Mathematics Assessment: Practices and Reporting Methods

The curriculum, teaching, and assessment Standards of the National Council of Teachers of Mathematics advocate the use of various assessment techniques in mathematics, K-12. Practitioners from 35 school districts in two regions of a midwestern state completed a questionnaire and reported information about their assessment practices, reporting methods, and other issues that affect change in mathematics. The results of the survey suggest that current reporting practices are limiting the use of alternative classroom assessment practices. Delays in the use of alternative mathematics assessment practice can probably be expected until the reporting systems used in school districts become consistent with the assessment systems and teachers participate in ongoing professional development programs to learn more about alternative assessment practices.     

Changing Assessment Practices in Science and Mathematics

One recommendation of the national reform movements in mathematics and science is to change the way students are assessed. In this article five formats of alternative assessment forms are described, and suggestions are given on how these methods can be implemented within science and mathematics classrooms. Benefits of each type of assessment are discussed and potential pitfalls are elucidated. Samples of the five formats, journal writing, open-ended problems, portfolios, interviews and performance assessments, are included.     

Classroom Assessment: Teachers’ Conceptions and Practices in Mathematics

The purpose of this study was to explore teachers’ conceptions about assessment and their actual assessment practices. The sample for the study consisted of 269 mathematics teachers in Grades 1 through 4 in a southeastern state. The teachers completed a cross-sectional, Likert-scale survey. Using chi-square to examine the relationships between grade level and teachers’ conceptions and between grade level and teachers’ practices, the researchers found several significant differences, transcending an indication of relationship by chance.     

Reviewing Integrated Science and Mathematics: The Search for Evidence and Definitions From New Perspectives

Based upon current research needs indicated from recent literature reviews, this integrative review concentrates on two of the perceived major impediments to integrating science and mathematics: The lack of evidence to support integration and the lack of a definition for integration. Using mixed methodology, this review found quantitative evidence favoring integration from a meta-analysis of 31 studies of student achievement, qualitative evidence revealing the existence of multiple forms of integration, and historical evidence of publishing patterns from across the 20th century. The forms of integration were identified and defined; differential effects were identified both between forms and between science and mathematics when the forms were analyzed by effect size. Additional research implications and suggestions for future research were also identified.     

Examining Instructional Practices of Elementary Science Teachers for Mathematics and Literacy Integration

Integration of content in core disciplines is viewed as an important curricular component in promoting scientific literacy. This study characterized the current practices of a group of elementary teachers relative to their development of interdisciplinary links between science, mathematics, and literacy. A qualitative analysis of survey data showed that there were substantial differences in the use of a well‐developed process for integrating instruction. Teachers also lacked a conceptual connection to integration, showed contradictions in the importance placed on hands‐on experiences, used measurement as the primary interdisciplinary connection between mathematics and science, and did not use instructional strategies designed specifically for nonfiction/expository text. The findings underscore the need for professional development that assists teachers in changing their conceptual perspectives to integration while also building pedagogical knowledge related to integration of science, mathematics, and literacy.     

Prospective Teachers' Perspectives on Mathematics Teaching and Learning: Lens for Interpreting Experiences in a Standards‐Based Mathematics Course

In a mathematics course for prospective elementary teachers, we strove to model standards‐based pedagogy. However, an end‐of‐class reflection revealed the prospective teachers were considering incorporating standards‐based strategies in their future classrooms in ways different from our intent. Thus, we drew upon the framework presented by Simon, Tzur, Heinz, Kinzel, and Smith to examine the prospective teachers' perspectives on mathematics teaching and learning and to address two research questions. What perspectives on the learning and teaching of mathematics do prospective elementary teachers hold? How do their perspectives impact their perception of standards‐based instruction in a mathematics course and their future teaching plans? Qualitative analyses of reflections from 106 prospective teachers revealed that they viewed mathematics as a logical domain representative of an objective reality. Their instructional preferences included providing firsthand opportunities for elementary students to perceive mathematics. They did not take into account the impact of a student's conceptions upon what is learned. Thus, the prospective teachers plan to incorporate standards‐based strategies to provide active experiences for their future elementary students, but they fail to base such strategies upon students' current mathematical conceptions. Throughout, the need to address prospective teachers' underlying perspectives of mathematics teaching and learning is stressed.     

Book review of Thinking Practices in Mathematics and Science Learning

The goal of this project, an animation-based tutor for algebra word problems, is to build instructional software to improve estimation, reasoning, and problem-solving skills. In this study, we focused on the 2nd (task completion) module, which uses tank-filling problems in which the unknown variable is the time it will take to fill a tank. Students build on the answers to a simple (no leak) problem to estimate, and then calculate, answers to problems in which there is a leak in the bottom of the tank, a leak in the side of the tank, and a delay in starting one of the pipes. We evaluated the software in an intermediate algebra class consisting of a diverse group of students at a community college. We use the findings to discuss what works (estimation and the solution of the simpler problems) and to make recommendations on how to improve use of the decomposition method for solving the more complex problems.     

New Perspectives on the Gender Stereotyping of Mathematics

Historically, mathematics has been stereotyped as a male domain, and there is considerable evidence to support this belief. In the last 30 years, mathematics education researchers have uncovered a range of factors contributing to the documented achievement and participation differences that favored males and sought to redress them. Mathematics as a male domain, one of the subscales of the Fennema-Sherman Mathematics Attitude Scales (1976), has been used widely to assess the extent to which mathematics is stereotyped as a masculine sphere. It has been argued that some of the items comprising the subscale are anachronistic and that the subscale scores can no longer be interpreted reliably. In this article we outline the development of two new instruments-the mathematics as a gendered domain instrument and the who and mathematics instrument-that have been designed to overcome the limitations of the original Fennema-Sherman mathematics as a male domain subscale. We also present findings from the administration of the two instruments in Australia, where they were developed, and in the United States, the site of the trials of the original Fennema-Sherman scales. The results indicate that females feel more strongly than males about some aspects of gender stereotyping in mathematics although, in general, most students feel that mathematics is gender neutral.     

New Perspectives on the Gender Stereotyping of Mathematics

Historically, mathematics has been stereotyped as a male domain, and there is considerable evidence to support this belief. In the last 30 years, mathematics education researchers have uncovered a range of factors contributing to the documented achievement and participation differences that favored males and sought to redress them. Mathematics as a male domain, one of the subscales of the Fennema-Sherman Mathematics Attitude Scales (1976), has been used widely to assess the extent to which mathematics is stereotyped as a masculine sphere. It has been argued that some of the items comprising the subscale are anachronistic and that the subscale scores can no longer be interpreted reliably. In this article we outline the development of two new instruments-the mathematics as a gendered domain instrument and the who and mathematics instrument-that have been designed to overcome the limitations of the original Fennema-Sherman mathematics as a male domain subscale. We also present findings from the administration of the two instruments in Australia, where they were developed, and in the United States, the site of the trials of the original Fennema-Sherman scales. The results indicate that females feel more strongly than males about some aspects of gender stereotyping in mathematics although, in general, most students feel that mathematics is gender neutral.     

Measuring Reform Practices in Science and Mathematics Classrooms: The Reformed Teaching Observation Protocol

The National Science Foundation has funded 22 Collaboratives for Excellence in Teacher Preparation. Despite the remarkable allocation of resources to this effort, it has proven exceptionally difficult to demonstrate the effectiveness of collaborative reform. In large part, this has resulted because of the difficulty of defining and measuring reform. The Reformed Teaching Observation Protocol (RTOP) was designed by the Evaluation Facilitation Group of the Arizona Collaborative for Excellence in the Preparation of Teachers (ACEPT). It is a 25‐item classroom observation protocol that is (a) standards based, (b) inquiry oriented, and (c) student centered. This instrument has provided the definition for reform and the basis for evaluation of the ACEPT collaborative. The data upon which this report is based were collected over a period of more than 2 years from 153 public school, college, and university mathematics and science classrooms. A trained team of observers consisting of two faculty members and seven graduate students was able to achieve exceptionally high levels of interrater reliability. Internal consistency, as estimated by Cronbach's alpha, was also remarkably high. Correlation coefficients ranging from 0.88 to 0.97 between RTOP scores for classrooms, and mean normalized gain scores for students in those classrooms on achievement measures demonstrate that reform, as defined by ACEPT and measured by the RTOP, has been effective.     

Crutch or Catalyst: Teachers' Beliefs and Practices Regarding Calculator use in Mathematics Instruction

This study investigated K‐12 teachers' beliefs and reported teaching practices regarding calculator use in their mathematics instruction. A survey was administered to more than 800 elementary, middle and high school teachers in a large metropolitan area to address the following questions: (a) what are the beliefs and practices of mathematics teachers regarding calculator use? and (b) how do these beliefs and practices differ among teachers in three grade bands? Factor analysis of 20 Likert scale items revealed four factors that accounted for 54% of the variance in the ratings. These factors were named Catalyst Beliefs, Teacher Knowledge, Crutch Beliefs, and Teacher Practices. Compared to elementary teachers, high school teachers were significantly higher in their perception of calculator use as a catalyst in mathematics instruction. However, the higher the grade level of the teacher, the higher the mean score on the perception that calculator use may be a way of getting answers without understanding mathematical processes. The mean scores for teachers in all three grade bands indicated agreement that students can learn mathematics through calculator use and using calculators in instruction will lead to better student understanding and make mathematics more interesting. The survey results shed light on teachers' self reported beliefs, knowledge, and practices in regard to consistency with elements of the National Council of Teachers of Mathematics Principles and Standards for School Mathematics (2000) technology principle and the NCTM use of technology position paper (2003). This study extended previous research on teachers' beliefs regarding calculator use in classrooms by examining and comparing the results of teacher surveys across three grade bands.     

数学类课程中开放性作业的研究与实践

在数学类的课程教学中,针对某些开放性问题进行研究,对提升学生数学素质起到了重要作用.本文探讨了开放性作业的内容,组织形式和考核方式.通过教学实践和对学生的问卷调查发现,在大班教学中,开放性作业适合少数学生选做,进一步将研究成果在课堂上分享,所占分数为2-3分,以附加分数的形式计入总分,得到学生的认可度较高,教学效果较好.     

Student and Teacher Perspectives Across Mathematics and Science Classrooms: The Importance of Engaging Contexts

Substantial recent focus has been placed upon the competitiveness of American students in increasingly global economies and entrepreneurial enterprises. As concerns center on students’ educational preparedness and their efforts at continued learning, researchers acknowledge the importance of student engagement with school. In order to foster engaged learners, teachers must be able to determine and monitor their students’ levels of engagement. The current study examined the alignment of perceptions of engagement by students, teachers, and outside observers across middle and high school mathematics and science classrooms. Results indicated significant teacher‐student differences in perceptions of student cognitive engagement across mathematics and science classrooms with teachers consistently perceiving higher levels than students. Moreover, most effect sizes were moderate to large. A subsequent multi‐level analysis indicated that while teacher perceptions of student cognitive engagement were somewhat predictive of student reported cognitive engagement, academic engagement ratings by outside observers were not.     

在数学专业运筹学教学中引入计算机实践环节的探索

运筹学是应用数学的一个重要分支,在许多学科领域及工程问题中均有应用.自从运筹学在上世纪建立以来,其算法及理论的发展都受到了电子计算机的发展的极大影响.因此在本科运筹学教学中引入适当的计算机实践是非常必要的.这将有助于学生更加深入清晰地理解算法的工作步骤、算法建立的思想以及每个算法的优缺点.在本文中我们试着探讨了为何应在数学专业的运筹学教学中引入计算机实践环节,以及应当如何引入这些实践活动.