Mathematics Instruction in Safe Space: Prospective Elementary Teachers' Views of Mathematics Education

This article discusses findings from an investigation into 42 prospective elementary teachers' views of mathematics teaching and learning. The focus is on how the participants saw the roles of teachers, pupils,and mathematics in mathematics classes as they communicated their views in journals written during the study. I argue that some prospective teachers express a tendency to protect and relieve their future pupils from mathematics. I show how this tendency affects a shift from school being the place where teachers teach pupils toward a conception of a safe space where the teachers predominantly care for the children. This revised version was published online in September 2006 with corrections to the Cover Date.     

民族地区小学生数学学习态度调查研究

本研究以民族地区194名少数民族和汉族小学生为被试,使用问卷法对小学生数学学习态度的现状进行调查,结果表明：1．少数民族学生对“数学用处”和“得到教师的关注少”的认识比汉族学生低;2．四年级数学态度得分最高,五、六年级出现下降趋势;3．女生数学态度均分高于男生,但男女生的数学成绩没有差异;4．数学态度在“对教师的看法”上存在地域差异,市区学生得分优于郊区和山区学生;5．数学学习成绩好的,其数学态度得分较高。数学态度与数学学习成绩是双向影响的,这在民族地区的数学教学中应给予充分重视．     

Learning mathematics—letting the pupils have their say

Pupil voice is an emerging force for change and improvement in many UK schools, but what is not fully understood is how best to access pupil voice within the specific context of secondary mathematics departments. This paper presents a research project designed to use pupils as co-researchers in increasing knowledge about how to improve learning in mathematics. Pupils within the school were selected and trained as “Ambassadors” to understand and disseminate innovative ways of learning mathematics into their school environment and to act to allow the voice of all the pupils in their year group to be heard. The project was intended both to raise the pupils’ awareness of how learning mathematics could be different and to enable them to voice their newly informed opinions about how best they learned mathematics. The pupils’ current feelings about the way that they were taught mathematics were explored, but the focus of the project was on enabling the pupils to make informed decisions about how they felt their learning could be improved. The pupils’ awareness of different ways of learning mathematics was raised by introducing them to alternative teaching approaches. The data generated were initially analysed by the pupils themselves in order to inform their teachers about their views and subsequently constant comparison analysis resulted in the outcomes reported here. The outcomes indicate that the students could have an important role in enabling schools to develop their teaching and improve their pupils’ mathematical learning when that voice is both informed and authorised.     

跨年级小学生数学学力认知诊断测验的IRT垂直等值分析

本研究采用“共同题?锚测验”设计,使用R语言ltm程序包中的IRT两参数模型进行各年级小学生数学学力认知诊断测验和被试参数的估计,并使用equateIRT程序包进行跨年级小学生数学学力认知诊断测验各项参数的等值转换。结果表明,等值转换后各年级测验的题目难度和小学生数学学力均随年级增长而逐渐递增,不同学校、民族、性别学生的数学学力发展差异性特征均与理论假设相符。本研究验证了采用IRT垂直等值方法构建跨年级小学生数学学力发展水平垂直量表的可行性,为制定系统性补救教学方案和自适应题库建设提供了必要的实证证据。     

Learning biology and mathematics outdoors: effects and attitudes in a Swedish high school context

This research suggests that learning biology in an outdoor environment has a positive cognitive and affective impact on 13–15-year-old, Swedish high school pupils. Eighty-five pupils in four classes participated in a quasi-experimental design. Half the pupils, taking a biology course in ecology or diversity of life, had several lessons outdoors and the other half were taught indoors. All of the classes, but one, also had mathematics lessons outdoors once a week. Twenty-one pupils were interviewed five months after the course and all were positive towards the new learning environment they had experienced outdoors in biology and/or mathematics. They also valued the higher degree of interaction among the pupils. Other findings from the interviews were that the pupils from the outdoor classes showed a higher degree of long-term knowledge retention. They remembered both activities and contents better than the pupils in the indoor classes. An essay-type question assessing their biological understanding qualitatively according to the Structure of Observed Learning Outcome taxonomy revealed no differences between the groups. The results are discussed in the light of neurocognitive models of long-term memory.     

Contribution of mathematics in-service training course to the professional development of elementary school teachers in Israel

The present study aimed to explore the contribution of a mathematics in-service training course to elementary school teachers (1st–6th grades) in Israel. The study was conducted among 449 educators. They were required to respond to background questions. Moreover, they were asked to indicate their expectations from the in-service training course and, at its end, point out to what extent they benefitted from that course. The research findings illustrate that educators who teach mathematics at elementary school and attended the course are generally women in their 40s, holding a BEd degree and a teaching certificate not in mathematics, with an average 13-year seniority. The participating teachers indicated their wish to enrich their didactic knowledge in order to acquire varied tools for teaching mathematics to the entire pupil population as well as to gifted pupils and pupils with learning difficulties. Nevertheless, their demand to expand their mathematics knowledge was very limited. Based on the fact that most teachers have no mathematics education, this is a surprising finding as, in order to be a good teacher, one must be versed not only in Pedagogical Content Knowledge but also in Subject Matter Knowledge.     

Analysing Cognitive Learning Processes through Group Interviews of Successful High School Pupils: Development and Use of a Model

The focus is on the metacognitive awareness of ten high-achieving high school pupils in mathematics in Denmark and England and their understanding of their cognitive learning processes and strategies. Mainly unstructured focus group interviews investigate how they explain that they learn a mathematical concept that is new to them. I develop the ‘CULTIS model for analysis’ (Consciousness, Unconsciousness, Language, Tacit, Individual, Social), which consists of six themes in which various psychological theories of learning are expressed. The model uses the theories of Ernest, von Glasersfeld, Hadamard, Krutetskii, Mason, Piaget, Polya, Sfard, Skemp, and Vygotsky. The model is used to sort the pupils' explanations and it also functions as a tool with which to compare the pupils' explanations with the various theories. I conclude that the pupils can talk about their learning in their own words and each pupil refers to elements that the researcher associates with different theories. Half the pupils furthermore explain that how they are used to learning, or being taught, influences how they later on are able to learn. I also discuss metacognition and whether this can be an aid to enhance learning.     

Development of learning to learn skills in primary school

In Finland, schools’ effectiveness in fostering the development of transversal skills is evaluated through large-scale learning to learn (LTL) assessments. This article presents how LTL skills—general cognitive competences and learning-related motivational beliefs—develop during primary school and how they predict pupils’ CPS skills at the end of sixth grade. The six-year follow-up of 608 pupils shows that cognitive competences demonstrated in the beginning of the first grade in a learning preparedness test predict both later cognitive LTL competences and CPS, but their development is not fully determined by earlier individual differences in learning preparedness in the first grade. Motivational beliefs begin to be related to cognitive LTL performance gradually from age 10 on, and they may have a slightly stronger effect on CPS than on cognitive LTL performance. It is concluded that the development of CPS is partly depending on pupils’ initial learning preparedness and the development of their LTL skills.     

Academic continuity through online collaboration: mathematics teachers support the learning of pupils with chronic illness during school absence

Recent applications of technology to mathematics education have been designed with cognitive and constructivist theoretical perspectives in mind, viewing mathematical learning as the acquisition of knowledge through the construction of meanings and connections between concepts. With the advent of increasingly flexible communication technologies, there is both the need and opportunity to consider how they might be utilised, particularly since emergent socio-cultural theories advocate learning in mathematics as an inherently social activity where understanding is developed and negotiated collaboratively. The need to examine effective technology-facilitated learning arose in the context of a research project, currently underway in a number of secondary schools in the state of Victoria and funded by the Australian Research Council. It is investigating the learning needs of pupils who are absent from school for prolonged or intermittent periods owing to chronic illness yet continue with their school studies. An emerging understanding of the significant difference between computer-mediated contact for mere information exchange and communication for teaching and learning has led to a consideration of socio-cultural perspectives on effective mathematical learning and a focussed investigation of technologies able to facilitate them. Early data have demonstrated the potential of videoconferencing, online whiteboarding and interactive whiteboard application sharing, but which require particular resources, aligned infrastructure and teacher support. This article explores issues surrounding the use of such technologies for collaborative mathematical learning in a context where online interaction is being considered for the learning support of pupils unable to attend school.     

Supporting English second-language learners in disadvantaged contexts: learning approaches that promote success in mathematics

In the Australian context, children living in disadvantaged circumstances, whose second language is English, are one of the groups at risk of failing in mathematics. This paper explores the impact purposely developed learning activities (Representations, Oral Language and Engagement in Mathematics Learning activities) have on pupils' mathematics outcomes in the first three years of formal schooling (Foundation, Year 1 and Year 2). The participating group (n = 461) comprised 328 English language learners and 133 mainstream pupils. To ascertain the impact of the learning experience, pre- and post-tests were conducted at the commencement and completion of each school year. The results indicate that all of the children significantly improved, with English as a second language pupils showing the greatest gains, and achieving norm-referenced expectations for their age.     

Elementary school mathematics teachers’ perception of their professional needs

This study explored Israeli elementary school mathematics teachers’ perceptions of their professional needs, with the purpose of developing in-service training courses which cater to these needs. Eighty-four teachers responded to the questionnaire and were interviewed. The results indicate that the respondents’ main needs are associated with strengthening their didactical knowledge capability of dealing with emotional aspects that relate to pupils’ learning of mathematics. Although most of the respondents lack formal mathematics education, they attribute less importance to their need to enhance knowledge in the field. In light of Israeli pupils’ relatively low attainments in mathematics, the authors believe that the education system should require teachers to expand their mathematics knowledge and that only teachers with appropriate knowledge will be permitted to teach the discipline.     

The Equation, the Whole Equation and Nothing But the Equation! One Approach to the Teaching of Linear Equations

There exists an extensive range of research looking at the teaching and learning of linear equations, resulting in many papers highlighting a range of teaching approaches and illustrating a variety of significant cognitive problems and stumbling blocks to the learning of linear equations with understanding. Building on this literature, this paper presents some of the results of a case study which looked at the mathematics classroom of one particular teacher, Alwyn, trying to teach mathematics with meaning to less able pupils at secondary school level. Our interest here is those lessons which dealt specifically with the learning of linear equations, in which firstly a different approach was utilised and secondly many of the problems referred to in the literature were not present. We contrast this method with the teaching of linear equations to a variety of ability levels in several other classrooms that we have studied and we attempt through use of the Pirie-Kieren model, to analyse and account for the successful growth of understanding of the lower ability, year eight pupils in one particular classroom.     

Teachers’ Attitudes Towards Their Pupils’ Mathematical Errors

This article is mainly concerned with the didactical and the epistemological approach towards pupils’ errors in mathematics. The findings of an investigation into a representative sample of Cypriot teachers’ attitudes of errors in mathematics are presented. Although teachers draw on behaviourism as a way of understanding pupils’ errors, a considerable number of teachers seem to be influenced by both cognitive science error theory and the obstacle theory. Inferential statistical analysis revealed that teachers’ attitudes of errors are mainly associated with the attendance of a specialised INSET course in mathematics. Teachers who attended the INSET course in mathematics are generally more influenced by the cognitive science theory and the obstacle theory. Implications for the development of policy on teacher training are discussed. It is argued that with appropriate training teachers will be able to adopt a model of interpreting errors in mathematics based on the didactical and the epistemological approaches.     

Cognitive style and behaviour in secondary school pupils in Kuwait

The behaviours of 207 (92 male and 115 female) 15-18-year-old Kuwaiti secondary school pupils were assessed, together with a rating of their home educational background and cognitive style. The teachers of Arabic and of mathematics independently rated the behaviour of their pupils on a 31-item pupil behaviour questionnaire, and home educational background was rated by the Social Officer in each school. The position of pupils on the Wholist-Analytic and Verbal-Imagery dimensions of cognitive style was assessed by means of the Arabic version of the computer-presented Cognitive Styles Analysis . A factor analysis of the questionnaire items indicated four factors: conduct behaviour, learning behaviour, stability and physical well-being. With conduct and learning behaviour, the overall level was higher for females than males and also improved as parents' educational level increased. While conduct behaviour did not vary much with style, learning behaviour was significantly lower for the Analytics than the Wholists, and for the Imagers than the Verbalizers. For stability and physical well-being, there was a significant interaction between the educational level of parents and style in their effect on overall rating, with the greatest style effect when the parental educational level was low. The results are discussed in terms of their practical implications and the nature of the educational system within Kuwait.     

Teacher Talk and Pupil Thought

Classroom activities were observed and audiotaped in four schools for pupils with moderate learning difficulties. Teacher talk which encouraged reflection by pupils was identified and categorised. Characteristic teacher styles emerged which differed mainly in the extent to which they encouraged pupils to reflect, by justification, explanation and metacognition, for example. Teacher styles in classroom interaction were related to their views on the nature and source of their pupils’ learning difficulties and on the role of cognitive challenge in helping their school achievement.     

Contrasting Approaches to the Acquisition of Mathematical Skills: Japan and England

Abstract

International differences in mathematical attainment among secondary school age pupils are well documented, indicating that the performance of pupils in England lags behind that of many other countries: in particular, the average attainment of 13‐ and 18‐year‐old pupils in Japan is significantly higher than that of the corresponding cohort of pupils in England. The causes of the poor performance at secondary schools in England, it is argued, may be found partly in the inferior foundations of mathematical understanding at primary schools, aggravated by the ‘linearity’ of mathematical development and the cumulative effect of failure. Although cultural influences at the societal level and curriculum influences at the governmental level are largely beyond the control of the school, yet the other main influence on mathematical development, namely classroom practice, is determined to a great extent by the class teacher, and can be adapted to improve conceptual learning. By considering the current practice in the teaching of mathematics to six‐year‐olds in Japan within the context of available research evidence relating to effective teaching, and comparing this with the practice in England, areas of possible change can be identified which are largely within the control of the individual class teacher and through which standards of attainment might be improved. This article is based on observations of mathematics teaching to six‐year‐old children in Japan and England in 1995, in the state or public sector of education.     

Teachers’ organisational practices and their perceptions of the benefits of support by withdrawal for mathematics in Irish primary schools

This study was designed to ascertain the organisational practices of learning support teachers and their perceptions of the benefits, if any, of support by withdrawal for mathematics in Irish primary schools. The study reports on the views of a sample of 137 teachers who have postgraduate qualifications in learning support/special education from six designated centres for professional development in this area. As an organisational model for learning support in mathematics, small group/individual withdrawal was the most popular way in which additional support was organised, with 94% of the survey respondents in this study operating this system to some degree. Furthermore, key findings also highlighted that teachers perceived the following as advantages to out-of-class support: learning benefits; enhanced learning space; benefits for certain types of pupils; time benefits; positive contrast with mainstream class; assessment benefits; and greater use of concrete materials. In contrast, only 41% of teachers reported that they provide an in-class service for pupils and class teachers. Barriers to collaboration for in-class support are identified and teachers’ responses to these outlined. The study found large differences between teachers in the amount of support they provided to class teachers, the amount of non-contact time they had for planning and collaboration, and in the range of practices used to carve out time for collaboration. The implications of these findings are then discussed.     

小学生数学自主学习策略的教学

课堂教学是培养小学生数学自主学习能力的重要路径。就自主学习的认知策略而言,教学要遵循选择合适的认知策略、联系小学生的生活实际、循序渐进等原则;而就自主学习的元认知策略的教学而言,则可以采用思维外显化、程序化训练、系统训练等方法。     

试论小学数学几何图形概念教学的策略

在小学数学几何图形教学中,几何图形作为小学数学教材中重要的学习内容之一,对学生来说是有一定的学习难度的。教师要在实际教学中,针对小学生的个性化发展以及性格特征,合理地制定教学方案,并促使学生掌握正确的学习方式,使学生对几何图形有所认知。因此,本文对试论小学数学几何图形概念教学的策略进行分析研究,以期能够提升教师的教学质量,帮助小学生掌握更多的几何图形知识。     

Implication de l'analyse de productions d'élèves dans la formation des maîtres du primaire

Several studies have shown that pre-service teachers enter their pre-service training with already formed conceptions pertaining to teaching and learning. Pre-service teachers are considered as epistemologically complex beings with varying postures they adopt according to differing contexts. This article looks at the meanings pre-service teachers attribute to pupil errors and how these meanings evolved over the course of a 15 week course in mathematics education. Pre-service teachers were interviewed at two intervals during their first semester: at mid session and at the end. First, we identified the various kinds of analyses they brought to bear upon pupil errors as well as the interventions they suggested to rectify these errors. Then, we conducted a statistical analysis of the changes in the type of interventions suggested. Four trend indicators were created to help understand these changes. When analysing pupil's errors, pre-service teachers identified specific conceptual elements without situating them within a global context. This led to piecemeal interventions about how pupils ought to correct their errors. In conclusion, pre-service teachers' appear to base their analyses of pupils's errors upon an epistemological positioning ween title ‘former-pupil stance‘. This stance seems to create a resistance to changing their conceptions regarding the teaching and learning of mathematics. This revised version was published online in July 2006 with corrections to the Cover Date.