Preservice elementary school teachers’ knowledge of fractions: a mirror of students’ knowledge?

This research analyses preservice teachers’ knowledge of fractions. Fractions are notoriously difficult for students to learn and for teachers to teach. Previous studies suggest that student learning of fractions may be limited by teacher understanding of fractions. If so, teacher education has a key role in solving the problem. We first reviewed literature regarding students’ knowledge of fractions. We did so because assessments of required content knowledge for teaching require review of the students’ understanding to determine the mathematics difficulties encountered by students. The preservice teachers were tested on their conceptual and procedural knowledge of fractions, and on their ability in explaining the rationale for a procedure or the conceptual meaning. The results revealed that preservice teachers’ knowledge of fractions indeed is limited and that last-year preservice teachers did not perform better than first-year preservice teachers. This research is situated within the broader domain of mathematical knowledge for teaching and suggests ways to improve instruction and student learning.     

Preservice teachers’ self efficacy and knowledge of emotional and behavioural disorders

This research examined preservice teachers’ knowledge of emotional and behavioural disorders (EBD) and their sense of efficacy. The participants included a convenience sample of 230 undergraduate general education and special education preservice teachers enrolled in teacher education classes. The age of the participants ranged from 19 to 51 with a mean age of 23.37 years (SD = 6.8 years). The Teacher Self Efficacy Scale (TSES) (long form) and Knowledge of Emotional and Behavioural Disorders questionnaire were administered. The participants had higher efficacy in instructional strategies, classroom management, and instructional abilities than in student engagement. There was no significant correlation between field experience, additional coursework, and familiarity with a child with EBD and the preservice teachers’ knowledge of EBD or self efficacy. This information is relevant to teacher programme coordinators indicating that further efforts should be made to increase preservice teachers’ knowledge of EBD, and to provide strategies for identifying and working with students with EBD in classrooms.     

Pre-service secondary teachers’ pedagogical content knowledge for the teaching of mathematical modelling

Educational Studies in Mathematics - The article deals with the pedagogical content knowledge of mathematical modelling as part of the professional competence of pre-service teachers. With the help...     

Providing written feedback on students’ mathematical arguments: proof validations of prospective secondary mathematics teachers

Mathematics teachers play a unique role as experts who provide opportunities for students to engage in the practices of the mathematics community. Proof is a tool essential to the practice of mathematics, and therefore, if teachers are to provide adequate opportunities for students to engage with this tool, they must be able to validate student arguments and provide feedback to students based on those validations. Prior research has demonstrated several weaknesses teachers have with respect to proof validation, but little research has investigated instructional sequences aimed to improve this skill. In this article, we present the results from the implementation of such an instructional sequence. A sample of 34 prospective secondary mathematics teachers (PSMTs) validated twelve mathematical arguments written by high school students. They provided a numeric score as well as a short paragraph of written feedback, indicating the strengths and weaknesses of each argument. The results provide insight into the errors to which PSMTs attend when validating mathematical arguments. In particular, PSMTs’ written feedback indicated that they were aware of the limitations of inductive argumentation. However, PSMTs had a superficial understanding of the “proof by contradiction” mode of argumentation, and their attendance to particular errors seemed to be mediated by the mode of argument representation (e.g., symbolic, verbal). We discuss implications of these findings for mathematics teacher education.     

Novice teachers’ learning and knowledge building during the induction programme

This paper explores the Estonian novice teachers’ learning and knowledge building (LKB) practices in the extended professional community during the induction programme using well-known knowledge conversion model. The assumption in this study is that a teachers’ participation in the extended professional community facilitates their professional development. The survey was conducted with 101 novices after their induction programme. The patterns of novices’ LKB practices in the professional learning community during the induction programme were explored. The analysis showed to what extent extended professional community may be formed during the induction year. LKB practice patterns among the novices were identified. We discovered that many novices felt that there was insufficient support from other teachers and from university experts. It appears that it is challenging to develop a coherent induction programme as the extended professional community of educators, where different partners collaborate and share professional knowledge is challenging.     

Preservice teachers’ articulated noticing through pedagogies of practice

Elementary preservice teachers at six universities engaged in a task that provided them opportunities to articulate their professional noticing within video representations, written decompositions, and animated approximations of practice. The preservice teachers’ written accounts indicated that a majority attended to students or student thinking; however, when asked to illustrate their noticing through animation, focus shifted to the classroom teacher. Findings indicate the extent to which preservice teachers articulated specific mathematics concepts within and across pedagogies of practice and highlight the critical importance for selecting and utilizing multiple types of tasks to better understand preservice teacher noticing. Implications for eliciting and supporting preservice teacher noticing are discussed.     

Prediction assessments: Using video-based predictions to assess prospective teachers’ knowledge of students’ mathematical thinking

In order to evaluate the effectiveness of an experimental elementary mathematics field experience course, we have designed a new assessment instrument. These video-based prediction assessments engage prospective teachers in a video analysis of a child solving mathematical tasks. The prospective teachers build a model of that child’s mathematics and then use that model to predict how the child will respond to a subsequent task. In this paper, we share data concerning the evolution and effectiveness of the instrument. Results from implementation indicate moderate to high degrees of inter-rater reliability in using the rubric to assess prospective teachers’ models and predictions. They also indicate strong correlation between participation in the experimental course and prospective teachers’ performances on the video-based prediction assessments. Such findings suggest that prediction assessments effectively evaluate the pedagogical content knowledge that we are seeking to foster among the prospective teachers.     

Using video analysis to support prospective K-8 teachers’ noticing of students’ multiple mathematical knowledge bases

As part of a larger research project aimed at transforming preK-8 mathematics teacher preparation, the purpose of this study was to examine the extent to which prospective teachers notice children’s competencies related to children’s mathematical thinking, and children’s community, cultural, and linguistic funds of knowledge or what we refer to as children’s multiple mathematical knowledge bases. Teachers’ noticing supports students’ learning in deep and meaningful ways. Researchers designed and enacted a video analysis activity with prospective teachers in their mathematics methods course. The activity served as a decomposition of practice in order to support prospective teachers in engaging in an approximation of the practice of noticing. Our findings showed that prospective teachers evidenced noticing of mathematics teaching and learning as early as the mathematics methods course. We also found that the prompts and structure of the activity supported prospective teachers by increasing their depth of noticing and their foci in noticing, moving from attending primarily to teacher moves (and merely describing what they saw) to becoming aware of significant interactions (and interpreting effects of these interactions on learning). Implications for teacher educators interested in designing and enacting activities to support noticing are discussed.     

On the alignment of teachers’ mathematical content knowledge assessments with the common core state standards

Instruments designed to measure teachers’ knowledge for teaching mathematics have been widely used to evaluate the impact of professional development and to investigate the role of teachers’ knowledge in teaching and student learning. These instruments assess a mixture of content knowledge and pedagogical content knowledge. However, little attention has been given to the content alignment between such instruments and curricular standards, particularly in regard to how content knowledge and pedagogical content knowledge items are distributed across mathematical topics. This article provides content maps for two widely used teacher assessment instruments in the USA relative to the widely adopted Common Core State Standards. This common reference enables comparisons of content alignment both between the instruments and between parallel forms within each instrument. The findings indicate that only a small number of items on both instruments are designed to capture teachers’ pedagogical content knowledge and that the majority of these items are focused on curricular topics in the later grades rather than in the early grades. Furthermore, some forms designed for use as pre- and post-assessment of professional development or teacher education are not parallel in terms of curricular topics, so estimates of teachers’ knowledge growth based on these forms may not mean what users assume. The implications of these findings for teacher educators and researchers who use teacher knowledge instruments are discussed.

     

Prospective elementary teachers’ knowledge of fraction division

Prospective elementary teachers must understand fraction division deeply in order to meaningfully teach this topic to their future students. This paper explores the nature of the subject content knowledge of fraction division possessed by a group of Taiwanese prospective elementary teachers at the beginning of their mathematics methods course. The findings provide preliminary evidence that many prospective Taiwanese elementary teachers have developed the knowledge package of fraction division as described by Ma (Knowing and teaching elementary mathematics: Teachers?? understanding of fundamental mathematics in China and the United States. Lawrence Erlbaum Associates, Mahwah, 1999). The nature of various strategies used by these teachers provides further illustration of a secure common content knowledge that can serve as a benchmark for the development of mathematics courses for prospective teachers. However, the findings also show that the tasks of representing fraction division, through either word problems or pictorial diagrams, are challenging even for those highly proficient in elementary and middle school mathematics. The broader implications of this research for the international community are discussed, and recommendations for elementary teacher education programs are presented.     

Collaboration and the growth of teachers’ knowledge

Studies of teachers working together have exposed the capricious nature of collaborative activity: sometimes it seems to work well; at other times collaboration actually works against improvement. Success in collaborative relationships is best understood through an appreciation of how teachers form and use knowledge. The teachers’ knowledge perspective is used in this paper to interpret qualitative data from two successful collaborative relationships in schools. Evidence supports the contention that personal qualities, underscored by mutual trust and respect for knowledge, form the basis for successful relationships in teaching, operating in different ways, for different purposes, for different people. Providing that teachers are approached with respect, collaboration holds promise as a slow (but powerful) path towards educational change.     

Preservice teachers’ use of contrasting cases in mathematics instruction

Drawing comparisons between students’ alternative solution strategies to a single mathematics problem is a powerful yet challenging instructional practice. We examined 80 preservice teachers’ when asked to design a short lesson when given a problem and two student solutions—one correct and one incorrect. These micro-teaching events were videotaped and coded, revealing that fewer than half of participants (43%) made any explicit comparison or contrasts between the two solution strategies. Those who did were still not likely to use additional support strategies to draw students’ attention to key elements of the comparison. Further, correlations suggest that participants’ mathematical content knowledge may be related to whether participants’ showed contrasting cases but not to whether they used specific pedagogical cues to support those comparisons. While these micro-teaching events differ from the interactive constraints of a classroom, they reveal that participants did not immediately orient toward differing student solutions as a discussion opportunity, and that future instruction on contrasting cases must highlight the utility of this practice.     

Upper primary school teachers’ mathematical knowledge for teaching functional thinking in algebra

This article is based on a project that investigated teachers’ knowledge in teaching an important aspect of algebra in the middle years of schooling—functions, relations and joint variation. As part of the project, 105 upper primary teachers were surveyed during their participation in Contemporary Teaching and Learning of Mathematics, a research project funded by the Catholic Education Office, Melbourne (2008–2012). Analysis of the survey responses revealed that two-thirds of teachers demonstrated content knowledge on a pattern generalisation task appropriate for upper primary levels of schooling (8- to 12-year-old students), but less than half demonstrated reasonable pedagogical content knowledge (PCK). On a paired variable (function machine) task, only one quarter of teachers demonstrated appropriate PCK. Although two-thirds of the teachers indicated that they currently taught content from the “Patterns and Algebra” strand of the new Australian Curriculum, less than half were able to provide examples of appropriate learning experiences for students. More than two-thirds of teachers expressed concern about their ability to teach this area of mathematics. Implications for the professional learning of teachers to improve their mathematics knowledge for developing students’ functional thinking are presented.     

Developing preschool teachers’ knowledge of students’ number conceptions

This article describes a study that investigates preschool teachers’ knowledge of their young students’ number conceptions and the teachers’ related self-efficacy beliefs. It also presents and illustrates elements of a professional development program designed explicitly to promote this knowledge among preschool teachers. Results indicated that teachers’ knowledge of students’ number conceptions improved and that their related self-efficacy increased. Furthermore, prior to participating in the programs, a significant negative correlation between knowledge and self-efficacy was reported. However, no significant correlation was found after the program.     

Socio-cultural factors of teachers’ conceptions of knowledge: epistemic beliefs of Arab teachers

This research investigates aspects of epistemic beliefs of pre-service and in-service teachers in the United Arab Emirates, and how these beliefs might be related to factors such as teachers’ gender, location (where they live), and the subject they teach (humanities vs. science). A standardized Arabic version of the Epistemic Belief Inventory was administered to a group of 35 pre-service teachers and 52 in-service teachers. The results indicated that in-service teachers had more sophisticated beliefs than pre-service teachers on Certainty of Knowledge, Omniscient Authority, and Innate Ability dimensions. Also, male teachers were found to be less sophisticated than female teachers and teachers from rural areas tended to think in a simpler way than teachers from urban areas. Although there was no main effect of subject, it significantly modified the effects of gender and location. The data showed the multidimensional nature of epistemic beliefs, and were discussed in the context of Arab/Emirati culture.     

Under-threes’ mathematical learning – teachers’ perspectives

This project highlights preschool teachers’ views of toddlers’ learning in mathematics. The Swedish national curriculum covers even the youngest children who are 1–3?years old. Interesting questions are thus: what should mathematics be for this age group and how should preschool teachers work with maths to achieve the curriculum objectives? Data were collected through interviews with six preschool teachers working in four different preschools. The data show that the teachers emphasize the body as very important for the learning process, which means that for these children, it is not a matter of simply talking about mathematical concepts, but experiencing them bodily. The teachers also report that they now pay more attention than previously to what material the children use and how they interact with it. They are more aware of how they organize and offer the various materials in the preschool and how this influences the way children use them and, consequently, their learning processes.     

Tracing teachers’ transformation of knowledge in social media

Social media represent an increasingly important vehicle for informal professional development amongst teachers, and provide an illuminating means of tracing their collective knowledge building. The study reported here examines six large Facebook groups, created by and for teachers in Sweden, to exchange information about the teaching of mathematics and Swedish. Analysis establishes that professional knowledge was made available in 86% of discussion threads, most commonly relating to Shulman’s categories of Knowledge of Learners, Curricular Knowledge, and Pedagogical Content Knowledge. Participants acknowledged opportunities for professional learning in 88% of such threads, and showed new understanding in 11%, particularly in longer threads.     

Measuring teachers’ content knowledge of language and reading

In the context of a longitudinal, four-year study of reading instruction in low-performing, high-poverty urban schools, we surveyed teacher knowledge of reading-related concepts, and established a modest predictive relationship between teachers’ knowledge, classroom reading achievement levels, and teachers’ observed teaching competence. There were significant associations among these variables at the third and fourth grade levels. To obtain this result, measures of teacher content knowledge in language and reading were refined in a three-stage process. Our purpose was to explore the type and level of questions that would begin to discriminate more capable from less capable teachers, and that would have a predictive relationship with student reading achievement outcomes. After experimenting with measurement of K-2 teachers’ content knowledge (Form #1), we piloted a Teacher Knowledge Survey with 41 second and third grade teachers in one study site (Form #2). We then refined and expanded the Survey (Form #3) and administered it to 103 third and fourth grade teachers in both project sites. Teachers’ misconceptions about sounds, words, sentences, and principles of instruction were pinpointed so that professional development could address teachers’ needs for insight and information about language structure and student learning.