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.

     

Working like real mathematicians: developing prospective teachers’ awareness of mathematical creativity through generating new concepts

This paper describes the experience of a group of 17 prospective mathematics teachers who were engaged in a series of activities aimed at developing their awareness of creativity in mathematics. This experience was initiated on the basis of ideas proposed by the participants regarding ways creativity of school students might be developed. Over a period of 6 weeks, they were engaged in inventing geometrical concepts and in the examination of their properties. The prospective teachers’ reflections upon the process they underwent indicate that they developed awareness of various aspects of creativity while deepening their mathematical and didactical knowledge.     

Measuring and exploring the structure of teachers’ educational beliefs

This study explores the structure of French teachers’ educational beliefs. First, instruments to measure these beliefs are adapted and developed. Data is collected through focus group discussions and an online survey (n = 302). Factor analyses reveal a three-factor structure to measure teachers’ general beliefs about teaching and learning: ‘beliefs about student regulation of learning processes and knowledge construction’, ‘beliefs about teacher regulation of learning processes’, and ‘beliefs about knowledge reproduction’. Furthermore, a two-factor structure to measure teachers’ subject-specific beliefs about French language teaching is found: ‘traditional beliefs about French language teaching’ and ‘constructivist beliefs about French language teaching’. Second, this study explores the relationships between and within teachers’ general beliefs about teaching and learning and their subject-specific beliefs about French language teaching. Results show two distinct and independent belief structures: a traditional and a constructivist belief structure. Via cluster analyses, two clusters of teachers are found whose beliefs are consistent with these belief structures.     

Preservice teachers’ knowledge of proof by mathematical induction

There is a growing effort to make proof central to all students’ mathematical experiences across all grades. Success in this goal depends highly on teachers’ knowledge of proof, but limited research has examined this knowledge. This paper contributes to this domain of research by investigating preservice elementary and secondary school mathematics teachers’ knowledge of proof by mathematical induction. This research can inform the knowledge about preservice teachers that mathematics teacher educators need in order to effectively teach proof to preservice teachers. Our analysis is based on written responses of 95 participants to specially developed tasks and on semi-structured interviews with 11 of them. The findings show that preservice teachers from both groups have difficulties that center around: (1) the essence of the base step of the induction method; (2) the meaning associated with the inductive step in proving the implication P(k) ⇒ P(k + 1) for an arbitrary k in the domain of discourse of P(n); and (3) the possibility of the truth set of a sentence in a statement proved by mathematical induction to include values outside its domain of discourse. The difficulties about the base and inductive steps are more salient among preservice elementary than secondary school teachers, but the difficulties about whether proofs by induction should be as encompassing as they could be are equally important for both groups. Implications for mathematics teacher education and future research are discussed in light of these findings.

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...     

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.     

Pre-service teachers’ future-oriented mathematical identity work

Research on identity has been a growing domain in the contexts of teacher education and mathematics education; however, identity work has been explored to a much lesser extent, with a future orientation overlooked. In addition, earlier studies have not provided sufficient knowledge on how different elementary teacher education programs might facilitate pre-service teachers’ identity work. In this study, we compare future-oriented mathematical identity work through a narrative framework considering six pre-service teachers undergoing two different teacher education programs. All pre-service teachers reported having had negative experiences with mathematics during their school years. Based on the results we conclude that despite the striking similarities in pre-service teachers’ mathematical backgrounds, the ways in which these cases are conducting their identity work differ substantially. It seems that the main reasons for these differences are different emphases and pedagogical practices in mathematics education courses. Additionally, we further elaborate on our earlier conceptualisation of identity work.     

Beyond the binary: rethinking teachers’ understandings of and engagement with inclusion

This article presents research into Canadian elementary and secondary teachers’ understandings of inclusion. The research investigates how a sample of 120 teachers in the southern part of Ontario defined inclusion, and the extent to which they believed an inclusive classroom is an effective way to teach all students. The article draws upon literature into how inclusion is currently defined followed by research into the politics of diversity in inclusive education; the latter signals the socio-political aporia which attends many understandings of inclusion. The study employs Nancy Fraser’s conception of justice as requiring redistribution, recognition, and representation; Fraser’s approach also demands attention to issues of recognition as intimately connected with concerns about social status. The findings reveal teachers’ relative lack of attention to issues of resourcing, but considerable emphasis upon issues of representation. While issues of recognition are largely valued, there is a tendency to reify categories of student identity, rather than challenging concerns about the lack of social status attending such foci. The research reveals a push ‘beyond the binary’ of considering teachers’ practices as either inclusive or exclusive, and how teachers’ engagement with resource provision, recognition of learners, and representation of student needs exists along contingent and intersecting spectra.     

Differences between experts’ and students’ conceptual images of the mathematical structure of Taylor series convergence

Taylor series convergence is a complicated mathematical structure which incorporates multiple concepts. Therefore, it can be very difficult for students to initially comprehend. How might students make sense of this structure? How might experts make sense of this structure? To answer these questions, an exploratory study was conducted using experts and students who responded to a variety of interview tasks related to Taylor series convergence. An initial analysis revealed that many patterns of their reasoning were based upon certain elements and actions performed on elements from the underlying mathematical structure of Taylor series. A corresponding framework was created to better identify these elements and how they were being used. Some of the elements included using particular values for the independent variable, working with terms, partial sums, sequences, and remainders. Experts and students both focused on particular elements of Taylor series, but the experts demonstrated the efficiency and effectiveness of their reasoning by evoking more conceptual images and more readily moving between images of different elements to best respond to the current task. Instead of moving between images as dictated by tasks, students might fixate on “surface level” features of Taylor series and fail to focus on more relevant features that would allow them to more appropriately engage the task. Furthermore, how experts used their images, supports the idea that they were guided by formal theory, whereas students were still attempting to construct their understanding.     

Exploring the characteristics of ‘teachers for excellence’: teachers’ own perceptions

This article is concerned first with identifying teacher characteristics connected with excellence. It then analyses the results of a survey conducted among primary school teachers in one local authority area in Scotland. Teachers responded to a questionnaire which asked them to rate in importance 44 characteristics of excellence. The findings suggest that teachers have a clear view of excellence. They consistently described excellence in terms of personal qualities and interpersonal skills. Teachers gave high ratings to qualities which demonstrated their expectations of positive classroom ethos and positive relationships with students. Practitioners consistently rated characteristics related to classroom ‘relationships in action’ as essential characteristics for excellent teachers. These findings suggest a new emphasis on interpersonal skills in continuing professional development for teachers who are increasingly expected to develop students’ dispositions relating to openness to new thinking; self‐respect and a commitment to responsible participation in social, cultural, economic and political life.     

Contested pasts,complicated presents: Pre-service teachers’ developing conceptions of community

To prepare pre-service teachers to work with diverse student populations, many teacher educators have developed community-engaged projects. This study analyzes data collected from pre-service teachers in the U. S. South as they completed a community-engaged project, where they spent time learning about the community, created a virtual tour, and revised a lesson plan to align with the information gained. The project is offered as a mediational tool contributing to pre-service teachers’ conceptions of community and teaching. Findings suggest that pre-service teachers need explicit instruction about how to analyze communities and opportunities to learn with community members during teacher education.     

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.     

The role of the Extended Project Qualification in developing self-regulated learners: exploring students’ and teachers’ experiences

Abstract

Self-regulated learning (SRL) – autonomously planning, self-monitoring and self-reflecting on learning – is a strong predictor of academic success. Mastery of the strategies needed to become a self-regulated learner does not develop automatically in all students; thus, the classroom environment, including pedagogy and modes of assessment, plays a vital role in stimulating SRL. Indeed, the post-16 curriculum in England is often criticised for failing to promote SRL. The Extended Project Qualification (EPQ), however, is a post-16 project-based qualification that has drawn from SRL theories. Research indicates that the EPQ can bolster A-level and degree attainment, yet, the mechanisms underlying this effect remain unknown. This article reports on a qualitative investigation using focus groups and interviews to explore students’ and teachers’ experiences of the EPQ and its effects on general academic performance. The qualification was seen to promote SRL by building learner agency and self-awareness, and improving engagement. We argue that the EPQ offers solutions for many of the perceived deficits of the English post-16 curriculum and that the decline of the AS-level in England represents an opportunity for learners to fill the ‘fourth space’ with qualifications like the EPQ, which could empower them to pursue a broad and diverse education.     

Secondary teachers’ experiences with students with disabilities: examining the global landscape

This study contributes to the international conversation about the education for students with disabilities at the secondary level by providing some international data about secondary teachers’ training and experiences with inclusive practices. Teacher leaders from 20 different countries participated in this exploratory survey research about the context in which they work, their experiences and their training as related to their efficacy to educate individuals with disabilities. We acknowledge that efficacy and teacher training for inclusion are predominated by a discourse defined through Western best practice. We learned that despite variations in the countries’ concept of disability and their implementation of inclusive educational systems, the teachers had moderately high self-efficacy for inclusive practices yet had limited training and experience with students with special needs. Results are discussed in relation to the preparation of secondary teachers for educating students with special needs.     

In the eyes of the beholder: students’ and teachers’ judgments of students’ motivation

This study was designed to examine the nature and accuracy of teachers’ judgments about students’ motivation related to mathematics. Seventeen fourth- through sixth-grade teachers and 100 of their students participated in this study. Teachers were asked to rate six target students’ motivation four times during the school year. At each of these times students also rated their own motivation. Teachers’ ratings were more stable over time and across mathematics topics and were less differentiated across motivation dimensions than were students’ ratings. The findings suggest that teachers need to pay careful attention to evidence related to children's motivation that might contradict their perceptions and therefore support more effective interventions.