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Inquiry Teaching and Learning: The Best Math Class StudyAbstract：This research reports on prospective middle school teachers perceptions of a “best mathematics class” during their involvement in an inquiry-designed mathematics content course. Grounded in the prestigious Glenn Commission report (U.S. Department of Education, 2000), the study examined the prospective teachers perceptions of effective mathematics instruction both prior to and after completing the inquiry course. Pre-essay analysis revealed that students could be grouped into one of two categories: the Watch-Learn-Practice view and the Self as Initiator view. Post-essay analysis indicated that over two thirds of all students involved in the study changed their views of a best math class after the inquiry courses. The Watch-Learn-Practice groups changes focused on developing reasoning skills and learning how one “knows” in mathematics. The Self as Initiator group noted expanded roles for the students, particularly emphasizing the importance of going beyond basic requirements to think deeply about the why and how of mathematics and expanded views of the benefits of group learning.Before Its Too Late, the prestigious Glenn Commission report (U.S. Department of Education, 2000), boldly stated that the economic and democratic foundations of our country are in serious jeopardy unless we produce students who understand and can use fundamental ideas from mathematics. Characterizing todays students as “simply not world-class learners” (p. 4), Commission Chair Senator John Glenn suggested that the solution to poor student performance rests with the improvement of teaching: “after an extensive review of what is happening in our classrooms, the Commission has concluded that the most powerful instrument for changelies at the very core of educationwith teaching itself” (p. 5). The Commission asserted that teaching must change from the “numbingly predictable” (p. 20) teacher-centered, lecture-dominated approaches commonly used in todays classrooms to student-centered, inquiry-based methodologies based on the Principles and Standards for School Mathematics (Standards; National Council of Teachers of Mathematics, 2000) that engage students in the processes of learning and creating mathematics:In high-quality teaching, the process of inquiry, not merely “giving instruction,” is the very heart of what teachers do. Inquiry not only tests what students know, it presses students to put what they know to the test. It uses “hands on” approaches to learning, in which students participate in activities, exercises, and real-life situations to both learn and apply lesson content. It teaches students not only what to learn but how to learn. (p. 22)Partly in response to the Glenn Commission Report, the Middle Childhood Mathematics Inquiry Learning Project (Wanko, Johnson, Keiser, & Stonewater, 2001) was designed to implement an inquiry-based undergraduate mathematics curriculum for prospective middle school mathematics teachers enrolled at a large midwestern University. The longitudinal research component of the project examined the growth of the students as inquiry learners and prospective teachers, especially with regard to the development of their mathematical thinking (Wanko, 2003), their ability to apply inquiry skills in learning mathematics (Keiser, 2003), and what they perceived to be characteristics of effective mathematics instruction both before and after experiencing inquiry mathematics courses (Johnson, 2003; Stonewater, 2003). It is this last item preservice teachers perceptions of effective mathematics instruction that is the focus of the “Best Math Class” study reported here.This research reports preservice teachers perceptions of effective mathematics instruction both prior to and after completing an inquiry-designed mathematics course. The researchers were interested in knowing what these perceptions were and if these perceptions changed after the students experienced inquiry-based instruction. As is outlined in the following review of the literature, teachers perceptions of mathematics instruction influence how they design and deliver classroom instruction. Thus, if teaching is to move from the “numbingly predictable” status quo to the inquiry-oriented approaches advocated by the NCTM Standards and the Glenn Commission, a necessary, although not sufficient, condition for this change rests with teachers perceptions of a “best” math class. Certainly, these perceptions are indeed affected by the kinds of instruction they experience as students.Review of LiteratureNumerous researchers have connected teachers perceptions of mathematics instruction to the ways in which they teach (Robitaille, 1994; Sowder, Philipp, Armstrong, & Schappelle, 1998). In a study of teachers involved in mathematics education reforms, Simon, Tzur, Heinz, Kinzel, and Smith (2000) found that teachers who were involved in classes in which they conducted mathematical explorations, made hypotheses and proved the hypotheses tended “to relinquish their traditional teaching methods and design opportunities for students to perceive mathematics directly” (p. 596).In another study, Gellert (1999) reported, “In mathematics education research, it seems to be undisputed that the teachers philosophy of mathematics has a significant influence on the structure of their mathematics classes” (p. 24). Studying preservice elementary teachers in Germany, he found that although most of his subjects believed that mathematics classes should be fun and child-centered, they could not connect “well-meant starting points, e.g., integrating games and stories in mathematics classes, with mathematical themes” (p. 39). Schoenfeld (1983, 1985, 1989) argued that “tangible cognitive actions are often the result of consciously or unconsciously held beliefs about the task at hand” (1983, p. 330). Although his studies focused on high school students problem solving in geometry and not on teachers perceptions of mathematics, these studies provide evidence of the connection between beliefs or perceptions and actions.Gellert (1999) also suggested that teachers “learning experiences exert a tremendous influence on their design of mathematics classes” (p. 27). In another study of the learning experiences of teachers involved in a summer-long mathematics by inquiry course, Baxter-Magolda and Buckley (1996) reported that by the end of the course a majority of participants “experienced important epistemological growth in moving from acquiring mathematical concepts to realizing they could construct an understanding of them. This understanding of math structure is a necessary foundation for their ongoing work” (p. 65). Nine of the 16 teachers involved in the study changed their perceptions of mathematics from a static view, that is, that all mathematics is certain and known, to a more constructivist view that mathematics is created by the individual. Although this study did not examine the implications of the teachers epistemological development to their actual teaching practices, it seems likely in light of the studies discussed abovethat their actual classroom practices would change.Yet, according to the Glenn Commission, the actual learning experiences of students in this country are grounded in a basic teaching style that has remained essentially the same for two generations. The Commission described a typical mathematics class as one that begins with a review of previous material and homework, then moves to a low-level problem illustration by the teacher. This is followed by supervised student seatwork to imitate the teachers procedures, and ends with a checking of solutions and the assignment of homework. Confirming this description of the typical mathematics classroom, Bookman (1998) pointed out that although “the Standards propose a radical departure from the way things have been done in most schools in the United States, there is no evidence that they have been adopted in practice by the majority of high school math teachers” (p. 117). In fact, in a recent study (Horizon Research, 2002) 40% of middle school teachers surveyed said they were either unfamiliar or only somewhat familiar with the Standards, and when asked to rate the extent of their agreement with the NCTM Standards, 25% said they disagreed, strongly disagreed, or had no opinion. When asked to rate the extent to which the Standards had been implemented, 21% said implementation had occurred only minimally or not at all.As the research points out, teachers perceptions of mathematics and mathematics teaching are a contributing factor to why, as the Glenn Commission said, “Too many of todays mathematics and science classrooms come up short” (p 21).Research QuestionsThe purpose of the Best Math Class study was to examine students perceptions of their ideal or best math class prior to experiencing inquiry-based instruction and to determine if these descriptions changed as they moved through the inquiry-based mathematics curriculum. Specifically, two questions guided the Best Math Class study:1. Prior to enrolling in a university-level inquiry mathematics course, what characteristics do prospective middle school mathematics teachers attribute to their idea of a best math class?2. Do these characteristics change after the students experience an inquiry-based course?During the first phase of the Best Math Class study, an essay instrument was developed to assess students perceptions of a best math class. The results of Phase 1 are discussed in this paper. In Phase 2 of the study, a Likert-style questionnaire was designed based on student responses to the essay instrument and is used as a second means of assessing student perceptions. Finally, Phase 3 of the study uses the essay, the questionnaire, and student interviews to assess benchmark perceptions of a best math class and changes in these perceptions as students progress through the inquiry-based mathematics curriculum. Phases 2 and 3 of the study are currently underway.MethodologyBest Math Class EssayIn order to assess student perceptions of a best math class, a free form, open-ended essay instrument was developed (see appendix). Students were asked to address three components of their view of a best math class: the role of the instructor in helping them learn; their role in learning; and how they expect their learning to be assessed. The role of instructor, student, and assessment were chosen as focus questions for the essay based on Perrys (1970) theory of college student intellectual development. In this theory, Perry described nine sequential and hierarchical stages of development. At each stage the role of instructor, student, and assessment are key descriptors of how students view the teaching and learning environment and their own ways of knowing. Additionally, in a small sample study of teachers, Stonewater and Oprea (1988) reported consistency between these teachers beliefs and their thought structures as predicted by assessed position along the Perry scheme. Thus, focusing the essay on these three components was consistent with issues Perry described as central to college student intellectual development.探究式教学和学习：最好的数学课研究摘 要：本研究报告是基于未来中学教师在参与调查设计数学内容的课程期间对“最好的数学课”的看法。基于著名的格伦委员会报告（美国教育部，2000），该研究致力于未来教师对有效的数学教学之前和之后完成的探究过程。预分析表明学生可以分为两类：观看学习实践观和自我引发的观点。后文分析表明，在参与研究的超过三分之二的学生改变了他们的一个最好的数学课探究性课程观。观察学习实践组的变化主要集中在发展推理能力和学习在数学中如何“知道”。作为发起者的Self指出，扩大学生的作用，特别强调超越基本要求深入思考为什么以及如何扩展小组数学学习的效益视野。在为时已晚之前，著名的格伦委员会报告（美国教育部，2000）大胆地指出，我国的经济和民主基础是在严重的危险之中，除非我们培养那些理解并懂得使用数学基本思想的学生。描述今天的学生为“不”的世界级的学习者”（4页），委员会主席参议员约翰格伦建议学生表现不佳的解决方案取决于教学质量的提高：“在一个广泛的审查后什么是在我们的教室里发生的，委员会认为，改变最有力的工具是在教育的核心教学本身”（5页）。该委员会认为，教学必须改变，从“非常可预测的”（20页）以教师为中心，讲授为主的方法，常用于今天的课堂以学生为主体，探究基于学校数学的原则和标准方法（标准；国家数学教师委员会，2000），吸引学生注重学习的过程和数学创造：高质量的教学，探究的过程，不仅是“给予指导，更是教师用心去做什么。询问不仅测试学生知道什么，它把学生把他们所知道的进行测试。它用“动手”的学习方法，使学生参与活动，锻炼，和现实生活中的情况来学习和应用课的内容。它不仅教导学生要学什么，而且包括怎么学。（22页）部分原因是格伦委员会的报告，儿童数学探究学习项目（Wanko, Johnson, Keiser, & Stonewater, 2001）进行了设计并实现了一个基于探究的本科数学课程为未来的中学数学教师参加一个大的中西部大学。该项目的纵向研究部分检查学生的成长作为探究学习者和教师，特别是关于他们数学思维的发展（威高，2003），他们在数学学习中的应用探究技能能力（凯瑟，2003），和他们认为是有效的数学教学的特点是之前和之后经历探究数学课程（约翰逊，2003；Stonewater，2003）。正是这最后一项职前教师对数学有效教学的重点，是“最好的数学班”研究报告。本研究报告职前教师对数学有效教学先于完成调查设计数学课程。研究人员知道这些观念，如果这些观念改变后，学生经历了探究式教学的兴趣。正如在下面的文献综述，教师对数学教学的感知影响他们如何设计和提供课堂教学。因此，如果教学是从“非常可预测”的现状转变为由NCTM标准倡导和格伦委员会提出的探究的方法是必要的，尽管不是充分条件，这种变化取决于教师的“最好”的数学课的看法。当然，这些看法确实是受他们以经验对不同类型学生进行指导的影响。文献回顾众多的研究者与教师的数学教学方式存在感知（罗比塔耶，1994；索德，菲利普，阿姆斯壮，和schappelle，1998）。在参与数学教育改革的研究中，Simon, Tzur, Heinz, Kinzel, and Smith (2000)发现参与实施数学课堂探索的教师，假设和证明的假设往往“放弃传统的教学方法和学生直接感知数学设计的机会”（596页）。在另一项研究中，盖勒特（1999）指出，“数学教育研究中，教师的数学观对数学课的结构的有着显著的影响似乎是无可争议的”（24页）。留学德国的职前小学教师，他发现，虽然他大部分的学生认为数学课堂应该是有趣的，以儿童为中心的，但是他们无法连接“善意的出发点，例如，搭配数学主题将游戏和故事融合在数学课，”（39页）。Schoenfeld（1983，1985，1989）认为，“有形的认知行为往往是自觉或不自觉地认为手头的任务的结果”（1983，p. 330）。虽然他的研究主要集中在高中学生的几何解题而不是教师对数学的看法，这些研究提供了信仰或观念和行为之间的联系的证据。