Dissecting success stories on mathematical problem posing: a case of the Billiard Task

“Success stories,” i.e., cases in which mathematical problems posed in a controlled setting are perceived by the problem posers or other individuals as interesting, cognitively demanding, or surprising, are essential for understanding the nature of problem posing. This paper analyzes two success stories that occurred with individuals of different mathematical backgrounds and experience in the context of a problem-posing task known from past research as the Billiard Task. The analysis focuses on understanding the ways the participants develop their initial ideas into problems they evaluate as interesting ones. Three common traits were inferred from the participants' problem-posing actions, despite individual differences. First, the participants relied on particular sets of prototypical problems, but strived to make new problems not too similar to the prototypes. Second, exploration and problem solving were involved in posing the most interesting problems. Third, the participants' problem posing involved similar stages: warming-up, searching for an interesting mathematical phenomenon, hiding the problem-posing process in the problem's formulation, and reviewing. The paper concludes with remarks about possible implications of the findings for research and practice.     

再论中小学"数学情境与提出问题"的数学学习

辩证唯物主义认识论，现代数学观和建构主义学习观指导下的“设置数学情境与提出数学问题”教学实验，旨在培养学生的数学问题意识，提高学生的提出数学问题和解决数学问题能力，增强学生的创新意识和实践能力，创设数学情境，就是呈现给学生刺激性数学信息，引起学生学习数学的兴趣，启迪思维，激起学生的好奇心，发现欲，产生认知冲突，诱发质疑猜想，唤醒强烈的问题意识，从而使其发现和提出数学问题，解决数学问题。     

三论中小学"数学情境与提出问题"的数学学习

中小学“情境-问题”数学学习是指中小学生在教师的引导下，从熟悉的或感兴趣的数学情境出发，通过积极思考、主动探究、提出问题、分析问题和解决问题，从而获取数学知识、思想方法和技能技巧并应用数学知识的过程。中小学“数学情境与提出问题”的数学学习活动，既重视学生数学问题意识的培养，又重视学生数学应用能力的培养；既关注数学知识的发现过程，又关注数学问题的解决过程；既强调学习内容的开放性，又强调学习过程的探究性。     

AN INVESTIGATION ON CHINESE TEACHERS�� REALISTIC PROBLEM POSING AND PROBLEM SOLVING ABILITY AND BELIEFS

In the present study, which is a part of a research project about realistic word problem solving and problem posing in Chinese elementary schools, a problem solving and a problem posing test were administered to 128 pre-service and in-service elementary school teachers from Tianjin City in China, wherein the teachers were asked to solve 3 contextually challenging division-with-remainder (DWR) word problems and pose word problems according to 3 symbolic expressions. Afterwards, they were also given 2 questionnaires wherein they had to evaluate 3 different pupil reactions to, respectively, 1 problem solving item and 1 problem posing item about DWR. First, our results revealed that teachers behaved quite ‘realistically’ not only when solving and posing DWR problems themselves but also when evaluating elementary school pupils’ DWR problem solving and problem posing performance. Second, we found a correspondence between teachers’ own performance on the tests and their evaluations of pupils’ reactions. Third, the present study provides some further insight into the role of one of the instructional factors that is generally considered responsible for the strong and worldwide tendency among elementary school children to neglect real-world knowledge and realistic considerations in their endeavours to solve and pose mathematical word problems, namely the teachers’ conceptions and beliefs about this topic.     

游泳场馆救生员风险管理问题研究——以福建省为例

以风险管理理论与方法为基础,采用文献资料调研、专家访谈、问卷调查、现场调查、逻辑分析、数理统计等方法,结合福建省游泳场馆救生员现状,研究救生员风险及游泳场馆溺死问题,旨在建立游泳场馆安全管理体系。结果表明,游泳场馆救生员风险源包括13个风险因素,福建省游泳场馆救生员面临7个主要风险因素,游泳场馆可以采用风险避免、风险控制、风险转移和风险自留四种方法应对救生员风险。     

我国中小学数学课程中“问题提出”的演变——基于课程标准(教学大纲)的分析

中小学数学课程标准(教学大纲)中对“问题提出”的相关要求影响教科书的编写、教师的教和学生的学。采用内容分析法,对10份涉及“问题提出”的中小学数学课程标准(教学大纲)进行编码,通过定量、定性统计分析发现,中小学数学课程文件中对“问题提出”重视程度逐渐提高,建构起了从宏观到微观的“问题提出”体系;内容表述由单一逐步转向综合;目标要求由模糊笼统趋向于明确具体。未来还需要提高数学课程标准中“问题提出”的实践性;增强课程标准和教科书中“问题提出”的一致性;促进教师成为好的“问题提出”者。     

山东省独立学院篮球课教学现状及发展对策研究

本研究运用了文献资料法、问卷调查法、专家访谈法、数理统计法,以山东省12所独立学院作为调查对象,对其中所涉及到的问题进行调查和理性分析,对所存在的问题及原因进行客观的辨析和深入研究,并据此提出促进山东省独立学院篮球教学的对策和意见。     

How do mathematicians learn math?: resources and acts for constructing and understanding mathematics

In this paper, we present an analytic framework for investigating expert mathematical learning as the process of building a network of mathematical resources by establishing relationships between different components and properties of mathematical ideas. We then use this framework to analyze the reasoning of ten mathematicians and mathematics graduate students that were asked to read and make sense of an unfamiliar, but accessible, mathematical proof in the domain of geometric topology. We find that experts are more likely to refer to definitions when questioning or explaining some aspect of the focal mathematical idea and more likely to refer to specific examples or instantiations when making sense of an unknown aspect of that idea. However, in general, they employ a variety of types of mathematical resources simultaneously. Often, these combinations are used to deconstruct the mathematical idea in order to isolate, identify, and explore its subcomponents. Some common patterns in the ways experts combined these resources are presented, and we consider implications for education.     

What makes a problem mathematically interesting? Inviting prospective teachers to pose better problems

School students of all ages, including those who subsequently become teachers, have limited experience posing their own mathematical problems. Yet problem posing, both as an act of mathematical inquiry and of mathematics teaching, is part of the mathematics education reform vision that seeks to promote mathematics as an worthy intellectual activity. In this study, the authors explored the problem-posing behavior of elementary prospective teachers, which entailed analyzing the kinds of problems they posed as a result of two interventions. The interventions were designed to probe the effects of (a) exploration of a mathematical situation as a precursor to mathematical problem posing, and (b) development of aesthetic criteria to judge the mathematical quality of the problems posed. Results show that both interventions led to improved problem posing and mathematically richer understandings of what makes a problem ‘good.’     

数学问题提出的实证研究述评

问题提出是指从一个数学情境中创造新问题或在解决问题过程中对问题的再阐述.在当今数学教育的课程改革中,问题提出成为许多数学家和数学教育家关注的焦点,许多国家都把培养学生的问题提出能力作为一项重要的课程目标.国内外众多学者对数学问题提出进行了大量的实证研究,主要集中在以下几个方面:问题提出与数学理解、问题提出与问题解决、问题提出的认知策略、问题提出的教学实验.这些研究的不足之处在于:研究内容较窄;缺乏认知方面的研究;研究方法比较单一;对学生差异性教学关注不够,等等.     

Learning to pose mathematical problems: Exploring changes in preservice teachers' practices

Learning to pose mathematical tasks is one of the challenges of learning to teach mathematics. How and when preservice teachers may learn this essential practice,however, is not at all clear. This paper reports on a study that examined the changes in the problem posing strategies of a group of elementary preservice teachers as they posed problems to pupils. It reports that their later problem posing practices significantly differed from their earlier ones. Rather than posing traditional single steps and computational problems, these preservice teachers ventured into posing problems that had multiple approaches and solutions, were open-ended and exploratory, and were cognitively more complex. Their problem posing style also changed. Rather than making adaptations that made students' work easier or narrowed the mathematical scope of the problem, their adaptations became less leading and less focused on avoiding pupils' errors. Posing problems to an authentic audience, engaging in collaborative posing, and having access and opportunities to explore new kinds of problems are highlighted as important factors in promoting and supporting the reported changes. This revised version was published online in August 2006 with corrections to the Cover Date.     

初中生"提出数学问题"的现状与对策

教育观念的偏差、教学方式的单一、教育评价的片面、问题意识的淡薄,造成了我国初中学生提出数学问题能力的欠缺.在数学教学中,应探讨培养初中学生提出数学问题能力的具体对策,比如:营造宽松和谐的问题环境;精心创设问题情境;培养学生的问题意识;开展合作交流学习;教给学生提出数学问题的具体方法.     

美国数学教育中的提出问题研究综述

在数学活动中,"提出问题"是指通过对情境的探索产生新问题,或在解决问题过程中对问题的再阐述.随着全美数学教师联合会<学校数学课程与评价标准>的颁布,提出问题成为美国数学课程的重要组成部分.最近20多年以来,美国在提出问题的教学研究方面取得了大量的研究成果.这些成果对于我国新课程下提出问题由课程理念向教学现实的转移具有借鉴和参考价值.     

Problem-posing research in mathematics education: looking back, looking around, and looking ahead

In this paper, I comment on the set of papers in this special issue on mathematical problem posing. I offer some observations about the papers in relation to several key issues, and I suggest some productive directions for continued research inquiry on mathematical problem posing.     

Balancing competing demands: Enhancing the mathematical problem posing skills of prospective teachers through a mathematical letter writing initiative

Responding to mathematical problems is a core activity in classrooms. The problems that teachers select determine the mathematical content, processes and nature of mathematical inquiry occurring in classrooms and thereby contribute to the development of mathematical skills and dispositions. Selecting, designing or reformulating mathematical problems is a critical skill, then, for prospective and practising teachers. This study explores the influence of a mathematical letter writing initiative in developing the problem posing skills of 28 prospective primary teachers. We examine the characteristics of mathematical problems designed by prospective teachers, and their understandings of what constitutes a good mathematical problem, prior to and following completion of a 12-week letter writing initiative with 10–11-year-old children. Analysis of the data reveals the benefits of engaging in the initiative as evidenced in improvements in several problem characteristics. There was an increase in the number of multiple approach and multiple solution problems and in the level of cognitive demand of problems posed. The challenge of posing non-traditional problems, alongside the competing demands of building in opportunities for success, may have diminished participants’ ability to evaluate and attend to the cognitive demand of problems.

     

Ways of thinking associated with mathematics teachers’ problem posing in the context of division of fractions

Twenty-four mathematics teachers were asked to think aloud when posing a word problem whose solution could be found by computing 4/5 divided by 2/3. The data consisted of verbal protocols along with the written notes made by the subjects. The qualitative analysis of the data was focused on identifying the structures of the problems produced and the associated ways of thinking involved in constructing the problems. The results suggest that success in doing the interview task was associated with perception the given fractions as operands for the division operation and, at the same time, the divisor 2/3 as an operator acting over 4/5. The lack of success was associated with perception of division of fractions as division of divisions of whole numbers and using the result of division of fractions as the only reference point. The study sheds new light on the teachers’ difficulties with conceptualization of fractions.     

甘肃省篮球裁判员晋升培养网络教学模式的构建

运用文献资料、专家访谈、数理统计和逻辑分析等研究方法,针对甘肃省行政区域狭长、裁判员集中授课培训学习困难和信息的传递不流畅等问题进行了调查,从教学模式的设计依据、设计原则、教学目标、教学设计等方面构建和探索适合甘肃省篮球裁判员晋升培养的教学模式.研究结果表明:网络教学模式能够解决这些困难,是现行篮球裁判员晋升培养培训方式的有益补充,具有可操作性和可行性.     

论中小学“数学情境与提出问题”的数学学习

创新源于问题，问题源于情境，在中小学数学教育中，应着力抓住创设数学情境与提出数学问题的“情境－问题”学习链，努力培养学生的创新意识和实践能力，以利于我国创造型人才的成长。     

Identification of effective visual problem solving strategies in a complex visual domain

Students in complex visual domains must acquire visual problem solving strategies that allow them to make fast decisions and come up with good solutions to real-time problems. In this study, 31 air traffic controllers at different levels of expertise (novice, intermediate, expert) were confronted with 9 problem situations depicted on a radar screen. Participants were asked to provide the optimal order of arrival of all depicted aircrafts. Eye-movements, time-on-task, perceived mental effort, and task performance were recorded. Eye-tracking data revealed that novices use inefficient means-end visual problem solving strategies in which they primarily focus on the destination of aircraft. Higher levels of expertise yield visual problem solving strategies characterized by more efficient retrieval of relevant information and more efficient scan paths. Furthermore, experts' solutions were more similar than intermediates' solutions and intermediates' solutions were more similar than novices' solutions. Performance measures showed that experts and intermediates reached better solutions than novices, and that experts were faster and invested less mental effort than intermediates and novices. These findings may help creating eye-movement modeling examples for the teaching of visual problem solving strategies in complex visual domains.     

专家型数学教师数学教育观及影响因素探究——一个个案研究

采用半结构式访谈和问卷调查深入探讨了1位专家型数学教师的数学观,数学学习观和数学教学观及其相关影响因素。本研究发现该专家型数学教师主要认为:(1)数学是培养学生思维和数学能力的载体,是源于生活和用于生活的,是学生学习、考试的一个科目;(2)学习数学需一定天赋,学生自主参与、归纳总结是数学学习最佳的方式,数学思维和分析解决问题的能力是学生数学学习的主要部分和应达到的水平之一;(3)数学教学的目标在于培养学生的数学思维、分析解决问题的能力,成功的数学教学应注重学生的参与和课后落实。中国传统文化、数学教育传统、新课程理念以及教师的工作环境等都对其观念系统有一定的影响。