上海大专学生对极限的理解的研究

通过问卷调查、课堂提问和课后访谈的方式研究上海大专学生对极限的概念、计算以及存在性这三方面的理解.学生难以描述极限的概念,对未定式极限的计算存在困难,难以理解函数在某点极限存在性与函数在这点定义无关.     

Assessing Students' Understanding of Functions in a Graphing Calculator Environment

In a classroom environment in which continual access to graphing calculators is assumed, items that have been used to assess students' understanding of functions often are no longer appropriate. This article describes strategies for modifying such items, including requiring students to explain their reasoning, using calculator-active items, analyzing graphs and tables, and using real contexts.     

Investigating High School Students' Science Experiences and Mechanics Understanding

This research study was designed to provide an introductory examination of how high school students' out‐of‐school science experiences, particularly those relevant to the physical sciences, relate to their learning of Newtonian mechanics. A factor analysis of the modified Science Experiences Survey (SES; Mason & Kahle, 1988 ) was performed, leading to three factors: Learning Attributes Related to Science, Physical Science Experiences, Nature Experiences. The students' learning of Newtonian mechanics was measured by their gain score from a pre‐instruction/post‐instruction administration of the Force Concept Inventory (FCI; Hestenes, Wells, & Swackhamer, 1992 ). An analysis of variance showed that females and males in honors physics courses demonstrated similar gain scores, while males in non‐honors courses demonstrated larger gains (p < 0.05) than the females. When the students' total SES and SES factor scores were correlated with their FCI pretest and gain scores, the SES Physical Science Experience score was found to be significantly related to the FCI pretest score (p = 0.01). No other correlations were significant.     

High School Precalculus Students' Understanding of Slope as Measure

Twenty-two high school precalculus students were interviewed to examine their understanding of slope as measure. The students examined and discussed real-world situations involving slope: physical situations involving slope as a measure of steepness and functional situations involving slope as a measure of rate of change. For the various physical situations, students measured steepness with angles instead of ratios. Overall, they demonstrated a better understanding of slope in functional situations, but many students had trouble interpreting slope as a measure of rate of change. Instruction should focus on helping students form connections and providing opportunities for students to communicate their understanding.     

Assessing Students' Mathematical Communication

Assessment of students' mathematical communication through the use of open-ended tasks and scoring procedures is addressed, as is the use of open-ended tasks to assess students' mathematical communication by providing students opportunities to display their mathematical thinking and reasoning. Also, two scoring procedures (quantitative holistic scoring procedure and qualitative analytic scoring procedure) are described for examining students' communication skills.     

Examining the Effects of a Reformed Junior High School Science Class on Students' Math Achievement

Though national standards emphasize the importance of connections between math and science, few empirical studies exist to support the notion that student achievement increases from such integration. This paper examines an eighth‐grade science class that integrated mathematics into science through the use of technology. In a setting of action research, the effects of such integration were examined. This paper reports that integrating mathematics into the science class positively affected students' achievement in their math class and describes the circumstances under which the integration occurred.     

Assessing Elementary Understanding of Multiplication Concepts

This article summarizes the basic concepts of multiplication and provides some evidence that the traditional third‐grade curriculum and instruction emphasizing memorization of multiplication facts produces much less understanding of the basic concepts of multiplication than a standards‐based curriculum and instruction emphasizing construction of number sense and meaning for operations. This study also describes a collection of assessment tasks that provided meaningful evidence of children's understandings of basic multiplication concepts, including understandings of the relationships between multiplication and addition.     

Two Teaching Methods and Students' Understanding of Sound

The purpose of this study was to examine fifth grade students' ideas related to sound and to compare the Learning Cycle teaching approach with a textbook/demonstration method of instruction to determine whether one method is more effective in facilitating conceptual change. Thirty-four fifth grade students were randomly selected and assigned to the two treatment groups. To assess the students' understanding of specific sound concepts, an interview protocol was administered to both groups before and immediately after instruction. Students were given a numerical rating corresponding to their levels of understanding. The numerical values for both groups at the pre- and post-interview assessments were analyzed by analysis of variance (ANOVA). Students who were taught using the Learning Cycle had a significantly better understanding.     

Students' Understanding of the Particulate Nature of Matter

The particulate nature of matter is identified in science education standards as one of the fundamental concepts that students should understand at the middle school level. However, science education research in indicates that secondary school students have difficulties understanding the structure of matter. The purpose of the study is to describe how engaging in an extended project‐based unit developed urban middle school students' understanding of the particulate nature of matter. Multiple sources of data were collected, including pre‐ and posttests, interviews, students' drawings, and video recordings of classroom activities. One teacher and her five classes were chosen for an indepth study. Analyses of data show that after experiencing a series of learning activities the majority of students acquired substantial content knowledge. Additionally, the finding indicates that students' understanding of the particulate nature of matter improved over time and that they retained and even reinforced their understanding after applying the concept. Discussions of the design features of curriculum and the teacher's use of multiple representations might provide insights into the effectiveness of learning activities in the unit.     

Exploring Students' Conceptual Understanding of the Averaging Algorithm

Conceptual understanding of arithmetic average includes both an understanding of the computational algorithm and the statistical aspects of the concept. This study focused on the examination of 250 sixth-grade students' understanding of the arithmetic average by assessing their understanding of the computational algorithm. The results of the study showed that the majority of the students knew the “add-them-all-up-and-divide” averaging algorithm, but only about half of the students were able to correctly apply the algorithm to solve a contextualized average problem. Students were able to use various solution strategies and representations to solve the average problem. Those who used algebraic and arithmetic representations were better problem solvers than those who used pictorial and verbal representations. This study not only suggests that the average concept is more complex than the simplicity suggested by the computational algorithm, but also indicates the need for teaching the concept of average, both as a statistical idea for describing and making sense of data sets and as a computational algorithm for solving problems.     

Contrasting Cases of Calculus Students' Understanding of Derivative Graphs

This study adds momentum to the ongoing discussion clarifying the merits of visualization and analysis in mathematical thinking. Our goal was to gain understanding of three calculus students' mental processes and images used to create meaning for derivative graphs. We contrast the thinking processes of these three students as they attempted to sketch antiderivative graphs when presented with derivative graphs. These students constructed different and idiosyncratic images and representations leading to different understandings of derivative graphs. Our results suggest that the two students whose cognitive preferences were strongly visual or analytic and who did not synthesize visual and analytic thinking experienced different difficulties associated with their preferred modes for mathematical representation and thinking. Even the student who did synthesize these modes to some extent, to good effect, experienced difficulty when he did not do so. We discuss pedagogical implications for these results in a final section.     

Using Lotteries to Improve Students' Number Sense and Understanding of Probability

This article examines ways in which a discussion of lotteries can be integrated into a classroom lesson on combinatorics and probability and be used to enhance the teaching and learning of number sense and probability of winning various lottery games, and facts and myths about lotteries. Instructional tips for incorporating discussions of lotteries into lesson plans are also given.     

Middle Grades Students' Algebraic Understanding in a Reform Curriculum

The National Council of Teachers of Mathematics' Curriculum and Evaluation Standards in 1989 was pivotal in mathematics reform. The National Science Foundation funded several curriculum projects to address the vision described in the Standards. This study investigates students' learning in one of these Standards‐based curricula, the Connected Mathematics Project (CMP). The authors of CMP believe that the teaching and learning of algebra is an ongoing activity woven through the entire curriculum, rather than being parceled into a single grade level. The content of the study investigates students' ability to symbolically generalize functions. The data regards the solutions of four performance tasks dealing with three different types of relationships—linear, quadratic, and exponential situations—completed by five pairs of eighth‐grade students. The major finding claims that middle to high achieving students who had 3 years in the CMP curriculum demonstrated achievement in five strands of mathematical proficiency of a significant piece of algebra.     

Middle School Students' Understanding of Number Sense Related to Percent

This study examined middle school students' understanding of percent, focusing on number sense. Participants in the study were 106 seventh-grade and 93 eighth-grade students. Students were given a written test that included 21 multiple-choice questions and an open-ended item. Research interviews were conducted with 28 selected students. Students performed better interpreting a quantity expressed as a percent given a pictorial continuous region than when a pictorial discrete set of circles was given. Students had difficulty interpreting a quantity expressed as a percent of a number. The strategies used by students to make comparisons about percents represent a wide range of correct and incorrect approaches to the questions. In addition to the use of 50% and 100% as common reference points, students successfully applied fractional relationships, estimation and mental computation to make comparisons. A variety of inappropriate strategies which included computational procedures and numerical comparisons were also employed, some of which resulted in the correct multiple-choice response.