The Interaction Between The Use of Ict And Mathematics Teachers’ Professional Knowledge Base for Teaching

With the increased availability of information and communications technology (ICT) in schools, it is important to examine how teachers use it in their instruction. This paper describes a study that involved a number of secondary mathematics teachers who had been using the new technology in their teaching for a number of years. It reports on some of the factors affecting teacher’ use of the new technology identified by analysing the data collected through questionnaires and teachers’ interviews. A possible theoretical framework to account for teachers’ learning about ICT and teachers’ incorporation of ICT into their planning for teaching mathematics is presented.     

Reflections on the promise and complexity of mathematics coaching

If students are to develop mathematical proficiency, then mathematics teaching must both change and improve. In an effort to provide site-based professional development addressing the mathematical content and pedagogical demands that teachers encounter in reality of public schooling, many school districts are turning to elementary mathematics coaches. Knowledgeable coaches can have a significant positive impact on teachers, yet this study documents substantial variance in the amount of coaching delivered and in the nature of activity that coaches undertake within schools. Coaches are frequently responsive to the needs of individual teachers. If this support is primarily marked by shared teaching or provision of instructional materials, it may not transform either instruction or teacher knowledge. Similarly if coaches assume duties that primarily address an administrator’s needs, they will have less time to enhance a school’s mathematics program. Coaches need to engage teachers in fundamental dialogue about mathematical content, mathematical learning, and student understanding. It may be that this dialogue and the effectiveness of a coach’s work with individual teachers would benefit from a coach’s concurrent work with grade-level teams. When a coach leads a grade-level team through discussion of targeted goals and approaches, the coach may facilitate individual teacher learning while building collective learning. When coupled with the support of a principal, this partnership may foster instructional change across a school.     

Making mathematics and science integration happen: key aspects of practice

The integration of mathematics and science teaching and learning facilitates student learning, engagement, motivation, problem-solving, criticality and real-life application. However, the actual implementation of an integrative approach to the teaching and learning of both subjects at classroom level, with in-service teachers working collaboratively, at second-level education, is under-researched due to the complexities of school-based research. This study reports on a year-long case study on the implementation of an integrated unit of learning on distance, speed and time, within three second-level schools in Ireland. This study employed a qualitative approach and examined the key aspects of practice that impact on the integration of mathematics and science teaching and learning. We argue that teacher perspective, teacher knowledge of the ‘other subject’ and of technological pedagogical content knowledge (TPACK), and teacher collaboration and support all impact on the implementation of an integrative approach to mathematics and science education.     

The mathematical knowledge and beliefs of elementary mathematics specialist-coaches

This paper reports on one aspect of a larger research project conducted in the United States that designed and implemented an elementary mathematics, specialist-coach preparation program and evaluated the effect of qualified specialist-coaches on student achievement. The paper discusses a conceptual framework for coaching in which a specialist-coach is to serve as a “more knowledgeable other” for a community of practice in a school, and ultimately to impact both the knowledge and professional practice of teachers and the school’s mathematics program as a whole. Specialist-coaches have unique opportunities and challenges in this daunting task, and the paper discusses one program designed to prepare well-respected teachers for the transition to the role and responsibilities of a specialist-coach. The reported analyses document changes in specialist-coaches’ mathematical content knowledge, mathematical knowledge for teaching, and beliefs regarding mathematics teaching and learning over the preparation program and during the specialist-coaches’ first years of service in a school. These specialist-coaches’ mathematical content knowledge grew and their beliefs became more aligned with a Making Sense perspective during the preparation program, and their changed state persisted throughout 2–3 years of service as specialist-coaches. Evidence addressing the specialist-coaches’ mathematical knowledge for teaching was mixed, but suggested that growth occurred both during the preparation program and in their first year of coaching, stabilizing in the years following.     

A touch of mathematics: coming to our senses by observing the visually impaired

During the last three decades, the impact of visualization processes on the teaching and learning of mathematics has been extensively researched. However, considerably less work has been devoted to haptic processes. In this paper, we describe and analyze the role of haptic processes from data collected in mathematics lessons taught in a school for blind students. The analysis builds upon studies from perception and also from teaching experiences in order to examine the teacher’s and the students’ hand movements and metaphors when handling solids of revolution and communicating verbally with each other about their insights. We highlight the powerful combination of the visual and the haptic components of their interactions for the conceptualization of mathematical experiences, and we also note the critical role mathematical language plays in supporting the teaching–learning processes in this context. Finally, we consider important educational implications not only for blind people, but for all students and teachers of mathematics.     

School mathematics curriculum materials for teachers’ learning: future elementary teachers’ interactions with curriculum materials in a mathematics course in the United States

This report describes ways that five preservice teachers in the United States viewed and interacted with the rhetorical components (Valverde et al. in According to the book: using TIMSS to investigate the translation of policy into practice through the world of textbooks, Kluwer, 2002) of the innovative school mathematics curriculum materials used in a mathematics course for future elementary teachers. The preservice teachers’ comments reflected general agreement that the innovative curriculum materials contained fewer narrative elements and worked examples, as well as more (and different) exercises and question sets and activity elements, than the mathematics textbooks to which the teachers were accustomed. However, variation emerged when considering the ways in which the teachers interacted with the materials for their learning of mathematics. Whereas some teachers accepted and even embraced changes to the teaching–learning process that accompanied use of the curriculum materials, other teachers experienced discomfort and frustration at times. Nonetheless, each teacher considered that use of the curriculum materials improved her mathematical understandings in significant ways. Implications of these results for mathematics teacher education are discussed.     

Changed views on mathematical knowledge in the course of didactical theory development—independent corpus of scientific knowledge or result of social constructions?

The study tries to show one line of how the German didactical tradition has evolved in response to new theoretical ideas and new—empirical—research approaches in mathematics education. First, the classical mathematical didactics, notably ‘stoffdidaktik’ as one (besides other) specific German tradition are described. The critiques raised against ‘stoffdidaktik’ concepts [for example, forms of ‘progressive mathematisation’, ‘actively discovering learning processes’ and ‘guided reinvention’ (cf. Freudenthal, Wittmann)] changed the basic views on the roles that ‘mathematical knowledge’, ‘teacher’ and ‘student’ have to play in teaching–learning processes; this conceptual change was supported by empirical studies on the professional knowledge and activities of mathematics teachers [for example, empirical studies of teacher thinking (cf. Bromme)] and of students’ conceptions and misconceptions (for example, psychological research on students’ mathematical thinking). With the interpretative empirical research on everyday mathematical teaching–learning situations (for example, the work of the research group around Bauersfeld) a new research paradigm for mathematics education was constituted: the cultural system of mathematical interaction (for instance, in the classroom) between teacher and students.     

A coloured window on pre-service teachers’ conceptions of rational numbers

In undergraduate mathematics courses, pre-service elementary school teachers are often faced with the task of re-learning some of the concepts they themselves struggled with in their own schooling. This often involves different cognitive processes and psychological issues than initial learning: pre-service teachers have had many more opportunities to construct understandings and representations than initial learners, some of which may be more complex and engrained; pre-service teachers are likely to have created deeply-held–and often negative–beliefs and attitudes toward certain mathematical ideas and processes. In our recent research, we found that pre-service teachers who used a particular computer-based microworld, one emphasising visual representations of and experimental interactions with elementary number theory concepts, overcame many cognitive and psychological difficulties reported in the literature. In this study, we investigate the possibilities of using a similarly-designed microworld that involves a set of rational number concepts. We describe the affordances of this microworld, both in terms of pre-service teacher learning and research on pre-service teacher learning, namely, the helpful “window” it gave us on the mathematical meaning-making of pre-service teachers. We also show how their interactions with this microworld provided many with a new and aesthetically-rich set of visualisations and experiences.     

Expanding in-service mathematics teachers’ horizons in creative work using technology

This article describes the experiences gained from a seminar in the teaching of mathematical reasoning and problem solving designed to prepare in-service high school mathematics teachers to teach genuine mathematical activity in a computer-based environment. Presented with a set of unfamiliar tasks and activities, the participants were encouraged to investigate each of them, using the Geometer's Sketchpad software, and then to justify their assertions accordingly. In the exploratory process the student teachers make the major mathematical contributions while the teacher plays the role of facilitator. The mathematics teachers began to realize the power of technology in teaching mathematics and were pleased to return to their classrooms with a great number of experiences and ideas for immediate use.     

Conception of expert mathematics teacher: a comparative study between Hong Kong and Chongqing

This paper reports the similarities and differences in how “expert mathematics teacher” is conceptualized by mathematics educators in Hong Kong and Chongqing, two cities in China which share similar but different cultural and social backgrounds. Thirty-seven mathematics education researchers, school principals with mathematics education background, and mathematics teachers were interviewed on their perceptions of expert mathematics teacher. It is found that in both cities an expert mathematics teacher should have a profound knowledge base in mathematics, teaching, and students; strong ability in teaching; and a noble personality and a spirit of life-long learning. As for differences, an expert mathematics teacher should have the ability to conduct research, mentor other teachers, and have profound knowledge of examination and educational theories in Chongqing. These attributes were not found in Hong Kong. These similarities and differences are discussed, and relevant social and cultural factors in the two contexts are examined.     

Teaching and Learning Mathematics With Interactive Spreadsheets

Learning to teach mathematics at the middle and secondary levels should include many opportunities for teachers to learn how to use technology to better understand mathematics themselves and promote students' learning of mathematical concepts with technology-enabled pedagogy. This article highlights work done in a variety of preservice and in-service mathematics teacher education courses to help teachers use commonly available spreadsheets as an interactive exploratory learning tool. Several examples of teachers' subsequent use of spreadsheets in their own teaching are also discussed.     

Challenging Preservice Teachers' Mathematical Understanding: The Case of Division by Zero

Preservice elementary school teachers' fragmented understanding of mathematics is widely documented in the research literature. Their understanding of division by 0 is no exception. This article reports on two teacher education tasks and experiences designed to challenge and extend preservice teachers' understanding of division by 0. These tasks asked preservice teachers to investigate division by 0 in the context of responding to students' erroneous mathematical ideas and were respectively structured so that the question was investigated through discussion with peers and through independent investigation. Results revealed that preservice teachers gained new mathematical (what the answer is and why it is so) and pedagogical (how they might explain it to students) insights through both experiences. However, the quality of these insights were related to the participants' disposition to justify their thinking and (or) to investigate mathematics they did not understand. The study's results highlight the value of using teacher learning tasks that situate mathematical inquiry in teaching practice but also highlight the challenge for teacher educators to design experiences that help preservice teachers see the importance of, and develop the tools and inclination for, mathematical inquiry that is needed for teaching mathematics with understanding.     

Exploring the Use of New Representations as a Resource for Teacher Learning

An important goal of mathematics education reform is to support teacher learning. Toward this end, researchers and teacher educators have investigated ways in which teachers learn about mathematical content, pedagogical strategies, and student thinking as they implement reform. This study extends such work by examining how one elementary school and one high school teacher learned from students' interpretations of new conceptually based representations contained in instructional materials aligned with the Principles and Standards for School Mathematics ( National Council of Teachers of Mathematics, 2000 ). Results indicated that teaching with new representations provided a rich context for teacher learning at both the elementary and high school level, and three dimensions were identified along which such learning occurred. The results suggest that pedagogical content knowledge with respect to representations is an important facet of teacher cognition that should be studied in greater depth.     

Teachers' Considerations of Students' Thinking During Mathematics Lesson Design

Teachers' abilities to design mathematics lessons are related to their capability to mobilize resources to meeting intended learning goals based on their noticing. In this process, knowing how teachers consider Students' thinking is important for understanding how they are making decisions to promote student learning. While teaching, what teachers notice influences their decision‐making process. This article explores the mathematics lesson planning practices of four 4th‐grade teachers at the same school to understand how their consideration of Students' learning influences planning decisions. Case study methodology was used to gain an in‐depth perspective of the mathematics planning practices of the teachers. Results indicate the teachers took varying approaches in how they considered students. One teacher adapted instruction based on Students' conceptual understanding, two teachers aimed at producing skill‐efficient students, and the final teacher regulated learning with a strict adherence to daily lessons in curriculum materials, with little emphasis on student understanding. These findings highlight the importance of providing professional development support to teachers focused on their noticing and considerations of Students' mathematical understandings as related to learning outcomes. These findings are distinguished from other studies because of the focus on how teachers consider Students' thinking during lesson planning. This article features a Research to Practice Companion Article . Please click on the supporting information link below to access.     

Secondary-school teachers’ noticing of aspects of mathematics teaching talk in the context of one-day workshops

In a research project with one-day teacher education workshops for secondary-school mathematics teachers, our study explores the potential of tool-supported discussions in helping them to notice important and critical aspects of mathematics teaching talk. Mathematical practices of naming and explaining in teaching talk, students’ content learning challenges, and noticing processes of identifying, interpreting and deciding are the components of our framework and the tools that guided the design and implementation of three workshops on linear equations, fractions and plane isometries. The data was collected during the discussions with the seven teachers and the teacher educator throughout these workshops. The coding of the discussions allowed us to see discourse moves that reveal the teachers’ noticing of: (i) challenges in the identification of mathematical naming, (ii) mathematical explaining that voices the students’ learning, (iii) classroom practice in relation to mathematical naming and explaining.     

Mathematics teacher education advanced methods: an example in dynamic geometry

We present a research work about an innovative national teacher training program in France: the Pairform@nce program, designed to sustain ICT integration. We study here training for secondary school teachers, whose objective is to foster the development of an inquiry-based approach in the teaching of mathematics, using investigative potentialities of dynamic geometry environments. We adopt the theoretical background of the documentational approach to didactics. We focus on the interactions between teachers and resources: teachers’ professional knowledge influences these interactions, which at the same time yield knowledge evolutions, a twofold process that we conceptualise as a documentational genesis. We followed in particular the work of a team of trainees; drawing on the data collected, we analyse their professional development, related with the training. We observe intertwined evolutions and stabilities, consistent with ongoing geneses.     

Crutch or Catalyst: Teachers' Beliefs and Practices Regarding Calculator use in Mathematics Instruction

This study investigated K‐12 teachers' beliefs and reported teaching practices regarding calculator use in their mathematics instruction. A survey was administered to more than 800 elementary, middle and high school teachers in a large metropolitan area to address the following questions: (a) what are the beliefs and practices of mathematics teachers regarding calculator use? and (b) how do these beliefs and practices differ among teachers in three grade bands? Factor analysis of 20 Likert scale items revealed four factors that accounted for 54% of the variance in the ratings. These factors were named Catalyst Beliefs, Teacher Knowledge, Crutch Beliefs, and Teacher Practices. Compared to elementary teachers, high school teachers were significantly higher in their perception of calculator use as a catalyst in mathematics instruction. However, the higher the grade level of the teacher, the higher the mean score on the perception that calculator use may be a way of getting answers without understanding mathematical processes. The mean scores for teachers in all three grade bands indicated agreement that students can learn mathematics through calculator use and using calculators in instruction will lead to better student understanding and make mathematics more interesting. The survey results shed light on teachers' self reported beliefs, knowledge, and practices in regard to consistency with elements of the National Council of Teachers of Mathematics Principles and Standards for School Mathematics (2000) technology principle and the NCTM use of technology position paper (2003). This study extended previous research on teachers' beliefs regarding calculator use in classrooms by examining and comparing the results of teacher surveys across three grade bands.     

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.     

Measuring pre-service teachers’ self-efficacy in tutoring children in primary mathematics: an instrument

This article reports on the use of Rasch modelling to develop and validate an instrument measuring self-efficacy in tutoring children in primary mathematics (SETcPM). In response to the literature on teacher efficacy, the 20-item instrument aims to inform teacher educators, and is designed for novice pre-service teachers (nPSTs) preparing to teach mathematics in primary school (grades 1–7, ages 6–13). To ensure that the tasks of teaching are imaginable for nPSTs, the instrument targets the core activity of teaching mathematics: helping a generic child with mathematics tasks. We propose that SETcPM is measurable for the intended population and represents a central part of self-efficacy in teaching mathematics (SETM). Understanding the initial SETcPM of novices and mapping its development over the course of their training programme can contribute to a better understanding of SETM, and allow teacher educators to tailor their support.