A comparison of Hungarian and English teachers' conceptions of mathematics and its teaching

This paper reports on a statistical study of English and Hungarian teachers' conceptions of mathematics and its teaching. A questionnaire was developed and distributed to teachers of mathematics in 200 English and40 Hungarian schools teaching children in the 11–14 age range. Factor analyses identified four conceptions of mathematics and five of mathematics teaching. These were compared with those yielded by an earlier study involving the same English teachers and found to be consistent indicating the existence of similar conceptions in different educational systems. Differences and similarities in the strengths with which those conceptions are held were suggestive of both global and national conceptual traditions. The significant similarity to emerge concerned teachers from both countries sharing, with similar strengths, a general conception of mathematics teaching incorporating the teaching of mathematical skills, a variety of classroom approaches including investigations and problem-solving, and a recognition that mathematics provides an essential lifetool. Multi-dimensional scaling indicated that English teachers have their perspectives informed by two underlying, and possibly conflicting, traditions– pedagogic relevance and mathematical utility. The Hungarians appeared concerned only with notions of pedagogic relevance – those practices perceived to facilitate effective learning of a subject which is untainted by utilitarian perspectives. This revised version was published online in July 2006 with corrections to the Cover Date.     

Mastery mathematics: Changing teacher beliefs around in-class grouping and mindset

Internationally, reform efforts in teaching of mathematics have found it difficult to change practice. This study used classroom video stimulated recall interviews with Primary teachers in England to investigate their beliefs during implementation of a textbook-based South Asian mastery approach to teaching mathematics. The self-reported beliefs of the teachers showed their support for change in practice, from in-class grouping by prior attainment to whole class teaching with everyone exploring the same problem, and suggested conceptions of malleable intelligence in the specific domain of school maths.     

The emerging of teachers' conceptions of new subjects inserted in mathematics programs: the case of informatics

The changes in mathematical curricula induced by the introduction of informatics in school represent the general framework of this research. In particular we focus on the teacher's role by analysing the different choices taken by mathematics teachers when faced with a curriculum reform induced by the introduction of informatics in secondary school courses (age 14–16). Our hypothesis is that these choices are the consequence of conceptions teachers have about informatics and its teaching in relation to the teaching of mathematics. Thus, through a case study research method, we focus on mathematics teachers' conceptions of informatics and its teaching. An attempt is made at outlining a typology of these conceptions, based on the different orientations identified.     

Prospective teachers' initial conceptions about pupils' understanding of science and mathematics

Earlier reports have shown that prospective teachers' conceptions about teaching science to a high degree are resistant and do not change substantially during the teacher‐training programme. In our investigation we elucidate the prospective teachers' initial conceptions about pupils' understanding of science and mathematics. We applied ‘The Lesson Preparation Method' and used a phenomenographic approach in order to reveal the range of conceptions that the prospective teachers hold. A third of the prospective teachers did not consider pupils' conceptions when planning lessons. The rest of the 32 participants expressed awareness; some of the prospective teachers even referred to subject‐specific teaching experience. Also regarding the prospective teachers' conceptions about pupils' knowledge and beliefs, as well as about pupils' difficulties, there was a significant diversity. By raising these issues about pedagogical content knowledge the prospective teachers' conceptions can be extended and developed during the education.     

Evaluating prospective elementary teachers' understanding of science and mathematics in a model preservice program

Two cohorts of students preparing to become elementary teachers participated in a model program in science and mathematics. These students were compared to other students taking similar courses on their conceptual understandings of science and mathematics, their investigative proficiencies, and their beliefs about effective methods of teaching these subjects. Instruments included newly developed tests of understanding, existing standardized achievement tests, and instruments devised for the evaluation to elicit conceptions of appropriate ways to teach science and mathematics to elementary children. Results from individual courses indicated that students participating in the model program developed more thorough understandings and more reform-minded beliefs related to teaching science and mathematics. Issues associated with the assessment and evaluation of innovative programs in science and mathematics are discussed, and recommendations for teacher preparation are offered.     

Learning environment,self-efficacy for teaching mathematics,and beliefs about mathematics

Learning environment research has a long history of significant relationships between the learning environment and student outcomes. This study investigated relationships between the learning environments of mathematics courses in a teacher education program and two outcomes, mathematics teaching self-efficacy and beliefs about mathematics. These two outcomes have been repeatedly shown to influence the future teaching practices of preservice teachers but, to date, their relationships with learning environment have been investigated neither with preservice teachers nor at all in the United Arab Emirates or wider Gulf region. The learning environment was found to be significantly related to preservice teachers’ mathematics teaching self-efficacy and beliefs about mathematics. Learning environments perceived more favourably by preservice teachers were associated with higher self-efficacy for teaching mathematics, but also with more-traditional beliefs, making these findings important for higher education institutes and teacher educators.

     

Building teacher capacity for implementing curricular coherence: mathematics teacher professional development tasks

Improving mathematics education in the United States has taken many forms. Our work has focused on two aspects: the content knowledge of teachers and a well-articulated coherent curriculum. Our aim was teacher “capacity building” that is enabling teachers to teach to coherent and significant mathematical curricular goals and describe the implementation in a large-scale project based at Michigan State University. We highlight the design, structure and use of mathematics teacher learning tasks that were intended to improve teachers’ capacity to teach to these goals and note how the teachers’ perceptions of the structure and sequencing of mathematics itself affect the ways they organize mathematics in their teaching and the ways they teach.     

The influences of conceptions of mathematics and self‐directed learning skills on university students' achievement in mathematics

This study tested the mediating role of self‐directed learning skills (SDL) between students’ conceptions of mathematics and their achievement in mathematics using a structural equation model. Data were collected using the “Conceptions of Mathematics Questionnaire” and the “Self‐Rating Scale of Self‐Directed Learning”, together with students’ achievement in mathematics. A sample of 440 first year university students at King Saud University participated in the study. The findings confirm the moderating role of students’ self‐directed learning skills between their conceptions of mathematics and their achievement in mathematics. Students who have a highly fragmented conception of mathematics tended to have low SDL skills, and in turn low mathematics achievement (partial mediation), whereas students who have a highly cohesive conception of mathematics tended to have high self‐directed learning skills, and in turn high mathematics achievement (full mediation). Mathematics educators should be aware that students’ conceptions of mathematics may be influential, but not sufficient to predict achievement in mathematics. Equipping students with appropriate conceptions of mathematics and self‐directed learning skills is key to enhancing their performance in mathematics.     

The Interfacing of Preservice and Inservice Experiences of Reform-Based Teaching: A Longitudinal Study

This article contrasts four elementary teachers who were graduates of a teacher education program that incorporated a reform-based mathematics methods course. The report provides results from a four-year longitudinal study that extended from the time that the participants were preservice teachers until the end of their second year of teaching. The article provides background information of each teacher, vignettes from her teaching, excerpts from interviews, and an analysis of each teacher's case. Results from the case studies indicate that two of the four teachers sustained their cognitively-based conceptions about mathematics teaching and learning, and implemented these conceptions into practice. The analysis suggests that there were several factors that influenced the teachers' conceptions and the choices they made in their teaching: personal commitment, professional strength, curriculum, planning, assessment, beliefs, knowledge, and support from school administration. The article concludes with implications for teaching and questions about the nature of what might be required in the beginning years of teaching if new teachers are expected to implement reform-based mathematics teaching practices. This revised version was published online in August 2006 with corrections to the Cover Date.     

Teaching the language of mathematics at three levels of an English-medium primary school

This study, situated in a multilingual, English-medium educational context, draws on theory from mathematics and language education to capture teachers’ perspectives on the place of language in their mathematics pedagogy. The benchmark study explored this topic through surveying and interviewing teachers. Additionally, it sought to relate teachers’ views to their practice by focusing on observing three teachers’ mathematics lessons at primary one, three, and five. Findings are that mathematics teachers placed importance on teaching language, being specifically concerned with language as input and comprehension. They taught vocabulary and reading skills in supportive ways explicitly yet differently at the three grade levels. Particularly at the lower levels, teachers contextualised language in the concrete examples employed for mathematics teaching. At all three levels, prominence was given to teaching pupils how to read word problems as well as how to solve them. However, at primary three, a tension was observed between the two aims of teaching mathematical vocabulary and teaching the reading skills for word problems. This paper illustrates the tension and discusses its possible causes.     

A Study of Prospective Primary Teachers’ Conceptions of Teaching And Learning School Geometry

There are two basic guiding referents in the present investigation: the teaching and learning of school-level geometry (basic concepts of Euclidian geometry and measure) and the analysis of the conceptions of prospective primary teachers in Spain. The work assumes that such conceptions appear and develop during school years, and that consequently, for the prospective teachers to learn to teach mathematics, account must be taken of the requirements that arise from these conceptions themselves and from the prospective teachers’ knowledge of school-level mathematics. The objective of the study was to describe and analyse prospective primary teachers’ conceptions of school-level geometry and it teaching and learning. To this end, we considered the hypothesis that the students’ memories and expectations provide information with which to characterize their conceptions of the field of geometry and its teaching/learning at the primary school level.     

Using mathematics strategies in early childhood education as a basis for culturally responsive teaching in India

The objective of this small study was to elicit responses from early childhood teachers in India on mathematics learning strategies and to measure the extent of finger counting technique adopted by the teachers in teaching young children. Specifically, the research focused on the effective ways of teaching mathematics to children in India, and examined teachers’ approach to number counting. In India, children were taught by their parents or by their teachers to use fingers to count. The qualitative study conducted by the researcher further enriched the topic with first‐hand comments by the teachers. Although the finger counting method was not the only process that teachers would adopt, it was embedded in the culture and taken into consideration while infusing mathematics skills. The teachers confirmed adopting the Indian method of finger counting in their teaching strategy; some specified that the method helped children to undertake addition and subtraction of carrying and borrowing, as counting by objects could not be available all the time. Although the study is limited by its small sample to the unique mathematics learning experience in India, it provides readers with a glimpse of culturally responsive teaching methods and an alternative mathematics teaching strategy.     

The role of university-based induction in beginning elementary teacher enactment of effective mathematics teaching

The purpose of this paper is to describe the school-based contextual constraints perceived by beginning elementary teachers as they attempted to enact their visions of effective mathematics teaching and the components of a mathematics-specific, university-based induction program designed to support elementary teachers in their teaching of mathematics. A significant number of beginning teachers in the program indicated that curriculum pacing guides and grade level teachers often discouraged their enactment of effective mathematics teaching. We describe the components of the induction program in general and then discuss the specific ways that program mentors, who were elementary teachers with expertise in mathematics teaching, supported beginning teachers in navigating the constraints identified. Data suggest that supporting teachers’ pedagogical agency to navigate local school contexts should be an additional consideration in teacher preparation and induction programs.     

Conceptions of mathematics and how it is learned: The perspectives of students entering university

This paper reports results from an investigation to identify the conceptions of mathematics held by beginning university students and their approaches to the study of mathematics. Phenomenographic techniques were used to analyze responses to a questionnaire administered to approximately 300 students. An analysis of the results identified a structural relationship between students' conceptions of mathematics and their approaches to learning it, with the majority of students viewing mathematics as a necessary set of rules and procedures to be learned by rote. The results of this research have implications for the ways in which teaching and learning are constituted within universities.     

Worldviews,religions, and beliefs about teaching and learning: perception of mathematics teachers with different religious backgrounds

Beliefs about mathematics education and their influences on teaching practices have been widely investigated in recent decades. There have been numerous empirical studies on the influences of religions on teachers’ and students’ beliefs about subjects such as sciences and language. However, the influences of worldviews in general and religions in particular, as one of the major sources of beliefs in relation to mathematics education, are under-researched. The current study is a first step to unpacking the relationship between teachers’ religions and their beliefs about mathematics teaching and learning. By means of semi-structured interviews with mathematics teachers of different religious backgrounds, teachers’ perceptions on the connection between their personal religious beliefs and their beliefs about teaching and learning are investigated. In-depth analyses of the perceptions of three mathematics teachers reveal the complex relationship between teachers’ religious beliefs and their teaching beliefs. First, there are some common values shared by different religions, which influence the beliefs about mathematics teaching and learning as well as education in general. Second, religion is a rich belief system, and the teachers appear to apply only a portion of their religious beliefs to guide their teaching. It is also possible that a teacher is influenced by more than one religion or cultural tradition. Despite its subtleties, our study provides evidence to support the alignment between teachers’ personal religious beliefs and their beliefs about mathematics teaching and learning.     

Structural and conceptual interweaving of mathematics methods coursework and field practica

This paper describes a study of observed relationships between the design of a preservice elementary mathematics methods course with accompanying field practicum and changes in the extent to which participating prospective teachers identified themselves with the mathematics reform movement after becoming practicing teachers. The curriculum of the course with its embedded field practicum experiences was designed to support prospective teachers in interweaving methods coursework and pedagogical instruction with classroom practice. University and elementary school structures were interwoven by conducting weekly mathematics methods course sessions at a school site followed immediately by related practicum experiences in classrooms in the school. The interweaving of conceptualizations about mathematics teaching and learning was facilitated as the methods instructor provided professional development for classroom teachers simultaneously with the methods course, encouraging the teachers to create “reform-friendly” classroom environments. Survey data from 68 prospective teachers involved in the project were analyzed to identify and describe relationships between course/practicum experiences and the degree to which they as practicing teachers, 1–3 years following the course/practicum, identified with a reform mathematics perspective, as well as to examine factors influencing those relationships.     

A capstone mathematics course for prospective secondary mathematics teachers

This article describes an innovative capstone mathematics course that links college mathematics with school mathematics and pedagogy. It describes how college juniors in a secondary mathematics teacher preparation program engage in leadership experiences that enable them to learn mathematics for teaching while developing student-centered pedagogical and assessment strategies. Examples show how students teach the class, complemented by excerpts from student journals containing students?? perceptions of their learning. Preliminary research indicates the effectiveness of this course for preparing future mathematics teachers.     

Building on student mathematical thinking in whole-class discourse: exploring teachers’ in-the-moment decision-making,interpretation, and underlying conceptions

Often, mathematics teachers do not incorporate whole-class discourse of students’ various ideas and solution methods into their teaching practice. Particularly complex is the in-the-moment decision-making that is necessary to build on students’ thinking and develop their collective construction of mathematics. This study explores the decision-making patterns of five experienced Dutch mathematics teachers during their novice attempts at orchestrating whole-class discourse concerning students’ various solution methods. Our goal has been to unpack the complexity of their in-the-moment decision-making during whole-class discourse through lesson observations and stimulated recall interviews. We investigated teacher decision-making adopting a model that combines two perspectives, namely (1) we explored student-teacher interaction with regard to building on student thinking and (2) we explored how the teachers based decisions during such interaction upon their own personal conceptions and interpretation of student thinking. During these novice attempts at orchestrating whole-class discourse, the teachers created many situations for students to articulate their thinking. We found that at certain instances, teachers’ in-the-moment decision-making resulted in opportunities to build on student thinking that were not completely seized. During such instances, the teachers’ decision-making was shaped by the teachers’ own conceptions of the relevant mathematics and by teacher conceptions that centered around student understanding and mathematical goals. Our findings suggest that teachers might be supported in their novice attempts at whole-class discourse by explicit discussion of the mathematics and of their conceptions with regard to student understanding and mathematical goals.

     

Linking university mathematics classroom environments to student achievement: The mediation of mathematics beliefs

This study examined the relationships between university mathematics classroom environments, mathematics beliefs, and achievement in the context of mainland China. As a type of affect in mathematics learning, mathematics beliefs including mathematics conceptions and efficacy were examined as mediators. Based on analysis of a sample of 1443 undergraduate students from a university in northern China, the study not only found that the mathematics classroom environment was associated with both affective and cognitive mathematics outcomes, but also confirmed the mediating role of mathematics beliefs. Specifically, a desirable classroom environment was related to students’ cohesive conceptions, enhanced self-efficacy, and higher mathematics achievement. In contrast, students’ perceived difficulty of learning mathematics was connected to their fragmented conceptions, reduced self-efficacy and lower mathematics achievement. Implications of the study are discussed.     

Investigating the Influence of a Mixed Face-to-Face and Website Professional Development Course on the Inquiry-Based Conceptions of High School Science and Mathematics Teachers

The purposes of this study are to report the influences of a mixed delivery professional development [PD] course involving face-to-face classes and the mentoring assisted inquiry-based teaching [MAIT] website that addressed the conceptual change and self-efficacy of high school mathematics and science teachers’ conceptions of inquiry-based teaching. Twenty-five in-service high school science and mathematics teachers participated in a 9 weeks, 4 h per week PD course. Data collection included the pre-post Inquiry Teaching Efficacy Questionnaire [ITEQ], teachers’ reflective journals posted on the MAIT website, teachers’ assignments requested by the course instructor, and follow-up teachers’ interviews. Findings revealed that both mathematics and science teachers’ conceptions of inquiry teaching show significant improvements after the course. The mixed delivery PD course had the same influence on science and mathematics teachers. The authentic video clips and lesson plans provided on the MAIT website and the specific feedback toward teachers’ performance are the main factors that shaped teachers’ inquiry-based teaching conceptions.