Teaching units as the integrating core of mathematics education

How to integrate mathematics, psychology, pedagogy and practical teaching within the didactics of mathematics in order to get unified specific theories and conceptions of mathematics teaching? This problem—relevant for theoretical and empirical studies in mathematics education as well as for teacher training—is considered in the present paper. The author suggests an approach which is based on teaching units (Unterrichtsbeispiele). Suitable teaching units incorporate mathematical, pedagogical, psychological and practical aspects in a natural way and therefore they are a unique tool for integration.     

Undergraduate Students’ Conceptions of Mathematics: An International Study

In this paper, we report on an international study of undergraduate mathematics students’ conceptions of mathematics. Almost 1,200 students in five countries completed a short survey including three open-ended questions asking about their views of mathematics and its role in their future studies and planned professions. Responses were analysed starting from a previously-developed phenomenographic framework (Reid et al., 2003) which required only minor modification. Students’ conceptions of mathematics ranged from the narrowest view as a focus on calculations with numbers, through a notion of mathematics as a focus on models or abstract structures, to the broadest view of mathematics as an approach to life and a way of thinking. Broader conceptions of mathematics were more likely to be found in later-year students (p<0.001) and there were significant differences between universities (p<0.001). The information obtained from the study not only confirms previous research, but also provides a basis for future development of a monitoring questionnaire.     

数学实验的理论研究与实践

中学数学教育应重视数学实验,应将数学实验作为课程内容的一部分来设计.作为课程内容的数学实验,目的是以实验为载体,展示数学的探索发现过程,使学生亲历这个过程,从中发现数学、体验数学、理解数学、运用数学,培养创新意识和探索精神。作为课程内容的数学实验应体现活动化、操作化特征,注意返璞归真,在注意揭示数学概念定理的形成和发展过程以及展示数学问题的解决过程的同时,注意与基本的数学思想、数学方法挂钩,有机地和数学知识教学相互结合、相互促进。     

UNIVERSITY STUDENTS’ VIEWS OF THE ROLE OF MATHEMATICS IN THEIR FUTURE

We report on an international study about mathematics students’ ideas of how they will use mathematics in their future study and careers. This builds on our previous research into students’ conceptions of mathematics. In this paper, we use data from two groups of students studying mathematics: those who participated in an in-depth interview and those who completed an open-ended questionnaire. We found that their responses could be grouped into four categories: don’t know; procedural skills; conceptual skills; and professional skills. Although some students held clear ideas about the role of mathematics, many were not able to articulate how it would be used in their future. This has implications for their approach to learning and our approach to teaching.     

Mathematical Modelling: the Interaction of Culture and Practice

Using a sociocultural approach we analyse the results of a Mexican/British project which investigated the ways in which mathematics is used in the practice of school science and the role of spreadsheets as a mathematical modelling tool. After discussing the different school cultures experienced by two groups of pre-university 16–18 year old students we analyse how these different cultures influenced their practice of mathematics, as well as their work with mathematical spreadsheet modelling activities. There were clear differences between the two groups of students in their preference for external representations, in their understanding of the kind of answers they were expected to produce and in the way they conceived the role of mathematics in the practice of science. Although students preferences for a particular representation were not significantly modified by the use of a spreadsheet as a modelling resource, at the end of the study the students recognised the value of using a more diverse set of representations. The results obtained suggest the possibility of enhancing students' capability to shift between a wider range of representations, including graphical, algebraic and numeric ones, using a modelling approach embedded in a computer environment such as a spreadsheet. This revised version was published online in July 2006 with corrections to the Cover Date.     

Affective productions of mathematical experience

In underscoring the affective elements of mathematics experience, we work with contemporary readings of the work of Spinoza on the politics of affect, to understand what is included in the cognitive repertoire of the Subject. We draw on those resources to tell a pedagogical tale about the relation between cognition and affect in settings of mathematical learning. Our interest is first captured in the way in which one teacher’s priority of establishing an inclusive learning community occasionally harboured what appeared to be pedagogically restrictive conceptions of mathematics. Yet, the classroom practices that produced these conceptions promoted the students’ motivation and provided meaningful access to mathematical learning within the classroom collectivity. In a second example, the postponement of scientific encapsulation in bodily imitations of planetary movement kept alive a shared dynamic sense of an elliptical orbit. In both of these cases, we draw on Spinoza’s work to show how the affectivity of classroom practice constituted conceptions of cognition and of mathematical activity crucially linked to the imperatives of participation.     

Making abstract mathematics concrete in and out of school

We adopt a neo-Vygotskian view that a fully concrete scientific concept can only emerge from engaging in practice with systems of theoretical concepts, such as when mathematics is used to make sense of outside school or vocational practices. From this perspective, the literature on mathematics outside school tends to dichotomise in- and out-of-school practice and glamorises the latter as more authentic and situated than academic mathematics. We then examine case study ethnographies of mathematics in which this picture seemed to break down in moments of mathematical problem solving and modelling in practice: (1) when amateur or professional players decided to investigate the mathematics of darts scoring to develop their “outing” strategies and (2) when a prevocational mathematics course task challenged would-be mathematics teachers’ concept of fractions. These examples are used to develop the Vygotskian framework in relation to vocational and workplace mathematics. Finally, we propose that a unified view of mathematics, outside and inside school, on the basis of Vygotsky’s approach to everyday and scientific thought, can usefully orientate further research in vocational mathematics education.     

Conceptions of mathematics and how it is learned: The perspectives of students entering university

This paper reports results from an investigation to identify the conceptions of mathematics held by beginning university students and their approaches to the study of mathematics. Phenomenographic techniques were used to analyze responses to a questionnaire administered to approximately 300 students. An analysis of the results identified a structural relationship between students' conceptions of mathematics and their approaches to learning it, with the majority of students viewing mathematics as a necessary set of rules and procedures to be learned by rote. The results of this research have implications for the ways in which teaching and learning are constituted within universities.     

数学生活化的含义与功能及其情境标准

数学生活化是指以生活化的素材与方式展开数学教学,包含了"生活化"和"数学化"两个维度.数学生活化是激发数学学习兴趣的途径,促进数学深刻理解的阶梯,培养抽象概括能力的素材,积累数学活动经验的载体,形成合理数学观念的基石.生活化的情境应该体现数学知识本质,引发学生数学思维,指向数学教学目标\源自真实生活情境,基于学生知识经验.     

Examining and elaborating upon the nature of elementary prospective teachers’ conceptions of partitive division with fractions

The purpose of this study was to examine and elaborate upon elementary prospective teachers’ (PSTs) conceptions of partitive division with fractions. We examined the degree to which PSTs’ conceptions were connected (i.e., capable of translating between representations correctly; aware that partitive division generates a unit rate for its quotient) and flexible (i.e., capable of differentiating between opportunities to partition or iterate (or both) when solving a partitive division task; aware that partitioning or iterating (or both) could be associated with the operation of division, as appropriate). Seventeen PSTs participated in task-based interviews prior to instruction in a mathematics content course for teachers. These PSTs demonstrated disconnected conceptions of partitive division with fractions when they incorrectly translated between representations and either inconsistently or did not express awareness that the purpose of the task was to generate a unit rate. These PSTs demonstrated rigid conceptions of partitive division with fractions such that they did not express awareness that the process of iterating could be associated with the operation of division, even when they obtained a correct answer by iterating. Results extend prior research by looking beyond PSTs’ performance on tasks to elaborate upon PSTs’ conceptions of the operation of partitive division. This study contributes new insights into PSTs’ conceptions that can be used by mathematics teacher educators to inform the design of future instructional interventions.     

磨·模·魔——小学数学教学中渗透模型思想的思考

《全日制义务教育数学课程标准(实验稿)》明确提出,在数学教学中应当引导学生感悟建模过程,发展"模型思想"。在小学,进行数学建模教学具有鲜明的阶段性、初始性特征,即要从学生熟悉的生活和已有的经验出发,引导他们经历将实际问题初步抽象成数学模型并进行解释与运用的过程,进而对数学和数学学习获得更加深刻的理解。就其教学实施的一般程序而言,教师先行琢磨、通过教学不断建模、学生在体验和感悟中为之着魔是小学数学建模教学的关键所在。     

The emerging of teachers' conceptions of new subjects inserted in mathematics programs: the case of informatics

The changes in mathematical curricula induced by the introduction of informatics in school represent the general framework of this research. In particular we focus on the teacher's role by analysing the different choices taken by mathematics teachers when faced with a curriculum reform induced by the introduction of informatics in secondary school courses (age 14–16). Our hypothesis is that these choices are the consequence of conceptions teachers have about informatics and its teaching in relation to the teaching of mathematics. Thus, through a case study research method, we focus on mathematics teachers' conceptions of informatics and its teaching. An attempt is made at outlining a typology of these conceptions, based on the different orientations identified.     

The Role of Intuitive Rules in Science and Mathematics Education

During the last two decades many researchers in mathematics and science education have studied students’ conceptions and ways of reasoning in mathematics and science. Most of this research is content‐specific. It was found that students hold alternative ideas that are not always compatible with those accepted in science. It was suggested that in the process of learning science or mathematics, students should restructure their specific conceptions to make them conform to currently accepted scientific ideas. In our work in mathematics and science education it became apparent that some of the alternative conceptions in science and mathematics are based on the same intuitive rules. We have so far identified two such rules: “More of A, more of B”, and “Subdivision processes can always be repeated”. The first rule is reflected in subjects’ responses to many tasks, including all classical Piagetian conservation tasks (conservation of number, area, weight, volume, matter, etc.) in all tasks related to intensive quantities (density, temperature, concentration, etc.) and in all tasks related to infinite quantities. The second rule is observed in students’, preservice and inservice teachers’ responses to tasks related to successive division of material and geometrical objects and in seriation tasks. In this paper, we describe and discuss these rules and their relevance to science and mathematics education.     

数学教育领域中的三个新“教条”——关于数学课程改革深入发展的再思考

"以学生为本"、"关注过程"与"知识形态的必要转变"是现今在数学教育领域中具有广泛影响的3个理念,但在实践中又经常看到一些简单化的理解,从而在一定程度上蜕化成了"教条".为了避免这一现象,应当坚持"教学的双中心"与"过程与结果并重",并应更加强调教学工作的再创造性.     

Mathematics learning difficulties in primary education: teachers’ professional knowledge and the use of commercially available learning packages

The present study builds on teachers’ professional knowledge about mathematics learning difficulties. Based on the input of 918 primary school teachers, an attempt is made to develop an overview of difficult curriculum topics in primary school mathematics. The research approach builds on new conceptions about the professional identity of teachers and earlier conceptions that point at the critical relevance of teachers’ pedagogical content knowledge. It is also found that the adoption of a specific commercially available learning package (CALP: manuals and exercise books used in the classroom) plays a mediating role.     

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.     

Prospective teachers' initial conceptions about pupils' understanding of science and mathematics

Earlier reports have shown that prospective teachers' conceptions about teaching science to a high degree are resistant and do not change substantially during the teacher‐training programme. In our investigation we elucidate the prospective teachers' initial conceptions about pupils' understanding of science and mathematics. We applied ‘The Lesson Preparation Method' and used a phenomenographic approach in order to reveal the range of conceptions that the prospective teachers hold. A third of the prospective teachers did not consider pupils' conceptions when planning lessons. The rest of the 32 participants expressed awareness; some of the prospective teachers even referred to subject‐specific teaching experience. Also regarding the prospective teachers' conceptions about pupils' knowledge and beliefs, as well as about pupils' difficulties, there was a significant diversity. By raising these issues about pedagogical content knowledge the prospective teachers' conceptions can be extended and developed during the education.     

数学活动经验及其对教学的影响

数学活动经验是数学学习的产物,从维度上分为数学思想方法、数学思维方法、数学活动过程中的体验;又表现为静态和动态两个层面。重视数学活动经验的教学意义在于:扩展学生的认知结构,提高教学设计的实效性,彰显个性化的学习,生成课程资源。其教学策略是:让学生亲历数学活动,增加交流的机会,开展反思与评价。     

Reflecting on the Use of Writing to Promote Mathematical Learning: An Examination of Preservice Mathematics Teachers' Perspectives

We examine preservice mathematics teachers' conceptions of writing as a tool for learning mathematics before and after participation in and reflection on writing tasks. We describe the use of two targeted activities incorporated into a secondary methods course: writing to learn mathematics (WTLM) and reflection on that writing. Prior to participation in these activities, the preservice teachers expressed reluctance toward the use of writing in mathematics and uncertainty as to how writing could be useful in mathematics, while accepting that some possible benefits might exist for students' procedural learning. Following participation in these activities, the preservice teachers expressed a willingness to accept writing as a useful tool for supporting an expanded view of teaching and learning mathematics. Specifically, the preservice teachers considered writing as a way to build connections between mathematics and other subjects, a means to assess student understanding of mathematics, and a beneficial support for student conceptual learning.     

Intercorporeality and ethical commitment: an activity perspective on classroom interaction

In this article, we present a sociocultural alternative to contemporary constructivist conceptions of classroom interaction. Drawing on the work of Vygotsky and Leont??ev, we introduce an approach that offers a new perspective through which to understand the specifically human forms of knowing that emerge when people engage in joint activity. To this end, we present two concepts: space of joint action and togethering. The space of joint action allows us to capture the collective and sensuous or intercorporeal dimensions of thought and feeling in interaction. We resort to the concept of togethering to capture the ethical commitment participants make to engage in and produce activity. These concepts are illustrated through a discussion of concrete episodes from an elementary mathematics classroom.