Showing papers in "Journal of Mathematics Teacher Education in 2015"
TL;DR: In this article, a case study was conducted to examine 186 episodes of struggles in middle school students as they engaged in tasks focused on proportional reasoning, and a Productive Struggle Framework was developed to capture the episodes of struggle episodes from initiation, to interaction and to resolution.
Abstract: Prior studies suggest that struggling to make sense of mathematics is a necessary component of learning mathematics with understanding. Little research exists, however, on what the struggles look like for middle school students and how they can be productive. This exploratory case study, which used episodes as units of analysis, examined 186 episodes of struggles in middle school students as they engaged in tasks focused on proportional reasoning. The study developed a classification structure for student struggles and teacher responses with descriptions of the kinds of student struggle and kinds of teacher responses that occurred. The study also identified and characterized ways in which teaching supported the struggles productively. Interaction resolutions were viewed through the lens of (a) how the cognitive demand of the task was maintained, (b) how student struggle was addressed and (c) how student thinking was supported. A Productive Struggle Framework was developed to capture the episodes of struggle episodes from initiation, to interaction and to resolution. Data included transcripts from 39 class session videotapes, teacher and student interviews and field notes. Participants were 327 6th- and 7th-grade students and their six teachers from three middle schools located in mid-size Texas cities. This study suggests the productive role student struggle can play in supporting “doing mathematics” and its implications on student learning with understanding. Teachers and instructional designers can use this framework as a tool to integrate student struggle into tasks and instructional practices rather than avoid or prevent struggle.
135 citations
TL;DR: The authors explored the development of teacher noticing as a way to help teachers learn to broaden their views of algebra, pay attention to a wide range of student algebraic thinking, and reason about students' ideas in substantive ways.
Abstract: Learning algebra is critical for students in the USA today, yet many students in the USA struggle in algebra classes. Researchers claim that one reason for these difficulties is that algebra classes often focus on symbol manipulation and procedures above, and many times at the expense of, a more conceptual understanding of the content. Teaching algebra in a more conceptual way, however, can be quite challenging for teachers. This study explores the development of teacher noticing as a way to help teachers learn to broaden their views of algebra, pay attention to a wide range of student algebraic thinking, and reason about students’ ideas in substantive ways. This study takes place in the context of a video club in which seven preservice teachers watched and discussed video excerpts from algebra classes over an 8-week period. A framework for noticing student algebraic thinking was created and used to structure the discussions in the video club. In addition, a new, online video-tagging tool was used to document the development of the teachers’ ability to notice student algebraic thinking. Results suggest that participating in the video club helped teachers more consistently attend to substantive student algebraic thinking and to reason about this thinking they noticed in deeper ways.
98 citations
TL;DR: In this article, the authors investigate ways of supporting teachers in converting challenging mathematics tasks into classroom lessons and supporting students in engaging with those tasks and propose a set of lessons built around challenging tasks.
Abstract: The following is a report on an investigation into ways of supporting teachers in converting challenging mathematics tasks into classroom lessons and supporting students in engaging with those tasks. Groups of primary and secondary teachers, respectively, were provided with documentation of ten lessons built around challenging tasks. Teachers responded to survey items in both Likert and free format style after teaching the ten lessons. The responses of the teacher participants indicate that the lesson structure we proposed was helpful, and the elements of the lessons suggested to teachers were both necessary and sufficient for supporting students in engaging with the challenging tasks. Implications for teacher educators and curriculum developers are offered.
88 citations
TL;DR: In this paper, the authors examined the relationship of CCK to SCK for those holding a bachelors degree or higher in mathematics and argued that SCK is different from CCK.
Abstract: The construct “mathematical knowledge for teaching” (MKT) has received considerable attention in the mathematics education community in recent years. The development and refinement of the MKT construct, including the components of common content knowledge (CCK) and specialized content knowledge (SCK), came from research into elementary teachers’ practices. In this article, we argue that various issues arise as these constructs are used in research on secondary and post-secondary teachers. For example, elementary teachers typically differ from teachers of higher grades in their content preparation. What then is the relationship of CCK to SCK for those holding a bachelors degree or higher in mathematics? The MKT construct is based on CCK being knowledge held or used by an average mathematically literate citizen and that SCK is different. However, among those teaching in secondary and post-secondary contexts, what should be considered CCK? Is conceptual understanding of the CCK among those with bachelor’s degrees or higher level mathematics the same as SCK? We examine these questions as well as others that arose from our examination of definitions of CCK and SCK as we attempted to utilize those definitions to characterize the nature of MKT at secondary and undergraduate levels. We illustrate these issues with data from two instructional settings.
65 citations
TL;DR: The authors examined a video club in which four student teachers utilized the Mathematical Quality of Instruction (MQI) analysis framework to code each other's lessons and to discuss their coding in facilitated group sessions.
Abstract: The ability to notice important events in a teaching situation and make decisions about these events is a key component of teaching well. Prospective teachers tend to notice more superficial aspects of classroom practice, such as class management. This article examines a video club in which four student teachers utilized the Mathematical Quality of Instruction (MQI) analysis framework to code each other’s lessons and to discuss their coding in facilitated group sessions. We found that participants became better able to notice important aspect of mathematics more generally and MQI components more specifically. They also adopted a less evaluative stance toward what they noticed. Finally, their self-reported beliefs and practices altered; they credited their participation in the video club for their attempts to incorporate more opportunities for students to engage with mathematical content and ideas.
63 citations
TL;DR: The authors studied the relationship between teachers' beliefs about mathematics teaching and learning and their classroom practices and found that non-mathematics beliefs and contextual factors took precedence in certain classroom situations, while contextual factors had an intervening influence on the actualization of beliefs.
Abstract: Researchers in the field of mathematics education have focused on beliefs as a significant area of study because of the influence of beliefs on what is taught and learned. Much of the research in this area speaks about inconsistency between teachers’ beliefs about mathematics teaching and learning and their classroom practices. In this case study, I look beyond two elementary teachers’ perceived inconsistencies to gain a better understanding of the nature of their beliefs and how they are organized. Data were gathered from individual and focus group interviews, classroom observations, email communications, and researcher memos over the course of 3 years. Results showed that non-mathematics beliefs and contextual factors took precedence in certain classroom situations and contextual factors had an intervening influence on the actualization of beliefs. Several theoretical, methodological and practical implications of the findings are discussed.
57 citations
TL;DR: In this paper, the authors explored effective mathematics teaching as constructed in Finnish and Swedish teacher educators' discourses, based on interview data from teacher educators as well as data from feedback discussions between teacher educators and prospective teachers in Sweden and Finland, and found that several aspects of the recent international reform movements are visible in the discourses in both countries.
Abstract: This article explores effective mathematics teaching as constructed in Finnish and Swedish teacher educators’ discourses. Based on interview data from teacher educators as well as data from feedback discussions between teacher educators and prospective teachers in Sweden and Finland, the analysis shows that several aspects of the recent international reform movements are visible in the discourses in both countries. However, the Swedish teacher educators tend to conceptualize effective teaching as interactions with individual children, building on students’ ideas and emanating mathematics from everyday situations, while the Finnish teacher educators stress the importance of a clear presentation of mathematics, routines and homework as well as specific goals for every lesson. The results of this cross-cultural study cannot be generalized to the two countries but rather show interesting conceptualizations of effective teaching, adding to international theory building.
47 citations
TL;DR: In this paper, the authors investigated how teachers' mathematical knowledge for teaching and their pedagogical beliefs contribute to their performance in providing explanations and selecting and using tasks, as studied in a teaching simulation.
Abstract: Attempts to understand what contributes to teaching quality have been channeled in different directions, with two main research streams focusing on either teacher knowledge or teacher beliefs. Few are the studies that have attended to both the cognitive and the affective domain simultaneously, trying to unpack how both jointly contribute to teaching quality. Situated at the nexus of these two domains, this study aims to understand how teachers’ mathematical knowledge for teaching and their pedagogical beliefs contribute to their performance in providing explanations and selecting and using tasks, as studied in a teaching simulation. Using a multiple-case approach and examining the development of three prospective teachers’ knowledge and beliefs over a content-and-methods course sequence, the study documents how limitations in either knowledge or beliefs can mediate the effect of the other component on prospective teachers’ performance. Implications for teacher preparation and in-service education are drawn and directions for future studies are offered.
46 citations
TL;DR: The authors examined elementary-and middle-grade teachers' understanding of the mathematical learning progression as they participated in a 6-month professional learning project and found that teachers developed vertical connections and representational fluency from their planning, teaching, observation and debriefs.
Abstract: This study examines elementary- and middle-grade teachers’ understanding of the mathematical learning progression as they participated in a 6-month professional learning project. Teachers participated in a professional development project that consisted of a 1-week summer content-focused institute with school-based follow-up Lesson Study cycles in the fall that focused on the vertical articulation of algebraic concepts across grade levels. The following research examines how a vertical team of teachers from multiple grades designed, taught and learned from the Lesson Study cycle. The video analysis from the research lessons and the teachers’ reflections revealed teachers’ developing vertical connections and representational fluency from their planning, teaching, observation and debriefs. In addition, Lesson Study afforded teachers opportunities to deepen their understanding of the mathematical learning progression through observation and analysis of students’ thinking through a situated school-based professional development experience.
40 citations
TL;DR: In this article, the authors presented the impact of place-based mathematics education on teachers' perceptions, skills, and practices and implemented curriculum, and resulted in a culturally responsive, place-Based mathematics education.
Abstract: Transforming teachers and their approach to teaching Indigenous students requires partnerships with the Indigenous community, planning, small steps, and funds. This article illustrates how teachers can change when funds are available to assist schools and communities to implement appropriate and effective professional development, to establish partnerships between school and community, to revise teaching approaches and curriculum, and to value family and Aboriginal cultural heritage. The larger study involved four schools in a Stronger Smarter Learning Community in a small rural city, the whole city community, and the interaction among the schools. Interviews with principals, teachers, Aboriginal students, and their community highlighted the increasing interaction between the Aboriginal community and the schools, the increasing warmth and welcome extended both ways, and the impact that these approaches are having on curriculum, teaching, and learning. This article presents the impact in one of the schools involved in the mathematics project. The findings illustrate how the projects facilitated changing teachers’ perceptions, skills, and practices and implemented curriculum, and resulted in a culturally responsive, place-based mathematics education.
38 citations
TL;DR: This article reported on the development of attitudes toward mathematics among pre-service elementary teachers in relation to their experiences as K-12 learners of mathematics and experiences within a teacher education program and found that significant changes in attitudes occurred over the duration of mathematics methods coursework and student teaching.
Abstract: This article reports on the development of attitudes toward mathematics among pre-service elementary teachers (n = 146) in relation to their experiences as K-12 learners of mathematics and experiences within a teacher education program. Using a combination of the Rasch Rating Scale Model and traditional parametric analyses, results indicate that significant changes in attitudes occurred over the duration of mathematics methods coursework and student teaching. Further, these changes can be explained, in part, by various teacher education experiences. In particular, having a student teaching experience with meaningful mathematics instruction played a significant role for those who entered with negative attitudes toward mathematics.
TL;DR: In this article, the authors analyse the problem-solving related beliefs, competence and classroom practice of three Cypriot upper-primary teachers and conclude that all participants exhibited consistency and inconsistency in the ways in which their belief, competence, and practice interacted.
Abstract: In this article, we analyse the problem-solving-related beliefs, competence and classroom practice of three Cypriot upper-primary teachers. Data derived from semi-structured interviews focused on teachers’ beliefs about the nature of mathematical problems, problem-solving, and their competence as both problem-solvers and teachers of problem-solving; clinical interviews during which teachers solved a context-free geometrical problem, and observations of a lesson during which teachers introduced that problem to students of grade six. Analyses, structured by a framework derived from key problem-solving literature, indicated firstly, that the framework was an effective tool, sensitive to variation within and across the data from teachers, and secondly, that all participants, in largely explicable ways, exhibited consistency and inconsistency in the ways in which their beliefs, competence and practice interacted. Some implications for further research are discussed.
TL;DR: In this paper, the authors report data from a different type of learning experience, that of a clinical simulation with a standardized individual, and highlight teachers' diagnostic, explanatory, mathematical, and instructional repertoires, as they guide a standardized student through two different graphing problems.
Abstract: Field placements serve as the traditional ‘clinical’ experience for prospective mathematics teachers to immerse themselves in the mathematical challenges of students. This article reports data from a different type of learning experience, that of a clinical simulation with a standardized individual. We begin with a brief background on medical education’s long-standing use of standardized patients, and the recent diffusion of clinical simulations to teacher and school leader preparation contexts. Then, we describe a single mathematics simulation and report data from prospective mathematics teachers’ interactions with a standardized student on the issue of iconic interpretation. Findings highlight teachers’ diagnostic, explanatory, mathematical, and instructional repertoires, as they guide a standardized student through two different graphing problems. Implications focus on the trends in teachers’ instructional decisions, contextualized explanations, and the use of clinical simulations to enhance mathematics teacher development.
TL;DR: This paper examined how prospective teachers' conceptions changed while working on these tasks in two settings (a teaching experiment with 6 PTs and a mathematics methods course with 33 PTs) and modified the tasks on the basis of the results.
Abstract: Several researchers have documented prospective teachers’ (PTs’) conceptions of various mathematical topics. However, less is known about how PTs’ conceptions develop. To address this gap, I designed two tasks with the goals of addressing the PTs’ initial conceptions of multidigit whole numbers and helping them develop more sophisticated ones. I examined how PTs’ conceptions changed while working on these tasks in two settings (a teaching experiment with 6 PTs and a mathematics methods course with 33 PTs) and modified the tasks on the basis of the results. Consistent with prior findings, this study showed that PTs entered with limited conceptions. This study showed further that (a) well-designed tasks (addressing the PTs’ incoming conceptions as well as focusing on the desired conceptions) can help PTs develop content knowledge, (b) conceptual difficulties may persist even with well-designed tasks, and (c) artifacts of children’s mathematical thinking can be used to develop mathematical content knowledge. Instructional implications are discussed.
TL;DR: In this paper, the authors explored the relationship between participating in a graduate course aimed at enhancing teachers' theoretical and practical knowledge of mathematical creativity and one teacher's changing perspectives regarding mathematical creativity, and tasks that may occasion mathematical creativity.
Abstract: This study explores the relationship between participating in a graduate course aimed at enhancing teachers’ theoretical and practical knowledge of mathematical creativity and one teacher’s changing perspectives regarding mathematical creativity and tasks that may occasion mathematical creativity. Results indicated that perceptions of creativity may include ideas about how creativity is characterized as well as among which students it may be promoted. These perceptions were closely related to the task features, cognitive demands, and affective issues, the teacher associated with tasks that may occasion mathematical creativity. The teacher’s reflections on her participation in the course indicated that both theoretical and practical elements of the course impacted on her changing perspectives. Also discussed are the advantages and limitations of providing professional development by means of university-based graduate courses.
TL;DR: In this paper, the authors explored the contribution of elementary mentor teachers to the learning-to-teach (Te2Teach) experiences of prospective teachers in a third-space learning context.
Abstract: Studies of mathematics teacher preparation frequently lament the divide between the more theoretically based university methods course and the practically grounded classroom field experience. In many instances, attempts to mediate this gap involve creating hybrid or third spaces, which seek to dissipate the differences in knowledge status as individuals from the university and from K-12 classrooms work together in support of prospective teacher (PST) learning. However, what is missing in the literature on these third-space enactments is an exploration of the contributions of different contexts (i.e., methods and the field) to PST learning and an articulation of the synergistic knowledge arising in the third space. This exploratory study draws on Lampert’s three-pronged teacher–child–content model to examine the possible contributions of elementary mentor teachers (MTs) to the learning-to-teach-mathematics experiences of PSTs. More specifically, we focus on a third-space learning context in which university-based teacher educators, MTs, and PSTs collaborated to conduct and analyze task-based problem-solving interviews of elementary children. Our analysis identified ways that MTs could potentially enhanced the learning-to-teach context as well as moments when MTs’ contributions introduced problematic ideas about children and teaching. Finally, we explore the benefits and complexities of leveraging these MT contributions to create a third-space learning opportunity.
TL;DR: In this article, a turn-by-turn analysis of mathematical discourse in the classroom has been carried out to study how single turns affect each other to form patterns in one teacher's practice, and two main repeating patterns were identified: one between student explanations and the teacher's focusing actions, and the other between teacher's progressing actions and students' teacher-led responses.
Abstract: Mathematical discourse in the classroom has been conceptualised in several ways, from relatively general patterns such as initiation–response–evaluation (Cazden in classroom discourse: the language of teaching and learning, Heinemann, London, 1988; Mehan in learning lessons: social organization in the classroom. Cambridge, MA: Harvard University Press, 1979) to concepts for more fine-grained description such as the ‘Advancing Children’s Mathematics’ framework (Fraivillig et al. in J Res Math Educ 30(2):148, 1999). This article suggests a framework to be used for detailed studies of mathematical discourse on a turn-by-turn basis. This framework was used to study how single turns affect each other to form patterns in one teacher’s practice. The method used belongs to conversation analysis: studying single turns and characterise these according to their role in the conversation. Two main repeating patterns were identified: one between student explanations and the teacher’s focusing actions, and the other between the teacher’s progressing actions and students’ teacher-led responses. The findings also included other connections that demonstrate how various student interventions (explanations, teacher-led responses, unexplained answers, partial answers, and initiatives) are followed by different types of teacher actions. One implication is that, by developing concepts capable of describing qualities of a discourse on a turn-by-turn basis, it then becomes possible to analyse when mathematical talk fosters delivery of facts and when it fosters mathematical argumentation, debate, and critique.
TL;DR: This article defined Mathematical Knowledge for Teaching (MKT) as a framework for teacher knowledge, which includes skills, habits, sensibilities as well as knowledge, but does not include affect-specific items (e.g., attitudes and beliefs).
Abstract: Teacher affect heavily influences instruction and learning (Cross 2009; Pajares 1992; Philipp 2007; Robertson-Kraft and Duckworth 2014). Teacher affect, which includes partially cognitive traits such as attitudes and beliefs as well as noncognitive traits such as emotion, motivation, and grit, is often defined in opposition to purely cognitive traits such as IQ or mathematical knowledge. As a consequence, teacher affect has often been studied in isolation from teacher cognition (Philipp 2007; Thompson 1992).By contrast, some researchers collapse cognitive and partially cognitive categories, grouping mathematics teacher knowledge and beliefs together as beliefs (e.g., Leatham 2006) or as knowledge (e.g., Beswick et al. 2012). Mathematical knowledge for teaching (MKT), one of the most widely used frameworks for teacher knowledge, is defined to comprise ‘‘skills, habits, sensibilities as well as knowledge’’ (Ball et al. 2008; p. 403). Although this definition of MKT seems to include components of teacher affect, measures of MKT do not include affect-specific items (e.g., Hill et al. 2004). As these examples suggest, teacher affect is recognized as a critical area for research, but researchers have wrestled with how to conceptualize and measure constructs that are sensitive to the content and context of instruction (e.g., Herbel-Eisenmann et al. 2006; Newton 2009) and that can even seem inconsistent with teachers’ own instructional practice (cf., Francis 2015). Moreover, researchers have struggled to clarify how teacher affect changes during teacher education or in the context of established instructional practice (Philipp 2007). Managing these complex interactions has led to definitions of teacher affect constructs that are distant from teaching itself.
TL;DR: In this paper, a qualitative study of the mathematics-focused lesson plans of elementary education prospective teachers provided data to determine the ways that the approach of literature integration in mathematics influenced prospective teachers' planned questions.
Abstract: Questioning is considered a powerful tool in mediating students’ knowledge construction and conceptual understanding. In this qualitative study, the mathematics-focused lesson plans of elementary education prospective teachers provided data to determine the ways that the approach of literature integration in mathematics influenced prospective teachers’ planned questions. All prospective teachers were required to incorporate children’s literature within the mathematics lessons they planned and presented during a field-based teaching experience. Analysis revealed variances in the numbers, types, and foci of prospective teachers’ planned questions. These findings allow speculation that the utilization of mathematics literature integration allowed many of the prospective teachers to create reform-oriented, constructivist mathematics-focused questions and experiences for their students.
TL;DR: In this paper, the authors investigated the relationship between US teachers' opportunities to learn about, knowledge of, motivation for, and instructional use of drawn models for representing multiplication and division of fractions.
Abstract: Past studies have suggested that in light of recent curriculum standards, many US teachers make limited use of drawn models in their mathematics instruction. To gain insight into this phenomenon, we investigated relationships between US teachers’ opportunities to learn about, knowledge of, motivation for, and instructional use of drawn models for representing multiplication and division of fractions. A national sample of 990 practicing middle-grade teachers was administered a three-part survey that contained a knowledge assessment; a professional history and teaching practices questionnaire that included questions about opportunities to learn to use drawn models; and a motivation questionnaire that measured teachers’ value, anxiety, and self-concept of ability for using such models in instruction. In regression models without motivation, opportunity to learn significantly predicted the teachers’ knowledge, frequency of use, and purpose for use of drawn models. In structural equation models that included motivation, knowledge and motivation substantially accounted for relationships between the teachers’ opportunity to learn and their self-reported use of drawn models in instruction. These findings are consistent with the general hypothesis that teacher’ opportunities to learn teaching practices indirectly affect their instructional practices. Teachers’ knowledge and motivation also play a central role.
TL;DR: The authors explored in-service high school mathematics teachers' conception of various forms of complex numbers and ways in which they transition between different representations of these forms, and found that teachers who taught courses such as geometry and international baccalaureate were able to draw from their teaching experiences as they attempted the interview tasks.
Abstract: This study explores in-service high school mathematics teachers’ conception of various forms of complex numbers and ways in which they transition between different representations of these forms. One 90-min interview was conducted with three high school mathematics teachers after they completed three professional development sessions, each 4 h, on complex numbers. Results indicate that, in general, these teachers did not necessarily have a dual conception of complex numbers. However, they demonstrated varying conceptions with different forms of complex numbers. Teachers worked at an operational level with the exponential form of complex numbers, but there was no evidence to indicate that they had a structural conception of this form. On the other hand, two teachers were very comfortable with the Cartesian form and exhibited a process/object duality by translating between different representations of this form. These results indicate that high school teachers need more opportunities to help them develop a dual conception of each form (multiple duals), which in turn can result in developing a dual conception of complex numbers. An interesting phenomenon that we found was that teachers who taught courses such as geometry and international baccalaureate were able to draw from their teaching experiences as they attempted the interview tasks. This particular observation may suggest that teachers’ teaching assignments coupled with appropriate professional development activities could facilitate their understanding of these concepts.
TL;DR: In this article, an interpretative phenomenological analysis investigating the lived experiences of a mathematician and a mathematics teacher educator as they team-taught a mathematics content and mathematics methods course for prospective secondary mathematics teachers is presented.
Abstract: Collaborations between mathematicians and mathematics teacher educators are increasingly being expected, and realized, within the context of mathematics teacher education. Most research related to collaborative efforts between members of the mathematics and mathematics education communities has focused on the products, rather than the process of collaboration. In this article, I present the results of an interpretative phenomenological analysis investigating the lived experiences of a mathematician and a mathematics teacher educator as they team-taught a mathematics content and mathematics methods course for prospective secondary mathematics teachers. I present extracts from interviews to illustrate the instructors’ perceptions that through collaboration and participation in the practice of the “other,” they were able to increase the awareness of their own practice and the practices characterizing their respective communities. The results of this study illustrate the potential of collaboration across these communities as a form of professional development for mathematics and mathematics education faculty.
TL;DR: In this article, the sets of mental computation strategies used by prospective elementary teachers to compute sums and differences of whole numbers were investigated, and the analysis led to the identification of the strategy ranges used by the participants, as well as descriptions of changes pre/post in those strategy ranges.
Abstract: This study investigated the sets of mental computation strategies used by prospective elementary teachers to compute sums and differences of whole numbers. In the context of an intervention designed to improve the number sense of prospective elementary teachers, participants were interviewed pre/post, and their mental computation strategies were analyzed. The analysis led to the identification of the strategy ranges used by the participants, as well as descriptions of changes pre/post in those strategy ranges. This article illustrates how strategy ranges, as an analytic tool, afford useful descriptions of the repertoires of mental computation strategies that individuals use.
TL;DR: The authors examined one teacher's opportunities to develop a coherent understanding of proportional situations through connection making in professional development and the ways in which those experiences were evidenced in her own classroom practice teaching the same task from PD.
Abstract: In this study, we examine one teacher’s opportunities to develop a coherent understanding of proportional situations through connection making in professional development (PD) and the ways in which those experiences were evidenced in her own classroom practice teaching the same task from PD. Data from both settings were analyzed using a framework for connection making that highlighted the ways in which the teacher or facilitator promoted discussion, used representations, promoted multiple approaches, and scaffolded learning. Our findings suggest that this teacher treated pedagogy and mathematical content as separable, which led to problematic implementation of the types of teaching practices that PD was intended to foster. We provide suggestions for addressing this shortcoming in future professional development.
TL;DR: In this article, the authors define proclivities for teaching mathematics as the beliefs, knowledge, and dispositions that are actionable in the flow of instruction, and argue that growth in this area contributes to positive change in mathematics instruction.
Abstract: This article presents a form of teacher reasoning that we call proclivities for teaching mathematics. We define proclivities for teaching mathematics as the beliefs, knowledge, and dispositions that are actionable in the flow of instruction, and we argue that growth in this area contributes to positive change in mathematics instruction. Proclivities for teaching mathematics encompass four domains: productive disposition, openness to new ideas, thought processes, and organization of mathematical thought; in this article, we focus on productive disposition. The article provides a rationale for the construct of proclivities for teaching mathematics, and describes an initial attempt to develop measures for these proclivities. Analyses of data from a sample of 63 teachers indicate that measures of proclivities for teaching mathematics are associated with increases in teacher knowledge, self-efficacy, and instructional improvement over time, and although the dataset is small, these positive indicators suggest that further exploration of this construct is warranted.
TL;DR: In 2001, the National Research Council (NRC) put forth two constructs of mathematical proficiency: procedural fluency (knowing how and when to apply procedures) and conceptual understanding (holding deep and rich connections among ideas), adaptive reasoning (the capacity to reason logically and to justify one's reasoning) as discussed by the authors.
Abstract: Constructs are the cornerstones on which the structure of any field rests. Constructs do not have an independent existence in the world waiting to be discovered; instead, they must be created within a field for the purpose of learning to see something in a new way, for example, enabling a community of teachers, teacher educators, and researchers to shift, often subtly, the focus of their attention. In 2001, the National Research Council (NRC) put forth two constructs, one embedded in the other. First, the construct of mathematical proficiency was put forth to broaden the way we view successful learning of mathematics. Mathematical proficiency was conceptualized as the interrelationship among procedural fluency (knowing how and when to apply procedures), conceptual understanding (holding deep and rich connections among ideas), adaptive reasoning (the capacity to reason logically and to justify one’s reasoning),
TL;DR: Speer et al. as discussed by the authors examined the definitions of common content knowledge and specialized content knowledge, and their relationship to typical characteristics of elementary teachers implicit in those definitions and compared those characteristics with characteristics typical of secondary and post-secondary teachers in their content preparation.
Abstract: Mathematics teachers’ knowledge for teaching mathematics has received significant attention in mathematics education research. It has been considered from different perspectives, with different constructs to describe it, resulting in a complex landscape of what it is about and what it entails. While category-based perspectives based on content knowledge and pedagogical content knowledge have received the most visibility in studies of the mathematics teacher, they provide a limited or biased representation of this knowledge. For example, teachers’ ways of knowing, thinking, and holding knowledge are important aspects of what is needed to teach mathematics and what we need to understand in order to help teachers to enhance their practice and students’ learning. Thus, ongoing research of this knowledge is needed to reveal details of it and issues associated with it from different perspectives and contexts. The articles in this issue of the Journal of Mathematics Teacher Education contribute to our understanding of this knowledge in a variety of ways. Collectively, they deal with topics that include: unpacking issues of mathematics knowledge for teaching (MKT) as a construct based on specific categories; supporting the development of teachers’ knowledge in using challenging mathematics tasks in their teaching; understanding prospective teachers’ knowledge in terms of how their conceptions of number develop during their engagement in well-designed tasks; and understanding teachers’ knowledge as beliefs. Natasha Speer, Karen King, and Heather Howell draw attention to issues that arise when the MKT construct derived from research into elementary teachers’ practices is used in research on secondary and post-secondary teachers who typically differ from elementary teachers in their content preparation. In particular, they examined the definitions of common content knowledge (CCK) and specialized content knowledge (SCK) and their relationships to typical characteristics of elementary teachers implicit in those definitions and compared those characteristics with characteristics typical of secondary and post-