Mathematics engagement in an Australian lower secondary school

The importance of actively engaging in mathematics discourse in order to learn mathematics is well recognized. In this paper, I use Basil Bernstein’s concepts of pedagogic discourse to document and analyse academic learning time of students in Years 8 and 9 at a suburban lower secondary school: in particular, for what proportion of class time students reported being academically engaged, their explanations for this engagement and how they felt about the discourse. It was found that many students had disengaged from mathematical endeavour as a result of the failure of the instructional discourse either to engage students or to serve the purpose of developing discipline-specific content knowledge. The reasons for this relate to the overemphasis on mundane mathematics resulting in some students lacking the cognitive tools to engage with the concepts and having neither the intrinsic nor instrumental motivation to persist with secondary school esoteric mathematics. The implications for mathematics curriculum development are discussed.     

Teachers' beliefs about school mathematics and mathematicians' mathematics and their relationship to practice

There is broad acceptance that mathematics teachers’ beliefs about the nature of mathematics influence the ways in which they teach the subject. It is also recognised that mathematics as practised in typical school classrooms is different from the mathematical activity of mathematicians. This paper presents case studies of two secondary mathematics teachers, one experienced and the other relatively new to teaching, and considers their beliefs about the nature of mathematics, as a discipline and as a school subject. Possible origins and future developments of the structures of their belief systems are discussed along with implications of such structures for their practice. It is suggested that beliefs about mathematics can usefully be considered in terms of a matrix that accommodates the possibility of differing views of school mathematics and the discipline.     

Do Advanced Mathematics Skills Predict Success in Biology and Chemistry Degrees?

The mathematical preparedness of science undergraduates has been a subject of debate for some time. This paper investigates the relationship between school mathematics attainment and degree outcomes in biology and chemistry across England, a much larger scale of analysis than has hitherto been reported in the literature. A unique dataset which links the National Pupil Database for England (NPD) and Higher Education Statistics Agency (HESA) data is used to track the educational trajectories of a national cohort of 16-year olds through their school and degree programmes. Multilevel regression models indicate that students who completed advanced mathematics qualifications prior to their university study of biology and chemistry were no more likely to attain the best degree outcomes than those without advanced mathematics. The models do, however, suggest that success in advanced chemistry at school predicts outcomes in undergraduate biology and vice versa. There are important social background differences and the impact of the university attended is considerable. We discuss a range of possible explanations of these findings.     

Instructing First-Grade Children on Patterning Improves Reading and Mathematics

Research Findings: In each of 16 public school classrooms serving multiethnic low-income neighborhoods, 2 first graders were assigned to be taught patterning, 2 to be taught reading, 2 to be taught mathematics, and 2 to be taught social studies for 15-min sessions 3 days per week for 6 months. Assignment within each classroom was randomized. The children's mean age was 6 years, 5.19 months. Patterning instruction included instruction on symmetrical patterns, patterns with increasing numbers of elements, and patterns involving the rotation of an object through 6 or 8 positions. In May, the 120 children still available were tested on patterns, reading, and mathematics. Patterning instruction had large, fully mediated effects on both reading and mathematics. There were no significant differences on some individual scales, but on others children who received patterning instruction scored best, often by grade equivalents of 4 to 8 months. Extant explanations of the efficacy of patterning instruction are reviewed, and the potential importance and limitations of its role in early education are discussed. Practice or Policy: Patterning instruction should be extended to more complex patterns than the alternations currently used in elementary schools in order to produce important improvements in both reading and mathematics.     

Mathematics assessment scenarios for at-risk students

In this paper, we describe the work by a team of university and public school mathematics educators to design mathematics assessment tasks that are effective for students who have been identified at risk in mathematics, through their failure on a state mathematics assessment. We have developed assessment tasks that have the very best chance to elicit the most mathematical knowledge possible, but are closely tied to the school curriculum and practical enough to be administered by classroom teachers. We offer some perspectives on the need for alternative mathematics assessment for at-risk students, along with the dangers inherent in this undertaking. Some practical design considerations are discussed and examples of student responses to a mathematics assessment scenario are given. Finally, policy issues related to authentic assessment are discussed.     

Understanding number sequences leads to understanding mathematics concepts

Ninety-six first grade students in an urban school system were tested in October and May on reading, mathematics, and their understanding of sequences of letters and numbers. A time lag analysis was subsequently conducted. In such analyses, cross-correlations between the first measurement of one variable and the second measurement of another are compared. The larger of the correlations indicates the direction of the relationship; i.e., which variable is most likely to be causal. Correlations of the fall scores on the number sequences with spring scores on the mathematics concepts scale were significant, while correlations of the fall mathematics concepts scores with spring number sequence scores were negligible. This indicates that understanding such complex sequences has a directional effect on understanding mathematics concepts. Fall–spring cross-correlations for the letter sequences and reading test, although significant, did not differ, and hence provided no indication of the direction of the relationship. Potential explanations were discussed.     

基于MQI评估系统的大班数学集体活动质量分析及其启示

数学是学前儿童学习和入学准备的重要领域,也是学前教育质量监测的重要方面。MQI评估系统作为一种评估数学课堂教学质量的框架,体现了最新数学教育观念,突出了数学的学科特性和认知要求,强调儿童的主动参与和生成性的教学过程,可同时对教师的教和儿童的学进行评估。本研究使用MQI评估系统对我国幼儿园大班数学集体活动的质量进行考察,结果表明除数学意义建构和数学语言之外,数学丰富度领域各维度表现水平不高;教师对儿童行为进行处理和反馈的水平较低;大多数活动片段中不存在数学表达和语言上的错误及不严密性问题;儿童参与有意义的解释或者提问和推理较少,参与活动的认知要求不高。幼儿园大班数学集体活动质量受到集体教学中"教"与"学"的关系、教师对数学领域知识和儿童数学发展目标的认知、教师评价儿童数学思维的能力及师幼互动技巧、教师培训与考核的侧重点等因素的影响。我国学前教育界应重视数学教育活动的过程性质量,关注儿童数学学习过程性能力的发展,并应在借鉴MQI评估系统的基础上开发适合我国幼儿园数学教育实践的质量评估系统。     

The National Curriculum and Standards in Primary Schools: A comparative perspective

A comparison is made of the standards in primary school mathematics in England and Cyprus, drawing upon national curricula, content of textbooks and data from international comparisons of attainment. Standards are conceptualised in three ways: expected, planned and realised. It is demonstrated that high expected and high planned standards, as set in national curricula and textbooks, are not associated with high realised standards. Four possible explanations are explored and policy issues associated with national standard setting are discussed. The value of setting national curriculum standards and targets by reference to comparative international data is raised.     

The Dutch gender gap in mathematics: Small for achievement, substantial for beliefs and attitudes

In general, studies on gender and mathematics show that the advantage held by boys over girls in mathematics achievement has diminished markedly over the last 40 years. Some researchers even argue that gender differences in mathematics achievement are no longer a relevant issue. However, the results of the Trends in Mathematics and Science Study of 2003 (TIMSS-2003), as well as the participation rates of girls in (advanced) mathematics courses, show that in some countries, such as the Netherlands, gender equity in mathematics is still far from a reality. Research on gender and mathematics is often limited to the relationship between gender differences in attitudes toward mathematics and gender differences in mathematics achievement. In school effectiveness research, theories and empirical evidence emphasize the importance of certain school and class characteristics (e.g., strong educational leadership, safe and orderly learning climate) for achievement and attitudes. However, there is little information available at to whether these factors have the same or a different influence on the achievement of girls and boys. This study used the Dutch data from TIMSS-2003 to explore the relationship between school- and class characteristics and the mathematics achievement and attitudes for both girls and boys in Grade 4 of the primary school. The explorations documented in this paper were guided by a conceptual model of concentric circles and involved multilevel analyses. Interaction effects with gender were assessed for each influencing factor that turned out to have a significant effect. The results of these analyses provide additional insight into the influence that non-school-related and school-related factors have on the mathematics achievement and attitudes of girls and boys.     

Proficiency in the explanation of procedures: A test of the intuitive understanding of teachers of undergraduate mathematics

A discussion of production system approaches to the representation of learned procedures, and of stage models of the acquisition of procedures, yielded four general principles applicable to the construction of effective explanations. The intuitive appreciation of these principles by tertiary teachers was tested in two studies. In the first, separate verisons of an explanation of a standard undergraduate mathematics problem solution procedure were prepared, these versions differing in the extent to which they exemplified the derived principles. Subjects ranked the explanations in terms of their perceived effectiveness. In the second study, subjects of varying experience at teaching tertiary level mathematics were video recorded while they individually explained three standard undergraduate problems. Their explanations were scored for the extent to which some awareness of the general principles was evidenced. In general terms the findings from these two studies suggested that mathematics teachers might be well able to discriminate between explanations which differ in their compliance with the derived general principles, but that they were not well skilled at employing these principles in explanations which they generated themselves.     

The role that mathematics plays in college- and career-readiness: evidence from PISA

Many studies have found a strong relationship between the mathematics students study in school and their performance on an academic or school mathematics assessment but not on an assessment of mathematics literacy (ML). With many countries, like the USA, placing emphasis on finishing secondary education being mathematically literate and prepared for college or career, this raises the question about the relationship between the mathematics studied in school and any ML students may have. The Programme for International Student Assessment (PISA) ML assessment is embedded in real-world contexts that provide an important window on how ready students are to tackle the situations and problems that await them whether they intend to pursue further education beyond high school or intend to go directly into the labour force. In this paper, we draw upon the PISA 2012 data to investigate the extent to which the cumulative exposure to rigorous mathematics content, such as that embedded in college- and career-ready standards, is associated with ML as assessed in PISA. Results reveal that both exposure to rigorous school mathematics and experiencing the instruction of this mathematics through real-world applications are significantly related to all the real-world contextualized PISA ML scores.     

The computer as a cultural influence in mathematical learning

My starting point in this paper is that there is a cultural gap between the mathematics that children do as part of their everyday experience and the mathematics that they learn at school; my thesis is that the computer has (perhaps uniquely) the potential to bridge this divide. The paper will examine the cultural impact-both actual and potential-of the computer on children's mathematical education; at the ways in which the introduction of the computer does and will changes the ambient space in which children learn mathematics.I begin with a brief discussion of the cultural context of mathematics learning and the relationship between informal, everyday mathematical activity, and formal, school mathematies. This perspective leads to a closer examination of what it means to do mathematics, and on the relationship of a technology to the mathematics embedded within a given culture. I discuss the issue of injecting meaning into mathematical activity, and then examine some ways in which the computer might offer a solution to this central problem. Next, I give some examples of the influence of the computer on the culture of the mathematics classroom. Finally, I suggest some of the outstanding issues of research and curriculum development which remain.This paper is based on substantially the same data as is discussed in an article inCultural Dynamics.     

Learning landscapes

This article explores the metaphor of learning landscapes, a tool developed in order to map children's experiences of, and attitudes to, learning (mathematics) before and after the transfer from primary to secondary school. Firstly, the continuing problems surrounding school transfer and why a re‐examination of this is required are considered. Secondly, as background to the metaphor, the sociocultural influences which impinge on educational processes are discussed. Following that, the core of the article develops the metaphor, seeking to demonstrate its pertinence to the study of children's schooling, the learning of mathematics and school transfer. The article also questions the potential of some of the current educational reforms to deliver sustained improvement.     

Parents' representations of their children's mathematics learning in multiethnic primary schools

There is a growing concern that governmental calls for parental involvement in children's school mathematics learning have not been underpinned by research. In this article the authors aim to offer a contribution to this debate. Links between children's home and school mathematical practices have been researched in sociocultural studies, but the origins of differences within the same cultural group are not well understood. The authors have explored the notion that parents' representations of school mathematics and associated practices at home may play a part in the development of these differences. This article reports an analysis of interviews with parents of 24 children of Pakistani and White origin enrolled in primary schools in England, including high and low achievers in school mathematics. The extent to which the parents represented their own school mathematics and their child's school mathematics as the ‘same’ or ‘different’ are examined. In addition, ways in which these representations influenced how they tried to support their children's learning of school mathematics are examined. The article concludes with reflections on the implications of the study for education policy.     

Exploring Israeli high school graduates’ explanations for the spread of the coronavirus

Educational Studies in Mathematics - The aim of this study is to explore Israeli high school graduates’ mathematical explanations for the spread of the coronavirus, given that the mathematics...     

Investigating Elementary and Middle School Students’ Subjective Well-Being and Mathematical Performance in Shanghai

Students in Shanghai ranked at the top in mathematics on the past two assessments of the Program for International Student Assessment (PISA). However, even though Asian students in general, and Chinese students in particular, ranked at the top in mathematics, we know little about Shanghai students’ subjective well-being (SWB). This paper reports two studies that investigated the SWB of elementary and middle school students in Shanghai, as well as their mathematics performance. It is found that the mean scores of elementary school students on all scales of SWB are significantly higher than those of middle school students, representing a large drop for students when moving from elementary to middle school. Furthermore, students’ mathematics performance is moderately correlated with their SWB. And for Grade 6, students’ SWB affects their mathematics performance, especially from academic self-concept, attentiveness in the classroom, and relationship with teachers.     

Achievement goal orientations,learning strategies,and mathematics achievement: A comparison of Chinese Miao and Han students

This study compared the relationships between the goal orientations, learning strategies, and mathematics achievement of Chinese Miao and Han students. A total of 211 Han and 321 Miao fifth‐ and sixth‐grade students from Qiandongnan participated in the study. The results suggest that, in both samples, mastery orientation positively predicted the use of surface/deep learning strategies and mathematics achievement, and that performance‐approach orientation significantly predicted the use of surface strategies. Ethnic differences were also detected. The positive relationship of the performance‐approach orientation to deep strategies was stronger among the Han than the Miao students. Overall, our findings confirm that goal orientations have an important influence on Chinese students’ strategy use and achievement and that ethnic differences affect the patterns of these relationships. Possible explanations for the results and the practical implications for school practitioners are discussed.     

The Emerging Pracitces of a Novice Teacher: The Roles of His School Mathematics Images

The relationship between teachers' beliefs about mathematics and its teaching and learning, and their classroom practices, has been investigated in many studies. The results of these studies are by no means unanimous, and the purpose of the present report is to contribute to a further understanding of the significance of one teacher's beliefs and images of school mathematics on his teaching practices. The article presents a novice teacher whose images of school mathematics were strongly influenced by the current reform of mathematics education. The study focuses on how the teacher coped with the complexities of the classroom. On the basis of the teacher's role within the classroom interactions, the construct of critical incidents of practice is developed, and some of the results of previous research in the field are re-examined. This revised version was published online in August 2006 with corrections to the Cover Date.