The impact of a conceptual model-based mathematics computer tutor on multiplicative reasoning and problem-solving of students with learning disabilities

The study explored the impact of Please Go Bring Me-COnceptual Model-based Problem Solving (PGBM-COMPS) computer tutoring system on multiplicative reasoning and problem solving of students with learning disabilities. The PGBM-COMPS program focused on enhancing the multiplicative reasoning and problem solving through nurturing fundamental mathematical ideas and moving students above and beyond the concrete level of operation. This is achieved by taking advantages of the constructivist approach from mathematics education and explicit conceptual model-based problem solving approach from special education. Participants were three elementary students with learning disabilities (LD). A mixed method design was employed to investigate the effect of the PGBM-COMPS program on enhancing students’ multiplicative reasoning and problem solving. It was found that the PGBM-COMPS program significantly improved participating students’ problem solving performance not only on researcher developed criterion tests but also on a norm-referenced standardized test. Qualitative and quantities data from this study indicate that, in addition to nurturing fundamental concept of composite units, it is necessary to help students to understand underlying problem structures and move toward mathematical model-based problem representation and solving for generalized problem solving skills.     

The application of automated reasoning to formal models of combinatorial optimization

Many formalisms have been proposed over the years to capture combinatorial optimization algorithms such as dynamic programming, branch and bound, and greedy. In 1989 Helman [9] presented a common formalism that captures dynamic programming and branch and bound type algorithms. The formalism was later extended to include greedy algorithms. In this paper, we describe the application of automated reasoning techniques to the domain of our model, in particular considering some representational issues and demonstrating that proofs about the model can be obtained by an automated reasoning program. The long-term objective of this research is to develop a methodology for using automated reasoning to establish new results within the theory, including the derivation of new lower bounds and the discovery (and verification) of new combinatorial search strategies.     

The use of concrete learning objects taken from the history of mathematics in mathematics education

This study aimed to reveal the effects of teaching with concrete learning objects taken from the history of mathematics on student achievement. Being a quasi-experimental study, it was conducted with two grade 8 classes in a secondary school located in Trabzon. The experimental group consisted of 27 students and the control group consisted of 25. Data were collected by using worksheets, an achievement exam and written opinion forms. The data from the achievement exam were analysed by using the Mann-Whitney U-test while the data from written opinion forms were analysed through content analysis. The Mann–Whitney U-test results showed a significant difference between the mean ranks of the experimental and control groups in favour of the former. Findings from the written opinion forms suggested that the students found the activities to be instructive and fun, enjoyed using concrete models in their classes, and learned from discovering the rules. It was also found that students had previously not engaged in similar activities and had only experienced the history of mathematics through the life stories and works of mathematicians and the representation of ancient numbers at the beginning of each unit.     

The potential of recreational mathematics to support the development of mathematical learning

ABSTRACT

A literature review establishes a working definition of recreational mathematics: a type of play which is enjoyable and requires mathematical thinking or skills to engage with. Typically, it is accessible to a wide range of people and can be effectively used to motivate engagement with and develop understanding of mathematical ideas or concepts. Recreational mathematics can be used in education for engagement and to develop mathematical skills, to maintain interest during procedural practice and to challenge and stretch students. It can also make cross-curricular links, including to history of mathematics. In undergraduate study, it can be used for engagement within standard curricula and for extra-curricular interest. Beyond this, there are opportunities to develop important graduate-level skills in problem-solving and communication. The development of a module ‘Game Theory and Recreational Mathematics’ is discussed. This provides an opportunity for fun and play, while developing graduate skills. It teaches some combinatorics, graph theory, game theory and algorithms/complexity, as well as scaffolding a Pólya-style problem-solving process. Assessment of problem-solving as a process via examination is outlined. Student feedback gives some indication that students appreciate the aims of the module, benefit from the explicit focus on problem-solving and understand the active nature of the learning.     

The origins of mathematics education research in the UK: a tribute to Brian Griffiths

The meeting of the British Society for Research into Learning Mathematics held at the University of Southampton on 21st June 2008 was dedicated to the memory of Brian Griffiths, who had died earlier that month. At the meeting it was suggested that I write a paper tracing the development of research in mathematics education in the UK up to the writing of Mathematics: Society and Curricula, a book that Brian and I co-authored, published in 1974. This paper is dedicated to Brian's memory: I hope that it will be considered a fitting tribute to a great friend and colleague, and to an outstanding mathematician and mathematics educator.     

Learning to teach high school mathematics: Patterns of growth in understanding right triangle trigonometry during lesson plan study

“Lesson plan study” (LPS), adapted from the Japanese Lesson Study method of professional development, is a sequence of activities designed to engage prospective teachers in broadening and deepening their understanding of school mathematics and teaching strategies. LPS occurs over 5 weeks on the same lesson topic and includes four opportunities to revisit one's own ideas and the ideas of others. In this paper, we describe one prospective teacher's growth in understanding right triangle trigonometry as she participated in LPS. This study is part of a much larger study investigating how prospective secondary teachers learn to teach mathematics within the context of LPS. Results of this study indicate that Image Saying, an activity for growth in understanding from the Pirie-Kieren model [Pirie, S., & Kieren, T. (1994). Growth in mathematical understanding: How can we characterize it and how can we represent it? Educational Studies in Mathematics, 26, 165-190], is critical to prospective teachers’ growth in understanding school mathematics. Multiple opportunities and contexts within which to share understanding of school mathematics led to significant growth in understanding of right triangle trigonometry which in turn led to growth in understanding of teaching strategies. That is, the results of this study indicate that growth in understanding school mathematics (what to teach) leads to growth in understanding teaching strategies (how to teach) as prospective teachers participate in LPS.     

Teaching routines and student-centered mathematics instruction: The essential role of conferring to understand student thinking and reasoning

We compare two lessons with respect to how a teacher centers student mathematical thinking to move instruction forward through enactment of five mathematically productive teaching routines: Conferring To Understand Student Thinking and Reasoning, Structuring Mathematical Student Talk, Working With Selected and Sequenced Student Math Ideas, Working with Public Records of Students’ Mathematical Thinking, and Orchestrating Mathematical Discussion. Findings show that the lessons differ in the enactment of teaching routines, especially Conferring to Understand Student Thinking and Reasoning which resulted in a difference in student-centeredness of the instruction. This difference highlights whose mathematics was being centralized in the classroom and whether the focus was on correct answers and procedures or on students’ mathematical thinking and justifying.     

A proportional view: The mathematics of James Glenie (1750–1817)

The mathematical work of James Glenie (1750–1817) was published at irregular intervals during a turbulent life. His ideas, mostly deriving from his time as an Assistant in Mathematics at St Andrews University in Scotland, were developed intermittently over a period of thirty-seven years. His mathematical achievements, underestimated by previous historians, were deeply rooted in Euclidean geometry and his own generalized theory of proportion. Among them are many new geometrical constructions and proofs, a novel demonstration of the binomial theorem, and an alternative approach to the differential calculus.     

Integration of ECQFD,TRIZ, and AHP for innovative and sustainable product development

Automotive organizations need to adopt sustainability principles to survive in a competitive environment. The rapidly changing marketplace also means that organizations need to include innovation in product development. We propose a model that integrates environmentally conscious quality function deployment (ECQFD), the theory of inventive problem-solving (TRIZ), and an analytical hierarchy process (AHP) for innovative and sustainable product development of automotive components. The voice of the customer (VOC) was captured and translated to engineering characteristics using ECQFD. Design options were identified using ECQFD and correlated with TRIZ to identify innovative design alternatives. Selection of the best design alternatives under many criteria is a typical multicriteria decision-making problem. We used AHP to identify the best design in terms of innovation and sustainability. These design changes were then incorporated in the component. A case study involving design of an automotive component demonstrates the applicability of our approach.     

Reply to the note on article “The evidential reasoning approach for multiple attribute decision analysis using interval belief degrees”

In a very recent note by Gao and Ni [B. Gao, M.F. Ni, A note on article “The evidential reasoning approach for multiple attribute decision analysis using interval belief degrees”, European Journal of Operational Research, in press, doi:10.1016/j.ejor.2007.10.0381], they argued that Yen’s combination rule [J. Yen, Generalizing the Dempster–Shafer theory to fuzzy sets, IEEE Transactions on Systems, Man and Cybernetics 20 (1990) 559–570], which normalizes the combination of multiple pieces of evidence at the end of the combination process, was incorrect. If this were the case, the nonlinear programming models we proposed in [Y.M. Wang, J.B. Yang, D.L. Xu, K.S. Chin, The evidential reasoning approach for multiple attribute decision analysis using interval belief degrees, European Journal of Operational Research 175 (2006) 35–66] would also be incorrect. In this reply to Gao and Ni, we re-examine their numerical illustrations and reconsider their analysis of Yen’s combination rule. We conclude that Yen’s combination rule is correct and our nonlinear programming models are valid.     

The effectiveness of resources created by students as partners in explaining the relevance of mathematics in engineering education

First-year engineering students often struggle to see the relevance of theoretical mathematical concepts for their future studies and professional careers. This is an issue, as students who do not see relevance in fundamental parts of their studies may disengage from these parts and focus their efforts on other subjects they think will be more useful to them. In this study, we surveyed engineering students enrolled in a first-year mathematics subject on their perceptions of the relevance of the individual mathematical topics taught. Surveys were administered at the start of semester when some of these topics were unknown to them, and again at the end of semester when students had not only studied all these topics but also watched a set of animated videos. These videos had been produced by higher-year students to explain where they had seen applications of the mathematical concepts presented in the first year. We notice differences between the perceived relevance of topics for future study and for professional careers, with relevance to study rated higher than relevance to careers. We also find that the animations are seen as helpful in understanding the relevance of first-year mathematics. The majority of students indicated that lecturers with students as partners should work collaboratively to produce future videos.     

The wholes that are greater than the sum of their parts: employing cooperative learning in mathematics teachers’ education

This paper presents a study on employing different types of cooperative learning (CL) settings in mathematics teacher education based on multiple research data. The study analyzes mechanisms in which CL contributes to the development of teacher knowledge of three kinds: subject matter knowledge, pedagogical content knowledge, and curricular content knowledge. Additionally, it reflects on interactions between different kinds of teachers’ knowledge in the process of development. Two wholes, which are greater than the sum of their parts, are analyzed in this paper. The first is a course whose design combines different CL settings, hence enhances different mechanisms for teachers’ professional development. The second whole, teachers’ “collaborative mind,” is presented as an outcome of the first. To exemplify possible contributions of employing CL in teacher education, this paper first zooms in on the structure of one particular course for mathematics teachers and then focuses on one particular mathematical activity.     

Validating DEA as a ranking tool: An application of DEA to assess performance in higher education

There is a general interest in ranking schemes applied to complex entities described by multiple attributes. Published rankings for universities are in great demand but are also highly controversial. We compare two classification and ranking schemes involving universities; one from a published report, ‘Top American Research Universities’ by the University of Florida's TheCenter and the other using DEA. Both approaches use the same data and model. We compare the two methods and discover important equivalences. We conclude that the critical aspect in classification and ranking is the model. This suggests that DEA is a suitable tool for these types of studies.     

L_(18)的一个案例及其研究

本文对正交设计Ｌ１８（６×３７）的一个案例及其分析方法重新进行了研究，阐述了正交试验设计中一些值得注意的问题     

我国普通高等教育发展水平的全局主成分分析

本文根据 1995年、1997年和 1999年共同构成的关于我国普通高等教育发展水平的时序立体数据表 ,通过全局主成分分析得到公共主成分子空间L(F1,F2 ,… ,Fm) ,并在该子空间对我国高等教育发展背景和现状进行统计分析 ,给出了排名和变化趋势。     

The activity structure of technology-based mathematics lessons: a case study of three teachers in English secondary schools

ABSTRACT

This article examines patterns of classroom organisation and interaction associated with the use of a particular type of digital technology – the dynamic software GeoGebra – in the lessons of an opportunity sample of three English secondary-school mathematics teachers. The concept of activity structure is used to organise this study, further informed by the concept of instrumental orchestration. While the case study analysis identifies structures already reported in those earlier papers, it also draws attention to the prevalence of a Predict-and-test format in tasks carried out by students at the computer. This study also shows how synthesising the activity structure and instrumental orchestration frameworks may be productive.     

VaR风险控制体系的建立与应用

目前VaR作为一种新的风险控制工具得到越来越广泛的应用,投资组合理论则一直沿用经典的σ2风险控制体系,虽说有人已经将VaR引入到了投资组合应用中来,但其风险控制尚未脱离对σ2的分解.将在引入股票相对价格的基础上构建了VaR风险控制体系,将投资风险VaRP分解为大盘指数风险VaRI和股票相对价格的风险VaRS之和,并给出了此风险控制体系在投资组合方面的基本应用方法.     

占座效应的数学建模与统计分析

利用因果图模型及其可识别性理论对占座效应 ,即占座行为对座位使用率的影响进行了建模 .并利用这一模型对实际例子中的占座效应进行了分析 ,最终得出占座行为对实例中座位使用率的影响程度