Cabinetmakers’ Workplace Mathematics and Problem Solving

This study explored what kind of mathematics is needed in cabinetmakers’ everyday work and how problem solving is intertwined in it. The informants of the study were four Finnish cabinetmakers and the data consisted of workshop observations, interviews, photos, pictures and sketches made by the participants during the interviews. The data was analysed using different qualitative techniques. Even though the participants identified many areas of mathematics that could be used in their daily work, they used mathematics only if they were able to. The cabinetmakers’ different mathematical skills and knowledge were utilized to their skill limit. Cabinetmakers were found to constantly face problem solving situations along with the creative processes. Being able to use more advanced mathematics helped them to solve those problems more efficiently, without wasting time and materials. Based on the findings, the paper discusses the similarities and differences between problem solving and creative processes. It is suggested that the combination of craftsmanship, creativity, and efficient problem solving skills together with more than basic mathematical knowledge will help cabinetmakers in adapting and surviving in future unstable labour markets.     

Institutions influencing mathematics students' private work: A factor of academic achievement

The aim of the research presented in this paper is to contribute to our knowledge about problem solving in mathematics. My purpose in this paper is to compare, from this point of view, two very different institutions in the French tertiary education system, with the intention to interpret the chronic inequality of performance in problem solving between populations of mathematics students coming from these institutions. Problem solving knowledge and skills are not an explicit objective of teaching and their development depends largely on the student's private mathematical activity. This hypothesis is the reason why the inquiry aims at comparing mathematics students' ways of working as they study in both institutions. The results of the research are interpreted, on the institutional level, as effects of differences between the two teaching systems. This revised version was published online in July 2006 with corrections to the Cover Date.     

Mathematical Problem Solving with Technology: the Techno-Mathematical Fluency of a Student-with-GeoGebra

This study offers a view on students’ technology-based problem solving activity through the lens of a theoretical model which accounts for the relationship between mathematical and technological knowledge in successful problem solving. This study takes a qualitative approach building on the work of a 13-year-old girl as an exemplary case of the nature of young students’ spontaneous mathematical problem solving with technology. The empirical data comprise digital records of her approaches to two problems from a web-based mathematical competition where she resorted to GeoGebra and an interview where she explains and describes her usual problem solving activity with this tool. Based on a proposed model for describing the processes of mathematical problem solving with technologies (MPST), the main results show that this student’s solving and expressing the solution are held from the early and continuing interplay between mathematical skills and the perception of the affordances of the tool. The analytical model offers a clear picture of the type of actions that lead to the solution of each problem, revealing the student’s ability to deal with mathematics and technology in problem solving. By acknowledging this as a case of a human-with-media in solving mathematical problems, the students’ efficient way of merging technological and mathematical knowledge is portrayed in terms of her techno-mathematical fluency.     

论数学课题探究教学

数学课题探究教学是指,在教师指导下,围绕某一课题,运用探究的方法主动获取数学知识,独立、自主地解决数学问题,培养科学精神和创造性思维与能力的一种实践活动。它具有内容的开放性和方法的多样性等特点。其内容主要有数学基本概念和规律、数学中的综合问题、现实中的数学问题以及科学前沿中的数学问题等。数学课题探究教学包括教师示范指导和诱导、学生探究、师生共探三个基本环节。     

Scaffolding 6th graders’ problem solving in technology-enhanced science classrooms: a qualitative case study

In response to the calls to improve and deepen scientific understanding and literacy, considerable effort has been invested in developing sustainable technology-enhanced learning environments to improve science inquiry. Research has provided important guidance for scaffolding learning in mathematics and science. However, these reports have provided relatively little insight into how the different types of scaffolds can (or should) be implemented in dynamic, everyday classroom settings. In this qualitative case study, we examined how students solve scientific problems in technology-enhanced classrooms and how peer-, teacher-, and technology-enhanced scaffolds influenced student inquiry. The results indicated that students manifested distinct inquiry patterns when solving scientific problems and integrated different types of scaffolds to facilitate inquiry activities. These findings suggest that to support scientific inquiry in problem-solving contexts, technology-enhanced scaffolds are effective when supported by clear project goals, relevant evidence, peer- and teacher-assessments, and exemplars of knowledge articulation.     

数学应用题教学之我见

应用题是数学的重要组成部分,也是教学的重点和难点,学生应用题解题能力可以在一定程度上代表数学知识掌握水平。教师应把教授学生怎样解答应用题作为教学重点,巩固学生的数学基础知识,培养学生的数学思维和逻辑思维能力。文章对数学教学中教师怎样对学生进行应用题教学进行论述。     

Mathematical knowledge for teaching teachers: knowledge used and developed by mathematics teacher educators in learning to teach via problem solving

In this study, we report on what types of mathematical knowledge for teaching teachers (MKTT) mathematics teacher educators (MTEs) use and develop when they work together and reflect on their teaching in a Community of Practice while helping prospective primary teachers (PTs) generate their own mathematical knowledge for teaching in learning mathematics via problem solving. Two novice MTEs worked with an experienced MTE and reflected on the process of learning to teach via problem solving and supporting PTs in developing deep understandings of foundational mathematical ideas. Taking a position of inquiry as stance, we examined our experiences teaching mathematics content courses for PTs via problem solving. We found that all of the MTEs used and developed some MKTT through (a) understanding and deciding on the mathematical goals of both the individual lessons and the two-course sequence as a whole, (b) choosing and facilitating tasks, and (c) using questions to scaffold PTs learning and engage them in mathematical processes such as making conjectures, justifying their reasoning, and proving or disproving conjectures.     

天津学生数学学习的性别差异——来自PISA的结果

数学学习的性别差异一直是人们关心的问题。PISA 2012测评结果显示,虽然天津男生与女生数学学习成绩不存在差异,但男女生数学学习的驱动力、动机和自我信念还是有明显差异。与男生相比,女生学习的坚持性、问题解决的开发性、对自身解决数学问题能力的自信更差,而数学焦虑更强,更倾向于将数学学习失败的责任归咎于自身以外的因素。     

在宽严相济的刑事政策下如何解决自侦案件“询问难”的问题

造成检察机关自侦案件“询问难99存在多方面的原因。解决“询问难”对于检察机关具有重要意义。从立法上赋予检察机关自侦部门机动侦查权、建立污点证人制度、提高询问能力和水平、建立拒绝提供证据罚则,都有利于检察机关“询问难”问题的解决。     

论有效渗透小学数学思想方法

数学课堂中,教师不仅要关注学生习得的基本知识和基本技能,更要适时采用不同策略有效渗透一些数学思想方法,培养学生的思维能力。现结合小学数学广角教学实践,从情境创设、知识形成、问题思考、知识应用及课外实践五个维度进行了渗透策略的研究,让学生获得更广泛的数学活动经验,领悟分析问题、解决问题的学习方法,进而实现渗透目的。     

Infusing assessment into mathematics content courses for pre-service elementary school teachers

The present study aims to explore the use of assessment in mathematics content courses for future elementary school teachers. Analysis of self assessment data on mathematical understanding and peer assessment data on oral mathematical presentation showed that pre-service teachers had a balanced understanding of procedural knowledge and problem solving. Conceptual understanding was not in the structure of pre-service teachers’ mathematical knowledge. Understandings of conceptual knowledge, procedural knowledge, and problem solving had no meaningful effects on gains in mathematics performance. Aspects of oral mathematical presentation were associated with improved understanding of procedural knowledge and in particular conceptual knowledge. The result of the study calls for a conceptual approach to mathematical knowledge and sufficient mathematical problem solving in college-level mathematics content courses and in particular the infusion of assessment into college-level mathematics education for pre-service teachers.     

Data and graph interpretation practices among preservice science teachers

The interpretation of data and construction and interpretation of graphs are central practices in science, which, according to recent reform documents, science and mathematics teachers are expected to foster in their classrooms. However, are (preservice) science teachers prepared to teach inquiry with the purpose of transforming and analyzing data, and interpreting graphical representations? That is, are preservice science teachers prepared to teach data analysis and graph interpretation practices that scientists use by default in their everyday work? The present study was designed to answer these and related questions. We investigated the responses of preservice elementary and secondary science teachers to data and graph interpretation tasks. Our investigation shows that, despite considerable preparation, and for many, despite bachelor of science degrees, preservice teachers do not enact the (“authentic”) practices that scientists routinely do when asked to interpret data or graphs. Detailed analyses are provided of what data and graph interpretation practices actually were enacted. We conclude that traditional schooling emphasizes particular beliefs in the mathematical nature of the universe that make it difficult for many individuals to deal with data possessing the random variation found in measurements of natural phenomena. The results suggest that preservice teachers need more experience in engaging in data and graph interpretation practices originating in activities that provide the degree of variation in and complexity of data present in realistic investigations. © 2005 Wiley Periodicals, Inc. J Res Sci Teach 42: 1063–1088, 2005     

“问题解决”在数学应用题教学中的运用

在小学数学应用题教学中,教师可以运用“问题解决”的策略。在实际运用中给予学生具体的方法指导,从而拓宽学生的解题思路,活跃学生的思维,培养学生运用数学知识解决实际问题的能力。     

Case studies of teaching problem solving

Summary Some important results that relate to classroom learning and teaching of problem solving emerge from these case studies. These are now summarized as follows. In terms of the students' potential learning experiences of problem solving, it was found that the students were mainly witnessing their teachers' demonstrations of using rules or algorithms for solution to problems. Repeated practice of solving the sorts of problems that occur in examinations was also emphatically included as part of the learning experience. The students were not exposed to a range of strategies that could possibly be used to solve the same problems. There was no explicit teaching of important problem solving skills such as translation skills (comprehending, analyzing, interpreting, and defining a given problem) and linkage skills (concept relatedness between two concepts or using cues from the problem statements to associate ideas, concepts, diagrams, etc. from memory). When teachers solve problems they use, in general, several strategies to solve the same class of problems and they are very careful and explicit about translating problem statements, making relevant linkages and checking. These absences in the teachers' teaching of problem solving (and hence in the students' range of learning experiences) are particularly interesting because they are part of the teachers' own repertoire of skills. Accordingly, it may not be too difficult to get teachers to include them in their teaching. This would mean that the students' range of learning experiences for problem solving would be very much strengthened.     

Mathematics anxiety and working memory: Longitudinal associations with mathematical performance in Chinese children

The link between mathematics anxiety and mathematical performance in young children remains inconclusive. The present study examined the longitudinal associations between mathematics anxiety and mathematical performance (calculation and story problem solving) in 246 Chinese children followed from second to third grade. Multiple regression analyses showed that mathematics anxiety made independent contributions to mathematical performance beyond non-verbal intelligence, working memory, number skills, general and test anxieties. However, mathematics anxiety does not affect all children and all kinds of mathematical performance equally. Mathematics anxiety has a more pronounced impact on mathematical problems that require more processing resources, as opposed to simple arithmetic problems and straightforward story problems and children who are higher in working memory are more vulnerable to its deleterious impacts.     

The acquisition of problem-solving skills in mathematics: How animations can aid understanding of structural problem features and solution procedures

In this paper the augmentation of worked examples with animations for teaching problem-solving skills in mathematics is advocated as an effective instructional method. First, in a cognitive task analysis different knowledge prerequisites are identified for solving mathematical word problems. Second, it is argued that so called hybrid animations would be most effective for acquiring these prerequisites, because they show the continuous transition from a concrete, but superficial problem representation to a more abstract, mathematical problem model that forms a basis for solving a problem. An experiment was conducted, where N = 32 pupils from a German high school studied either only text-based worked examples explaining different problem categories from the domain of algebra or worked examples augmented with hybrid animations. Learners with hybrid animations showed superior problem-solving performance for problems of different transfer distance relative to those in the text-only condition.     

Predictors of well‐structured and ill‐structured problem solving in an astronomy simulation

This study compared the problem‐solving skills required for solving well‐structured problems and ill‐structured problems in the context of an open‐ended, multimedia problem‐solving environment in astronomy. Two sets of open‐ended questions assessed students' abilities for solving well‐structured and ill‐structured problems. Generalized, rubric scoring systems were developed for assessing problem‐solving skills. Instruments were also developed and administered to assess cognitive and affective predictors of problem‐solving performance. By regressing the scores on the cognitive and affective predictors onto students' scores on the well‐structured and ill‐structured problems, we concluded that solving well‐structured and ill‐structured problems require different component skills. Domain knowledge and justification skills were significant predictors of well‐structured problem‐solving scores, whereas ill‐structured problem‐solving scores were significantly predicted by domain knowledge, justification skills, science attitudes, and regulation of cognition. Implications for problem solving in science education are presented. © 2003 Wiley Periodicals, Inc. J Res Sci Teach 40: 6–33, 2003     

An investigation of relationships between students’ mathematical problem-posing abilities and their mathematical content knowledge

The importance of students’ problem-posing abilities in mathematics has been emphasized in the K-12 curricula in the USA and China. There are claims that problem-posing activities are helpful in developing creative approaches to mathematics. At the same time, there are also claims that students’ mathematical content knowledge could be highly related to creativity in mathematics, too. This paper reports on a study that investigated USA and Chinese high school students’ mathematical content knowledge, their abilities in mathematical problem posing, and the relationships between students’ mathematical content knowledge and their problem-posing abilities in mathematics.     

数学课堂教学中数据分析核心素养的培育

培养学生数据分析素养是新时代教学变革对数学教学提出的新要求,同时,在数学课堂教学中培育学生的数据分析核心素养也是新时代教师面临的挑战。21世纪初各国提出的核心素养之一即包括数学与科学技术素养,我国数学课程标准也明确将数据分析列入数学学科核心素养。因此,数据分析的数学课堂教学应突破传统、单一的教学理论与方法,创设新维度的课堂教学及模式。数据分析知识体系是数学课堂教学及其建构的重要载体,建构的基本框架包括问题与情境、材料与数据、活动与经验、知识与技能、理论与方法五个维度。教师应通过数学问题解决教学、数据情境教学、案例式教学、探究式教学与开放式教学、数学思想方法的教学等多元化路径培育学生的数据分析核心素养。