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This paper brings together research on slope from mathematics education and research on line of best fit from statistics education by considering what knowledge of slope students transfer to a novel task involving determining the placement of an informal line of best fit. This study focuses on two students who transitioned from placing inaccurate to accurate lines of best fit during a task-based interview. The analysis focuses on describing shifts in slope reasoning that accompanied the change to accurately placed lines, and investigates factors that may have influenced the shift in reasoning. The results have implications for the teaching of both slope and the line of best fit. [ABSTRACT FROM AUTHOR]

With the increasing emphasis on integrating engineering into K-12 classrooms to help meet the needs of our complex and multidisciplinary society, there is an urgent need to investigate teachers' engineering-focused professional development experiences as they relate to teacher learning, implementation, and student achievement. This study addresses this need by examining the effects of a professional development program focused on engineering integration, and how teachers chose to implement engineering in their classrooms as a result of the professional development. 198 teachers in grades 3-6 from 43 schools in 17 districts participated in a yearlong professional development program designed to help integrate the new state science standards, with a focus on engineering, into their teaching. Posters including lesson plans and student artifacts were used to assess teachers' engineering practices and the implementation in their classrooms. Results indicated that the majority of the teachers who participated in the professional development were able to effectively implement engineering design lessons in their classrooms suggesting that the teachers' success in implementing engineering lessons in their classroom was closely related to the structure of the professional development program. [ABSTRACT FROM AUTHOR]

This collective case study analyzes the use of manipulatives in math lessons developed and taught by 4 groups of elementary teachers (K-8) involved in lesson study as part of a professional development program. The study found that in three of four lessons studied manipulative use was turned into an end in and of itself rather than a tool, and that in the fourth lesson manipulative use hindered rather than helped student learning. These problems with manipulative use by teachers in the lessons provide helpful guidance for planning of future professional development for math teaching. Our conclusion contains recommendations for successful implementation of manipulatives for both teachers and professional developers. Most importantly, we stress the need to emphasize the link between pedagogy and content, not the specific use of manipulatives. [ABSTRACT FROM AUTHOR]

In this article, the author discusses integrated STEM learning experiences for helping students in making connections between discipline areas, applying content area knowledge and advancing student understanding and knowledge as a whole. Topics include integration of mathematics content has been a challenge for teachers creating lessons, topics such as water quality, energy conservation and robotics, and challenges identified by the teachers such as difficulty in using pedagogical strategies.

Research suggests the importance of mathematics knowledge for teaching ( MKT) for enabling elementary school teachers to effectively teach mathematics. MKT involves both mathematical content knowledge ( M- CK) and mathematical pedagogical content knowledge ( M- PCK). However, there is no consensus on how best to prepare elementary preservice teachers ( PSTs) to achieve M- CK and M- PCK. This study builds on research related to MKT by investigating influences of mathematics content courses designed specifically for elementary PSTs ( IMPACT courses- Impact of Mathematics Pedagogy and Content on Teaching) on their attitudes (i.e., confidence and motivation) toward M- CK and M- PCK. Results suggest that the PSTs who participated in these IMPACT courses not only acquired high levels of confidence and motivation toward M- CK, but also showed significant and greater gains in attitudes toward M- PCK, after taking the required mathematics methods course, than their counterparts. Further, the findings suggest that these IMPACT courses provided a mathematical foundation that allowed the PSTs to engage in mathematics teaching methods better than those PSTs who did not have such a foundation. These results suggest potential course experiences that may enhance M- CK and M- PCK for elementary PSTs. [ABSTRACT FROM AUTHOR]

In this qualitative research study, we sought to understand teachers' conceptions of integrated mathematics. The participants were teachers in the first year of implementation of a state-mandated, high school integrated mathematics curriculum. The primary data sources for this study included focus group and individual interviews. Through our analysis, we found that the teachers had varied conceptions of what the term integrated meant in reference to mathematics curricula. These varied conceptions led to the development of the Conceptions of Integrated Mathematics Curricula Framework describing the different conceptions of integrated mathematics held by the teachers. The four conceptions-integration by strands, integration by topics, interdisciplinary integration, and contextual integration-refer to the different ideas teachers connect as well as the time frame over which these connections are emphasized. The results indicate that even when teachers use the same integrated mathematics curriculum, they may have varying conceptions of which ideas they are supposed to connect and how these connections can be emphasized. These varied conceptions of integration among teachers may lead students to experience the same adopted curriculum in very different ways. [ABSTRACT FROM AUTHOR]

This study investigates the conceptions and use of inquiry during classroom instruction among beginning secondary science teachers. The 44 participants were beginning secondary science teachers in their first year of teaching. In order to capture the participants' conceptions of inquiry, the teachers were interviewed and observed during the school year. The interviews consisted of questions about inquiry instruction, while the observations documented the teachers' use of inquiry. All of the interviews were transcribed or coded in order to understand the conceptions of inquiry held by the teachers, and all of the observations were analyzed in order to determine the presence of inquiry during the lesson. The standard for assessing inquiry came from the National Science Education Standards. A quantitative analysis of the data indicated that the teachers frequently talked about implementing 'scientific questions' and giving 'priority to evidence.' This study found a consistency between the way new teachers talked about inquiry and the way they practiced it in their classrooms. Overall, our observations and interviews revealed that the beginning secondary science teachers tended to enact teacher-centered forms of inquiry, and could benefit from induction programs focused on inquiry instruction. [ABSTRACT FROM AUTHOR]

This study examined students' accuracy of measurement estimation for linear distances, different units of measure, task context, and the relationship between accuracy estimation and logical thinking. Middle school students completed a series of tasks that included estimating the length of various objects in different contexts and completed a test of logical thinking ability. Results found that the students were not able to give accurate estimations for the lengths of familiar objects. Students were also less accurate in estimating in metric units as compared to English or novel units. Estimation accuracy was dependent on the task context. There were significant differences in estimation accuracy for two- versus three-dimensional estimation tasks. There were no significant differences for estimating objects with different orientations or embedded objects. For the tasks requiring the students to estimate in English units, the embedded task and the three-dimensional tasks were correlated with logical thinking. For estimation tasks with novel units, three-dimensional and two-dimensional estimation tasks were significantly correlated with the logical thinking. In order to interact effectively with our environment it is essential to possess an intuitive grasp of both dimension and scale and to be able to manipulate such information. Estimation, approximating and measuring are all components of such intuition (Forrester, Latham, & Shire, 1990, p. 283). [ABSTRACT FROM AUTHOR]

In Florida, recent legislative changes have granted community colleges the ability to offer baccalaureate degrees in education, frequently to non-traditional students. Based on information obtained from the literature covering preservice teachers' math knowledge, teachers' efficacy beliefs about math, and high-stakes mathematics testing, a study examined a population of preservice teachers in a new Florida teacher preparation program. The research investigated relationships surrounding non-traditional preservice teachers' characteristics such as: ages, high-stakes math failures, lower division mathematics history, and math methods course performance, in relation to their efficacy beliefs about mathematics. Results revealed that preservice teachers' ages, lower division mathematics history, and math methods course performance, did have a significant relationship with their math efficacy beliefs, as measured by the Mathematics Teaching Efficacy Beliefs Instrument (MTEBI); the variable of high-stakes math failures did not. Additionally, a multiple regression model including the aforementioned variables did predict preservice teachers' MTEBI scores, but did not generalize to the greater population. The findings from this study can assist new teacher preparation programs in isolating variables that identify preservice teachers who are at risk for poor mathematical attitudes; can posit avenues for fostering positive math beliefs in preservice teachers; and can recommend further research in this area. [ABSTRACT FROM AUTHOR]

Translation errors and conceptual misunderstandings made by students translating among graphical, tabular, and symbolic representations of linear functions were examined. The study situated student errors in the context of the 'Translation-Verification Model' developed specifically for the purpose of explaining student behavior during the process of translating relationships from one mathematical representation to another. Three distinct error types were identified to explain student performance. An examination of the error types revealed that specific translation errors tend to occur at different stages of the translation process. Translation errors are also related to 'attribute density,' the amount of information inherently encoded in a given representation. The findings of the study have implications for teaching linear relationships-student weaknesses and strengths are identified. [ABSTRACT FROM AUTHOR]

Using data from 503 inservice elementary teachers, this study investigated the relationship between teachers' background characteristics, teachers' beliefs about manipulatives, and the frequency with which teachers use manipulatives as part of their mathematics instruction. Findings from the study show that teachers' grade level and beliefs about manipulatives are important predictors of teachers' use of manipulatives in their mathematics instruction. [ABSTRACT FROM AUTHOR]

The purpose of the research was to improve the effectiveness of instruction in constructivist pedagogy in a college elementary mathematics education course through intentional integration of instruction in mathematics content. Instructors of this course previously used examples involving mathematics content on an ad hoc basis in an attempt to illuminate desirable constructivist pedagogy but discovered that they were ineffective because students experienced difficulty with the mathematics content itself. An instrument was created to assess students' mathematics content knowledge required to understand these examples. Based on the outcome of the assessment, intentional instruction of mathematics content using anchoring examples was integrated with pedagogical instruction. Results showed significant improvement in mathematics content knowledge and confidence in that knowledge with a better understanding of constructivist pedagogy. [ABSTRACT FROM AUTHOR]

Previous research (e.g., Woodward & Baxter 1997) found that Standards-based mathematics teaching provides. marginal or no benefits for low achievers, in contrast with positive effects for middle and high ability students. A randomized quasi-experiment in 52 Canadian schools found that low achieving grade 7 and 8 students who received support consisting of placement on a learning continuum, instruction focused on their specific learning needs, and concrete materials to represent mathematical constructs, benefited from teaching that emphasized construction over transmission of knowledge. Treatment students showed small but statistically significant improvements over controls in student achievement, and controversially, in mathematical beliefs, and attitudes. The latter finding raised issues of the appropriate balance between Type I and Type II error in educational research. [ABSTRACT FROM AUTHOR]

This study describes 5th, 8th, and 11th-grade students' teachers' and medical professionals' conceptions of flu and microbial illness. Participants constructed a concept map on "flu" and participated in a semi-structured interview The results showed that these groups of students, teachers and medical professionals held and structured their conceptions about microbes differently. A progression toward more accurate and complete knowledge existed across the groups but this trajectory was not always a predictable, linear developmental path from novice to expert. Across the groups, participants were most knowledgeable about symptoms of microbial illness, treatments of symptoms, and routes of transmission for respiratory illnesses. This knowledge was tightly linked to participants' prior experiences with colds and flu. There were typically large gaps in participants' (children and teachers) understandings of vaccines, immune system responses, treatments (including the mechanisms of pain medications and the functions of antibiotics), and transmission of non-respiratory microbial illness. A common misconception held by students was the belief that antibiotics can cure viral infections. [ABSTRACT FROM AUTHOR]

This article is about an investigation of six middle school science teachers' beliefs and instructional practice about the coherence of the science they teach as articulated by National Science Education Standards (NRC, 1996). Many well intentioned reform efforts focus on improving content knowledge of teachers, but many class- room teachers regularly miss opportunities to provide conceptual connections within the science ideas building the sense of coherence in .science. This investigation involved a quasi experimental study to examine the efficacy of a method for collecting data about middle school science teachers' thinking about science and to determine if they teach science coherently. The teachers were surveyed, interviewed, provided concept maps about their thinking of the science they taught, and observed to investigate whether their practice reflects their beliefs. An examination of the teachers' beliefs stated and unstated curriculum, the connections among topics and the nature of science revealed that one, the observation tool may have merit for identifying the content and connections among science topics, and two, that teachers' stated beliefs consistent with the National Science Education Standards' vision for coherent science, did not match their demonstrated practice. The content taught could he characterized in three ways: coherent content and few connections, coherent content and connections, and not coherent content. This indicates for this group of middle school science teachers that knowing how they think about science and how those beliefs' are reflected in their teaching is complex. This study can inform teacher education and professional development efforts about the need to move beyond just content enhancement to examine prior beliefs' about the connections of concepts within science. [ABSTRACT FROM AUTHOR]

This study investigated the mathematics beliefs and content knowledge of 103 elementary pre-service teachers in a developmental teacher preparation program that included a two course mathematics methods sequence. Pre-service teachers' pedagogical beliefs became more cognitively-oriented during the teacher preparation program with these changes occurring during the two methods courses. Pedagogical beliefs remained stable during student teaching. The pre-service teachers also significantly increased their personal efficacy for teaching mathematics throughout the program with these shifts occurring across both methods courses and into student teaching. Pedagogical beliefs and teaching efficacy beliefs were not related at the beginning of the program, but, in general, were positively related throughout the program. In addition, the pre-service teachers' pedagogical beliefs were positively related to their specialized content knowledge for teaching mathematics at the end of the program. [ABSTRACT FROM AUTHOR]

Although the reform literature in mathematics and science is replete with calls for the integration of math and science, there remain precious few empirical studies examining the prerequisite skills, beliefs, knowledge bases, and experiences necessary for teachers to implement integrated instruction. The initial intent of this study was to examine the content knowledge, pedagogical content knowledge, attitudes, and beliefs (with respect to the integration of mathematics and science) that prospective secondary mathematics and science teachers bring to their respective preparation programs. This study then explored a collaborative model intended to create meaningful educational opportunities within the context of teacher preparation programs to foster preservice teachers' desire and ability to pursue connected teaching in the classroom. Presented in the article are findings related to the initial beliefs and experiences prospective teachers bring to the preparation process, the results of their collaborative work together in the creation of interdisciplinary units connecting mathematics and science topics, and their ongoing efforts to work together once engaged in schools for their student teaching internships. Moreover, this article proposes subtle shifts in both the conceptualization of, and language used to describe, the integration of mathematics and science. Building on sociocultural theories, this article proposes the use of connections and pedagogical context knowledge as levers to promote integrated mathematics and science instruction. [ABSTRACT FROM AUTHOR]

Many K-8 preservice teachers have not experienced learning mathematics in a standards-based classroom. This article describes a mathematics content course designed to provide preservice teachers experiences in learning mathematics that will help build a solid foundation for a standards-based methods course. The content course focuses on developing preservice teachers' mathematical knowledge, as well as helping them realize what it means to learn mathematics that is taught using the pedagogy in the Principles and Standards for School Mathematics (National Council of Teachers of Mathematics, 2000). Furthermore, findings are presented from a study on this course that describe students' pre-and postcourse beliefs, attitudes, and perceptions of what it means to learn and teach mathematics. These findings provide evidence that the students in the study are beginning to understand what is meant by a standards-based classroom. Data were collected from surveys and interviews. Quotes from the students who aspire to be elementary teachers are used throughout the article to support the points. [ABSTRACT FROM AUTHOR]

This study compares the effectiveness of two forms of a knowledge mobilization task on preservice elementary teachers' (n = 65) performance in solving a triangle fraction problem. The study then identifies the source of the successful solutions by linking solutions to earlier activities. One group worked with the triangle fraction task individually; a second worked with the triangle fraction task in a social constructivist setting; a control group had no knowledge mobilization pretask. Although there was no significant difference in the frequency of successful solutions among treatment groups, a chi-square analysis found that the social-constructivist pretask group applied fewer ideas from the manipulative lessons as solutions to the posttask than did the comparison groups. The social constructivist group was, however, most successful at generating novel solutions to the triangle problem. The potential benefits of individual and socially constructed knowledge mobilization tasks are discussed. [ABSTRACT FROM AUTHOR]