Mathematics teachers’ support and retention: using Maslow's hierarchy to understand teachers’ needs

As part of a larger study, four mathematics teachers from diverse backgrounds and teaching situations report their ideas on teacher stress, mathematics teacher retention, and their feelings about the needs of mathematics teachers, as well as other information crucial to retaining quality teachers. The responses from the participants were used to develop a hierarchy of teachers’ needs that resembles Maslow's hierarchy, which can be used to better support teachers in various stages of their careers. The interviews revealed both non content-specific and content-specific needs within the hierarchy. The responses show that teachers found different schools foster different stress levels and that as teachers they used a number of resources for reducing stress. Other mathematics-specific ideas are also discussed such as the amount of content and pedagogy courses required for certification.     

Impacting prospective teachers’ beliefs about mathematics

This study investigated: (1) the changes in the beliefs about mathematics held by 25 prospective elementary teachers as they went through a university mathematics course that aimed, among other things, to promote a problem-solving view about mathematics; and (2) the possible factors that accounted for the observed changes. The course incorporated specific features that prior research suggested reflect successful mechanisms for belief change (e.g., cognitive conflict). The data included students’ reflections, and responses to prompts and interview questions. Analysis of the data revealed the following major trends: (1) a movement towards a problem-solving view from the more traditional Platonist and instrumentalist views; and (2) no change in students’ initial views. Activities creating cognitive conflict, as well as the implementation of instruction valuing group collaboration and explanations, appear to have played important roles in the process of belief change. The findings have implications for research on teacher beliefs and teacher education.     

Pre-service mathematics teachers’ concept images of radian

This study investigates pre-service mathematics teachers’ concept images of radian and possible sources of such images. A multiple-case study was conducted for this study. Forty-two pre-service mathematics teachers completed a questionnaire, which aims to assess their understanding of radian. Six of them were selected for individual interviews on the basis of theoretical sampling. The data indicated that participants’ concept images of radian were dominated by their concept images of degree. As the data in this study suggested, pre-service mathematics teachers were reluctant to accept trigonometric functions with the inputs of real numbers but rather they use value in degrees. More interestingly, they have two distinct images of π : π as an angle in radian and π as an irrational number.     

Primary teachers’ subject matter knowledge: decimals

The main objective of this study was to investigate primary teachers’ subject matter knowledge in the domain of decimals and more elaborately to investigate their performance and difficulties in reading scale, ordering numbers, finding the nearest decimal and doing operations, such as addition and subtraction. The difficulties in these particular areas are analysed and suggestions are made regarding their causes. Further, factors that influence this knowledge were explored. The sample of the study was 63 primary teachers. A decimal concepts test including 18 tasks was administered and the total scores for the 63 primary teachers ranged from 3 to 18 with a mean and median of 12. Fifty per cent of the teachers were above the mean score. The detailed investigation of the responses revealed that the primary teachers faced similar difficulties that students and pre-service teachers faced. Discrepancy on teachers’ knowledge revealed important differences based on educational level attained, but not the number of years of teaching experience and experience in teaching decimals. Some suggestions have been made regarding the implications for pre- and in-service teacher training.     

Zimbabwean in-service mathematics teachers’ understanding of matrix operations

In Zimbabwe, school pupils study matrix operations, a topic that is usually covered as part of linear algebra courses taken by most mathematics undergraduate students at university. In this study we focused on Zimbabwean teachers who were studying the topic at university while also teaching the topic to their high school pupils. The purpose of the study was to explore the mental conceptions of matrix operations concepts of a sample of 116 in-service mathematics teachers. The Action Process Object Schema (APOS) theoretical framework describes the development in understanding of mathematics concepts through the hierarchical growth of mental constructions called action, process, object and schema. The results showed that many of the participants had interiorized actions on matrix operations of addition, scalar multiplication and matrix multiplication into processes. However, more than 50% of the participants struggled with scalar multiplication of a row matrix by a column matrix. In terms of notational errors, some participants could not distinguish between brackets that denote a matrix and that of a determinant, while some used the equal sign as an operator symbol and not as one denoting equivalence between two objects. It is recommended that future in-service teacher programs should try to create more structured opportunities to allow participants to engage more deeply with these concepts.     

Kindergarten teachers’ accounts of their developing mathematical practice

This study explores kindergarten teachers?? accounts of their developing mathematical practice in the context of their participation in a developmental research project. Observations and interviews were analysed to elaborate the accounts as regards orchestrating mathematical activities in the kindergarten. A co-learning agreement was established as collaboration between the kindergarten teachers and researchers. The study reveals that the kindergarten teachers argue that they have been empowered in developing an inquiry stance towards mathematics and mathematical activities. Taking an inquiry stance, they claim, has increased their awareness of the mathematics involved in activities, and enabled them to be more explicit when communicating mathematical ideas to children. An adjusted didactic triangle within the kindergarten setting is proposed based on these results.     

Prospective teachers’ unified understandings of the structure of identities

United States curriculum standards advise mathematics teachers to teach students to attend to structure and understand how mathematical concepts are related. This requires teachers to have a structural perspective and a coherent, unified understanding of mathematical structures that span curricula. This study explores Prospective Secondary Mathematics Teachers’ (PSMTs) unified understandings of identities and characterizes the structural features of identities that PSMTs attend to. I contribute a theoretical framework of three ways in which PSMTs reason about identities: a do-nothing element, a result of undoing something, and a coordination with inverse, binary operation, and set. I classify the level of coherence of their identity schemas demonstrated as they reasoned about the structural connections among additive, multiplicative, and compositional identities. I illustrate how having unified, coherent understandings of identities can lead PSMTs to reason productively about inverse and identity functions, while having incoherent understandings of identities can lead to inaccurate reasoning about inverse and identity functions. I conclude with teaching implications for fostering PSMTs’ unified understandings of algebraic concepts.     

Prospective elementary teachers’ claiming in responses to false generalizations

When faced with a false generalization and a counterexample, what types of claims do prospective K-8 teachers make, and what factors influence the type and prudence of their claims relative to the data, observations, and arguments reported? This article addresses that question. Responses to refutation tasks and cognitive interviews were used to explore claiming. It was found that prospective K-8 teachers’ claiming can be influenced by knowledge of argumentation; knowledge and use of the mathematical practice of exception barring; perceptions of the task; use of natural language; knowledge of, use of, and skill with the mathematics register; and abilities to technically handle data or conceptual insights. A distinction between technical handlings for developing claims and technical handlings for supporting claims was made. It was found that prudent claims can arise from arguer-developed representations that afford conceptual insights, even when searching for support for a different claim.     

Prospective mathematics teachers’ understanding of the base concept*

The purpose of this study is to analyze what kind of conceptions prospective mathematics teachers¹1. prospective mathematics teachers (hereafter referred to as PMTs)View all notes(PMTs) have about the base concept²2. base concept (hereafter referred to as BC)View all notes(BC). One-hundred and thirty-nine PMTs participated in the study. In this qualitative research, data were obtained through open-ended questions, the semi-structured interviews and pictures of geometric figures drawn by PMTs. As a result, it was determined that PMTs dealt with the BC in a broad range of seven different images. It was also determined that the base perception of PMTs was limited mostly to their usage in daily life and in this context, they have position-dependent and word-dependent images. It was also determined that PMTs named the base to explain the BC or paid attention to the naming of three-dimensional geometric figures through the statement: ‘objects are named according to their bases’. At the same time, it was also determined that PMTs had more than one concept images³3. concept images (hereafter referred to as CIs)View all noteswhich were contradicting with each other. According to these findings, potential explanations and advices were given.     

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.     

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.     

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.     

The missing link in teachers’ knowledge about common fractions division

The current study explored the difficulties teachers encounter when teaching common fractions division, focusing on teachers’ knowledge concerning this issue. Nine teachers who study towards a M.Ed. degree in mathematics education demonstrated the algorithms they apply in order to solve fractions division problems, described how they teach the subject, and attempted to explain a student's mistake, in understanding a word problem involving dividing by fraction. The findings indicate there is a missing link in the teachers’ pedagogical capability, stemming from insufficient content knowledge. They presented different solution algorithms and reported using constructivist teaching methods, yet the methods they described couldn't lead a student to understand the logic behind the algorithm they teach (invert-and-multiply – multiplication by an inverse number, in accordance with the requirements of the curriculum). Furthermore, the participating teachers did not possess specialized mathematics content knowledge (SCK) and knowledge of content and students (KCS), enabling them to identify the source of a student's misconception.     

Developing prospective teachers’ covariational reasoning through a model development sequence

Forming part of a wider research study, the current study investigated prospective middle school mathematics teachers’ ways of covariational reasoning on tasks involving simultaneously changing quantities. As the introductory theme of a larger unit on derivative, a model development sequence on covariational reasoning was designed and experimented with 20 participants in a mathematical modeling course offered to prospective teachers. The participants’ developing abilities of covariational reasoning were documented under three categories: (i) identifying the variables, (ii) ways of coordinating the variables, and (iii) ways of quantifying the rate of change. The results revealed significant improvement in the prospective teachers’ ways of identifying and coordinating the variables, and in quantifying the rate of change. Moreover, the results indicated that preference for a particular way of thinking in identifying and coordinating the variables determined the prospective teachers’ way of quantifying the rate of change and thereby their level of covariational reasoning.     

Characterizing student mathematics teachers’ levels of understanding in spherical geometry

This article presents an exploratory study aimed at the identification of students’ levels of understanding in spherical geometry as van Hiele did for Euclidean geometry. To do this, we developed and implemented a spherical geometry course for student mathematics teachers. Six structured, task-based interviews were held with eight student mathematics teachers at particular times through the course to determine the spherical geometry learning levels. After identifying the properties of spherical geometry levels, we developed Understandings in Spherical Geometry Test to test whether or not the levels form hierarchy, and 58 student mathematics teachers took the test. The outcomes seemed to support our theoretical perspective that there are some understanding levels in spherical geometry that progress through a hierarchical order as van Hiele levels in Euclidean geometry.     

Examining middle school pre-service teachers’ knowledge of fraction division interpretations

This study investigated 11 pre-service middle school teachers’ solution strategies for exploring their knowledge of fraction division interpretations. Each participant solved six fraction division problems. The problems were organized into two sets: symbolic problems (involving numbers only) and contextual problems (involving measurement interpretation and the determination of unit rate interpretation). Results showed that most of the participants exhibited a great amount of procedural knowledge as they applied algorithms to obtain the correct answers to the symbolic problems. They also exhibited a great amount of conceptual understanding of how and why they obtained the correct answers to the contextual problems. However, the pre-service middle school teachers neither provided interpretations to the symbolic problems nor accepted that the contextual problems involved fraction division operation. The results suggest that the measurement and rate concepts were often unlinked to fraction division.     

Preservice teachers’ pictorial strategies for a multistep multiplicative fraction problem

Previous research has documented that preservice teachers (PSTs) struggle with understanding fraction concepts and operations, and misconceptions often stem from their understanding of the referent whole. This study expands research on PSTs’ understanding of wholes by investigating pictorial strategies that 85 PSTs constructed for a multistep fraction task in a multiplicative context. The results show that many PSTs were able to construct valid pictorial strategies, and the strategies were widely diverse with respect to how they made sense of an unknown referent whole of a fraction in multiple steps, how they represented the wholes in their drawings, in which order they did multiple steps, and which type of model they used (area or set). Based on their wide range of pictorial strategies, we discuss potential benefits of PSTs’ construction of their own representations for a word problem in developing problem solving skills.     

Improving teachers�� expertise in mathematics instruction through exemplary lesson development

This study examined how two selected expert teachers improved their expertise in mathematics instruction through participating in the development of exemplary lessons throughout the years. The main data for this study included the lesson designs at two crucial stages (with relevant video-taped lessons), teachers?? reflection reports, written surveys, and a phone interview. These two case studies showed that the teachers continuously developed their proficiency in the following four aspects: obtaining a better understanding of content knowledge; becoming more skillful in addressing difficult content points; having a more purposeful organization of problem sequences; and developing more comprehensive and feasible instructional objectives. Both teachers appreciated the learning experience from outside experts?? critical feedback, collaborative teaching experiments, self-reflection on teaching, and helping other teachers. They also realized a tension between exemplary lesson development and the reality of examination-driven teaching.     

Ways of promoting the sustainability of mathematics teachers’ professional development

This paper deals with the sustainable effectiveness of professional development programmes. Based on a review of literature and research findings, the following questions are raised: What is regarded as an effective way of promoting mathematics teachers’ sustainable professional development? Which levels of impacts are aimed at? What are the factors promoting the effectiveness of professional development programmes? Regarding these questions, the article links theoretical considerations with research findings from a case study. A secondary mathematics teacher, taking part in a teacher professional development programme in 2002, was revisited in 2005 and 2010 to gather data regarding the sustainable impact of the programme. The case study’s results provide information about the teacher’s professional growth and lead to a discussion of implications for mathematics teachers’ professional development and teacher education in general.