The Effects of a Teacher In‐Service on Low‐Achieving Grade 7 and 8 Mathematics Students

Previous research (e.g., Woodward & Baxter, 1997 ) found that Standards‐based mathematics teaching provides marginal or no benefits for low achievers, in contrast with positive effects for middle and high ability students. A randomized quasi‐experiment in 52 Canadian schools found that low achieving grade 7 and 8 students who received support consisting of placement on a learning continuum, instruction focused on their specific learning needs, and concrete materials to represent mathematical constructs, benefited from teaching that emphasized construction over transmission of knowledge. Treatment students showed small but statistically significant improvements over controls in student achievement, and controversially, in mathematical beliefs, and attitudes. The latter finding raised issues of the appropriate balance between Type I and Type II error in educational research.     

Teaching mathematics using Steplets

This article evaluates online mathematical content used for teaching mathematics in engineering classes and in distance education for teacher training students. In the EU projects Xmath and dMath online computer algebra modules (Steplets) for undergraduate students assembled in the Xmath eBook have been designed. Two questionnaires, a compulsory student project and teaching in front of class show that using Steplets turn mathematics teaching from drill to understanding. The Steplets use algorithms developed for the Mathematica programming language.     

How can self-regulated learning support the problem solving of third-grade students with mathematics anxiety?

The study compares 140 third-grade Israeli students (lower and higher achievers) who were either exposed to self-regulated learning (SRL) supported by metacognitive questioning (the MS group) or received no direct SRL support (the N_MS group). We investigated: (a) mathematical problem solving performance; (b) metacognitive strategy use in three phases of the problem-solving process; and (c) mathematics anxiety. Findings indicated that the MS students showed greater gains in mathematical problem solving performance than the N_MS students. They reported using metacognitive strategies more often, and showed a greater reduction in anxiety. In particular, the lower MS achievers showed these gains in the basic and complex tasks, in strategy use during the on-action phase of the problem solving process and a decrease in negative thoughts. The higher achievers showed greater improvement in transfer tasks and an increase in positive thoughts towards mathematics. Both the theoretical and practical implications of this study are discussed.     

How Emergent Models May Foster the Constitution of Formal Mathematics

This article deals with the role that so-called emergent models can play in the process of constituting formal mathematics. The underlying philosophy is that formal mathematics is something that is, or should be, constituted by the students themselves. In the instructional design theory for realistic mathematics education, models always have been employed to foster a process in which formal mathematics is reinvented by the students themselves. This article describes how the use of models became more and more explicated over time and developed into the notion of emergent models. The design of an instructional sequence, which deals with flexible mental computation strategies for addition and subtraction up to 100, is taken as an instance for elaborating what is meant by emergent models and what role they play in fostering the constitution of formal mathematics. The analysis shows that there are 3 interrelated processes. First. at a more holistic level, there is a global transition in which “the model” initially emerges as a model of informal mathematical activity and then gradually develops into a model for more formal mathematical reasoning. Second, the transition from “model of” to “model for” involves the constitution of anew mathematical reality that can be denoted formal in relation to the original starting points of the students. Third, in the series of instructional activities, there is not 1 model, but the model actually is shaped as a series of signs, in which each new sign comes to signify activity with a previous sign in a chain of signification.     

Preservice Elementary Teachers' Use of Mathematics in a Project‐Based Science Approach

The use of a project‐based science (PBS) approach to teaching encourages students to integrate mathematics and science in meaningful ways as they create projects. As a beginning study of how students use mathematics in such an approach, an analysis of 23 projects developed by preservice elementary teachers enrolled in an elementary science course was conducted. Findings showed that students made a number of different types of mathematical errors and underutilized data representation and summary forms. Implications included the importance of developing methods for supporting the use of mathematical tools in utilizing a project‐based approach and considering ways that such tools mediate scientific thinking.     

How Emergent Models May Foster the Constitution of Formal Mathematics

This article deals with the role that so-called emergent models can play in the process of constituting formal mathematics. The underlying philosophy is that formal mathematics is something that is, or should be, constituted by the students themselves. In the instructional design theory for realistic mathematics education, models always have been employed to foster a process in which formal mathematics is reinvented by the students themselves. This article describes how the use of models became more and more explicated over time and developed into the notion of emergent models. The design of an instructional sequence, which deals with flexible mental computation strategies for addition and subtraction up to 100, is taken as an instance for elaborating what is meant by emergent models and what role they play in fostering the constitution of formal mathematics. The analysis shows that there are 3 interrelated processes. First. at a more holistic level, there is a global transition in which "the model" initially emerges as a model of informal mathematical activity and then gradually develops into a model for more formal mathematical reasoning. Second, the transition from "model of" to "model for" involves the constitution of anew mathematical reality that can be denoted formal in relation to the original starting points of the students. Third, in the series of instructional activities, there is not 1 model, but the model actually is shaped as a series of signs, in which each new sign comes to signify activity with a previous sign in a chain of signification.     

A comparative investigation of the presence of psychological conditions in high achieving eighth graders from TIMSS 2007 Mathematics

This study is a second analysis of Trends in International Mathematics and Science Study (TIMSS) 2007 background questionnaires to investigate high achieving eighth-grade students’ possession of three elements of Krutetskii’s (The Psychology of Mathematical Abilities in Schoolchildren, The University of Chicago, 1976) psychological conditions. Analyses were made between high achieving eighth graders and their lower achieving counterparts, and between Korean high achievers and their high achieving peers across ten high performing TIMSS participating countries. After reviewing 44,643 students across selected countries, we conclude that a larger percentage of students with mathematical talent demonstrate positive attitudes toward mathematics, value mathematics, and have self-confidence in their ability to learn mathematics than their peers without high achievement in mathematics. However, a larger portion of high achieving Korean students displayed low self-confidence and valued mathematics less than their peers from other high performing countries. Findings from this study will provide insight into some educational issues in science, technology, engineering and mathematics education.     

Which mathematics should we teach engineering students? An empirically grounded case for a broad notion of mathematical thinking

While many engineering educators have proposed changes to theway that mathematics is taught to engineers, the focus has oftenbeen on mathematical content knowledge. Work from the mathematicseducation community suggests that it may be beneficial to considera broader notion of mathematics: mathematical thinking. Schoenfeldidentifies five aspects of mathematical thinking: the mathematicscontent knowledge we want engineering students to learn as wellas problem-solving strategies, use of resources, attitudes andpractices. If we further consider the social and material resourcesavailable to students and the mathematical practices studentsengage in, we have a more complete understanding of the breadthof mathematics and mathematical thinking necessary for engineeringpractice. This article further discusses each of these aspectsof mathematical thinking and offers examples of mathematicalthinking practices based in the authors' previous empiricalstudies of engineering students' and practitioners' uses ofmathematics. The article also offers insights to inform theteaching of mathematics to engineering students.     

Teaching mathematical modelling through project work

The paper presents and analyses experiences from developing and running an inservice course in project work and mathematical modelling for mathematics teachers in the Danish gymnasium, e.g. upper secondary level, grade 10~12. The course objective is to support the teachers to develop, try out in their own classes, evaluate and report a project based problem oriented course in mathematical modelling. The in-service course runs over one semester and includes three seminars of 3, 1 and 2 days. Experiences show that the course objectives in general are fulfilled and that the course projects are reported in manners suitable for internet publication for colleagues. The reports and the related discussions reveal interesting dilemmas concerning the teaching of mathematical modelling and how to cope with these through «setting the scene» for the students modelling projects and through dialogues supporting and challenging the students during their work. This is illustrated and analysed on the basis of two course projects.     

A locus for transnational exchanges: European mathematical journals for students and teachers, 1860s–1914

While there were a few mathematical journals aimed at teachers and students as early as the 1840s, it was only in the late 19th century that they became more numerous in Europe. This article is based on the analysis of a corpus of European mathematical journals published between the 1860s and World War I, selected in the first place because they were aimed at high school teachers and high school or/and first two years university students, which are often referred to as “intermediate journals”. All these journals had focused on the teaching of mathematics and, as such, they were shaped by the educational context of the country in which they were published. However, leafing through theses journals, one is struck by the fact that the mathematics they published was in fact highly commensurable, and can see that they were the locus of transnational exchanges on mathematical knowledge. This article shows that several aspects of “internationalisation” were in fact at stake in mathematical journals for students: making knowledge from elsewhere available and of publicizing to the whole world the mathematics produced in one country; making people from different countries collaborate. Finally, it focuses on the effects of transnational exchanges between journals for teachers and students: what was the mathematical knowledge that was circulated through them, and in what respect was it different from that published in other mathematical journals?     

Applying Cognition-Based Assessment to Elementary School Students' Development of Understanding of Area and Volume Measurement

As part of a discussion of cognition-based assessment (CBA) for elementary school mathematics, I describe assessment tasks for area and volume measurement and a research-based conceptual framework for interpreting students' reasoning on these tasks. At the core of this conceptual framework is the notion of levels of sophistication. I provide details on an integrated set of levels for area and volume measurement that (a) starts with the informal, preinstructional reasoning typically possessed by students, (b) ends with the formal mathematical concepts targeted by instruction, and (c) indicates cognitive plateaus reached by students in moving from (a) to (b).     

Factors shaping mathematics lecturers’ service teaching in different departments

In this article we focus on university lecturers’ approaches to the service teaching and factors that influence their approaches. We present data obtained from the interviews with 19 mathematics and three physics lecturers along with the observations of two mathematics lecturers’ calculus courses. The findings show that lecturers’ approaches to teaching the same topic vary across departments; that is, they consciously privilege different aspects of mathematics, set different questions on examinations and follow different textbooks while teaching in different departments. We discuss factors influencing lecturers’ decision of what (mathematics) to teach in different departments and offer educational implications for service mathematics teaching in terms of students’ mathematical needs and the role of mathematics for client students.     

Examining Secondary Mathematics Teachers' Opportunities to Develop Mathematically in Professional Learning Communities

To make progress toward ambitious and equitable goals for students’ mathematical development, teachers need opportunities to develop specialized ways of knowing mathematics such as mathematical knowledge for teaching (MKT) for their work with students in the classroom. Professional learning communities (PLCs) are a common model used to support focused teacher collaboration and, in turn, foster teacher development, instructional improvement, and student outcomes. However, there is a lack of specificity in what is known about teachers’ work in PLCs and what teachers can gain from those experiences, despite broad claims of their benefit. We discuss an investigation of the work of secondary mathematics teachers in PLCs at two high schools to describe and explicate possible opportunities for teachers to develop the mathematical knowledge needed for the work of teaching and the ways in which these opportunities may be pursued or hindered. The findings show that, without pointed focus on mathematical content, opportunities to develop MKT can be rare, even among mathematics teachers. Two detailed images of teacher discussion are shared to highlight these claims. This article contributes to the ongoing discussion about the affordances and limitations of PLCs for mathematics teachers, considerations for their use, and how they can be supported.     

How Informal Out-of-School Mathematics Can Help Students Make Sense of Formal In-School Mathematics: The Case of Multiplying by Decimal Numbers

This article presents a teaching experiment on the relationship between informal out-of-school and formal in-school mathematics, and the ways each can inform the other in the development of abstract mathematical knowledge. This study concerns the understanding of some aspects of the multiplicative structure of decimal numbers. It involved a series of classroom activities in upper elementary school, using suitable cultural artifacts and interactive teaching methods. To create a substantially modified teaching/learning environment, new sociomathematical norms (Yackel & Cobb, 1996) were also introduced. The focus was on fostering a mindful approach toward realistic mathematical modeling, which is both real-world based and quantitatively constrained sense-making (Reusser & Stebler, 1997). In addition, procedures not commonly used in ordinary teaching activities, such as estimation and approximation processes, were also introduced.     

Learning study – promoting and hindering factors in mathematics teaching

This article focuses on presenting success factors for a group of teachers in carrying out a learning study in mathematics at their school. The research questions are: what are the actions of the school teaching community during development projects? What factors enable a group of teachers to carry out a learning study at their school? Activity theory provides a holistic framework to investigate relationships among the components present in a learning study. The results are based on analysis of interviews with teachers, students, principal organizers of schools and project coordinators, videotaped lessons, students’ tests and minutes taken at meetings of mathematics projects. The results show that the skills of facilitators, the time devoted to collaborative work, the link to learning theory and avoiding overly comprehensive content when teaching lessons are important promoting factors in mathematics teaching. The findings raise important questions about the way in which teacher work within universities.     

Can't do maths--understanding students' maths anxiety

The number of students continuing with their mathematics educationpost GCSE level has declined in recent years and hence studentsentering Engineering degrees are reducing. The University ofBirmingham recognized this problem and introduced the Suiteof Technology programme (STP) which no longer requires studentsto have A-level mathematics. Therefore lecturers at universityare now faced with teaching A-level mathematics in order togive the students the mathematical skills for their technologydegree. With little experience of teaching at this level, lecturersfrequently face the challenge of choosing the most appropriatelevel for a lecture that encourages students to engage withand learn a subject that they are novices in. It turned outthat some students have a mathematics anxiety and hence thebiggest challenge for a lecturer is supporting the studentsovercoming this fear of mathematics. Choosing the appropriatestarting level for any lecture and the fear of mathematics wasacknowledged during a peer learning group meeting as part ofthe Post Graduate Certificate in Learning and Teaching in HigherEducation (PGCLTHE) at the University of Birmingham. In orderto explore this further, a session of mathematics was taughtby a Civil Engineering lecturer to fellow peers who do not havean engineering background. This article describes the mathematicsteaching session, reflections from the lecturer and the learners,and the impact that this had on teaching mathematics to undergraduates.Further, the article explores the difficulties and challengesexperienced by lecturers when teaching mathematics as a servicesubject.     

“Everything is Math in the Whole World”: Integrating Critical and Community Knowledge in Authentic Mathematical Investigations with Elementary Latina/o Students

This critical ethnographic study of an after-school mathematics club for elementary-aged Latina/o youth focuses on connecting critical, community, and mathematical knowledge in the context of authentic, community-based investigations. We present cases of two extended projects to highlight tensions and dilemmas that emerged, particularly tensions related to ensuring rich mathematics in the contexts of projects that were personally and socially meaningful to the students. Our analysis offers insights into critical mathematics education with elementary aged students, and has the potential to counter dominant deficit perspectives of Latina/o youth. Additionally, the findings of this study inform critical approaches to teaching mathematics in schools attended by marginalized students in order to reverse prevalent trends of our educational system failing these students.     

Paraprofessionals in Cyprus and England: perceptions of their role in supporting primary school mathematics

Paraprofessionals increasingly work alongside teachers in many countries, with research suggesting they undertake pedagogic roles for which they are not formally prepared. We investigate this from the perspective of paraprofessionals supporting individual children with special needs in primary schools in Cyprus and England and develop a typology to conceptualise their views of their role in mathematics lessons in relation to children, teachers and mathematical processes. All perceive themselves as explaining mathematical ideas and dealing with difficulties. Some report having major or sole responsibility for teaching and planning mathematics. The vast majority feel able to do their job with only informal preparation, often linking this to the low level of mathematics involved. We argue that the current situation is contrary to the subject knowledge literature. Expectations placed on paraprofessionals and the mathematical experiences of the children they support arouse concerns.     

Validating proofs and counterexamples across content domains: Practices of importance for mathematics majors

Validating proofs and counterexamples across content domains is considered vital practices for undergraduate students to advance their mathematical reasoning and knowledge. To date, not enough is known about the ways mathematics majors determine the validity of arguments in the domains of algebra, analysis, geometry, and number theory—the domains that are central to many mathematics courses. This study reported how 16 mathematics majors, including eight specializing in secondary mathematics education, who had completed more proof-based courses than transition-to-proof classes evaluated various arguments. The results suggest that the students use one of the following strategies in proof and counterexample validation: (1) examination of the argument's structure and (2) line-by-line checking with informal deductive reasoning, example-based reasoning, experience-based reasoning, and informal deductive and example-based reasoning. Most students tended to examine all steps of the argument with informal deductive reasoning across various tasks, suggesting that this approach might be problem dependent. Even though all participating students had taken more proof-related mathematics courses, it is surprising that many of them did not recognize global-structure or line-by-line content-based flaws presented in the argument.     

Orchestrating Productive Mathematical Discussions: Five Practices for Helping Teachers Move Beyond Show and Tell

This article presents a teaching experiment on the relationship between informal out-of-school and formal in-school mathematics, and the ways each can inform the other in the development of abstract mathematical knowledge. This study concerns the understanding of some aspects of the multiplicative structure of decimal numbers. It involved a series of classroom activities in upper elementary school, using suitable cultural artifacts and interactive teaching methods. To create a substantially modified teaching/learning environment, new sociomathematical norms (Yackel &; Cobb, 1996) were also introduced. The focus was on fostering a mindful approach toward realistic mathematical modeling, which is both real-world based and quantitatively constrained sense-making (Reusser &; Stebler, 1997). In addition, procedures not commonly used in ordinary teaching activities, such as estimation and approximation processes, were also introduced.