A Learning Model That Stimulates Geometric Thinking Through Use of PCLogo and Geometer's Sketchpad

Summary

PCLogo and Geometer's Sketchpad are powerful tools that may be used by mathematics teachers who want to integrate technology with geometry instruction in the elementary classroom. The purpose of this study was to examine the usefulness of PCLogo and Geometer's Sketchpad to stimulate thinking about geometric concepts in elementary age children. We used a collective case study design that included four cases, two girls (ages 8 and 10) and two boys (ages 10 and 11). All participants were trained to use PCLogo and Geometer's Sketchpad to construct geometric shapes and measure the attributes of the shapes. After the training, participants used these technologies as tools to stimulate thinking about geometric concepts. As a result of our observations of children's performances as they reasoned about geometric concepts, we developed a learning model for teaching children about geometry.     

Assessment from a Sociocultural Perspective: Narratives from a first grade classroom

This study focuses on observations of classroom conversation as an approach to assessment of relationships between a teacher's teaching and pupils' learning and identity‐development processes. Detailed observation notes from two conventional conversation situations from a first grade classroom are written down as narratives and analysed within a sociocultural theoretical framework. Three significant themes emerge: (1) How the teacher sees the pupils, (2) How she connects with them, and (3) How she wanders on together with them. Together these themes function as strong “process motors” demonstrating how a teacher's support and scaffolding add positive influences to children's learning and personal growth. The study concludes that writing down information from classroom observations in a narrative genre manufactures excellent opportunities for revealing, describing, interpreting, and evaluating significant relationships between a teacher's teaching and pupils' learning and identity development.     

K-12 Mathematics and the Web

Abstract

The Web offers numerous learning resources and opportunities for K-12 mathematics education. This paper discusses those resources and opportunities. Discussion includes (a) asynchronous and synchronous communication tools, (b) the use of data sets to make connections between mathematics concepts and real-world applications, and (c) interactive environments that promote active thinking by allowing students to manipulate mathematical systems, observe patterns, form conjectures, and validate findings.     

Coteaching in early childhood clinical field experiences: a cross-case study of learning affordances

ABSTRACT

Coteaching, a model for learning to teach, places teacher candidates alongside clinical educators in classrooms. Learning occurs through shared practice and on-going explication of thinking and reflection. This cross-case study of six dyads from an undergraduate early childhood inclusive preservice education program examines ways that coteaching afforded opportunities for developing collaborative and adaptive expertise. It was found that opportunities for learning these skills were afforded through coteaching student teaching experiences, although in different ways, and that limited affordances outnumbered strong learning opportunities. Implications for further development of the early childhood coteaching teacher education model, and for professional development, are discussed.     

An Account of a Teacher's Perspective on Learning and Teaching Mathematics: Implications for Teacher Development

This report presents an account of one teacher's mathematics teaching and a perspective that underlies his teaching. Nevil was a fifth grade teacher participating incurrent mathematics education reforms in the United States. Through the account, we make distinctions about teachers' thinking and practice that can inform teacher education efforts. We constructed an account by analyzing four sets of classroom observations and interviews. We observed that Nevil decomposed his understandings of the mathematics into smaller components and connections among those components. He created situations that he believed made those components and connections transparent and attempted to elicit those connections from the students. This account illustrates a practice that is different both from traditional practice and the type of practice that we would envision as a goal for teacher development. We contribute two important aspects of mathematics teacher development from traditional to reform-oriented teaching. In particular, we describe teachers' perspectives – assimilatory structures that constrain and afford (a) the sense they make of professional development opportunities and (b) their potential learning in teacher education settings. This revised version was published online in August 2006 with corrections to the Cover Date.     

Research Practice into/influencing Mathematics Teaching and Learning Development: Towards a Theoretical framework based on co-learning partnerships

This paper addresses issues linking research into the classroom teaching and learning of mathematics with the growth of knowledge in mathematics teaching, developments in the practice of teaching and the enhanced learning of mathematics by students in classrooms. A basic premise is that research promotes development. The paper considers both insider andoutsider research and co-learning between teachers and educators in promoting classroom inquiry. Through a consideration of elements of theory such as knowledge and inquiry in teaching and of learning as knowledge growth through research/inquiry leading to enhancement of students' learning of mathematics, a framework is suggested. Its purposes include analysis of a research project's contribution to teaching development and conceptualization of research which has teaching development as one of its aims. Use of the framework is exemplified through its application to reports of three mathematics education research projects in the public domain. A brief afterword links the framework to concepts in activity theory.     

A Model for Staff Development in Information Technology within Initial Teacher Training Institutions

ABSTRACT

This paper reflects on the uptake of information technology (IT) as a delivery strategy in the learning and teaching processes and suggests that it is no longer satisfactory from the pupils' perspective to leave this to informal staff development strategies. The ability to use IT appropriately and effectively in the classroom must be thoroughly embedded in the pedagogy of teacher training institutions. For this to happen, these institutions need to adopt interventionist strategies for staff development so that learning through IT is a process that all student teachers experience, evaluate and learn to manage in the classroom situation. This paper examines a project for staff development in IT in a particular college and, following on the project's success, offers it as a possible model for successful staff development in IT.     

Communicating to Learn Multivariable Calculus: Students' Blackboard Presentations as a Means for Enhancing Mathematics Learning

This paper reports observations and reflections from a pedagogic intervention intended to enhance mathematics students' learning of course content in a course in multivariable calculus. The intervention design required students to structure, practice, and deliver an oral presentation that was a synthesis of the teacher's weekly lectures, for the benefit of themselves and their peers. The intervention appears to have incentivized students to engage with and learn mathematics contents in an effective way, and it also helped the mathematics teacher to reflect on, adapt, and develop his teaching strategies. To conclude, it would seem that both students and teachers in mathematics have much to gain from giving students opportunities to develop their own voice in learning contexts otherwise relying on books and lectures.     

Pedagogical Content Knowledge: How do student teachers identify and describe the causes of their pupils' learning difficulties?

This study examines the pedagogical content knowledge of student teachers of biology during their teaching practice in school. The research focuses on the assessment of the teachers' ability to identify their pupils' learning difficulties and characterise their presumed sources. Diaries, kept by 40 student teachers in the course of the two stages of their teaching practice (comprising observation of their mentors in action and their own teaching experience), provided the data for a qualitative and quantitative analysis of the findings. A difference between the two stages was found only with regard to the identification of difficulties: the student teachers identified learning difficulties in most of the lessons they observed, but only in half of the lessons they taught. Their characterisation of the sources of the pupils' difficulties and their recurrence were similar during both stages. The sources of the difficulties were defined according to four categories: the pupil's cognitive and affective characteristics, the type of content, the teacher's methods, and factors inherent in the lesson. The characteristics of the pupil were considered the most frequent source of the difficulties. The study points to the need to increase the teacher educators' awareness of the important role of didactic processes, aimed at exposing the student teachers to their pupils' learning difficulties and help them deal with them effectively.     

Are small classes better? Understanding relationships between class size,classroom processes and pupils' learning

Twelve years ago Blatchford and Mortimore's authoritative review of class size research appeared in this journal. They concluded that a major problem with class size research was the lack of detailed studies of complex classroom processes that might mediate class size effects on pupils' learning. This article reviews two UK class size reviews and quantitative, qualitative and mixed method class size research. Evidence from research, and insights from 30 years of classroom‐based inquiry, form the basis for the development of theoretical models of relationships between class size, classroom processes and pupils' learning. Recent research evidence from secondary school classrooms calls into question simple one‐way relationships between class size and pupils' learning. Politicians are challenged to face up to the complexities involved and to be open to more flexible approaches to reforming the organisation of teaching and learning in schools that go beyond expensive programmes of crude across‐the‐board class size reductions. Further class size research is recommended that incorporates sophisticated qualitative methods in order to adequately understand and represent the kinds of teacher and pupil expertise involved in promoting and maximising opportunities for high quality learning in different large and small class contexts in primary and secondary schools.     

Facilitating metacognitive talk: a research and learning tool

This paper describes a research tool which aims to gather data about pupils' views of learning and teaching, with a particular focus on their thinking about their learning (metacognition). The approach has proved to be an adaptable and effective technique to examine different learning contexts from the pupils' perspective, while also acting as an aid to reflective dialogue between pupils and teachers as part of the teaching and learning process. A range of templates have been created as psychological or semiotic tools. They form the basis of a mediated interview by providing an image of the learning environment or activity on which the research is focused. The image then becomes the stimulus for a three‐way interaction between the researcher (or the teacher), the pupil and the template. This paper provides examples from a number of research projects where the technique has been used to gather data in classrooms.     

Teachers’ innovative change within countrywide reform: a case study in Rwanda

This article presents practical perspectives on mathematics teacher change through results of collaborative research with two mathematics secondary school teachers in order to improve the teaching and learning of mathematics in Rwanda. The 2006 national mathematics curriculum reform stresses pedagogies that enhance problem-solving, critical thinking and argumentation. Teachers need to use new teaching strategies. This article is a case study looking at issues around developing teachers’ use of interactions in mathematics classrooms independently of the national programme. Outputs of the study include teachers’ awareness of the need for change and their increased flexibility to accept learners’ autonomy in shifting from teacher-centred to learner-centred pedagogy. Geometer’s Sketchpad challenged teachers’ practice and then provoked reflection to improve student learning.     

Computer Programming

Summary

Computers can be powerful aids to mathematics teaching and learning. Changes brought about by the availability of these tools and the demands of an increasingly technological society impact curricular content and pedagogy in mathematics education as well as the very nature of mathematical thinking and understanding. This article presents ways in which technology is changing mathematics education, guidelines for appropriate technology use in the mathematics classroom, the impact of computers on mathematics learning, common uses of computers in mathematics education, and issues and concerns related to technology use in mathematics.     

Critical Conversations on Social Justice in Undergraduate Mathematics

Abstract

This article explores how critical conversations engage undergraduate mathematics faculty in a community of practice that enhances their knowledge about teaching and learning mathematics for social justice. More broadly, critical conversations are defined as a cooperative learning strategy that can be used to identify, explore, and respond to various interests and issues situated across differing values and beliefs. We present a case study of a critical conversation that took place at a 2016 Mathematics for Social Justice workshop organized by a group of junior faculty. Participant reflections situate perspectives that can help novice and experienced instructors design conversations about teaching mathematics for social justice. Specifically, individual and group reflections highlight the importance of: (i) framing and reflecting on the conversation; (ii) exploring implications and content connections; and (iii) identifying barriers. Implications for faculty members and mathematics departments are provided.     

The role of a children's mathematical thinking experience in the preparation of prospective elementary school teachers

Historically, content preparation and pedagogical preparation of teachers in California have been separated. Recently, in integrating these areas, many mathematics methodology instructors have incorporated children's thinking into their courses, which are generally offered late in students’ undergraduate studies. We have implemented and studied a model for integrating mathematical content and children's mathematical thinking earlier, so that prospective elementary school teachers (PSTs) engage with children's mathematical thinking while enrolled in their first mathematics course. PSTs’ work with children in the Children's Mathematical Thinking Experience (CMTE) may enhance their mathematical learning. Preliminary study results indicate that the sophistication of CMTE students’ beliefs about mathematics, teaching, and learning increased more than the sophistication of beliefs held by students enrolled in a reform-oriented early field experience and that experiences considering children's mathematical thinking provided PSTs with increased motivation for learning mathematics.     

Theory,Practice and Reflection: A pre-service primary mathematics education programme

This paper describes a case study investigating the impact of a primary mathematics method programme on pre-service primary teachers' capacity, and willingness to learn and teach mathematics for relational understanding. The programme, a collaborative endeavour between primary schools and university tutors, enables student teachers to practice in an on-going, integrated and consistent manner what they are learning in theory, and to reflect on their experiences. The findings from test, questionnaire and interview data indicate that the weekly cycle of theory, practice and reflection in which student teachers engage enhances their capacity, and provides an incentive for them to learn and teach mathematics for relational understanding.     

Learning Experientially through Case Studies? A Conceptual Analysis

In this conceptual article three points are made. First, reasons are offered for why experiential learning approaches are often implemented incompletely in higher education teaching. Secondly, it is argued that the case studies approach to teaching in higher education, if properly facilitated, is an effective way to provide students with the opportunity to become involved in all four phases of Kolb's experiential learning cycle. Thirdly, drawing on previous studies, it is proposed that experiential learning is likely to foster students' learning on a higher-order level, such as their critical thinking ability and propensity for self-direction in learning. The article concludes with suggestions for research to be done by those who teach in institutions of higher education to corroborate the arguments presented.     

Teachers' Interpretations of Effective Whole-class Interactive Teaching in Secondary Mathematics Classrooms

Prompted by ongoing concerns over standards of numeracy, eight mathematics teachers from four secondary schools in south Wales formed a teacher inquiry group to research the impact of introducing whole-class interactive teaching strategies into their own practice. Possible teaching approaches and activities were developed and agreed during group meetings. These were trialled by the teachers during their normal mathematics lessons. Lesson observations, interviews with teachers and discussions at the group meetings, however, indicated that the implementation of the approaches varied significantly across teachers. This paper analyses the variation in interpretations and teaching styles, and discusses their impact on the quality of the classroom discourse. The determining features discerned within the classroom dialogue included the extent to which pupils were encouraged to reflect on their mathematical knowledge, and the ways in which teachers were able to scaffold pupils' learning.     

The Information Challenges Approach to Teaching Information Management

This article describes one approach to teaching Information Management (IM) at honours degree level that the authors helped develop and deliver at Napier University Business School, namely 'Information Challenges' . Information Challenges provides a flexible framework upon which to teach the diverse and rapidly changing area of IM, as well as learning about current thinking and practice from leading academics and practitioners from industry and commerce. This article identifies the main areas of study within the field of IM and explores the main teaching and learning methods that were adopted and implemented within the Information Challenges approach. The main lessons learnt from developing and applying the Information Challenges approach will be highlighted, along with a number of issues that may emerge in the future.     

Critical thinking and the disciplines reconsidered

This paper argues that Moore's specifist defence of critical thinking as ‘diverse modes of thought in the disciplines’, which appeared in Higher Education Research & Development, 30(3), 2011, is flawed as it entrenches relativist attitudes toward the important skill of critical thinking. The paper outlines the critical thinking debate, distinguishes between ‘top-down’, ‘bottom-up’ and ‘relativist’ approaches and locates Moore's account therein. It uses examples from one discipline-specific area, namely, the discipline of Literature, to show that the generalist approach to critical thinking does not ‘leave something out’ and outlines why teaching ‘generic’ critical thinking skills is central to tertiary education, teaching and learning, and employment opportunities for students. The paper also defends the assessment of critical thinking skills.