Distinguishing between mathematics classrooms in Australia, China, Japan, Korea and the USA through the lens of the distribution of responsibility for knowledge generation: public oral interactivity and mathematical orality

The research reported in this paper examined spoken mathematics in particular well-taught classrooms in Australia, China (both Shanghai and Hong Kong), Japan, Korea and the USA from the perspective of the distribution of responsibility for knowledge generation in order to identify similarities and differences in classroom practice and the implicit pedagogical principles that underlie those practices. The methodology of the Learner’s Perspective Study documented the voicing of mathematical ideas in public discussion and in teacher–student conversations and the relative priority accorded by different teachers to student oral contributions to classroom activity. Significant differences were identified among the classrooms studied, challenging simplistic characterisations of ‘the Asian classroom’ as enacting a single pedagogy, and suggesting that, irrespective of cultural similarities, local pedagogies reflect very different assumptions about learning and instruction. We have employed spoken mathematical terms as a form of surrogate variable, possibly indicative of the location of the agency for knowledge generation in the various classrooms studied (but also of interest in itself). The analysis distinguished one classroom from another on the basis of “public oral interactivity” (the number of utterances in whole class and teacher–student interactions in each lesson) and “mathematical orality” (the frequency of occurrence of key mathematical terms in each lesson). Classrooms characterized by high public oral interactivity were not necessarily sites of high mathematical orality. In particular, the results suggest that one characteristic that might be identified with a national norm of practice could be the level of mathematical orality: relatively high mathematical orality characterising the mathematics classes in Shanghai with some consistency, while lessons studied in Seoul and Hong Kong consistently involved much less frequent spoken mathematical terms. The relative contributions of teacher and students to this spoken mathematics provided an indication of how the responsibility for knowledge generation was shared between teacher and student in those classrooms. Specific analysis of the patterns of interaction by which key mathematical terms were introduced or solicited revealed significant differences. It is suggested that the empirical investigation of mathematical orality and its likely connection to the distribution of the responsibility for knowledge generation and to student learning ourcomes are central to the development of any theory of mathematics instruction and learning.     

Landscapes of Investigation

According to many observations, traditional mathematics education falls within the exercise paradigm. This paradigm is contrasted with landscapes of investigation serving as invitations for students to be involved in processes of exploration and explanation. The distinction between the exercise paradigm and landscapes of investigation is combined with a distinction between three different types of reference which might provide mathematical concepts and classroom activities with meaning: references to mathematics; references to a semi-reality, and references to a real-life situation. The six possible learning milieus are illustrated by examples. Moving away from the exercise paradigm and in the direction of landscapes of investigation may help to abandon the authorities of the traditional mathematics classroom and to make students the acting subjects in their learning processes. Moving away from reference to pure mathematics and in the direction of real life references may help to provide resources for reflection on mathematics and its applications. My hope is that finding a route among the different milieus of learning may provide new resources for making the students both acting and reflecting and in this way providing mathematics education with a critical dimension.     

如何提高高等数学课堂教学的质量

数学的课堂教学过程是在教师的传授和指导下进行学习、掌握数学知识、技能、思想、方法的一种认识过程.影响数学课堂教学质量的因素是多方面的,本文仅就提高高等数学课堂教学质量谈几点个人的认识.     

Representation in Computational Environments: Epistemological and Social Distance

Computational environments have the potential to provide new representational resources and new ways of supporting teaching and learning of mathematics. In this paper, we seek to characterize relationships between the representations offered by particular technologies and other representations commonly available in the classroom context, using the notion of ‘distance’. Distance between representations in different media may be epistemological, affecting the nature of the mathematical concepts available to students, or may be social, affecting pedagogic relationships in the classroom and the ease with which the technology may be adopted in particular classroom or national contexts. We illustrate these notions through examples taken from cross-experimentation of computational environments in national contexts different from those in which they were developed. Implications for the design and dissemination of computational environments for use in learning mathematics are discussed.     

计算机及数学软件在高职数学教学中的应用

阐述了计算机与数学软件运用于高等职业教育数学教学中的设想与实践探索．把数学实验作为传统教学、计算机和数学软件相结合的一种实践,贯穿于日常的教学过程,并列举了对应的课堂教学具体实例加以展示．最后指出,高职数学教学中应该加强这种教学模式的设计和开发．     

Conceptions of Turkish mathematics teachers about the effectiveness of classroom teaching

This study investigated how Turkish mathematics teachers evaluate the effectiveness of classroom teaching in terms of improving students’ mathematical proficiency. To this purpose, teachers were asked to evaluate a mathematics lesson as presented them in a vignette. By means of cluster analysis, the participants’ evaluations of the lesson were described in five thematic dimensions, which could be further assembled into two overriding categories: students’ understanding of the subject, and teachers’ classroom practices. The overall aim of the current paper is to propose a preliminary model of the framework that Turkish mathematics teachers use to evaluate a mathematics lesson.     

Humans-with-media and continuing education for mathematics teachers in online environments

This paper begins by situating online mathematics education in Brazil within the context of research on digital technology over the past 25?years. I argue that Brazilian research on technology in mathematics education can be divided into four phases, and then present an example that ??blends?? aspects of the second and third phases. Phase two can be characterized by research with software designed to address traditional mathematics topics, such as functions, while the third phase is characterized by online courses. The data presented show creative solutions for a problem designed for collectives of humans-with-function-software. The paper is analyzed from a perspective that emphasizes the role of different technologies as teachers and professors collaborate to produce knowledge about the use of mathematical software in regular face-to-face classrooms. A model of online education is presented. Finally, the paper discusses how technology may change collaboration and teaching approaches in continuing education, as it allows for greater integration of online learning with teachers?? classroom activities in schools. In this case, the online platform plays an active role in the learning collective composed of humans-with-media.     

Justification as a teaching and learning practice: Its (potential) multifacted role in middle grades mathematics classrooms

Justification is a core mathematics practice. Although the purposes of justification in the mathematician community have been studied extensively, we know relatively little about its role in K-12 classrooms. This paper documents the range of purposes identified by 12 middle grades teachers who were working actively to incorporate justification into their classrooms and compares this set of purposes with those documented in the research mathematician community. Results indicate that the teachers viewed justification as a powerful practice to accomplish a range of valued classroom teaching and learning functions. Some of these purposes overlapped with the purposes in the mathematician community; others were unique to the classroom community. Perhaps surprisingly, absent was the role of justification in verifying mathematical results. An analysis of the relationship between the purposes documented in the mathematics classroom community and the research mathematician community highlights how these differences may reflect the distinct goals and professional activities of the two communities. Implications for mathematics education and teacher development are discussed.     

Mimicry of proofs with computers: the case of Linear Algebra

In common teaching practice the habit of connecting mathematics classroom activities with reality is still substantially delegated to wor(l)d problems. During recent decades, a growing body of empirical research has documented that the practice of word problem solving in school mathematics does not match this idea of mathematical modelling and mathematization. If we wish to construct ‘real problems arising from real experiences of the child’ following the spirit of these new suggestions, we have to make changes. On the one hand we have to replace the type of activity in which we delegate the process of creating an interplay between reality and mathematics by substituting the word problems with an activity of realistic mathematical modelling, i.e. of both real-world based and quantitatively constrained sense-making; and, on the other hand, to ask for a change in teacher beliefs; furthermore, a directed effort to change the classroom socio-math norms will be needed. This paper discusses some classroom activities that takes these factors into account.     

Mathematics in Children's Thinking

The human mind inevitably comprehends the world in mathematical terms (among others). Children's informal and invented mathematics contains on an implicit level many of the mathematical ideas that teachers want to promote on a formal and explicit level. These ideas may be innate, constructed for the purpose of adaptation, or picked up from an environment that is rich in mathematical structure, regardless of culture. Teachers should attempt to uncover the mathematical ideas contained in their students' thinking because much, but not all, of the mathematics curriculum is immanent in children's informal and invented knowledge. This mathematical perspective requires a focus not only on the child's constructive process but also on the mathematical content underlying the child's thinking. Teachers then can use these crude ideas as a foundation on which to construct a significant portion of classroom pedagogy. In doing this, teachers should recognize that children's invented strategies are not an end in themselves. Instead, the ultimate goal is to facilitate children's progressive mathematization of their immanent ideas. Children need to understand mathematics in deep, formal, and conventional ways.     

Learning and Assessing Mathematics through Reading and Writing

Students learn mathematics more effectively and more deeply when reading and writing is directed at learning mathematics. Although reading and writing in mathematics may necessitate more skills and practice to master, the mathematical learning derived from reading and writing mathematics far outweighs the burden it places on teachers and students. This paper explores the vital role of purposeful reading and writing in the mathematics classroom and outlines some techniques to promote successful integration of reading and writing in mathematics.     

Proof validation and modification in secondary school geometry

Proof validation is important in school mathematics because it can provide a basis upon which to critique mathematical arguments. While there has been some previous research on proof validation, the need for studies with school students is pressing. For this paper, we focus on proof validation and modification during secondary school geometry. For that purpose, we employ Lakatos’ notion of local counterexample that rejects a specific step in a proof. By using Toulmin’s framework to analyze data from a task-based questionnaire completed by 32 ninth-grade students in a class in Japan, we identify what attempts the students made in producing local counterexamples to their proofs and modifying their proofs to deal with local counterexamples. We found that student difficulties related to producing diagrams that satisfied the condition of the set proof problem and to generating acceptable warrants for claims. The classroom use of tasks that entail student discovery of local counterexamples may help to improve students’ learning of proof and proving.     

An experiment to discover mathematical talent in a primary school in Kampong Air

It is often said that many pupils have hidden talent in mathematics. This hidden ability is rarely seen in a normal classroom teaching and learning situation if the focus of the teacher is on drilling with routine exercises. To allow pupils to display their mathematical talent and to break from mental set and fixation in mathematics, they must be given opportunity to think by themselves with mininum cue or guidance. The pupils could be left entirely on their own to show their mathematical creativity even on mathematical topics which have not been exposed to them. With this approach, five non-routine questions were administered one at a time to a standard 5 class. One out of the 25 pupils in the class consistently exhibited mathematical creativity and talent is answering the questions. Her responses were shown and discussed in this paper.     

Cultural distance and identities-in-construction within the multicultural mathematics classroom

In this paper we present and exemplify our interpretation of some theoretical constructs that have proved useful to our understanding of the complexity of multicultural mathematics classrooms. Constructs such as culture, cultural distance, cultural conflict and identities-in-construction have oriented our study of the complexity of highly multicultural mathematics classrooms in Barcelona. The purpose of this paper is to discuss how cultural distance arising from the different meanings that students, being local or immigrant, inevitably bring to the mathematics classroom may turn into cultural conflicts when cultural interaction is not facilitated through classroom discourse. The lack of cultural interaction and communication may give rise to strong negative feedings and refusal to participate on the side of the students. Students' nonparticipation can be understood as an active response to cultural distance and negative opinions in order to safeguard the identities they (wish to) construct within a context that they perceive as hostile.     

Students’ ratings of teacher practices

In this paper, we explore a novel approach for assessing the impact of a professional development programme on classroom practice of in-service middle school mathematics teachers. The particular focus of this study is the assessment of the impact on teachers’ employment of strategies used in the classroom to foster the mathematical habits of mind and mathematical self-efficacy of their students. We describe the creation and testing of a student survey designed to assess teacher classroom practice based primarily on students’ ratings of teacher practices.     

Instances of mathematical thinking among low attaining students in an ordinary secondary classroom

This paper is a report of a classroom research project whose aim was to find out whether low attaining 14-year-old students of mathematics would be able to think mathematically at a level higher than recall and reproduction during their ordinary classroom mathematics activities. Analysis of classroom interactive episodes revealed many instances of mathematical thinking of a kind which was not normally exploited, required or expected in their classes. Five episodes are described, comparing the students’ thinking to that usually described as “advanced.” In particular, some episodes suggest the power of a type of prompt which can be generalized as “going across the grain.”     

The Development of Mathematical Discussions: An Investigation in a Fifth-Grade Classroom

Promoting discussion and argumentation of mathematical ideas among students are aspects of the vision for communication in recent school mathematics reform efforts. Having rich mathematical discussions, however, can present a variety of classroom challenges. Many factors influence classroom discussions and need to be addressed in ways that will assist teachers in creating more inquiry-based mathematics classrooms. The study presented here examined the development of mathematical discussions in a fifth-grade classroom over the course of a school year. Various aspects of the participants' interactions, teacher's pedagogy, and the classroom microculture were investigated. One major result is the evolution of student participation from nonactive listening to active listening and use of others' ideas to develop new conjectures. These changes were paralleled by changes in the teacher's role in the classroom and the nature of her questions, in particular.     

The Development of Mathematical Discussions: An Investigation in a Fifth-Grade Classroom

Promoting discussion and argumentation of mathematical ideas among students are aspects of the vision for communication in recent school mathematics reform efforts. Having rich mathematical discussions, however, can present a variety of classroom challenges. Many factors influence classroom discussions and need to be addressed in ways that will assist teachers in creating more inquiry-based mathematics classrooms. The study presented here examined the development of mathematical discussions in a fifth-grade classroom over the course of a school year. Various aspects of the participants' interactions, teacher's pedagogy, and the classroom microculture were investigated. One major result is the evolution of student participation from nonactive listening to active listening and use of others' ideas to develop new conjectures. These changes were paralleled by changes in the teacher's role in the classroom and the nature of her questions, in particular.     

FROM ONE LANGUAGE TO ANOTHER: A SEMIOTIC INTERPRETATION OF THE TRANSLATION OF MATHEMATICAL WORDS

The teaching and learning of Primary school mathematics in Malta involves the use of code-switching between the local language Maltese, and English Mathematical terms themselves are usually retained in English and teachers may use various strategies to share the meaning of these words with their pupils. One strategy that may be used in a bilingual situation is translation from one language to another. In this paper I explore how a teacher used this strategy to teach her 7 to 8-year-old pupils mathematical vocabulary related to the topic'Money and Shopping'. While Maltese equivalents for these words exist, it is the English versions that form part of the school mathematics register. I develop a semiotic model where a mathematical word is considered to be a sign, and the process of translation is viewed as a chain of signification from one language to another.     

Capturing the S in segregation: a simple model of flowing granular mixtures in rotating drums

Flowing granular mixtures in rotating cylindrical drums arise in numerous industrial settings and are of great technological significance worldwide. To date, the development of a robust mathematical model for this process remains an open research problem. However, simple mathematical models may be developed that capture some of the underlying mechanisms, including segregation. The key lies in the analysis of the flowing surface layer whose profile is characteristically S-shaped: past models have shown that capturing this characteristic is an essential ingredient in predicting segregation. Using secondary to first-year undergraduate level mathematics, an analysis is presented of the flowing surface layer, which leads to an S-shaped profile. This problem presents an ideal case study for teaching applied mathematics and modelling. Interesting simulations that aid the visualization of this problem can be downloaded from the web, and simple classroom demonstrations of this process can be assembled with no specialized equipment.