Reviving Pólya's “Look Back” in a Singapore school

This study is based on the stance that Pólya's “Look Back,” though understudied, remains relevant to Mathematics curricula that place emphasis on problem solving. Although the Singapore Mathematics curriculum adopts the goal of teaching Look Back, research about how it is carried out in actual classroom practice is rare. In our project, we focus on a redesign of a teacher development programme that is targeted to help teachers realize Pólya's original vision of Look Back in the classroom. We report the cases of two teachers who have participated in the teacher development programme: their thinking about Look Back (through interview records) and their interpretation of Look Back in their classroom practice (through video records). By bringing these two spheres of data together, we discuss their degree of ‘buy-in’ to Look Back in the overall problem solving enterprise.     

问题驱动 多元表征的概念教学探究——对偶线性规划教学案例设计

数学与应用数学(师范)专业中的《运筹学》具有跨学科、实践性的课程特点,目标在于培养职前教师用数学方法解决实际问题的能力.结合义务教育阶段新课程标准中"四基"的提出这一背景,本文将以线性规划部分(运筹数学)对偶线性规划概念的引入这一知识模块为例,探讨通过问题串形式进行问题驱动、多元表征的概念教学过程.即遵循问题驱动—兴趣驱动—问题意识发展—提出和解决新问题,依据数学与外部联系、数学内部联系两条主线设计教学和学习,探索如何通过问题驱动、多元表征的结构化教学过程引导学生的学习方式发生改变,增强探究学习的动机,发展问题解决能力.课堂教学实践证明效果优于以往单一的讲授式教学法,一定程度上提高了学生的学业成绩、应用问题的兴趣和问题解决意识.     

The model method: Students’ performance and its effectiveness

This study describes Singapore students’ (N = 607) performance on two tasks in a recently developed Mathematics Processing Instrument (MPI). The MPI comprised tasks sourced from Australia's NAPLAN and Singapore's PSLE. This study also examines students’ use of the model method to solve the two tasks. The model method is a visual problem-solving heuristic prevalently used in Singapore classrooms. The study found that students who solved the tasks using a visual method predominantly used the model method as a visual problem-solving strategy. Another interesting observation was the hindrance of successful problem solving caused by the persistence of prototypical images of model drawings. Implications include encouraging teachers to get their students to identify problem situations where the model method will both work and not work well, and making the role of the generator in the model method explicit in the mathematics textbooks.     

The preparation of primary mathematics teachers in Singapore: programs and outcomes from the TEDS-M study

This paper describes five aspects of primary mathematics teacher education in Singapore: (a) the teaching profession in Singapore, (b) the structure of pre-service teacher education programs offered by the National Institute of Education, (c) self-reports of Singapore future primary mathematics teachers about the opportunities to learn mathematics-related contents offered by these programs, based on the Teacher Education and Development Study in Mathematics (TEDS-M) survey, (d) the performance of these future teachers in mathematics content knowledge and mathematics pedagogical content knowledge assessed by the TEDS-M study, and (e) the relationships of opportunities to learn with this performance. The paper concludes with some suggestions about how to improve the quality of initial teacher preparation in the areas of recruitment and training.     

Enactment of school mathematics curriculum in Singapore: whither research!

The official curriculum for mathematics in Singapore schools is based on a framework that has mathematical problem solving as its primary goal. It is detailed and one may say that the gap between the designated curriculum and teacher intended curriculum is often very narrow. This is so as the main source of instructional materials is textbooks which are very closely aligned with the official national curriculum. There is a dearth of research on the enactment of the curriculum in Singapore schools, with the few research studies done so far appearing to cover only a narrow focus. The author’s view is that, even though only a few such studies have been published, schools have always been engaged in small-scale investigations, the findings of which are necessary to guide decisions on matters related to choice of textbooks and pedagogies for improved student learning. Considering all the published research and the investigative work undertaken by educators in Singapore, it may be said that the conceptual model proposed by Remillard and Heck is rigorous. In addition, the issues in this particular issue of ZDM offer educators, both classroom teachers and others, very good perspectives for research on the enactment of the school mathematics curriculum.     

A metacognitive-based instruction for Primary Four students to approach non-routine mathematical word problems

Many Primary Four students (fourth graders) in Singapore have difficulties initiating or persevering in the problem-solving process even though the curriculum has focused on problem solving since 1992. This study served to examine the role of metacognition in self-regulated problem solving. The study, a quasi-experimental pretest–posttest design involving a convenience sample of 63 students from two intact mixed-ability Primary Four classes, examined the impact of using a metacognitive scheme that focuses on the understanding and planning stages of Pòlya’s four-stage approach on students’ mathematical problem-solving behavior, performance and attitudes. The findings revealed that the metacognitive-based scheme had a positive impact on students’ understanding of the problem posed, solution planning, confidence in and personal control of problem-solving behavior and emotions. It had also helped them to initiate and persevere in the problem-solving process to achieve a higher level of problem-solving success. Limitations and instructional implications are discussed.     

Improving teacher expertise through master teacher work stations: a case study

Professional development is emphasized in China not only for junior teachers, but also for experienced teachers. In this study, we investigated a master teacher??s work station that has been established at a provincial level for training experienced ??key?? teachers across the whole province. Three types of data were collected to examine the focuses and approaches used in this master teacher??s work station, teacher participants?? learning experiences and feedback, and selected teachers?? expertise improvement. The results show that the master teacher emphasized teachers?? development of a deeper understanding of elementary mathematics content and its structure through intensive studies of a textbook series. His training led to very positive feedback from teacher participants regarding what they have learned, especially in elementary mathematics content and content connections as presented in the textbooks. The detailed analysis of two teachers?? knowledge learning revealed dramatic changes in their thinking and instruction about specific content topics. Teacher expertise improvement through this master teacher??s work station is then summarized and discussed.     

Problem Solving: Teachers' Perceptions,Content Area Models,and Interdisciplinary Connections

Features of common problem-solving models in mathematics and science, as well as those found in business and industry today, are discussed. Commonalties in the models are used to advance a case for interdisciplinary or integrated instruction in mathematics, science and technology. The Integrated Mathematics, Science and Technology (IMaST) program's problem-solving model is presented as an example of a curriculum project that draws upon the commonalties in the problem-solving models as a basis for a seventh grade integrated curriculum.     

Homogeneity or heterogeneity? Profiles of opportunities to learn in primary teacher education and their relationship to cultural context and outcomes

The curriculum of teacher education has been described as heterogeneous across countries and influenced by the context in which it is implemented. The present study investigates this potential heterogeneity by conducting latent class analysis of opportunities to learn mathematics, mathematics pedagogy and general pedagogy in primary teacher education as indicated by future teachers from 15 countries at the end of their training. The aim was to identify curricula profiles based on data from the “Teacher Education and Development Study: Learning to Teach Mathematics (TEDS-M)”. In each teacher education component, three groups of primary teachers were identified which differed quantitatively and qualitatively. Associations between these profiles and countries revealed broader cultural influences on OTL and thus shared (within a culture) and at the same time distinct (between cultures) visions of what primary teachers should know. Within countries, associations between curricula profiles and teacher education programs pointed to shared (within a program) and at the same time distinct (between programs) visions of what primary teachers should know at the end of their training. The OTL profiles were significantly related to outcomes of primary teacher education in terms of mathematics content knowledge, mathematics pedagogical content knowledge and general pedagogical knowledge.     

Reflection and Problem Solving: Integrating Methods of Teaching Mathematics and Science

An innovative teacher preparation course which integrates methods of teaching elementary mathematics and science was the context of this study. The course was developed as a prototype for the Teachers As Reflective Problem Solvers model for the preparation of elementary mathematics and science teachers. Data from 35 preservice elementary teachers' performance-portfolios were analyzed to reveal patterns of change in their reflections and problem-solving performance during the semester. Many of the students' reflections changed from task-focused learning to broader teaching applications. No relationship was discerned between changes in students' reflections and changes in their levels of problem-solving performance, although both increased during the semester. A significant correspondence was found, however, between students' perceptions of their problem-solving abilities and their actual performance in solving teaching problems in integrated mathematics and science contexts.     

Impact of the Science and Technology for Children Curriculum in the Oshkosh Area School District

This study examines the instructional impact of National Science Resources Center's Science and Technology for Children curriculum in the Oshkosh Area School District. Specifically, the instructional effectiveness of four physical‐science units for grades 1‐4 were investigated. Students were pre‐ and posttested using a multiple‐choice exam containing items adapted from the Third International Mathematics and Science Study, National Association of Educational Progress, TerraNova, and other widely recognized sources. Results were compared with existing instructional materials. Extensive resources for data interpretation include audiotaped lessons, classroom observations, interviews with students and teachers, student‐attitude surveys, and observations of teacher training. Results suggest that the adoption of this curriculum among experienced teachers in the district will provide little or no immediate gains on student achievement and potentially a slight decrease in student attitudes toward science.     

Opportunity to learn problem solving in Dutch primary school mathematics textbooks

In the Netherlands, mathematics textbooks are a decisive influence on the enacted curriculum. About a decade ago, Dutch primary school mathematics textbooks provided hardly any opportunities to learn problem solving. In this study we investigated whether this provision has changed. In order to do so, we carried out a textbook analysis in which we established to what degree current textbooks provide non-routine problem-solving tasks for which students do not immediately have a particular solution strategy at their disposal. We also analyzed to what degree textbooks provide ‘gray-area’ tasks, which are not really non-routine problems, but are also not straightforwardly solvable. In addition, we inventoried other ways in which present textbooks facilitate the opportunity to learn problem solving. Finally, we researched how inclusive these textbooks are with respect to offering opportunities to learn problem solving for students with varying mathematical abilities. The results of our study show that the opportunities that the currently most widely used Dutch textbooks offer to learn problem solving are very limited, and these opportunities are mainly offered in materials meant for more able students. In this regard, Dutch mainstream textbooks have not changed compared to the situation a decade ago. A textbook that is the Dutch edition of a Singapore mathematics textbook stands out in offering the highest number of problem-solving tasks, and in offering these in the materials meant for all students. However, in the ways this textbook facilitates the opportunity to learn problem solving, sometimes a tension occurs concerning the creative character of genuine problem solving.     

An investigation of preservice teachers’ use of guess and check in solving a semi open-ended mathematics problem

Open-ended problems have been regarded as powerful tools for teaching mathematics. This study examined the problem solving of eight mathematics/science middle-school teachers. A semi-structured interview was conducted with (PTs) after completing an open-ended triangle task with four unique solutions. Of particular emphasis was how the PTs used a specific heuristic strategy. The results showed that the primary strategy PTs employed in attempting to solve the triangle problem task was guess and check; however, from the PTs’ reflections, we found there existed misapplications of guess and check as a systematic problem-solving strategy. In order to prepare prospective teachers to effectively teach, teacher educators should pay more attention to the mathematical proficiency of PTs, particularly their abilities to systematically and efficiently use guess and check while solving problems and explain their solutions and reasoning to middle-school students.     

What is different across an ocean? How Singapore and US elementary mathematics curricula introduce and develop length measurement

This article compares the opportunity to learn length measurement in the USA and Singapore as revealed in the close analysis of some of their written elementary curriculum materials. Written curricula strongly influence students’ learning of mathematics, without completely determining it. The Trends in Third International Mathematics and Science study 2007 showed the relatively low performance of the US and Singapore fourth graders in measurement, which was attributed in part to the learning opportunities provided to the students. We examined and coded all instances of length measurement in three different US curricula and one Singapore curriculum through Grade 3, using a very detailed scheme that identified particular elements of conceptual, procedural and conventional knowledge and the textual forms that present this knowledge. Results show strong emphasis on measurement procedures, across all grades and curricula, in both countries. However, in numerous ways, the Singapore curriculum is more focused, organizationally, procedurally and conceptually. US curricula provide more diverse access to conceptual knowledge where Singapore materials focus on independent work involving procedures, within and across grade levels. Limitations of the curricula in both countries are discussed.     

Korean and American Elementary School Teachers' Beliefs about Mathematics Problem Solving

Elementary school teachers in South Korea and the United States completed a beliefs and practices questionnaire pertaining to mathematical problem-solving instruction. Although both groups of teachers shared a general approach to teaching with a focus on problem-solving strategies, many differences were apparent. Korean teachers rated themselves and their students higher in problem-solving ability than American teachers. Korean teachers perceived their mathematics textbook as a more valuable source for problem-solving instruction and word problems. Korean teachers more strongly agreed that students should know the key-word approach for solving problems. American teachers reported more frequent use of calculators, manipulatives, and small group instruction. The results indicate that American teachers may more often use instructional techniques that are aligned with current recommendations for mathematics instruction.     

Student Reactions to Standards-Based Mathematics Curricula: The Interplay Between Curriculum,Teachers, and Students

As standards-based mathematics curricula are used to guide learning, it is important to capture not just data on achievement but data on the way in which students respond to and interact in a standards-based instructional setting. In this study, sixth and seventh graders reacted through letters to using one of two standards-based curriculum projects (Connected Mathematics Project or Six Through Eight Mathematics). Letters were analyzed by class, by teacher, and by curriculum project. Findings suggest that across classrooms students were positive toward applications, hands-on activities, and working collaboratively. The level of students' enthusiasm for the new curricula varied much from class to class, further documenting the critical role teachers play in influencing students' perceptions of their mathematics learning experiences. The results illustrate that, while these curricula contain rich materials and hold much promise, especially in terms of their activities and applications, their success with students is dependent on the teacher.     

A Vision of Learning to Teach for the 21st Century

Glenda Lappan received her Ed. D. in mathematics and education from the University of Georgia in 1965 and has since worked at Michigan State University. From 1989–91 she took leave to serve as the program director for Teacher Preparation at the National Science Foundation. From 1997–2001 she is on leave to serve as president of the National Council of Teachers of Mathematics (NCTM). Her research and development interests are in the connected areas of students' learning of mathematics and mathematics teacher professional growth and change at the middle and secondary levels. She has published over a hundred scholarly papers and numerous books for middle grades students and teachers. She is currently the codirector of the Connected Mathematics Project II, which is funded by the National Science Foundation to revise and continue development of a complete middle school curriculum for teachers and for students. She served as the chair of the middle school writing group for the NCTM Curriculum and Evaluation Standards for School Mathematics (1989), and as Chair of the Commission that developed the NCTM Professional Standards for Teaching Mathematics (1991). She served on the NCTM Board of Directors from 1989 to 1992 and from 1997–2001. She has been a member of many national advisory boards, including the following: Glenn T. Seaborg Center for Teaching and Learning Science and Mathematics, the Ford Foundation/University of Pittsburgh QUASAR Project, the NSF/University of Maryland Teacher Preparation Collaborative, and the NSF/San Diego State University Mathematics for Elementary Teacher Preparation Materials Development Project.     

Online paths in mathematics teacher training: new resources and new skills for teacher educators

This paper considers the work carried out by online teacher educators and their professional development. We use the theoretical perspective of the documentational approach that focuses, in this case, on the interaction between teacher educators and the resources they use for their online training work. We thus study the following issues: (1) What kinds of resources do online teacher educators need, and how are such resources modified according to the educators?? specific skills and needs? (2) What specific skills are needed for setting up online training for mathematics teachers and how do these skills evolve as teacher education resources are used? We consider both questions simultaneously, while presenting results from a study within a specific teacher training programme in France that proposes ??training paths?? on a national platform. These ??paths?? are resources designed for teacher educators. We follow the appropriation of two training paths by two educator teams. The ways in which these educator teams were able to appropriate the paths give insights into the teacher educators?? skills and, as well, into the resources they need. By looking at their use of resources (as online mathematics teacher educators), we observe and analyse professional geneses, leading to the development of new skills.