Examining the didactic contract when handheld technology is permitted in the mathematics classroom

The use of mathematics analysis software (MAS) including handheld scientific and graphics calculators offers a range of pedagogical opportunities. Its use can support change in the didactic contract. MAS may become an alternative source of authority in the classroom empowering students to explore variation and regularity, manipulate simulations and link representations. Strategic use may support students to direct their own learning and explore mathematics, equipping them to share their findings with the teacher and the class with more confidence. This paper offers a framework for examining the impact of the use of MAS on the didactic contract. Lessons were observed in 12 grade 10 classes, with 12 different teachers new to MAS. MAS technology was used with a variety of didactic contracts, mostly traditional. The framework drew attention to many ways in which the teaching differed. Analysis of the didactic contract must consider both the teaching of mathematics and of technology skills, because these have different characteristics. In all classes, both teachers and students saw the teacher as having a responsibility to teach technology skills. Students saw technology skills as the main point of the lesson, but the teachers saw the lesson as primarily teaching mathematics—one of the mismatches which may need negotiation to adapt didactic contracts to teaching with MAS.     

Being a mathematics teacher in times of reform

Within research on mathematics teachers and/or their professional development, the concept of identity emerges as a critique of views of how teaching practice is related to teachers’ ‘internal states’ of knowledge and beliefs. Identity relates teachers’ professional lives to teaching practices and to the contexts in which the teaching and/or professional development occurs. However, what might count as the context still needs in-depth discussion. In order to contribute to the development of a theoretical framework for understanding mathematics teachers’ professional lives, we will draw on one remarkable teacher’s identity as a primary mathematics teacher in relation to one political, sociocultural, and pedagogical context. We use this teacher’s experience to discuss how education policies that create what Ball (2003) called ‘terrors of performativity’ tend to impede the formation of a balanced teacher identity.     

Perspectives on technology mediated learning in secondary school mathematics classrooms

The introduction of technology resources into mathematics classrooms promises to create opportunities for enhancing students’ learning through active engagement with mathematical ideas; however, little consideration has been given to the pedagogical implications of technology as a mediator of mathematics learning. This paper draws on data from a 3-year longitudinal study of senior secondary school classrooms to examine pedagogical issues in using technology in mathematics teaching — where “technology” includes not only computers and graphics calculators but also projection devices that allow screen output to be viewed by the whole class. We theorise and illustrate four roles for technology in relation to such teaching and learning interactions — master, servant, partner, and extension of self. Our research shows how technology can facilitate collaborative inquiry, during both small group interactions and whole class discussions where students use the computer or calculator and screen projection to share and test their mathematical understanding.     

Handling errors as they arise in whole-class interactions

There has been a long history of research into errors and their role in the teaching and learning of mathematics. This research has led to a change to pedagogical recommendations from avoiding errors to explicitly using them in lessons. In this study, 22 mathematics lessons were video-recorded and transcribed. A conversation analytic (CA) approach was then taken to examine how mathematical errors are treated by teachers and students when they arise in interaction. Despite pedagogical recommendations, in these interactions, errors continue to be predominantly treated as something to avoid. There is a tension between the affective aspects of managing errors in interactions and the cognitive aspects. Close examination of classroom interactions enable us to see how these tensions are managed both by teachers and students.     

Digital resources inviting changes in mid-adopting teachers’ practices and orchestrations

Digital resources offer opportunities to improve mathematics teaching and learning, but meanwhile may question teachers’ practices. This process of changing teaching practices is challenging for teachers who are not familiar with digital resources. The issue, therefore, is what teaching practices such so-called ‘mid-adopting’ mathematics teachers develop in their teaching with digital resources, and what skills and knowledge they need for this. To address this question, a theoretical framework including notions of instrumental orchestration and the TPACK model for teachers’ technological pedagogical content knowledge underpins the setting-up of a project with twelve mathematics teachers, novice in the field of integrating technology in teaching. Technology-rich teaching resources are provided, as well as support through face-to-face group meetings and virtual communication. Data include lesson observations and questionnaires. The results include a taxonomy of orchestrations, an inventory of skills and knowledge needed, and an overview of the relationships between them. During the project, teachers do change their orchestrations and acquire skills. On a theoretical level, the articulation of the instrumental orchestration model and the TPACK model seems promising.     

Teaching Science in Higher Education: Faculty Professional Development and Barriers to Change

The focus of this research was to better understand the change processes necessary for university science teaching reform to be successful. The professional development processes involved faculty cognitive perceptions of learning, teaching skills, and pedagogical knowledge, as well as faculty culture in teaching science courses. A series of faculty development programs were conducted at nine U.S. locations to explore, develop strategies, and implement changes in science classrooms. A review of research and these professional development experiences provided a base to carry out research activities related to understanding change in science faculty. Faculty participants in the program from 30 institutions were selected to be involved in the study. Ethnographic and case study approaches were used to collect and analyze data. Many faculty members encountered in this study had conceptions of the change process that inhibited successful action. These research efforts provide a predictive model for assisting faculty change and help determine which faculty professional development efforts may be successful in overcoming barriers to change in undergraduate science classrooms.     

Professional Development Strategically Connecting Mathematics and Science: The Impact on Teachers' Confidence and Practice

The press to integrate mathematics and science comes from researchers, business leaders, and educators, yet research that examines ways to support teachers in relating these disciplines is scant. Using research on science and mathematics professional development, we designed a professional development project to help elementary teachers improve their teaching of mathematics and science by strategically connecting these disciplines. The purposes of this study are: (a) to identify changes in teachers' confidence and practice after participating in the professional development and (b) to identify different ways to connect mathematics and science during the professional development. We use a Likert‐scale survey to assess changes in teachers' confidence related to teaching mathematics and science. In addition, we report on a thematic analysis of teachers' written responses to open‐ended questions that probed teachers' perceived changes in practice. We analyze field notes from observations of project workshops to document different types of opportunities for connecting mathematics and science. We conclude with implications for future professional development that connects mathematics and science in meaningful ways, as well as suggestions for future research.     

Elementary mathematics specialists in “departmentalized” teaching assignments: Affordances and constraints

In this article, we describe the experiences of three Elementary Mathematics Specialists (EMS) who were part of a larger project investigating the impact of EMS certification and assignment (self-contained or “departmentalized”) on teaching practices and student achievement outcomes. All three of the teachers were “departmentalized,” in the sense that each was responsible for teaching mathematics to at least two groups of students, and accordingly, did not teach all subjects as would a typical self-contained elementary teacher. Each teacher had recently earned an Elementary Mathematics Specialist certificate through completion of a 24-credit, graduate-level program designed to build pedagogical content knowledge and leadership capacity in mathematics. Through a series of observations and interviews over the course of one school year, we examined how the teachers described and navigated specific affordances and constraints they encountered in their particular contexts. Common affordances included opportunities to revise and learn from instruction, and constraints included reduced flexibility introduced by the need to schedule multiple classes of mathematics. Despite these common features, we found important differences between the three models of departmentalization, which we describe as team approach, class swap, and grade-level mathematics teacher. For example, some of the models provided more opportunities for collaboration while others made it difficult for teachers to address potential inequities in learning opportunities across sections. Despite the constraints of their respective models, we found evidence of the EMS-certified teachers drawing on professional expertise in mathematics to meet student needs.     

A case study of pedagogy of mathematics support tutors without a background in mathematics education

This study investigates the pedagogical skills and knowledge of three tertiary-level mathematics support tutors in a large group classroom setting. This is achieved through the use of video analysis and a theoretical framework comprising Rowland's Knowledge Quartet and general pedagogical knowledge. The study reports on the findings in relation to these tutors’ provision of mathematics support to first and second year undergraduate engineering students and second year undergraduate science students. It was found that tutors are lacking in various pedagogical skills which are needed for high-quality learning amongst service mathematics students (e.g. engineering/science/technology students), a demographic which have low levels of mathematics upon entering university. Tutors teach their support classes in a very fast didactic way with minimal opportunities for students to ask questions or to attempt problems. It was also found that this teaching method is even more so exaggerated in mandatory departmental mathematics tutorials that students take as part of their mathematics studies at tertiary level. The implications of the findings on mathematics tutor training at tertiary level are also discussed.     

Teachers’ mathematical problem-solving knowledge: In what way is it constructed during teachers’ collaborative work?

This research aims to analyze the type of mathematics problem-solving knowledge for teaching used when working collaboratively in a Lesson Study (LS) process and examine how dialogic interactions contribute to knowledge construction. Five meetings during one LS cycle of a group of eight Swiss primary teachers were video recorded, transcribed and coded with the help of qualitative data analysis software. This analysis is conducted by crossing theoretical frameworks from two different fields in education, namely mathematics education and dialogic analysis. The mixed-method uses quantitative analysis with Markov chains and cross-tables, as well as qualitative analysis at micro-, meso- and macro-levels. This research suggests that participants collectively use their mathematics and their problem-solving content knowledge to focus on pedagogical problem-solving knowledge, that they navigate between different knowledge levels and that the roles of teachers and facilitators are differentiated but are also coequal.     

Key Developmental Understandings in Mathematics: A Direction for Investigating and Establishing Learning Goals

Although mathematics educators seem to agree on the importance of teaching mathematics for understanding, what they mean by understanding varies greatly. In this article, I elaborate and exemplify the construct of key developmental understanding to emphasize a particular aspect of teaching for understanding and to offer a construct that could be used to frame the identification of conceptual learning goals in mathematics. The key developmental understanding construct is based on extant empirical and theoretical work. The construct can be used in the context of research and curriculum development. Using a classroom example involving fractions, I illustrate how focusing on key developmental understandings leads to particular, potentially useful types of pedagogical thinking and directions for inquiry.     

The Influences of Three Interventions on Prospective Elementary Teachers' Beliefs About the Knowledge Base Needed for Teaching Mathematics

To meet the challenge to reform mathematics education, effective opportunities to learn are needed to promote prospective elementary school teachers' development of the knowledge base that supports teaching for mathematical proficiency. This article describes three professional development interventions and their influence on prospective teachers' beliefs about mathematics, how children learn mathematics, and mathematics teaching. The three interventions consisted of problem‐solving journals, structured interviews, and peer teaching that were integrated in a PreK‐6 mathematics methods course. Results of precourse and postcourse survey data are included that measured 24 prospective teachers' beliefs about the knowledge base needed to teach elementary school mathematics. Data indicated that using these interventions and other course experiences facilitated change in the prospective teachers' beliefs, with a shift toward reform‐oriented mathematics education perspectives.     

Influence of gender,single-sex and co-educational schooling on students’ enjoyment and achievement in mathematics

This research investigates the influence that gender, single-sex and co-educational schooling can have on students’ mathematics education in second-level Irish classrooms. Although gender differences in mathematics education have been the subject of research for many years, recent results from PISA (Programme for International Student Assessment) show that there are still marked differences between the achievement and attitude of male and female students in Irish mathematics classrooms. This paper examines the influence of gender in more detail and also investigates the impact of single-sex or co-educational schooling. This is a follow on study which further analyses data collected by the authors when they designed a pedagogical framework and used this to develop, implement and evaluate a teaching intervention in four second-level Irish schools. The aim of this pedagogical framework was to promote student interest in the topic of algebra through effective teaching of the domain. This paper further analyses the quantitative data collected and investigates whether there were differences in students’ enjoyment and achievement scores based on their gender and whether they attended single-sex or co-educational schools.     

Assessment of teacher knowledge across countries: a review of the state of research

This review presents an overview of research on the assessment of mathematics teachers’ knowledge as one of the most important parameters of the quality of mathematics teaching in school. Its focus is on comparative and international studies that allow for analyzing the cultural dimensions of teacher knowledge. First, important conceptual frameworks underlying comparative studies of mathematics teachers’ knowledge are summarized. Then, key instruments designed to assess the content knowledge and pedagogical content knowledge of future and practicing mathematics teachers in different countries are described. Core results from comparative and international studies are documented, including what we know about factors influencing the development of teacher knowledge and how the knowledge is related to teacher performance and student achievement. Finally, we discuss the challenges connected to cross-country assessments of teacher knowledge and we point to future research prospects.     

Key Developmental Understandings in Mathematics: A Direction for Investigating and Establishing Learning Goals

Although mathematics educators seem to agree on the importance of teaching mathematics for understanding, what they mean by understanding varies greatly. In this article, I elaborate and exemplify the construct of key developmental understanding to emphasize a particular aspect of teaching for understanding and to offer a construct that could be used to frame the identification of conceptual learning goals in mathematics. The key developmental understanding construct is based on extant empirical and theoretical work. The construct can be used in the context of research and curriculum development. Using a classroom example involving fractions, I illustrate how focusing on key developmental understandings leads to particular, potentially useful types of pedagogical thinking and directions for inquiry.     

Construction of Teacher Knowledge in Context: Preparing Elementary Teachers to Teach Mathematics and Science

The purpose of this study was to further the understanding of how preservice teachers construct teacher knowledge and pedagogical content knowledge of elementary mathematics and science in a school‐based setting and the extent of knowledge construction. Evidence of knowledge construction (its acquisition, its dimensions, and the social context) was collected through the use of a qualitative methodology. The methods course was content‐specific with instruction in elementary mathematics and science. Learning experiences were based on national standards with a constructivist instructional approach and immediate access to field experiences. Analysis and synthesis of data revealed an extensive acquisition of teacher knowledge and pedagogical content knowledge. Learning venues were discovered to be the conduits of learning in a situated learning context. As in this study, content‐specific, school‐based experiences may afford preservice teachers greater opportunities to focus on content and instructional strategies at deeper levels; to address anxieties typically associated with the teaching of elementary mathematics and science; and to become more confident and competent teachers. Gains in positive attitudes and confidence in teaching mathematics and science were identified as direct results of this experience.     

Standards for professionalization of mathematics teachers: policy, curricula, and national teacher employment test in Korea

International comparative studies such as the Trend in International Mathematics and Science Study (TIMSS) and the OECD Programme for International Student Assessment (PISA) indicate that Korean students have consistently performed well. In addition, a recent study titled Mathematics Teaching in the 21st Century (MT21) compared prospective teachers’ knowledge and beliefs about teaching and learning in six participant countries, reporting that Korean prospective secondary mathematics teachers were better prepared than those in other countries. In this context, this study has examined the curricula for mathematics teacher education and teacher employment tests in order to investigate the social expectation for teacher professionalization in Korea, particularly focusing on teacher knowledge. The analysis shows that while elementary mathematics teacher education emphasizes pedagogical knowledge, the secondary mathematics education curricula are highly content knowledge oriented. However, the secondary mathematics teacher education includes various aspects of pedagogical content knowledge in its curricula and teacher employment test. This research also identifies the discourse concerning mathematics instruction for diversity and equity and the emphasis of reflective practice as the significant development of the current Korean teacher education.     

Pourquoi, pour qui enseigner les mathématiques?

The evolution of education in Belgium is described from 1830 to our days. The mastery of education has always been a subject of disputes between the political forces in presence. With the flow of years, more attention is paid to social considerations and the necessity of a more democratic educational system arises. Simultaneously the pedagogical ideas change and active methods of teaching are promoted, although not always used. Different schoolsystems still coexist. However, except for philosophical and religious education, the differences between them tend to diminish. This is in particular the case for mathematics teaching whose evolution is also described with some details.     

The relationship of theory and practice in mathematics teacher professional development: an activity theory perspective

This paper describes certain interactions between the activity of “teaching” and the activity of “researching” in which the teachers participate in a 52-h professional development course aiming to introduce them to research in mathematics education as a tool for inquiry in their teaching. The teachers are involved in different research tasks such as: reading and presenting research papers; analyzing classroom dialogues and tasks; and designing, implementing, and evaluating collaboratively a classroom teaching intervention. From an activity theory (AT) perspective, and in particular Engeström’s (2001) third generation AT, distinguish two activity systems, the activity system of research and the activity system of teaching, to identify links that the participating teachers make. These links indicate the development of an inquiring stance to mathematics teaching and learning as a means of professional learning.