Using Metaphors as a Tool for Examining Preservice Elementary Teachers' Beliefs About Mathematics Teaching and Learning

This study used metaphors as a tool to gain insight about preservice teachers' conceptualizations of the role of the teacher and the learner and held the view that the examination of these metaphors might provide an opportunity for teacher educators to reflectively and critically examine those beliefs. Thus, this research examined possible differences in the reflected beliefs of elementary preservice teachers as depicted in their metaphors about mathematics teaching and learning at three different points throughout their mathematics education methods courses. The results of this study indicated that elementary preservice teachers' beliefs primarily remained static throughout their mathematics methods courses despite ongoing experiences designed to challenge and extend those beliefs.     

Preservice Teacher Beliefs About Proofs

This article describes a study of backgrounds, beliefs, and attitudes of teachers about proofs. Thirty preservice elementary teachers enrolled in a mathematics content course and 21 secondary mathematics teachers in an abstract algebra course were surveyed. The study explored four issues: preservice teachers' experiences/exposure to proof, their beliefs about what constitutes a proof and the role of proof in mathematics, and their beliefs about when proof should be introduced in grades K-12. Results of the survey are described as a means for discussing the backgrounds and beliefs future teachers hold with regard to teaching proofs in their own classrooms. Finally, a short collection of sample explorations and questions, which could be used to encourage the thinking and writing of proofs in grades K-12, is provided. One of these questions was posed to 215 secondary students; examples of their reasoning and a discussion of the various techniques employed by the students are included.     

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.     

Envisioning my mathematics classroom: Validating the Draw-a-Mathematics-Teacher-Test Rubric

Teachers need the opportunity to reflect, rethink, and adapt as they continually develop their image of their role in their mathematics classrooms. Thus, the purpose of this research was to examine how the Draw-a-Mathematics-Teacher-Test (DAMTT) and rubric can be used to assess preservice elementary teachers’ images of and beliefs about their future mathematics classrooms and validate the Draw-a-Math-TeacherTest-Rubric (DAMTT-R). Results suggest that the DAMTT-R is a valid measure and yields consistent results. Additionally, analysis of preservice elementary teachers’ (PETs) DAMTT revealed that only slightly more than one-third (36.9%) drew a picture and described their classroom in such a way that it reflected beliefs aligned with student-centered pedagogic practices. While mathematics educators may aim for the majority of PETs to leave their programs having developed beliefs aligned with and supportive of student-centered pedagogic practices, the results of this study revealed that 25% of PETs held beliefs that align with teacher-centered pedagogic practices. Lastly, 38.1% of the PETs reflected beliefs about their pedagogic practices, as measured by the DAMTT and the DAMTT-R, aligned with a transition between teacher-centered and student-centered.     

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.     

Justification as a teaching and learning practice: Its (potential) multifacted role in middle grades mathematics classrooms

Justification is a core mathematics practice. Although the purposes of justification in the mathematician community have been studied extensively, we know relatively little about its role in K-12 classrooms. This paper documents the range of purposes identified by 12 middle grades teachers who were working actively to incorporate justification into their classrooms and compares this set of purposes with those documented in the research mathematician community. Results indicate that the teachers viewed justification as a powerful practice to accomplish a range of valued classroom teaching and learning functions. Some of these purposes overlapped with the purposes in the mathematician community; others were unique to the classroom community. Perhaps surprisingly, absent was the role of justification in verifying mathematical results. An analysis of the relationship between the purposes documented in the mathematics classroom community and the research mathematician community highlights how these differences may reflect the distinct goals and professional activities of the two communities. Implications for mathematics education and teacher development are discussed.     

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.     

United States teachers’ views of effective mathematics teaching and learning

This study investigates US teachers’ cultural beliefs concerning effective mathematics teaching using semi-structured interviews with 11 experienced teachers. For US teachers, effective teaching is student-centered. Cognitively appropriate mathematical content should be understood through many hands-on activities that allow students to explore by themselves the relationship between mathematical knowledge and their life experiences. Correspondingly, the US teachers view an effective teacher as a facilitator who is sensitive to student social and cognitive needs and is skillful at organizing collaborative learning. The result of this study helps researchers and educators understand the student-centered learning model in US classrooms.     

Impacts of professional development focused on teaching engineering applications of mathematics and science

With the recent national emphasis on preparing children for future careers in science, technology, engineering, and mathematics, K-12 teachers are being called upon to include engineering in their instruction. This study explores the impacts of a summer professional development (PD) program focused on the engineering applications of mathematics and science on in-service K-12 teachers' (a) personal engineering efficacy, (b) engineering teaching efficacy, and (c) perceived barriers to teaching engineering. This quantitative study revealed that a single engineering-focused PD could increase teachers' personal engineering efficacy and engineering teaching efficacy and reduce particular perceived barriers to teaching engineering. No differences existed in pre- to post workshop assessment scores based on grade level taught, gender, or years of teaching experiences. However, pre- to post workshop assessment differences existed between participants depending on the discipline they taught and wether or not they had previously used engineering activities in their classrooms. These findings suggest that a single engineering PD can have significant impacts on in-service teachers' personal engineering efficacy, engineering teaching efficacy, and perceived barriers to teaching engineering, but a one-size-fits-all approach to such PD is not equally effective for all participants.     

Voices from the Front Lines: Exemplary Science Teachers on Education Reform

The purpose of this study is to gain insight into the experiences that nationally award‐winning, exemplary science teachers have had over their career and examine the alignment of their responses with calls for K‐12 science education reform from a selection of prominent commissioned government reports since 1980. From an assessment of the alignment of exemplary teachers, calls for reform have had a limited effect and highlight the weakness of using national reports as a wide‐scale, nationalized approach to science education reform. Findings are focused on seven different areas of teacher development: classroom issues, teaching scientific inquiry, use of technology, preservice experiences, professional development of in‐service teachers, vertical articulation, and science education reform over time. Among other issues, the teachers indicated one of the biggest barriers to inquiry teaching is the pressure to conform to high‐stakes testing and the lack of examples of inquiry teaching during teacher education experiences.     

Teaching and learning of mathematics: what really matters to teachers and students?

The learner’s perspective study, motivated by a strong belief that the characterization of the practices of mathematics classrooms must attend to learner practice with at least the same priority as that accorded to teacher practice, is a comprehensive study that adopts a complementary accounts methodology to negotiate meanings in classrooms. In Singapore, three mathematics teachers recognized for their locally defined ‘teaching competence’ participated in the study. The comprehensive sets of data from the three classrooms have been used to explore several premises related to the teaching and learning of mathematics. In this paper the student interview data and the teacher interview data were examined to ascertain what do students attach importance to and what do teachers attach importance to in a mathematics lesson? The findings of the student interview data showed that they attached importance to several sub-aspects of the three main aspects, i.e., exposition, seatwork and review and feedback of their teachers’ pedagogical practices. The findings of the teacher interview data showed that they attached importance to student’s self assessment, teacher’s demonstration of procedures, review of prior knowledge and close monitoring of their student’s progress in learning and detailed feedback of their work. It was also found that teachers and students did attach importance to some common lesson events.     

Elementary preservice teachers' integration of engineering into STEM lesson plans

Science, technology, engineering, and mathematics (STEM) integration is a desired outcome according to Next Generation Science Standards. However, learning to teach integrated STEM content has been challenging for teachers. Consequently, the purpose of this qualitative study was to describe how 16 preservice teachers enrolled in a mathematics methods course created integrated STEM lesson plans that incorporated an authentic engineering problem. Content analysis of the completed integrated STEM lesson plans used the Quality K-12 Engineering Education Framework to identify any characteristics of engineering. We found that 15 of 16 preservice teachers demonstrated at least an emerging ability to create an integrated STEM lesson that contained an engineering problem, constraints, a prototype or model, model testing, and data collection and analysis related to the model. We concluded that giving preservice teachers opportunities to experience engineering design problems could better prepare them to design and implement integrated STEM education in their classrooms. The findings from this study have practical implications for mathematics methods teacher educators who teach the pedagogy behind STEM education. This study also has theoretical implications because socially situated learning theory was extended to Model-Eliciting Activities and connected them to the K-12 Framework for Quality Engineering Education.     

Supporting Preservice Teachers' Understanding of Place Value and Multidigit Arithmetic

This article provides an analysis of a teaching experiment conducted in the context of teacher education designed to support preservice teachers' understandings of place value and multidigit addition and subtraction. The experiment addresses the following research question: Can the results from research conducted in elementary mathematics classrooms guide preservice elementary teachers' development of conceptual understanding of the same concepts? In both cases, the students (e.g., elementary students and preservice teachers) participated in activities from an instructional sequence designed to support conceptual understanding of both place value and multidigit addition and subtraction. Analyses of the episodes from the teaching experiment document the learning of the preservice teachers and how that learning was supported by initial conjectures grounded in the research on elementary students' ways of reasoning.     

Linking Preservice Teachers' Mathematics Self‐Efficacy and Mathematics Teaching Efficacy to Their Mathematical Performance

This study examined preservice teachers' mathematics self‐efficacy and mathematics teaching efficacy and compared them to their mathematical performance. Participants included 89 early childhood preservice teachers at a Midwestern university. Instruments included the Mathematics Self‐Efficacy Scale (MSES), Mathematics Teaching Efficacy Beliefs Instrument (MTEBI), and the Illinois Certification Testing System (ICTS) Basic Skills Test. The results indicate that preservice teachers' mathematics self‐efficacy is positively correlated to their personal mathematics teaching efficacy. In addition, their mathematical performance is related to their mathematics self‐efficacy and mathematics teaching efficacy. In regard to affecting student outcomes, only those preservice teachers who are very confident in their ability to teach believe they can have an effect on their students. Implications on teacher education programs are discussed.     

Investigations and explorations in the mathematics classroom

In Portugal, since the beginning of the 1990s, problem solving became increasingly identified with mathematical explorations and investigations. A number of research studies have been conducted, focusing on students’ learning, teachers’ classroom practices and teacher education. Currently, this line of work involves studies from primary school to university mathematics. This perspective impacted the mathematics curriculum documents that explicitly recommend teachers to propose mathematics investigations in their classrooms. On national meetings, many teachers report experiences involving students’ doing investigations and indicate to use regularly such tasks in their practice. However, this still appears to be a marginal activity in most mathematics classes, especially when there is pressure for preparation for external examinations (at grades 9 and 12). International assessments such as PISA and national assessments (at grades 4 and 6) emphasize tasks with realistic contexts. They reinforce the view that mathematics tasks must be varied beyond simple computational exercises or intricate abstract problems but they do not support the notion of extended explorations. Future developments will show what paths will emerge from these contradictions between promising research and classroom reports, curriculum orientations, professional experience, and assessment frameworks and instruments.     

Models are a “metaphor in your brain”: How potential and preservice teachers understand the science and engineering practice of modeling

We investigated beginning secondary science teachers’ understandings of the science and engineering practice of developing and using models. Our study was situated in a scholarship program that served two groups: undergraduate STEM majors interested in teaching, or potential teachers, and graduate students enrolled in a teacher education program to earn their credentials, or preservice teachers. The two groups completed intensive practicum experiences in STEM‐focused academies within two public high schools. We conducted a series of interviews with each participant and used grade‐level competencies outlined in the Next Generation Science Standards to analyze their understanding of the practice of developing and using models. We found that potential and preservice teachers understood this practice in ways that both aligned and did not align with the NGSS and that their understandings varied across the two groups and the two practicum contexts. In our implications, we recommend that teacher educators recognize and build from the various ways potential and preservice teachers understand this complex practice to improve its implementation in science classrooms. Further, we recommend that a variety of practicum contexts may help beginning teachers develop a greater breadth of understanding about the practice of developing and using models.