions in the small scale. This paper introduces and discusses the epistemological, cognitive and didactical aspects of the process and the roles these activities can play in the learning and understanding of mathematics and mathematical modeling. It is suggested that a project oriented study of game and decision theory can develop situational literacy, which can be of interest for both mathematics education and general education.     

Values in Japanese mathematics education: their historical development

Individual mathematics teachers may value different aspects of teaching and learning mathematics, but at the same time their value systems are under the influence of socially shared values. This paper describes such values in Japanese mathematics education from a historical and normative perspective. After the introduction of Western mathematics into the modern school system in the Meiji period (1868?C1912), the people of Japan struggled to adapt and absorb it onto the foundation of Japanese tradition. In the subsequent development of Japanese mathematics education, the integration of both practical and theoretical aspects have been issues, alongside changes in educational focus and in society at large, which are symbolically represented by the enrolment rates at all of elementary, secondary, and tertiary education levels. Mathematics education in Japan has also been subject to international influences, such as the reform movement and the modernization movement, at critical junctures in its development. Key concepts such as mathematical ideas, mathematical thinking, and mathematical activities are traces of such historical efforts by the Japanese mathematics education community, and represent their socially shared values.     

The Recursive Paradigm: Suppose We Already Knew

The recursive paradigm is a central theme in discrete mathematics. Through examples, this paper explains the paradigm and provides some ideas for approaching the material with students. Repeated use of the phrase “Suppose We Already Knew” and the acronym SWAK help drive the paradigm home.     

Understanding the Affects of Audience on Mathemtaical Writing

This study formally explored how high school students addressed audience when they wrote mathematically. Student explicated the solving of a mathematics problem to their mathematics teacher and their English teacher. This study showed that students did change the style and language of their mathematical writing as their audience changed. It adds knowledge and support to current ideas of teacher presentation during routine daily instruction. Also included are implications this information may have on mathematics education.     

The potential of recreational mathematics to support the development of mathematical learning

ABSTRACT

A literature review establishes a working definition of recreational mathematics: a type of play which is enjoyable and requires mathematical thinking or skills to engage with. Typically, it is accessible to a wide range of people and can be effectively used to motivate engagement with and develop understanding of mathematical ideas or concepts. Recreational mathematics can be used in education for engagement and to develop mathematical skills, to maintain interest during procedural practice and to challenge and stretch students. It can also make cross-curricular links, including to history of mathematics. In undergraduate study, it can be used for engagement within standard curricula and for extra-curricular interest. Beyond this, there are opportunities to develop important graduate-level skills in problem-solving and communication. The development of a module ‘Game Theory and Recreational Mathematics’ is discussed. This provides an opportunity for fun and play, while developing graduate skills. It teaches some combinatorics, graph theory, game theory and algorithms/complexity, as well as scaffolding a Pólya-style problem-solving process. Assessment of problem-solving as a process via examination is outlined. Student feedback gives some indication that students appreciate the aims of the module, benefit from the explicit focus on problem-solving and understand the active nature of the learning.