Revealing German primary school students’ achievement in measurement

The focus of this study was to investigate primary school students’ achievement in the domain of measurement. We analyzed a large-scale data set (N = 6,638) from German third and fourth graders (8- to 10-year-olds). These data were collected in 2007 within the framework of the ESMaG (Evaluation of the Standards in Mathematics in Primary School) project carried out by the Institute for Educational Quality Improvement (IQB) at Humboldt University, Berlin, Germany. The data were interpreted using a classification scheme based on a conceptual–procedural distinction in measurement competence. The analyses with this classification revealed that grade, gender, and in particular figural reasoning ability are significantly related to overall measurement competence as well as on the sub-competencies of Instrumental knowledge and Measurement sense. The paper concludes with a discussion of the implications of the findings of this study for teaching and assessing measurement.     

A primary obstruction to topological embeddings¶and its applications

For a proper continuous map f:M→N between topological manifolds M and N with m≡ dimM < dimN≡m+k, a primary obstruction to topological embeddings θ(f) ∈H ^c _{m − k} (M; Z ₂) has been defined and studied by the authors in {9, 8, 2, 3], where H ^c _* denotes the singular homology with closed support. In this paper, we study the obstruction from the viewpoint of differential topology and give various applications. We first give some characterizations of embeddings among generic differentiable maps, which are refinements of the results in [9, 10]. Then we give a result concerning the number of connected components of the complement of the image of a codimension-1 continuous map with a normal crossing point, which generalizes the results in [6, 4, 5, 9]. Finally we give a simple proof of a theorem of Li and Peterson [20] about immersions of m-manifolds into (2m-1)-manifolds. Received: 3 December 1999 / Revised version: 10 October 2000     

Factors affecting primary health care centers’ economic and production efficiency

Little is known about the efficiency of health centers, despite their central role in primary health care strategy for several countries. This study evaluates the health centers in Greece, and identifies factors impeding the achievement of efficiency, with the aim of determining how their efficiency could be improved. Two alternative conceptual models are used to ensure the consistency of the efficiency results: one model is focusing on production efficiency and the other on economic efficiency. Subsequently a second stage analysis is performed to account for the impact of explanatory variables on efficiency. The use of DEA models alongside with bootstrap techniques allows calculating more accurately the efficiency scores that can reflect the performance of health centers more properly. The main drivers of health centers’ technical efficiency for both conceptual models were the location characteristics, the population growth, the mortality rate and the competition. The scale efficiency of health centers in production model is reflected by the size of their respective covered populations, the location characteristics and the mortality rate while the economic model is affected by their size, the location characteristics and the percentage of population working in agriculture. Determining how these variables influence on efficiency is essential for determining performance improvement strategies.     

A note “On some primary subgroups and the structure of finite groups”

We point out an error in the paper (Monatsh Math. doi:10.1007/s00605-013-0492-3, 2013) by Kang, and give new brief proofs of the main results.     

Measuring pre-service teachers’ self-efficacy in tutoring children in primary mathematics: an instrument

This article reports on the use of Rasch modelling to develop and validate an instrument measuring self-efficacy in tutoring children in primary mathematics (SETcPM). In response to the literature on teacher efficacy, the 20-item instrument aims to inform teacher educators, and is designed for novice pre-service teachers (nPSTs) preparing to teach mathematics in primary school (grades 1–7, ages 6–13). To ensure that the tasks of teaching are imaginable for nPSTs, the instrument targets the core activity of teaching mathematics: helping a generic child with mathematics tasks. We propose that SETcPM is measurable for the intended population and represents a central part of self-efficacy in teaching mathematics (SETM). Understanding the initial SETcPM of novices and mapping its development over the course of their training programme can contribute to a better understanding of SETM, and allow teacher educators to tailor their support.     

‘Excellent’ primary mathematics teachers’ espoused and enacted values of effective lessons

This paper reports a study that explored the characteristics of mathematics lessons that were espoused as effective by six ??excellent?? mathematics teachers and how they enacted their values in their classroom practice. In this study, we define espoused values as values that we want other people to believe we hold, and enacted values as values that we actually practice. Qualitative data were collected through video-recorded lesson observations (3 lessons for each teacher) and in-depth interviews with teachers after each observation. At the end of the project, stimulated-recall focus group interviews were used to allow teachers to define the meaning of an effective mathematics lesson as well as to recall and reflect on a 10-min edited video clip of one of their teaching lessons. The findings showed that these teachers shared five common characteristics of effective mathematics lessons: achieving teaching objectives; pupils?? cognitive development; affective achievement of pupils; focus on low-attaining pupils; and active participation of pupils in mathematics activities. These values were espoused explicitly as well as enacted in the lessons observed.     

Finite groups with two conjugacy class sizes of π-elements of primary and biprimary orders

We determine the structure of a finite group all of whose π-elements of order divisible by at most two primes have conjugacy class sizes 1 or n, where π is a set of primes and n > 1 is a positive integer.     

Solution representations of percentage change problems: the pre-service primary teachers’ mathematical thinking and reasoning

Solution representations can reveal how problem solvers communicate mathematical thinking and reasoning in problem-solving process. The present study examined the solution representations used by 20 pre-service teachers for the percentage change problems. The pre-service teachers were invited to solve a combination of simple and complex percentage change problems. The score for the majority of simple problems was 75% or above, but the score for the complex problems was below 75%. The highest percentage error occurred when the pre-service teachers encountered a percentage greater than 100% in the percentage change problems. Irrespective of their level of mathematics qualifications, the equation approach demonstrating two-step problem-solving process was the predominant strategy adopted by the pre-service teachers. The equation approach imposes low cognitive load and, therefore, is more accessible and efficient than the unitary approach. A few pre-service teachers used the unitary approach. The findings indicate that the pre-service teachers possessed relevant mathematical knowledge for percentage change problems. Furthermore, the inclusion of the equation approach in mathematics textbooks would provide an alternative perspective regarding the teaching and learning of percentage change problems.     

The flux of primary cosmic ray nuclei of atomic numberZ⩾2 at geomagnetic latitude 30°

The method of determining the primary flux of nuclei with atomic number Z?3 developed by Bradt and Peters, has been extended to include all nuclei of charge Z?2. It is shown that the method permits the identification of primary helium nuclei with an efficiencyν?90%. The primary α-particle flux obtained in this way is in very good agreement with that obtained by other methods and the relative flux values for nuclei of atomic number 3?Z?5 and 6?Z?9 agree with previous determinations.     

“He was poking holes …” A case study on figuring out metadiscursive rules through primary sources

An important aspect of participation in a new academic discourse pertains to the metadiscursive rules which govern that discourse. Researchers have documented the viability of using primary sources in undergraduate mathematics education for scaffolding students’ recognition of those rules. Our research explores the related question of whether the use of primary sources can support students’ learning of metadiscursive rules in a way that goes beyond mere recognition. We present a case study of one student’s “figuring out” of metadiscursive rules in a university Analysis course as a result of her experience with a Primary Source Project, illustrate evidence for three dimensions of “figuring out” (adoption, acceptance, awareness) that emerged from that case study, and discuss the implications of our findings for classroom instruction and future research.     

An in-service primary teacher’s implementation of mathematical tasks: the case of length measurement and perimeter instruction

The purpose of this paper is to examine the cognitive demand levels of tasks used by an in-service primary teacher during length measurement and perimeter instruction and to examine a possible link between these tasks and the teacher’s mathematical knowledge in teaching. For this purpose, a case study approach was used and the data was drawn from classroom observations, semi-structured interviews, and field notes. Specific tasks from length measurement and perimeter instruction were presented and analyzed according to the Mathematical Tasks Framework. Then, how these tasks gave information about the teacher’s mathematical knowledge in teaching in the length measurement and perimeter topics was examined according to the Knowledge Quartet model. According to the findings of the study, the tasks used during length measurement and perimeter instruction were mostly categorized as low-level tasks. In addition, teacher’s mathematical knowledge in teaching affected the implementation of the tasks.     

Homogeneity or heterogeneity? Profiles of opportunities to learn in primary teacher education and their relationship to cultural context and outcomes

The curriculum of teacher education has been described as heterogeneous across countries and influenced by the context in which it is implemented. The present study investigates this potential heterogeneity by conducting latent class analysis of opportunities to learn mathematics, mathematics pedagogy and general pedagogy in primary teacher education as indicated by future teachers from 15 countries at the end of their training. The aim was to identify curricula profiles based on data from the “Teacher Education and Development Study: Learning to Teach Mathematics (TEDS-M)”. In each teacher education component, three groups of primary teachers were identified which differed quantitatively and qualitatively. Associations between these profiles and countries revealed broader cultural influences on OTL and thus shared (within a culture) and at the same time distinct (between cultures) visions of what primary teachers should know. Within countries, associations between curricula profiles and teacher education programs pointed to shared (within a program) and at the same time distinct (between programs) visions of what primary teachers should know at the end of their training. The OTL profiles were significantly related to outcomes of primary teacher education in terms of mathematics content knowledge, mathematics pedagogical content knowledge and general pedagogical knowledge.     

The cultural dimension of beliefs: an investigation of future primary teachers’ epistemological beliefs concerning the nature of mathematics in 15 countries

Beliefs constitute a central part of a person’s professional competencies and are crucial to the perception of situations as they influence our choice of actions. This paper focuses on epistemological beliefs about the nature of mathematics of future primary teachers from an international perspective. The data reported are part of a larger sample originating from the TEDS-M study which compares primary mathematics teacher education in 15 countries. In this paper we examine the pattern of beliefs of future teachers aiming to teach mathematics at primary level. We explore whether and to what extent beliefs concerning the nature of mathematics are influenced by cultural factors, in our case the extent to which a country’s culture can be characterized by an individualistic versus collectivistic orientation according to Hofstede’s terminology. In the first part of the paper, the literature on epistemological beliefs is reviewed and the role of culture and individualism/collectivism on the formation of beliefs concerning the nature of mathematics will be discussed. In the empirical part, means and distributions of belief ratings will be reported. Finally, multilevel analyses explore how much of the variation of belief preferences between countries can be explained by the individualistic orientation of a country.