A revised theorization of the relationship between teachers’ conceptions of mathematics and its teaching

We assembled the ideas about mathematics and about its teaching which were expressed by mathematicians and mathematics educators into two pairs of ‘official’ (collective) conceptions: mathematics is either static or dynamic, and mathematics teaching is either closed or open. These polar conceptions produce a 4-pair relationship between the conceptions of mathematics and its teaching. The adherence to official conceptions was tapped by a questionnaire encompassing 176 Israeli high school mathematics teachers, aimed at examining the relationship between their conceptions of mathematics and its teaching. The majority of these teachers either hold a single conception in one of the domains or do not adhere to any conception, and a quarter of them hold either the static-closed or dynamic-open pairs of conceptions that prevail among teachers in other countries. Consequently, we define a conception of an entity as a comprehensive and homogenous set of ideas about a particular characteristic or feature of that entity. Reality is that teachers practice their profession without adhering to any official conception, and perhaps are (/to be?/) praised for their reluctance to blindly adopt the clear-cut rigid official conceptions of mathematics and its teaching while maintaining their individual and independent blends of ideas.     

Mathematics teachers’ conceptions about modelling activities and its reflection on their beliefs about mathematics

The current study examines whether the engagement of mathematics teachers in modelling activities and subsequent changes in their conceptions about these activities affect their beliefs about mathematics. The sample comprised 52 mathematics teachers working in small groups in four modelling activities. The data were collected from teachers' Reports about features of each activity, interviews and questionnaires on teachers' beliefs about mathematics. The findings indicated changes in teachers' conceptions about the modelling activities. Most teachers referred to the first activity as a mathematical problem but emphasized only the mathematical notions or the mathematical operations in the modelling process; changes in their conceptions were gradual. Most of the teachers referred to the fourth activity as a mathematical problem and emphasized features of the whole modelling process. The results of the interviews indicated that changes in the teachers' conceptions can be attributed to structure of the activities, group discussions, solution paths and elicited models. These changes about modelling activities were reflected in teachers' beliefs about mathematics. The quantitative findings indicated that the teachers developed more constructive beliefs about mathematics after engagement in the modelling activities and that the difference was significant, however there was no significant difference regarding changes in their traditional beliefs.     

Integration of Science and Mathematics: A Theoretical Model

Interest in interdisciplinary, integrated curriculum development continues to increase. However, teachers, who have been given primary responsibility for developing these materials, are often working with little guidance. At present there exists no clear definition of the meaning of integration of mathematics and science. A continuum model of integration is proposed as a useful tool for curriculum developers as they create new integrated mathematics and science curricula or adapt commercially prepared materials. On the continuum, activities range from mathematics or science involving no integration to those activities including balanced mathematics and science concepts. Several examples are given to illustrate the utility of the continuum model for analyzing integrated curricula. The continuum model is intended to be used by curriculum developers to clarify the relationship between the mathematics and science activities and concepts and to guide the modification of lessons.     

Rhetoric,Reality, and Possibilities: Interdisciplinary Teaching and Secondary Mathematics

The National Council of Teachers of Mathematics has proposed a broad core mathematics curriculum for all high school students. One emphasis in that core is on “mathematical connections” both among mathematical topics and between mathematics and other disciplines of study. It is suggested that mathematics should become a more integrated part of all students' high school education. In this article, working definitions for the terms curriculum, interdisciplinary, and integrated and a model of three categories of curriculum design based on the work of Harold Alberty are developed. This article then examines how a “connected” mathematics core curriculum might be situated within the different categories of curriculum organization. Examples from research on interdisciplinary education in high schools are presented. Issues arising from this study suggest the need for a greater emphasis on building and using models of curriculum integration both to frame and to give impetus to the work being done by teachers and administrators.     

Mathematics teaching practices with technology that support conceptual understanding for Latino/a students

We analyze how three seventh grade mathematics teachers from a majority Latino/a, linguistically diverse region of Texas taught the same lesson on interpreting graphs of motion as part of the Scaling Up SimCalc study (Roschelle et al., 2010). The students of two of the teachers made strong learning gains as measured by a curriculum-aligned assessment, while the students of the third teacher were less successful. To investigate these different outcomes, we compare the teaching practices in each classroom, focusing on the teachers’ use of class time and instructional format, their use of mathematical discourse practices in whole-class discussions, and their responses to student contributions. We show that the more successful teachers allowed time for students to use the curriculum and software and discuss it with peers, that they used formal mathematical discourse along with less formal language, and that they responded to student errors using higher-level moves. We conclude by discussing implications for teachers and mathematics educators, with special attention to issues related to the mathematics education of Latinos/as.     

Making Mathematical Connections Across the Curriculum: Activities to Help Teachers Begin

Ample evidence is available to support the contention that, for learning to be meaningful, concepts must be connected and integrated within the experiences of the learner. In mathematics, at least three kinds of connections are particularly beneficial: connections within mathematics, across the curriculum, and with real world contexts. The authors' work with preservice and inservice teachers has convinced them that teachers possess both the willingness and the capability to help students make meaningful connections, given encouragement and support. This article focuses on making mathematical connections across the curriculum; activities which help teachers learn how to design their own are shared.     

Undergraduate students’ initial conceptions of factorials

Counting problems offer rich opportunities for students to engage in mathematical thinking, but they can be difficult for students to solve. In this paper, we present a study that examines student thinking about one concept within counting, factorials, which are a key aspect of many combinatorial ideas. In an effort to better understand students’ conceptions of factorials, we conducted interviews with 20 undergraduate students. We present a key distinction between computational versus combinatorial conceptions, and we explore three aspects of data that shed light on students’ conceptions (their initial characterizations, their definitions of 0!, and their responses to Likert-response questions). We present implications this may have for mathematics educators both within and separate from combinatorics.     

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.     

Teachers’ views on creativity in mathematics education: an international survey

The survey described in this paper was developed in order to gain an understanding of culturally-based aspects of creativity associated with secondary school mathematics across six participating countries. All participating countries acknowledge the importance of creativity in mathematics, yet the data show that they take very different approaches to teaching creatively and enhancing students’ creativity. Approximately 1,100 teachers from six countries (Cyprus, India, Israel, Latvia, Mexico, and Romania) participated in a 100-item questionnaire addressing teachers’ conceptions about: (1) Who is a creative student in mathematics, (2) Who is a creative mathematics teacher, (3) In what way is creativity in mathematics related to culture, and (4) Who is a creative person. We present responses to each conception focusing on differences between teachers from different countries. We also analyze relationships among teachers’ conceptions of creativity and their experience, and educational level. Based on factor analysis of the collected data we discuss relevant relationships among different components of teachers’ conceptions of creativity as they emerge in countries with different cultures.     

The design of tasks that address applications to teaching secondary mathematics for use in undergraduate mathematics courses

This paper describes theoretical design principles emerging from the development of tasks for standard undergraduate mathematics courses that address applications to teaching secondary mathematics. While researchers recognize that mathematical knowledge for teaching is a form of applied mathematics, applications to teaching remain largely absent from curriculum resources for courses for mathematics majors. We developed various materials that contain applications to teaching that have been integrated into four standard undergraduate mathematics courses. Three primary principles influenced the design of the tasks that prepare future teachers to learn and apply mathematics in a manner central to their future work. Additionally, this paper provides guidance for instructors desiring to develop or implement similar applications. The process of developing these tasks underscores the importance of key features regarding the roles of human beings in the tasks, the intentional focus on advanced content connected to school mathematics, and the integration of active engagement strategies.     

The examination of a vignette activity sequence in a secondary mathematics methods course

Because of their brief nature, vignettes are a strategic way to highlight or explore complex instructional practices. Using a qualitative approach, we examined how the use of vignettes in a Vignette Activity Sequence contributed to secondary mathematics preservice teachers’ understanding of the Mathematical Practices and the Mathematics Teaching Practices. By examining three vignettes used in two iterations of a secondary mathematics methods course, the researchers found that preservice teachers were able to draw connections between the vignettes and their own teaching experiences. However, some misconceptions or incomplete understandings related to the practices were revealed. Preservice teachers sometimes provided vague evidence when identifying particular practices in the vignettes that did not clearly indicate if they understood the practices. Taken together, the researchers found the Vignette Activity Sequence to be a valuable formative assessment that could be used to inform instruction in a secondary mathematics methods course. These findings have implications for teacher preparation programs and mathematics teacher educators.     

A Procedure for Integrating Math and Science Units

During our work with pairs of elementary and middle school lead teachers, one mathematics and one science, we struggled to get beyond activities developed by AIMS and others. This article presents a procedure whereby two teachers engage in a semi-structured professional dialogue to create curriculum experiences which highlight mathematics and science connections.     

The Integration of Science,Mathematics, and Technology in a Discipline-Based Culture

The culture of the middle years of schooling in Western Australia, as in many parts of the world, is predominantly discipline based. This paper focuses on exceptions to this norm by describing examples of integrated teaching of science, mathematics, and technology in seventh- to ninth-grade classrooms. Several different forms of integration were found in the 16 Western Australian schools examined in this study, including thematic approaches, cross-curricular approaches, technology-based projects, and local community projects. Interviews with teachers in these schools raised several implementation issues, including the process of getting started, implications for teachers and students, implications for schedule structure, and implications for departmental structure. All the forms of integration observed in this study were through secondary means, in which the discrete subject discipline boundaries were being maintained. The deep culture of subject disciplines, underwritten by curriculum documents organized in terms of subjects, means that there may be few incentives for teachers to teach and students to learn in an integrated manner.     

Conditions and decisions of urban elementary teachers regarding instruction of STEM curriculum

The study was situated in a National Science Foundation supported Math Science Partnership between a private university and an urban school district. This study sought to understand the decision‐making process of elementary teachers as they implement an integrated science, technology, engineering, and mathematics (STEM) curriculum in their classrooms and the interactions that occur between the teachers and curriculum during that process. This qualitative study utilized a comparative case study approach to understanding the decision‐making process of three elementary teachers enacting the same lesson. Analysis of the interactions revealed that the teachers' perceptions of student ability, their pedagogical design capacity, and time were influences that impacted implementation. These findings have implications for STEM‐focused professional development of elementary teachers.     

Potential Benefits and Barriers to Integration

The rhetoric surrounding integration of mathematics and science abounds. Professional organizations’ standards and recommendations for reform in mathematics and science education each point out the need to make connections among various disciplines. However, some remain unconvinced, citing a lack of research supporting the assertion that integration improves student achievement. This article examines the current situation, discusses the growing body of related research, and examines the implementation issues related to integrated curriculum projects. The conclusion calls for mathematics and science educators to work collaboratively to address implementation issues surrounding reform of any kind and to explore further the possibilities of integration.     

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.     

Turnaround Students in High School Mathematics: Constructing Identities of Competence Through Mathematical Worlds

This study investigates prospective secondary teachers’ cognitive difficulties and mathematical ideas involved in making connections among representations. We implemented a three-week teaching unit to help prospective secondary mathematics teachers develop understanding of big ideas that are critical to formulating connections among representations, in the context of conic curves. Qualitative analysis of data showed that most undergraduate mathematics majors and minors in this study struggled with variation, the Cartesian Connection, and other affiliated ideas such as graph as a locus of points. Furthermore, they were unable to identify basic metric relations encoded in algebraic expressions such as the distance between points, which further compounded their difficulties in making connections among representations. We argue that mathematics teacher education needs more focus on these ideas so that their graduates can successfully teach these big ideas in their future instruction.     

Prospective Teachers' Perspectives on Mathematics Teaching and Learning: Lens for Interpreting Experiences in a Standards‐Based Mathematics Course

In a mathematics course for prospective elementary teachers, we strove to model standards‐based pedagogy. However, an end‐of‐class reflection revealed the prospective teachers were considering incorporating standards‐based strategies in their future classrooms in ways different from our intent. Thus, we drew upon the framework presented by Simon, Tzur, Heinz, Kinzel, and Smith to examine the prospective teachers' perspectives on mathematics teaching and learning and to address two research questions. What perspectives on the learning and teaching of mathematics do prospective elementary teachers hold? How do their perspectives impact their perception of standards‐based instruction in a mathematics course and their future teaching plans? Qualitative analyses of reflections from 106 prospective teachers revealed that they viewed mathematics as a logical domain representative of an objective reality. Their instructional preferences included providing firsthand opportunities for elementary students to perceive mathematics. They did not take into account the impact of a student's conceptions upon what is learned. Thus, the prospective teachers plan to incorporate standards‐based strategies to provide active experiences for their future elementary students, but they fail to base such strategies upon students' current mathematical conceptions. Throughout, the need to address prospective teachers' underlying perspectives of mathematics teaching and learning is stressed.     

PBL as a pedagogical approach for integrated STEM: Evidence from prospective teachers

Problem-based learning (PBL) and science, technology, engineering, and mathematics (STEM) are two acronyms widely visible in education literature today. However, few studies have explored these in connection with one another, specifically with regard to teacher preparation. This study investigated how 47 prospective elementary teachers developed PBL units and how they integrated STEM and other disciplines into those units. It also addressed the affordances and constraints of integrated STEM as perceived by the prospective elementary teachers. Data sources in this multimethod study included PBL units and interviews. Findings revealed that all of the units integrated at least two of the STEM disciplines, as well as literacy, in a variety of ways. The prospective teachers articulated perceived benefits of integrated STEM, such as: making connections across content areas, preparing students for the real world, teaching students that failure is not a bad thing, and providing future opportunities. They also addressed perceived barriers of integrated STEM, such as: having limited experience with the content, diminishing the effect of individual content areas, and needing better curriculum alignment. Overall, this study provides evidence that PBL can be a pedagogical approach to integrate STEM. Implications for teachers, teacher educators, and curriculum specialists are discussed.