The nature and development of middle school mathematics teachers’ knowledge

In this article, we report on the use of a teacher profiling instrument with 62 middle school teachers at the start of a 3-year professional learning programme. The instrument was designed to assess the aspects of teachers’ knowledge identified by Shulman (1987) refined by Ball et al. (2008) and extended to include teachers’ confidence to use and teach various topics in the middle school mathematics curriculum and their beliefs about mathematics teaching and learning. Based on a hierarchical coding of items, the application of the partial credit Rasch model revealed that the profile items were measuring a single underlying construct and suggested that the various facets of teacher knowledge develop together. We describe the characteristics of four levels of the hierarchical construct measuring teacher knowledge and understanding for teaching mathematics in the middle years of schooling, and discuss the unique affordances of a holistic view of teacher knowledge in contrast to considerations of multiple knowledge categories.     

Instructional practices related to prospective elementary school teachers’ motivation for fractions

This study was undertaken in order to better understand prospective elementary school teachers’ motivations for working with fractions before and after taking a course designed to deepen their understanding of mathematics, as well as what instructional practices might be related to any changes detected in their motivations. Eighty-five education students were given a motivation questionnaire at the beginning and end of the semester, and observations were made of the 9 days when fractions were taught. Three levels of teacher data were collected to understand instructional practices. Students’ ratings of the importance and usefulness of fractions (value), self-concept of ability, and anxiety were near the center of the scale at pre-test, with only value in the desired direction. At posttest, value and self-concept of ability increased while anxiety decreased, but these changes differed somewhat by instructor. In particular, reform-oriented practices, such as engaging students in high-level discourse, seemed to be associated with lowered anxiety.     

Pre-service teachers’ perceptions of middle school students

This report shares a study that explored the perceptions pre-service teachers have of middle school students. Participants were asked to complete the Adjective Checklist (ACL) by endorsing the words they considered most characteristic of a typical middle school student. Items most frequently endorsed indicated a predominantly negative perception (e.g. “awkward”, “confused” and “emotional”). Implications for pre-service teacher training are discussed.     

Prospective elementary teachers’ perceptions of real-life connections reflected in posing and evaluating story problems

This study explored prospective elementary teachers?? perspectives on real-life connections in the case of story problems. A series of tasks were designed to uncover the participants?? collective perceptions of real-life connections by speculating on participants?? emerging beliefs about real-life connections and how their beliefs were reflected in the story problems they posed or evaluated. Findings revealed the following ideas: (a) there were overly positive beliefs with insufficient specifics; (b) participants perceived utility and reality as critical components of real-life connections; and (c) there were vast discrepancies between their beliefs and the way they posed or evaluated story problems for real-life connections. These results will generate further discussions about the issues and challenges teacher education programs face, while providing suggestions for future research.     

Pre-service teachers’ conceptions of creativity in elementary school English

Widely thought to be something worth encouraging in young learners, creativity has popularly been associated more with music and art than with other areas of the curriculum. There have been many studies of creativity but few that focus explicitly on what counts as creative thinking in specific subject areas. The aim of the research reported here was to determine pre-service teachers’ conceptions of creativity within the curriculum for English. The study involved analyses of primary school trainee teachers’ responses to questionnaires and follow-up focus group discussion to identify their conceptions. A group of 48 trainees in the final year of an undergraduate degree in primary education leading to qualified teacher status in England completed the questionnaire. Of these, eight volunteered to participate in a follow-up focus group discussion to further explore ideas. Responses were analysed quantitatively and qualitatively. Conceptions of creativity in English were found to be limited, focused mainly on naïve views of story writing and dramatic activity. Responses indicated that they were often unable to distinguish clearly between the concept of creativity, an example of its occurrence in the classroom, and what feature of that example made it creative. Consequently, their limited constructions of creativity were confused. It is important that teachers in schools as well as those responsible for training teachers in universities are advised that trainees’ conceptions of creativity in English may be inadequate in several respects and that they may not recognise opportunities for creativity. Pre-service training programmes could well benefit from structured courses on the forms and applications of creativity.     

Developing teachers’ subject didactic competence through problem posing

Problem posing (not only in lesson planning but also directly in teaching whenever needed) is one of the attributes of a teacher’s subject didactic competence. In this paper, problem posing in teacher education is understood as an educational and a diagnostic tool. The results of the study were gained in pre-service primary school teacher education. Students were asked to pose problems containing some given data (namely fractions). The subsequent analysis of the problems posed by the students revealed shortcomings in their conceptual understanding of fractions. Classroom-based joint reflection became the means of reeducation.     

Prospective middle school mathematics teachers’ preconceptions of geometric translations

This article reports an analysis of 44 prospective middle school mathematics teachers’ pre-existing knowledge of rigid geometric transformations, specifically the geometric translations. The main data source for this study was the participants’ responses to the tasks that were presented during semi-structured clinical interviews. The findings of the study revealed that prospective teachers had difficulties recognizing, describing, executing, and representing geometric translations. The results indicated that teacher candidates held various views about the geometric translations: (1) translation as rotational motion, (2) translation as translational motion, and (3) translation as mapping. The results further revealed various interpretations of the vector that defines translations: (1) vector as a force, (2) vector as a line of symmetry, (3) vector as a direction indicator, and (4) vector as a displacement. Although many of the teacher candidates interviewed knew that a vector has a magnitude and a direction, this knowledge did not generally lead them to conclude that vectors define translations.     

Chinese elementary school teachers’ perceptions of students’ classroom behaviour problems

This study examined teachers’ perceptions of classroom behaviour problems in five provinces of the People’s Republic of China. Researchers surveyed 527 Chinese teachers from 27 elementary schools. Consistent with previous studies in China, teachers perceived non‐attention to be the most frequent and troublesome behaviour problem. Teachers’ perceptions of which behaviour problems were most difficult to tolerate and most negative in their effects on student development were also investigated. Approximately 45% of the teachers reported spending too much time on behaviour problems. Significant differences were found in the prevalence of teachers’ perceptions of student misbehaviour and of the time spent on classroom management; there were also differences in these perceptions according to students’ gender, type of school, classroom subject taught, and teachers’ level of experience. The implications of these findings for researchers and practitioners are addressed.     

The relationship between teachers’ empathy and perceptions of school culture

This research examined the relationship between teachers’ empathy and perceptions of their school’s culture. Teachers’ ability to change their school’s culture might be limited by their inability to interpret and respond appropriately to student behaviour. As teachers’ empathic abilities increase, it seems likely that they would be better able to understand and respond appropriately to their students. Teachers’ perspective‐taking was positively associated with their positive perceptions of student–peer relations, school norms and educational opportunities. Teachers’ personal distress was negatively related to student–peer relations. Empathy was unrelated to student–teacher relations. It is postulated that it takes more than just empathy to be able to negotiate the complex relationship between student and teacher. With teacher training programmes currently focusing on teacher dispositions, such programmes need to focus more on training future teachers to recognise and exercise their cognitive and emotional empathic capacities.     

Prospective elementary school teachers’ professional noticing of children’s early numeracy

The goal of this study is to develop the professional noticing abilities of prospective elementary school teachers in the context of the Stages of Early Arithmetic Learning. In their mathematics methods course, ninety-four prospective elementary school teachers from three institutions participated in a researcher-developed five-session module that progressively nests the three interrelated components of professional noticing—attending, interpreting, and deciding. The module embeds video excerpts of diagnostic interviews of children doing mathematics (representations of practice) to prepare the prospective teachers for similar work. The module culminates with prospective teachers implementing similar diagnostic interviews (approximations of practice) to gain experience in the three component skills of professional noticing. A pre- and post-assessment was administered to measure prospective teachers’ change in the three components. A Wilcoxon signed ranks test was conducted and found the prospective elementary school teachers demonstrated significant growth in all three components. Selected prospective elementary school teacher responses on the pre- and post-assessment are provided to illustrate sample growth in the prospective teachers’ abilities to professionally notice. These results, the first in an ongoing study, indicate the potential that prospective teachers can develop professional noticing skills through this module. Continued data collection and analysis from the ongoing study by these authors and future, longer-term emphasis on professional noticing for prospective teachers should be studied.     

An exploratory study of pre-service middle school teachers’ knowledge of algebraic thinking

Using algebraic habits of mind as a framework, and focusing on thinking about functions and how they work, we examined the relationship between 18 pre-service middle school teachers’ ability to use the features of the algebraic thinking (AT) habit of mind “Building Rules to Represent Functions” and their ability to recognize and interpret the features of the same AT habit of mind in middle school students. We assessed the pre-service teachers’ own ability to use the AT habit of mind Building Rules to Represent Functions by examining their solutions to algebra-based tasks in a semester-long mathematics content course. We assessed the pre-service teachers’ ability to recognize and interpret students’ facility with the AT habit of mind Building Rules to Represent Functions by analyzing their interpretations of students’ written solutions to algebra-based tasks and their ability to plan and analyze AT interviews of middle school students during a concurrent field-based education course. The data revealed that the pre-service teachers had a limited ability to recognize the full richness of algebra-based tasks’ potential to elicit the features of Building Rules to Represent Functions in students. The pre-service teachers’ own overall AT ability to Build Rules to Represent Functions was related to their ability to recognize the overall ability of students to Build Rules to Represent Functions, as exhibited during one-on-one interviews, but not to their ability to recognize the overall ability to Build Rules to Represent Functions exhibited exclusively in students’ written work. Implications for mathematics teacher education are discussed.     

The structure of prospective kindergarten teachers’ proportional reasoning

Lamon (Teaching fractions and ratios for understanding. Essential content knowledge and instructional strategies for teachers, 2nd edn. Lawrence Erlbaum Associates, Mahwah, 2005) claimed that the development of proportional reasoning relies on various kinds of understanding and thinking processes. The critical components suggested were individuals’ understanding of the rational number subconstructs, unitizing, quantities and covariance, relative thinking, measurement and “reasoning up and down”. In this study, we empirically tested a theoretical model based on the one suggested by Lamon (Teaching fractions and ratios for understanding. Essential content knowledge and instructional strategies for teachers, 2nd edn. Lawrence Erlbaum Associates, Mahwah, 2005), as well as an extended model which included an additional component of solving missing value proportional problems. Data were collected from 238 prospective kindergarten teachers. To a great extent, the data provided support for the extended model. These findings allow us to make some first speculations regarding the knowledge that prospective kindergarten teachers possess in regard to proportional reasoning and the types of processes that might be emphasized during their education.     

Integrating learner and learning concerns: prospective elementary science teachers’ paths and progress

This article describes three prospective teachers’ progress toward an integrated view of learner and learning concerns. Two prospective teachers formulated similar questions by the end of their first year about how to balance concerns for individual learners’ experiences in class with concerns that they learn content. However, the two arrived at this point from very different initial conceptions via different conceptual paths during the year. In contrast, the third prospective teacher (same program, same year) prepared to give up content learning goals if they interfered with learners’ enthusiasm about the lesson. Possible explanations and implications for teacher education are discussed.     

Examining and elaborating upon the nature of elementary prospective teachers’ conceptions of partitive division with fractions

The purpose of this study was to examine and elaborate upon elementary prospective teachers’ (PSTs) conceptions of partitive division with fractions. We examined the degree to which PSTs’ conceptions were connected (i.e., capable of translating between representations correctly; aware that partitive division generates a unit rate for its quotient) and flexible (i.e., capable of differentiating between opportunities to partition or iterate (or both) when solving a partitive division task; aware that partitioning or iterating (or both) could be associated with the operation of division, as appropriate). Seventeen PSTs participated in task-based interviews prior to instruction in a mathematics content course for teachers. These PSTs demonstrated disconnected conceptions of partitive division with fractions when they incorrectly translated between representations and either inconsistently or did not express awareness that the purpose of the task was to generate a unit rate. These PSTs demonstrated rigid conceptions of partitive division with fractions such that they did not express awareness that the process of iterating could be associated with the operation of division, even when they obtained a correct answer by iterating. Results extend prior research by looking beyond PSTs’ performance on tasks to elaborate upon PSTs’ conceptions of the operation of partitive division. This study contributes new insights into PSTs’ conceptions that can be used by mathematics teacher educators to inform the design of future instructional interventions.     

Developing prospective elementary teachers’ abilities to identify evidence of student mathematical achievement

This study investigated the effects of a classroom intervention on prospective elementary teachers’ ability to evaluate evidence of student achievement of mathematical learning goals. The intervention was informed by a framework for teacher education which aims to provide prospective teachers (PTs) with the skills needed to systematically learn from their own teaching practice. Prospective teachers (N = 160) participated in an intervention aimed at addressing their misconceptions about evidence of student learning. Results revealed that after the intervention, PTs were less likely to consider teacher behaviors to be evidence of student learning and more likely to discount student responses that were irrelevant to a specified learning goal. However, PTs were still likely to take procedural fluency as evidence of conceptual understanding and may have become overly skeptical of student understanding. Implications of the study suggest new ways of developing prospective teachers’ ability to systematically study and improve their teaching.     

Feedback during active learning: elementary school teachers’ beliefs and perceived problems

Giving feedback during active learning is an important, though difficult, task for teachers. In the present study, the problems elementary school teachers perceive and the beliefs they hold regarding this task were investigated. It appeared that teachers believe conditional teacher skills, especially time management, hinder them most from giving good feedback. The most widely held belief was that “feedback should be positive”. Teachers also believed that it is important to adopt a facilitative way of giving feedback, but they found this difficult to implement. Only some teachers believed goal-directedness and a focus on student meta-cognition were important during active learning and teachers did not perceive problems regarding these aspects. It was discussed whether teachers’ feedback behaviour was in line with these perceived problems and beliefs. The results give directions for the professional development of teachers to improve their feedback during active learning.