Understanding of Earth and Space Science Concepts: Strategies for Concept‐Building in Elementary Teacher Preparation

This research is concerned with preservice teacher understanding of six earth and space science concepts that are often taught in elementary school: the reason for seasons, phases of the moon, why the wind blows, the rock cycle, soil formation, and earthquakes. Specifically, this study examines the effect of readings, hands‐on learning stations, and concept mapping in improving conceptual understanding. Undergraduates in two sections of a science methods course (N= 52) completed an open‐ended survey, giving explanations about the above concepts three times: as a pretest and twice as posttests after various instructional interventions. The answers, scored with a three point rubric, indicated that the preservice teachers initially had many misconceptions (alternative conceptions). A two way ANOVA with repeated measures analysis (pretest/posttest) demonstrated that readings and learning stations are both successful in building preservice teacher's understanding and that benefits from the hands‐on learning stations approached statistical significance. Concept mapping had an additive effect in building understanding, as evident on the second posttest. The findings suggest useful strategies for university science instructors to use in clarifying science concepts while modeling activities teachers can use in their own classrooms.     

Developing Student Understanding: Contextualizing Calculus Concepts

This study looked at the practice of one high school teacher who provided students with concrete examples from their physics class to give them a contextually rich environment in which to explore the abstractions of calculus. Students discovered connections between the physics concepts of position, velocity, and acceleration and the calculus concepts of function, derivative, and antiderivative. The qualitative study sought to describe several critical aspects of understanding: students' ability to explain concepts and procedures, to apply concepts in a physics context, and to explore their own learning. It included 32 seniors at a large, urban, comprehensive, religious school in a midwestern stale. Samples of student work and reflections were collected by the teacher, as well as by students in individual portfolios. The teacher kept a reflective journal. This study suggests that making connections between calculus and physics can yield deep understandings of semantic as well as procedural knowledge.     

Prospective Elementary Teachers’ Understanding of Order of Operations

This study investigates how well 381 prospective elementary, early childhood, and special education majors solved four arithmetic problems that required using the order of operations. Self‐reported data show these students to be relatively able mathematically and confident in their ability, with no substantial dislike of mathematics. The percentage of answers that were incorrect that is attributable to order of operations ranged from 21.7% to 78.5%. Overall, fewer than half the subjects answered more than two questions correctly. Of those subjects who performed multiplication before addition, which indicates some knowledge of order of operations, 30.9% performed addition before subtraction and 38.0% performed multiplication before division rather than from left to right, which suggests that instead of using the correct order of operations, these students used the common mnemonic PEMDAS or “Please excuse my dear Aunt Sally “literally, performing multiplication before division and performing addition before subtraction, rather than from left‐to‐right. Furthermore, 78.5% of subjects used the incorrect order of operations to compute ?3².     

Using Analogies to Improve Elementary School Students' Inferential Reasoning About Scientific Concepts

Various scientific concepts were taught to students in the third through sixth grades. Some children were taught the concepts using instructional analogies. Each analogy explicitly compared the science concept to a more familiar topic. Other children received expository texts not containing analogies. Students were asked to recall the texts and to answer inference questions about the science concepts. Fourth- and sixth-grade students read the texts on their own in Experiment 1. Students who read the analogical text showed higher levels of performance on inference questions than students who received the non-analogical texts. In Experiment 2, texts were read aloud to third- and fifth-grade students. The analogical texts were read once, and the nonanalogical texts were read twice to equate the number of times students were exposed to the general principles governing the domains. As in Experiment 1, students who received the analogical texts demonstrated better inferential reasoning than students who received the non-analogical texts.     

Number Worlds: Visual and Experimental Access to Elementary Number Theory Concepts

Recent research demonstrates that many issues related to the structure of natural numbers and the relationship among numbers are not well grasped by students. In this article, we describe a computer-based learning environment called Number Worlds that was designed to support the exploration of elementary number theory concepts by making the essential relationships and patterns more accessible to learners. Based on our research with pre-service elementary school teachers, we show how both the visual representations embedded in the microworld, and the possibilities afforded for experimentation affect learners' understanding and appreciation of basic concepts in elementary number theory. We also discuss the aesthetic and affective dimensions of the research participants' engagement with the learning environment. This revised version was published online in July 2006 with corrections to the Cover Date.     

Elementary Teachers' Progressive Understanding of Inquiry through the Process of Reflection

What is inquiry? Although many teachers are using inquiry based curricula, often they have not engaged in answering or personalizing this question. This study examined teachers' changing definitions of inquiry over a semester using the process of guided reflection. Through inquiry experiences and reflection, these teachers developed and communicated a more sophisticated understanding of inquiry. The findings suggest that conscious consideration of what inquiry means assisted teachers in broadening their perceptions of inquiry in four distinct aspects: 1) inquiry is a coherent process consisting of particular actions, 2) inquiry exists on a continuum, 3) the goal of inquiry is science conceptual development, and 4) inquiry provides a context for building connections between those engaged in inquiry, science and other content areas, and science and life.     

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.     

借助几何图形理解高等数学中的抽象概念和结论

高等数学的很多内容比较抽象,学生不易理解.通过几个例子说明如何将抽象的数学概念和结论与几何图形有机的结合起来,加深对这些概念结论的理解,激发学生的学习兴趣.     

Focusing on Units to Support Prospective Elementary Teachers' Understanding of Division in Fractional Contexts

This theoretical paper proposes a way to extend the partitive and measurement interpretations of whole number division to fractional contexts focusing on the issue of units. We define the unit‐changing and unit‐keeping interpretations for division and suggest a stronger and earlier focus on the concept of units in courses for prospective elementary teachers. As we highlight the importance of the concept of unit, we use the proposed interpretations in the analysis of one example that emerged in our methods course, discuss the idea of inverting and multiplying, and suggest implications for teacher education.     

Applying Cognition-Based Assessment to Elementary School Students' Development of Understanding of Area and Volume Measurement

As part of a discussion of cognition-based assessment (CBA) for elementary school mathematics, I describe assessment tasks for area and volume measurement and a research-based conceptual framework for interpreting students' reasoning on these tasks. At the core of this conceptual framework is the notion of levels of sophistication. I provide details on an integrated set of levels for area and volume measurement that (a) starts with the informal, preinstructional reasoning typically possessed by students, (b) ends with the formal mathematical concepts targeted by instruction, and (c) indicates cognitive plateaus reached by students in moving from (a) to (b).     

Applying Cognition-Based Assessment to Elementary School Students' Development of Understanding of Area and Volume Measurement

As part of a discussion of cognition-based assessment (CBA) for elementary school mathematics, I describe assessment tasks for area and volume measurement and a research-based conceptual framework for interpreting students' reasoning on these tasks. At the core of this conceptual framework is the notion of levels of sophistication. I provide details on an integrated set of levels for area and volume measurement that (a) starts with the informal, preinstructional reasoning typically possessed by students, (b) ends with the formal mathematical concepts targeted by instruction, and (c) indicates cognitive plateaus reached by students in moving from (a) to (b).