Arguments constructed within the mathematical modelling cycle

Socio-cultural theories in mathematics education field recently emphasize the importance of the collective argumentation within small-group work. Since mathematical modelling tasks require a process in which students search for a solution for real life problems through small-group work, the arguments in this process become an issue of concern. This study examines the arguments constructed within the mathematical modelling cycle by considering the participants’ modelling processes. In this context, four primary pre-service mathematics teachers worked on a modelling task and their arguments were explained through the components of Toulmin’s argumentation schema. Findings revealed that the data and the claims of most of the arguments corresponded to the starting and ending points of the modelling transition in which the current arguments constructed. The existence of the arguments corresponded through warrant-claim originated from inquiring the assumptions in the modelling cycle. In addition, the participants made assumptions as warrants to support their arguments and as rebuttals to show the degree of certainty of claims in intra-group challenging situations. Both the warrants and the backings depended on modelling context as well as mathematics context.     

A report on the introduction of a multiple choice examination for a first year university engineering mathematics course

The paper is an account of the experience of introducing a multiple choice mathematics examination paper for the first year engineering mathematics course at Nottingham University. The form of the paper is described, as is a system for providing the candidates with feedback on their performance without the publication of the examination paper.     

Decision development in facilitated modelling workshops

While the facilitated modelling literature recognises the importance of the group process within facilitated modelling workshops, published empirical research rarely examines their dynamic nature. In this paper, we address this gap in the literature in two ways. First, we propose to locate facilitated modelling workshops as the main focus of investigation, and adopt decision development as the analytical lens. Second, we provide guidance regarding how to implement a research strategy that is informed by such a focus. We start by mapping the different conceptualisations of decision development that seem embedded within the facilitated modelling tradition, and contrast them with theoretical models from the group communication field. Our analysis identifies a number of potentially useful areas for the study of facilitated modelling workshops from a decision development perspective, and articulates a number tentative research questions and testable propositions amenable to empirical research. Central to our proposal are research methods for the study of dynamic group processes. We thus discuss the steps required to extract group process data from facilitated modelling workshops that are usable and open to analysis. This includes a review of issues regarding research design, coding scheme development, data coding and choice of analytical techniques. Finally, we offer conclusions and briefly discuss some feasibility issues related to the implementation of our proposal.     

An analysis of the form and content of quadrilateral definitions composed by novice pre-service teachers

Mathematical definitions are an important mathematical construct which has been noted as a challenging topic for both teachers and students. This study provides an analysis of the form and content of a set of 308 definitions of quadrilateral types provided by 44 novice pre-service elementary teachers (NPSTs) who had not yet studied geometry or definitions on the college level. Analysis of definition structure including necessary, sufficient, and minimal conditions, as well as hierarchical and partitional structure, provides insight into what NPSTs may think about the form of mathematical definitions in general. Analysis of the properties used in definitions of each shape type and the frequency of those properties reveals the shape types with which NPSTs are most and least familiar. These results are presented alongside possible applications and implications for instruction.     

Implementation and outcomes of inquiry-based learning in mathematics content courses for pre-service teachers

This mixed-methods study describes classroom characteristics and student outcomes from university mathematics courses that are based in mathematics departments, targeted to future pre-tertiary teachers, and taught with inquiry-based learning (IBL) approaches. The study focused on three two-term sequences taught at two research universities, separately targeting elementary and secondary pre-service teachers. Classroom observation established that the courses were taught with student-centred methods that were comparable to those used in IBL courses for students in mathematics-intensive fields at the same institutions. To measure pre-service teachers' gains in mathematical knowledge for teaching, we administered the Learning Mathematics for Teaching (LMT) instrument developed by Hill, Ball and Schilling for in-service teacher professional development. Results from the LMT show that pre-service teachers made significant score gains from beginning to end of their course, while data from interviews and from surveys of learning gains show that pre-service teachers viewed their gains as relevant to their future teaching work. Measured changes on pre-/post-surveys of attitudes and beliefs were generally supportive of learning mathematics but modest in magnitude. The study is distinctive in applying the LMT to document pre-service teachers' growth in mathematical knowledge for teaching. The study also suggests IBL is an approach well suited to mathematics departments seeking to strengthen their pre-service teacher preparation offerings in ways consistent with research-based recommendations.     

Developing teachers' models for assessing students' competence in mathematical modelling through lesson study

Applications and modelling have gained a prominent role in mathematics education reform documents and curricula. Thus, there is a growing need for studies focusing on the effective use of mathematical modelling in classrooms. Assessment is an integral part of using modelling activities in classrooms, since it allows teachers to identify and manage problems that arise in various stages of the modelling process. However, teachers’ difficulties in assessing student modelling work are a challenge to be considered when implementing modelling in the classroom. Thus, the purpose of this study was to investigate how teachers’ knowledge on generating assessment criteria for assessing student competence in mathematical modelling evolved through a professional development programme, which is based on a lesson study approach and modelling perspective. The data was collected with four teachers from two public high schools over a five-month period. The professional development programme included a cyclical process, with each cycle consisting of an introductory meeting, the implementation of a model-eliciting activity with students, and a follow-up meeting. The results showed that the professional development programme contributed to teachers’ knowledge for generating assessment criteria on the products, and the observable actions that affect the modelling cycle.     

Aims and Objectives in the Teaching of Mathematics to non‐Mathematicians∗

After a discussion on what is a non‐mathematician and what is an applied mathematician, aims that have been variously suggested are presented and discussed. Attention is drawn to the importance of an understanding of model building and of mathematics as the language of science, and this leads to a plea for co‐operation between the mathematician and the non‐mathematician. Reference is made to the relevance of the ‘New Mathematics’ and to the demand for mathematical rigour, and the paper closes with a brief discussion of the importance of detailed objectives.

     

Mathematics teachers’ conceptions about modelling activities and its reflection on their beliefs about mathematics

The current study examines whether the engagement of mathematics teachers in modelling activities and subsequent changes in their conceptions about these activities affect their beliefs about mathematics. The sample comprised 52 mathematics teachers working in small groups in four modelling activities. The data were collected from teachers' Reports about features of each activity, interviews and questionnaires on teachers' beliefs about mathematics. The findings indicated changes in teachers' conceptions about the modelling activities. Most teachers referred to the first activity as a mathematical problem but emphasized only the mathematical notions or the mathematical operations in the modelling process; changes in their conceptions were gradual. Most of the teachers referred to the fourth activity as a mathematical problem and emphasized features of the whole modelling process. The results of the interviews indicated that changes in the teachers' conceptions can be attributed to structure of the activities, group discussions, solution paths and elicited models. These changes about modelling activities were reflected in teachers' beliefs about mathematics. The quantitative findings indicated that the teachers developed more constructive beliefs about mathematics after engagement in the modelling activities and that the difference was significant, however there was no significant difference regarding changes in their traditional beliefs.     

Discursive positionings and emotions in modelling activities

Mathematical modelling is suggested as an activity through which students engage in meaningful mathematics. In the current research, the modelling activity of a group of four seventh-grade students was analysed using the discursive analysis framework. The research findings show that the positionings and emotions of the group members during their participation in the modelling activity changed as the activity proceeded. Overall, it can be said that three of the four group members acted as insiders, while the fourth acted as an outsider, and only, towards the end of the group's work on the activity, he acted as an insider. Moreover, the research findings point at four factors that affected the group members’ positionings and emotions during the modelling activity: the member's characteristics, the member's history of learning experiences, the activity characteristics and the modelling phases. Furthermore, the different positionings of the group members in the different modelling phases were accompanied by different emotions experienced by them, where being an insider and a collaborator resulted in positive emotions, while being an outsider resulted in negative emotions.     

Stochastic modelling of household behaviour before,during and after a consumer price promotion

A stochastic model is developed to study household behaviour with regard to purchase quantity, brand choice and purchase timing before, during and after a price change and a price promotion such as price-offs and price-cuts. The basic assumption of the model is that price promotion and levels of consumer inventory influence a household's purchase-timing and brand-switching decisions. The model incorporates market segments and brand switching on aggregated demand for the brands by the use of multivariate Markov processes. A transient stochastic model is employed to analyse the dynamic process of household behaviour before, during, and after a price promotion. The interpurchase time that is derived from the model does not require any assumptions and is not independently, identically distributed. An empirical analysis using the Information Resources Incorporated cracker data indicated that (1) price promotion does affect household purchase of the brand and (2) households with larger family size tend to purchase promoted items. We conjecture that households with larger family size take advantage of the lower price of the promoted brands while smaller households tend to remain loyal to one brand.     

A typological analysis: understanding pre-service teacher beliefs and how they are transformed

This article describes the beliefs and their transformations of members of a cohort of early-childhood, elementary and middle-level pre-service teachers (PSTs) as they professionally develop. A typological analysis of both quantitative and qualitative data collected between August 2011 and May 2013 was utilized to categorize how 40 PSTs’ beliefs transformed throughout their formal teacher preparation. Five typologies were identified, showing variation in how PST beliefs transform or remain static.

Among the findings, strong support related to the development of innovative beliefs during coursework coupled with at least one transformative experience where innovation was observed ‘working’ in the field were sufficient for the transformation to innovative beliefs, despite potential constraints by supervisors, cooperating teachers and/or mandated curricula (Typology 3). Another finding revealed disguised growth toward innovation among those in Typology 5, who reported being innovative and having productive beliefs but described extremely traditional practices. Implications call for improved connections between mathematics methods professors and field supervisors, particularly during clinical internships when PSTs are no longer enrolled in methods courses, to enhance PSTs’ productive struggle in their development of innovative beliefs (T3) and to increase opportunities for disconnects between innovative beliefs and traditional practices to be made explicit and negotiated (T5).     

Mathematical modelling of a diesel common-rail system

In diesel common-rail systems, the exact knowledge of the injection pressure is important to accurately control the injected diesel mass and thus the combustion process. This paper focuses on the mathematical modelling of the hydraulic and mechanical components of a common-rail system in order to describe the dynamics of the diesel rail pressure. Based on this model, an average model is derived to reduce the model complexity and to allow for a fast calculation of the mass flow into the rail for different crank shaft revolution speeds and openings of the fuel metering unit. The main purpose of this average model is to serve as a basis for a model-based (non-linear) controller design. The stationary accuracy of the models is validated by means of measurement data.     

Computer performance modelling: an overview

An overview of the state-of-the-art in the performance modelling of computer systems is presented in this paper. The various forms of modelling procedures used in the past are reviewed with a greater emphasis given to new approaches which include queueing networks for mutliple-resource system analysis. Aspects of accuracy and computational efficiency of the algorithms involved are also discussed.     

Mathematics identity and achievement among Black students

Employing data from the National Center of Educational Statistics' High School Longitudinal Study and utilizing critical race theory and intersectionality as theoretical frameworks, this article interrogates the relationship between mathematics identity and math success for a nationwide sample of Black secondary school students. More specifically, hierarchical regression modeling is employed to examine the relative impact of math identity, demographic variables, and school/parent social capital variables on the math grade point averages of this sample. The article ends with a discussion of specific steps for teaching mathematics that put the identity of those from traditionally marginalized communities at the center of mathematics instruction. Thus making experiences, histories, culture, and abilities essential elements of students' learning, that are to be supported and built upon.     

Dynamic process scheduling and resource allocation in distributed environment: an agent-based modelling and simulation

This paper addresses the issues concerning resource allocation and process scheduling in a dynamic environment, where resources are distributed and availability of them is uncertain. In this context, we introduce a new multi-agent-based resource allocation and process scheduling approach, where agents communicate and cooperate among themselves to produce an optimal schedule. A distributed constraint optimization problem-based model in accordance with Markov Decision Process is proposed in this regard. We overcome the hardship of existing centralized approach and our technique optimizes not only the process completion delay but also the number of resources being idle, which is much more beneficial. Apart from the theoretical approach, we take a case study in its practical application domain to validate our claim. Analysis and experimental results show that this proposed method outperforms the state-of-the-art methods and bridges the gap between theory and its applications.