数学与应用数学外文翻译1

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1、宁波大学科学技术学院本科毕业设计（论文）系列表格英文翻译1.1. Teachers the implementersIn a 2-year study in two Singapore schools to investigate pedagogical practices in the elementary mathematics classroom, Chang, Kaur, Koay, and Lee (2001) found that traditional teaching approaches predominated amongst the teachers. The typic

2、al teaching approach was expository, followed by students practicing routine exercises to consolidate the concepts, knowledge and skills. Chang et al.s study involved video taping five 1-h mathematics lessons for four teachers, two from an elite school and two from a local school. In another study,

3、Foong, Yap, and Koay(1996) described how a number of teachers expressed their concern over their perceived lack of skills for the teaching of mathematics using a problem solving approach. They found that teachers felt inadequately prepared to teach MPS when the examples were nonroutine problems that

4、 had several possible solutions. They doubted their ability to communicate the multiple concepts required by students to understand without being confused by the number of methods and heuristics suggested in the newly released syllabus, and the teachers expressed unease with the emphasis on open-end

5、ed problem solving. Such lack of confidence led to the general belief that there was an over reliance on textbooks and a narrowrange of problem types used in classroom examples, both contrary to the official syllabus (MOE,2000). Teachers development in giving problem solving instructions has been in

6、sufficiently explored by researchers(Chapman, 1999; Lester, 1994). In one study, Norton, McRobbie, and Cooper(2002)investigated how nine teachers responded to a reform curriculum (Board of Senior Secondary School Studies, 1992) in line with reforms initiated by the National Council for Teachers of M

7、athematics (NCTM, 1989, 2000) and the Australian Education Council (AEC,1990). They sought to find if teachers who were using an investigative approach that involved students actively engaging in: “making sense of new information and ideas” (Curriculum Council, 1998, p. 1), “investigating mathematic

8、al processes situated within meaningful contexts” (Australian Association of Mathematics Teachers Inc., 1996, p. 4), and “construction of meaning” (Anderson, 1994, p. 1). The teachers varied their pedagogical approaches differentially for students with different abilities, notwithstanding the stated

9、 goals of conceptual understanding for more able studentsand predominately calculational goals for less able students. Three teachers still favored the “show and tell” approach for both groups of students, while another three employed a mix of “explain” and “show and tell” approaches. Of the remaini

10、ng three, two used the investigative approach for the more able students and “show and tell” for the less able. Only one teacher out of the nine used an investigative approach as intended in the curriculum for both groups of students. The researchers observed that while the teachers expressed suppor

11、t for the investigative approach and the objective of teaching for conceptual understanding, other factors (particularly preparation for high-stake examinations)appeared to influence the goals and approaches adopted in classrooms. Teachers understanding of problem solving, their interpretations of h

12、ow to teach it and how much time to spend on it vary (Grouws, 1996). Possible conceptions of teaching problem solving include: teaching about, teaching for, and teaching via problem solving (Schroeder & Lester, 1989);problematizing mathematics as a way to think about problem solving (Hiebert et al.,

13、 1996). The teachers role in implementing a curriculum that emphasizes problem solving involves more than just expressions of support on the part of the teachers (Senger, 1999). Possible supportive ingredients in the process might include interventions that explore what approaches teachers could ado

14、pt, other suitable lesson formats or problem tasks.In studies of approaches to teaching problem solving, teachers were often “assigned” particular approaches by researchers who then proceeded to investigate the implementation and subsequent effects each approach has on students learning. In Sigurdso

15、n, Olson, and Masons (1994) study, the effects of classroom teaching that incorporated a problem-solving dimension on student learning of mathematics were investigated. Three approaches were implemented: algorithmic practice, teaching with meaning, and a problem-solving approach. The problem-solving

16、 approach involved teaching with meaning (Sigurdson & Olson, 1992) plus a daily insertion of 10 min of problem-process work.The three approaches were assigned to the 41 teachers in the study. Preparation of teachers involved 10 h of workshops for the algorithmic practice approach and 25 h of worksho

17、ps for the other two approaches, all spread over the implementation period of 5 months. The outcomes of their study were somewhat complex, with the analysis done along the three approaches and the students in each approach divided into low-, medium-, and high-achievers. They claimed, among other thi

18、ngs, that the meaning and problem-process approaches in teaching were important, resulting in more students learning with improved achievement and positive attitudes. They also noted that the higherachieving students benefited more. However, in another study about two classes, one high-ability and t

19、he other low, Holton, Anderson, Thomas, and Fletcher (1999) found that lower ability students seemed to benefit more from the introduction of problem solving lessons. While such research on problem solving has significant implications, the extent and the way in which problem solving is implemented i

20、n the classroom remains largely unexplored. Empirical data about the way that teachers taught before their involvement in the project, observations of pedagogical practices within in the classrooms during implementation, and the salient features of different approaches that impacted students was not

21、 collected. These aspects are important to a better understanding of the process of curriculum implementation. The current study aims to address these issues. In particular, this study posits links between describing what is happening in the classrooms and subsequent changes in teachers classroom pr

22、actices without imposing or assigning any particular approach for teachers to adopt and follow. We also address the following issue: how students learning of problem solving skills is impacted by the changes in the teachers classroom practices.1.2. Students the “attainers”Foong and Koay (1997) found

23、 interesting consequences of teachers lack of preparation in using the new approaches recommended in a revised syllabus. Using eight pairs of items, each pair consisting of a standardword problem typically found in textbooks and a realistic word problem where the student needed to consider the reali

24、ties of the context of the problem statements, the researchers found that students tended “to disregard the actual situation described” in word problems and “instead, go straight into exploring the possible combinations of numbers to infer directly the needed mathematical operations” (p. 73). Earlie

25、r, Koay and Foong (1996) many of the nearly 300 lower secondary studentsthat they examined failed to make connections between school mathematics and everyday life. These studies suggest the teaching of MPS did not apply mathematics in practical tasks and real world problems, as mandated in the inten

26、ded curriculum. Students attainments are falling short of the intentions.Cais (2003) exploratory study suggests that most students were “able to select appropriate solution strategies to solve” the tasks, and chose “appropriate solution representations to clearly communicate their solution processes

27、” (p. 733). He explored fourth, fifth and sixth grade students MPS skills, using four tasks which were mathematically rich, and were “embedded in different content areas and contexts, and allowed Singaporean students thinking from various perspectives.” Further, Singaporean students repeated top ran

28、king performance in mathematics on the Trends in International Mathematics and Science Study (TIMSS-2003) (Mullis, Martin, Gonzalez, & Chrostowski, 2004) suggests that the current syllabus (MOE, 2000) is working well.Such seemingly contrasting findings about students attainment warrant a need for a

29、closer look at teachers classroom practices and their possible impact on students learning.2. MethodThis study began by identifying the elements MPS pedagogical practice that exists in typical elementary mathematics classrooms. In particular, it addressed the main research question: What teachers cl

30、assroom practices support mathematical problem solving development in their students? Following the collection of a systematic, evidence base describing current mathematics instruction practices, an intervention was designed to raise the teachers awareness of MPS ideas and processes and to support a

31、n increased emphasis on the centrality of problem solving in the Singapore Mathematics Program. The intervention had threecomponents and followed a design research approach: First, teachers were interviewed to reviewsalient features of their classroom practices and prompted to give their own descrip

32、tions and interpretations of events. Second, we conducted a workshop which discussed Polyas (1957) four phases of problem solving understanding, planning, executing, and looking back. Third, after the workshop and some lapse of time, informal post-lesson interviews were conducted when we returned to

33、 observe the teachers again. These were short discussions immediately after observations. The purpose was to talk about the lesson, its degree of success (in terms of problem solving instruction) and the teachersown assessment of the outcomes. The intervention lead to a second research question: Giv

34、en a reflective intervention emphasizingMPSinstructional emphasis, do teachers change teaching strategies and does it result in increased students problem solving successes? From the analysis of the initial observations, we sawthat the teachers, when introducing problems, tended to read the problems

35、 quickly and proceed to immediately execute the solutions, with little or no strategic planning. They also did not reflect on the solution or its success. As the goal was to explore the explicit development of students metacognitive aspects of problem solving, when conducting the workshop, the impor

36、tance of explaining during the reading of the problem and the possibility of being more explicit about planning and reflecting were highlighted. The need to employ more rich and authentic problem tasks was also highlighted. Several examples were shown and their solutions wereworked through thoroughl

37、y following Polyas four phases.While it is not possible to trace the individual trajectory of each and every students learning, an adaptation of the Cobb, Stephan, McClain, and Gravemeijer (2001) approach was used to document the collective mathematical development of a classroom community over the

38、extended periods of time covered by instructional sequences. Cobbet al. resolved the issue about “the trajectory of . . . students learning” and the “significant qualitative differences in their mathematical thinking at any point in time,” by proposing “a hypothetical learning trajectory as consisti

39、ng of conjectures about the collective mathematical development of the classroom community” (p. 117). Likewise inthis study, it is difficult to ascertain the casual relations or direct impact between the intervention, possible teacher change and students learning outcomes. Hence instructional impact

40、 on students was viewed in terms of a collective mathematical development through their responses in a pre- and post-set of problem solving tasks.Kai Fai Ho，John G. Hedberg.Teachers pedagogies and their impact on studentsmathematical problem solvingJ. Journal of Mathematical Behavior,24(2005):238-252.3