Counter-examples for refinement of conjectures and proofs in primary school mathematics

The purpose of this study is to explore how primary school students reexamine their conjectures and proofs when they confront counter-examples to the conjectures they have proved. In the case study, a pair of Japanese fifth graders thought that they had proved their primitive conjecture with manipulative objects (that is, they constructed an action proof), and then the author presented a counter-example to them. Confronting the counter-example functioned as a driving force for them to refine their conjectures and proofs. They understood the reason why their conjecture was false through their analysis of its proof and therefore could modify their primitive conjecture. They also identified the part of the proof which was applicable to the counter-example. This identification and their action proof were essential for their invention of a more comprehensive conjecture.     

The Effects of Teachers and Schooling on the Vocational Choice of University Research Scientists

Thirty-five research scientists from the academic areas of Chemistry, Geology, Physics and Zoology were surveyed to obtain information regarding the effect of teachers and formal schooling on their decisions to become scientists. All subjects included in the study held the PhD. degree and were actively involved in research. Data analysis indicated that 63 percent of the subjects had identified a genuine interest in science by the time they were in the ninth grade, but only 9 percent of them attributed this interest to their elementary or junior high school teachers. The data revealed that 43 percent of the population were influenced to become scientists by one or more high school teachers. Also, one-third of the subjects had made the decision to become a scientist by the time of high school graduation. Most of the other two-thirds of the individuals studied decided to become scientists because of the influence of college professors in their freshman-and sophomore-level courses. The subjects were asked if particular events that occurred in junior or senior high school science classes positively influenced them to become scientists. Seventy-eight percent of the subjects responded that there were no classroom activities occurring that attracted them to science.     

PRIMARY TEACHER TRAINEES’ MATHEMATICS SUBJECT KNOWLEDGE AND CLASSROOM PERFORMANCE

UK government agencies have recently set in train requirements to ‘crack down’ on primary [elementary] school teacher trainees whose own knowledge of mathematics is weak. The responsibility to identify and support (or fail) them currently rests with training providers (mainly university schools of education). We describe one approach to this process, presenting some findings concerning what trainees find difficult and how their knowledge is related to their teaching competence. We flesh out these findings with a case study of a mathematically-strong trainee whose path to qualification was less than smooth.     

Explicit constructions and the arithmetic of local number fields

This is a survey of results obtained by members of the St. Petersburg school of local number theory headed by S.V. Vostokov during the past decades. All these results hardly fit into the title of the paper, since they involve a large circle of ideas, which are applied to an even larger class of problems of modern number theory. The authors tried to cover at least a small part of them, namely, those related to the modern approach to explicit expressions of the Hilbert symbol for nonclassical formal modules in the one- and higher-dimensional cases and their applications in local arithmetic geometry and ramification theory.     

Teachers Share Their Mathematics Backgrounds: Telling It Like It Was

This study focused on the mathematics backgrounds of 45 elementary mathematics teachers. A survey was administered which asked teachers to reflect on their elementary, secondary, and college mathematics experiences, the support they received and the confidence they have with their ability to teach elementary mathematics. Quantitative and qualitative data derived from the survey, revealed that sharing mathematics knowledge in elementary school; committed, enthusiastic teachers in high school; parental support; and practical uses of mathematics were positive influences for these teachers. Rote drill, seatwork, and ability grouping were negative influences. The majority of the teachers felt inservice staff development experiences helped make them confident teachers of elementary mathematics. The implications of the study for mathematics education are discussed.     

Teaching Algebraic Expressions to Young Students: The Three‐day Journey of “a+ 2”

This classroom scholarship report is based on the teaching experience using Davydov's mathematics curriculum, which was developed in the former Soviet Union. While “from arithmetic to algebra” is the normally accepted instructional sequence in school mathematics, Davydov's curriculum is laid out “from algebra to arithmetic,” focusing on algebraic thinking from the very beginning of the elementary grades. The purpose of this report is not to provide a definitive conclusion about which curriculum or sequence is better nor to address which instructional strategy is right in all circumstances. Rather, it is to explore how primary grade students develop their own conceptual understanding while confronting difficulties met within a specific context. This report provides actual classroom episodes from working with a group of first graders and describes dynamic interactions between the teacher and children while they discuss the use of algebraic expressions and understand the meaning behind them.     

The school bus routing problem: A review

This paper aims to provide a comprehensive review of the school bus routing problem (SBRP). SBRP seeks to plan an efficient schedule for a fleet of school buses where each bus picks up students from various bus stops and delivers them to their designated schools while satisfying various constraints such as the maximum capacity of a bus, the maximum riding time of a student in a bus, and the time window of a school. This class of problem consists of different sub-problems involving data preparation, bus stop selection, bus route generation, school bell time adjustment, and bus scheduling. In this paper, the various assumptions, constraints, and solution methods used in the literature on SBRP are summarized. A list of issues requiring further research is also presented.     

A third grader's way of thinking about linear function tables

This paper is inscribed within the research effort to produce evidence regarding primary school students’ learning of algebra. Given the results obtained so far in the research community, we are convinced that young elementary school students can successfully learn algebra. Moreover, children this young can make use of different representational systems, including function tables, algebraic notation, and graphs in the Cartesian coordinate grid. In our research, we introduce algebra from a functional perspective. A functional perspective moves away from the mere symbolic manipulation of equations and focuses on relationships between variables. In investigating the processes of teaching and learning algebra at this age, we are interested in identifying meaningful teaching situations. Within each type of teaching situation, we focus on what kind of knowledge students produce, what are the main obstacles they find in their learning, as well as the intermediate states of knowledge between what they know and the target knowledge for the teaching situation. In this paper, we present a case study focusing on the approach adopted by a third grade student, Marisa, when she was producing the formula for a linear function while she was working with the information of a problem displayed in a function table containing pairs of inputs-outputs. We will frame the analysis and discussion on Marisa's approach in terms of the concept of theorem-in action (Vergnaud, 1982) and we will contrast it with the scalar and functional approaches introduced by Vergnaud (1988) in his Theory of Multiplicative Fields. The approach adopted by Marisa turns out to have both scalar and functional aspects to it, providing us with new ways of thinking of children's potential responses to functions.     

Conditions and decisions of urban elementary teachers regarding instruction of STEM curriculum

The study was situated in a National Science Foundation supported Math Science Partnership between a private university and an urban school district. This study sought to understand the decision‐making process of elementary teachers as they implement an integrated science, technology, engineering, and mathematics (STEM) curriculum in their classrooms and the interactions that occur between the teachers and curriculum during that process. This qualitative study utilized a comparative case study approach to understanding the decision‐making process of three elementary teachers enacting the same lesson. Analysis of the interactions revealed that the teachers' perceptions of student ability, their pedagogical design capacity, and time were influences that impacted implementation. These findings have implications for STEM‐focused professional development of elementary teachers.     

Mathematics standards and curricula in the Netherlands

This paper addresses the question of what mathematics Dutch students should learn according to the standards as established by the Dutch Ministry of Education. The focus is on primary school and the foundation phase of secondary school. This means that the paper covers the range from kindergarten to grade 8 (4~14 years olds). Apart from giving an overview of the standards, we also discuss the standards' nature and history Furthermore, we look at textbooks and examination programs that in the Netherlands both have a key role in determining the intended mathematics curriculum. In addition to addressing the mathematical content, we also pay attention to the way mathematics is taught. The domain-specific education theory that forms the basis for the Dutch approach to teaching mathematics is called “Realistic Mathematics Education” Achievement scores of Dutch students from national and international tests complete this paper. These scores reveal what the standards bring us in terms of students' mathematical understanding. In addition to informing an international audience about the Dutch standards and curricula, we include some critical reflections on them.     

A note on conjugate distributions for copulas

A family of conjugated distributions for a given type of copulas is defined in this paper. Those copulas can be written as a mixture of d‐dimensional parameter exponential functions. The generalized Farlie–Gumbel–Morgenstern copula is an example of this representation. This family is used to illustrate the estimation technique with real data. Also, the applicability of Bayesian predictive approach is shown in an education policy issue by defining goals for the number of students per class that leads to improve their performance at school. Copyright © 2014 John Wiley & Sons, Ltd.     

Investigating changes in high-stakes mathematics examinations: a discursive approach

ABSTRACT

This article focuses on the theoretical-methodological question of how to identify reform-induced changes in school mathematics. The issue arose in our project The Evolution of the Discourse of School Mathematics (EDSM), in which we studied transformations in high-stakes examinations taken by students in England at the end of compulsory schooling. We have adopted a conceptualisation that draws on social semiotics and on a communicational approach, according to which school mathematics can be thought of as a discourse. Methods of comparing examinations of different years developed on the basis of this definition enable identification of subtle disparities that are nevertheless significant enough to make an important difference in students’ vision of mathematics, in their performance and, eventually, in their ability to cope with problems that can benefit from the use of mathematics. In this article, we present these methods and argue that they have wider application for comparative studies of school mathematics.     

Assigning mathematics instruction time in secondary schools: what are the influential factors?

Similar to countries such as the Netherlands and the United Kingdom, secondary schools in Ireland can decide how to allocate instruction time between curriculum subjects. Although there are national guidelines available from the Department of Education and Skills (DES), the majority of schools make their own decisions about how much time they allocate to different subjects. This results in variations between the amounts of time allocated to teaching mathematics in different schools and between different year and class groups within the same school. Decisions regarding time allocation are generally taken by the school management. This means that the ethos of the school and the individual opinions of school management can determine the amount of mathematics that students experience throughout their second level education. The aim of this study is to evaluate the most influential factors that management considers when assigning instruction time in Irish secondary schools. For the purpose of this research, seven possible factors were identified and 400 deputy principals from a stratified sample of secondary schools around Ireland were asked to select their top three. Timetabling constraints, the availability of mathematics teachers and the perceived importance of the subject were found to be the most influential factors.     

Assessing European primary school performance through a conditional nonparametric model

This paper uses a fully nonparametric framework to assess the efficiency of primary schools using data about schools in 16 European countries participating in PIRLS 2011. This study represents an original enterprise since most of the empirical research in the field is restricted to evaluations at regional or national level and focused on secondary education. For our purpose, we adapt the metafrontier framework to compare and decompose the technical efficiency of primary schools operating in heterogeneous contexts, which in our case is represented by different educational systems or countries. Similarly, we use an extension of the conditional nonparametric robust approach to test the potential influence of a mixed set of environmental school factors and variables representing cultural values of each country. Our results indicate that the intergenerational transmission of non-cognitive skills such as responsibility or perseverance are significantly related to school efficiency, whereas most school factors do not seem to have a significant influence on school performance.     

Elementary school teachers’ growth in inquiry-based teaching of mathematics

This article reports on a self-directed, school-based, practice-based professional development (PD) experience aimed at helping elementary school teachers to develop knowledge and expertise in inquiry-based teaching of mathematics. It discusses the characteristics of the self-directed orientation of this PD that supported the teachers’ learning, the nature of the inquiry-based knowledge they constructed, and the impact on their teaching. It highlights the centrality of agency, practical knowledge, and situated learning in this PD approach. The findings suggest that this approach can help mathematics teachers who want to be the architect of their own learning to transform their classrooms in meaningful and desirable ways.     

Clarifying the Place of Essential Topics and Unifying Principles in High School Biology

The content of high school biology courses is overwhelming in quantity. This coverage does not permit students to engage in inquiry whereby they construct knowledge in a meaningful manner. Initially, the study sought to identify a small list of essential topics as the focus of high school biology. With the help of seven science supervisors in a large metropolitan area in the Southwest who have taught high school biology for many years, the problem of course content was examined. After three separate contacts by mail to the supervisors using a Delphi technique, a view emerged: Teach students the big ideas of biology rather than specific topics, but use topics to develop the major themes and unifying principles. These findings need to be discussed with biology teachers to help them focus their teaching on a smaller body of course content that relates directly to the major ideas of biology.     

The Zoo Connection: A Cooperative Project Between Formal and Informal Educational Institutions

This project was a cooperative effort between university faculty, elementary school teachers, and members of the Education Department at the Indianapolis Zoo. The purpose of this project was to develop, evaluate, and disseminate a set of fourteen K-6 science lessons that could be used in conjunction with field trips to the zoo. These lessons, titled The Zoo Connection, follow the Learning Cycle teaching model. In addition to following the model, the development of the lessons was based on the premise that teachers should focus on a specific science concept or set of related concepts when visiting a zoo. Workshop sessions were conducted to introduce teachers to the materials and to provide them with strategies for implementing the materials in their science instruction. An evaluation was conducted for each workshop session to determine teachers' perceptions of the materials and to determine whether they felt prepared to use them with their students. The materials were also field-tested in several elementary schools to assess their effectiveness for presenting science concepts to elementary school children. Results indicate that teachers felt the workshops adequately prepared them to use the materials and that the materials were effective for presenting science concepts.     

Addition and subtraction of three-digit numbers: adaptive strategy use and the influence of instruction in German third grade

Empirical findings show that many students do not achieve the level of a flexible and adaptive use of arithmetic computation strategies during the primary school years. Accordingly, educators suggest a reform-based instruction to improve students’ learning opportunities. In a study with 245 German third graders learning by textbooks with different instructional approaches, we investigate accuracy and adaptivity of students’ strategy use when adding and subtracting three-digit numbers. The findings indicate that students often choose efficient strategies provided they know any appropriate strategies for a given problem. The proportion of appropriate and efficient strategies students use differs with respect to the instructional approach of their textbooks. Learning with an investigative approach, more students use appropriate strategies, whereas children following a problem-solving approach show a higher competence in adaptive strategy choice. Based on these results, we hypothesize that different instructional approaches have different advantages and disadvantages regarding the teaching and learning of adaptive strategy use.     

Altering the Bias in Differential School Effectiveness Using Data Envelopment Analysis

There is evidence that many schools are differentially effective in that they promote with different effectiveness the academic attainments of different groups of pupils, divided by such characteristics as prior academic attainments, socio-economic background and so on. Awareness of the extent and direction of a school's differential effectiveness is important. Measures of overall performance of a differentially effective school can be misleading and targets seeking to improve the overall performance of such a school may not address its underlying inadequacies in raising the attainments of specific groups of pupils. This paper develops a data envelopment analysis based method for setting targets for diffentially effective schools to enable them to alter the bias of their differential effectiveness. The method also identifies role model schools which a differentially effective school can emulate in seeking to redress the imbalances in its effectiveness.     

Learning to represent,representing to learn

This study explores how students learn to create, discuss, and reason with representations to solve problems. A summer school algebra class for seventh and eighth graders provided opportunities for students to create and use representations as problem-solving tools. This case study follows the learning trajectories of three boys. Two of the three boys had been low-achievers in their previous math classes, and one was a high achiever. Analysis of all three boys’ written work reveals how their representations became more sophisticated over time. Their small group interactions while problem-solving also show changes in how they communicated and reasoned with representations. For these boys, representation functioned as a learning practice. Through constructing and reasoning with representations, the boys were able to engage in generalizing and justifying claims, discuss quadratic growth, and collaborate and persist in problem-solving. Negotiating different student-constructed representations of a problem also gave them opportunities to act with agency, as they made choices and judgments about the validity of the different perspectives. These findings have implications for the importance of giving all students access to mathematics through representations, with representational thinking serving as a central disciplinary practice and as a learning practice that supports further mathematics learning.