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.     

Teachers’ views on dynamically linked multiple representations, pedagogical practices and students’ understanding of mathematics using TI-Nspire in Scottish secondary schools

Do teachers find that the use of dynamically linked multiple representations enhances their students’ relational understanding of the mathematics involved in their lessons and what evidence do they provide to support their findings? Throughout session 2008–2009, this empirical research project involved six Scottish secondary schools, two mathematics teachers from each school and students from different ages and stages. Teachers used TI-Nspire PC software and students the TI-Nspire handheld technology. This technology is specifically designed to allow dynamically linked multiple representations of mathematical concepts such that pupils can observe links between cause and effect in different representations such as dynamic geometry, graphs, lists and spreadsheets. The teachers were convinced that the use of multiple representations of mathematical concepts enhanced their students’ relational understanding of these concepts, provided evidence to support their argument and described changes in their classroom pedagogy.     

Doing Is Believing: Do Laboratory Experiences Promote Conceptual Change?

The present study examined whether notions students hold about scientific concepts are impacted by generating and testing hypotheses. Students enrolled in three ninth grade physical science classes were asked to select and test two student-generated hypotheses concerning flotation. Conceptual change was assessed by concept maps and word sorting activities administered at three points in the investigation (pre, mid, and post). Lawson's Classroom Test of Scientific Reasoning (CTSR) and the Integrated Test of Process Skills (TIPS) were used to assess scientific reasoning ability and proficiency with science process skills, respectively. Twenty-one of 63 students enrolled in the course completed the battery of assessment items and met the classroom attendance requirement. Those 21 students represented the data sample. Students selecting and investigating the hypothesis that weight determines an object's ability to float generally changed their notion of the relationship between weight and flotation. Students investigating the hypothesis that shape determines an object's ability to float did not exhibit conceptual change regarding shape.     

Nontraditional College Students and the Role of Collaborative Learning as a Tool for Science Mastery

This study investigates the way collaborative learning that occurs primarily outside the classroom affects college students' understanding of science. Collaborative learning is particularly important for the increasing number of nontraditional students who have limited time available for study groups and other peer learning activities occurring outside of class time. Using a national study of 4,644 college students of various academic majors, multiple linear regression was used to identify variables that enhance science learning. Time spent in peer learning settings, such as teaching science to peers and discussing science with peers, were the strongest predictors of understanding science; moreover, this finding was consistent even for nontraditional students who reported less frequency of engagement in such activities. The study suggests that science educators can enhance learning when they structure their courses to include peer learning that engages students with each other over science issues outside the classroom.     

Students’ multilingual repertoires-in-use for meaning-making: Contrasting case studies in three multilingual constellations

Mathematics education for multilingual classrooms calls for instructional approaches that build upon students’ multilingual resources. However, so far, students’ multilingual resources and the interplay of their components have only partly been disentangled and rarely compared between different multilingual contexts. This article suggests a conceptualization of multilingual repertoires-in-use as characterized by (a) what students use of certain languages, registers, and representations as sources for meaning-making in mathematics classrooms and (b) their processes of how they connect certain languages, registers, and representations. This qualitative learning-process study compares students’ multilingual repertoires-in-use in three contexts: Spanish-speaking foreign language learners of German in Colombia, Turkish- and German-speaking students born in Germany, and Arabic-speaking German language beginners recently immigrated to Germany. The analysis reveals the biggest differences not only in what the students use, but how they connect languages, registers, and representations. Some of these differences can partly be traced back to different classroom cultural practices. These findings suggest extending the conceptual framework for multilingual repertoires-in-use and including it in a social theoretical perspective. Thus, these findings have important practical consequences for multilingual mathematics classrooms: The instructional approach of relating languages, registers, and representations needs to be applied more flexibly, taking into account students’ different starting points. When doing so, students’ connection processes should be supported and explicated more systematically in order to fully exploit the students’ repertoires.     

Representation in Computational Environments: Epistemological and Social Distance

Computational environments have the potential to provide new representational resources and new ways of supporting teaching and learning of mathematics. In this paper, we seek to characterize relationships between the representations offered by particular technologies and other representations commonly available in the classroom context, using the notion of ‘distance’. Distance between representations in different media may be epistemological, affecting the nature of the mathematical concepts available to students, or may be social, affecting pedagogic relationships in the classroom and the ease with which the technology may be adopted in particular classroom or national contexts. We illustrate these notions through examples taken from cross-experimentation of computational environments in national contexts different from those in which they were developed. Implications for the design and dissemination of computational environments for use in learning mathematics are discussed.     

Instructional Supports for Representational Fluency in Solving Linear Equations with Computer Algebra Systems and Paper‐and‐Pencil

This research addresses the issue of how to support students' representational fluency—the ability to create, move within, translate across, and derive meaning from external representations of mathematical ideas. The context of solving linear equations in a combined computer algebra system (CAS) and paper‐and‐pencil classroom environment is targeted as a rich and pressing context to study this issue. We report results of a collaborative teaching experiment in which we designed for and tested a functions approach to solving equations with ninth‐grade algebra students, and link to results of semi‐structured interviews with students before and after the experiment. Results of analyzing the five‐week experiment include instructional supports for students' representational fluency in solving linear equations: (a) sequencing the use of graphs, tables, and CAS feedback prior to formal symbolic transpositions, (b) connecting solutions to equations across representations, and (c) encouraging understanding of equations as equivalence relations that are sometimes, always, or never true. While some students' change in sophistication of representational fluency helps substantiate the productive nature of these supports, other students' persistent struggles raise questions of how to address the diverse needs of learners in complex learning environments involving multiple tool‐based representations.     

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.     

Children's approaches to area measurement through different contexts

This study focuses on 12 years old children's approaches to area measurement in a project environment. These approaches are not explored through a specific set of mathematical tasks. The tasks, here, are defined through researchers' and children's interactions in a classroom. The children by working in small groups are asked to make a proposal about the location and the form of an area which would be given to them for their leisure activities. This environment defines different contexts where the children act and consider different aspects of the area measurement. These aspects are identified and compared among the three groups of children. The study has shown that the concept of area measurement carries different cultural dimensions for the children. Moreover, the children use those elements of the concept which fit in with their personal experience and the tasks they have to face.     

Mathematics textbooks and their use in English, French and German classrooms:

After a through review of the relevant literature in terms of textbook analysis and mathematics teachers' user of textbooks in school contexts, this paper reports on selected and early findings from a study of mathematics textbooks and their use in English, French and German mathematics classrooms at lower secondary level. The research reviewed in the literature section raises important questions about textbooks as representations of the curriculum and about their role as a link between curriculum and pedagogy. Teachers, in tunr, appear to exercise control over the curriculum as it is enacted by using texts in the service of their own perceptions of teaching and learning. The second and main part of the paper analyses the ways in which textbooks vary and are used by teachers in classroom contexts and how this influences the culture of the mathematics classroom. The findings of the research demonstrate that classroom cultures are shaped by at least two factors: teachers' pedagogic principles in their immediate school and classroom context; and a system's educational and cultural traditions as they develop over time. It is argued that mathematics classroom cultures need to be understood in terms of a wider cultural and systemic context, in order for shared understandings, principles and meanings to be established, whether for promotion of classroom reform or simply for developing a better understanding of this vital component of the mathematics education process.     

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.     

Motion, speed, and other ideas that “should be put in books”

This paper focuses on a portion of a research project involving a group of inner-city middle school students who used SimCalc simulation software over the course of an entire school year to investigate ideas relating to graphical representations of motion and speed. The classroom environment was one in which students openly defended and justified their thinking as they actively explored and solved rich mathematical problems. The activities, generally speaking, involved functions that were intended to tap students’ real world intuitions as well as prior mathematical skills and understandings about speed, motion, and other graphical representations that underlie the mathematics of motion. Results indicate that these students did build ideas related to those concepts. This paper will provide documentation of the ways in which these students interpreted graphical representations involving linear and quadratic functions that are associated with constant and linearly changing velocities, respectively.     

Revoicing in a Multilingual Classroom

The concept of revoicing has recently received a substantial amount of attention within the mathematics education community. One of the primary purposes of revoicing is to promote a deeper conceptual understanding of mathematics by positioning students in relation to one another, thereby facilitating student debate and mathematical argumentation. Our study reexamines revoicing in a multilingual high school algebra classroom; our findings challenge the assumption that revoicing is necessarily tightly connected with classroom argumentation. We demonstrate that a single discursive form, such as revoicing, can play a wide range of valuable functions within the classroom. More importantly, we investigate systematic differences in the ways that revoicing is used, by a particular teacher, across languages. Implications for policy and practice are discussed.     

High school students' use of representations in physics problem solving

Findings from physics education research strongly point to the critical need for teachers’ use of multiple representations in their instructional practices such as pictures, diagrams, written explanations, and mathematical expressions to enhance students' problem‐solving ability. In this study, we explored use of problem‐solving tasks for generating multiple representations as a scaffolding strategy in a high school modeling physics class. Through problem‐solving cognitive interviews with students, we investigated how a group of students responded to the tasks and how their use of such strategies affected their problem‐solving performance and use of representations as compared to students who did not receive explicit, scaffolded guidance to generate representations in solving similar problems. Aggregated data on students' problem‐solving performance and use of representations were collected from a set of 14 mechanics problems and triangulated with cognitive interviews. A higher percentage of students from the scaffolding group constructed visual representations in their problem‐solving solutions, while their use of other representations and problem‐solving performance did not differ with that of the comparison group. In addition, interviews revealed that students did not think that writing down physics concepts was necessary despite being encouraged to do so as a support strategy.     

Understanding and representing the arithmetic averaging algorithm: an analysis and comparison of US and Chinese students' responses

Analysing the responses of 311 sixth-grade Chinese students and 232 sixth-grade US students to two problems involving arithmetic average, this study explored students' understanding and representation of the averaging algorithm from a cross-national perspective. Results of the study show that Chinese students were more successful than US students in obtaining correct numerical answers to each of the problems, but US and Chinese students had similar cognitive difficulties in solving the second task. The difficulties were not due to their lack of procedural knowledge of the averaging algorithm, rather due to their lack of conceptual understanding of the algorithm. There were significant differences between the US and Chinese students in their solution representations of the two average problems. Chinese students were more likely to use algebraic representations than US students; while US students were more likely to use pictorial or verbal representations. US and Chinese students' use of representations are related to their mathematical problem-solving performance. Students who used more advanced representations were better problem solvers. The findings of the study suggest that Chinese students' superior performance on the averaging problems is partly due to their use of advanced representations (e.g. algebraic).     

Algebra Teachers' Utilization of Problems Requiring Transfer Between Algebraic,Numeric, and Graphic Representations

For students to develop an understanding of functions, they must have opportunities to solve problems that require them to transfer between algebraic, numeric, and graphic representations (transfer problems). Research has confirmed student difficulties with certain types of transfer problems and has suggested instructional factors as a possible cause. Algebra teachers (n= 28) were surveyed to determine the amount of class time they devote to different types of transfer problems and how many times these problems appear on their teacher‐made assessments. Results suggest that teachers dedicate less class time to graphic to numeric transfer problems than to any other type of transfer problem and that these problems appear less frequently on assessments. These are exactly the types of transfer problems that pose the most difficulty for students. It is conjectured that teachers' familiarity with these problems, combined with assumed student mastery, contribute to this mismatch.     

Understanding a Teacher's Reflections: A Case Study of a Middle School Mathematics Teacher

The purpose of this research was to understand how one teacher reflected on different classroom situations and to understand whether the teacher's approach to these reflections changed over time. For the purposes of this study, we considered reflection as the teacher's act of interpreting her own practices and students' thinking to make sense of student understanding and how teaching might relate to that understanding. We investigated a middle school mathematics teacher's reflection on her students while watching videotapes of her classroom and categorized the reflection as Assess, Interpret, Describe, Justify, and Extend. The results show a higher percentage of Extend instances in later interviews than in earlier ones indicating the teacher's increasing attention to her own teaching in how her students developed their understanding. In addition, her reflection became clearer and better integrated as defined by the Cohen and Ball's triangle of interactions.     

Inventing Creativity: An Exploration of the Pedagogy of Ingenuity in Science Classrooms

Concerns with the ability of U.S. classrooms to develop learners who will become the next generation of innovators, particularly given the present climate of standardized testing, warrants a closer look at creativity in science classrooms. The present study explored these concerns associated with teachers' classroom practice by addressing the following research question: What pedagogical factors, and related teacher conceptions, are potentially related to the demonstration of creativity among science students? Seventeen middle‐level, high school, and introductory‐level college science teachers from a variety of school contexts participated in the study. A questionnaire developed for this study, interviews, and classroom observations were used in order to explore potential areas of relatedness between pedagogical factors and manifestations of student creativity in science. Five categories ultimately emerged and described potential areas in which teachers would have to explicitly plan for creativity. These areas could inform the pedagogical considerations that teachers would have to make within their lesson plans and activities in order to support its manifestation among students. These provide a starting point for science teachers and science teacher educators to consider how to develop supportive environments for student creative thinking.