This longitudinal study tracks how 5- to 7-year-olds perform with early number skills. The aim of this study is to diagnose at-risk mathematics students by distinguishing the skills that, if not mastered by the end of Kindergarten, lead to greater difficulty in mathematics in 1st grade. This study’s methodology is mixed as it follows an exploratory and inductive path in light of its use of a hypothesis, an interpretive path in light of its interest in the individual student, and a positivist path in light of its focus on developing rules from analyzed data. An oral diagnostic test based on a comprehensive collection of early number skills was used to test students as Kindergarteners and again as 1st graders. The test results created benchmarks, revealing how the majority of the students performed with early number skills. The test results also revealed that each early number skill is highly, moderately, or minimally predictive in terms of student placement by the end of 1st grade. When comparing the individual skill scores of each Kindergarten student to his/her total test results of 1st grade, the predictive power of each skill emerged. Performing poorly with skills that are minimally predictive did not seem to have an impact on how the Kindergarten student finished in 1st grade; performing poorly with moderately predictive skills had a greater impact on 1st grade placement; performing poorly with highly predictive skills in Kindergarten increased the likelihood that the student would finish in the lower attaining group in 1st grade. A third result of the test showed that certain skills serve as preconditions for other skills; success with certain skills usually meant success with other skills. These connections between skills point to a learning model called in this study “simultaneous pathways,” indicating that there are connections between certain skills, and that students can be learning on several pathways simultaneously. The impact of the predictive power of early number skills is that diagnosis becomes more effective. Early diagnosis means early remediation which may prevent at-risk students from falling further behind their peers. The benchmarks developed by this research will help teachers assess their students because they will know the general skill level of Kindergarteners and 1st graders. This oral diagnostic test informs curriculum development. If test results show that students are missing the skills that are highly predictive, teachers can address those gaps in order to insure mastery.

The purpose of this study was to examine pre-service mathematics teachers&amp / #8217 / subject matter knowledge, pedagogical content knowledge, and the relationships between the two on multiplication and division of fractions. For this purpose, pre-service teachers&amp / #8217 / understanding of key facts, concepts, principles, and proofs, their knowledge on common (mis)conceptions and difficulties held by the elementary students, their strategies of teaching these concepts, and representations they use to reason their understanding on multiplication and division of fractions were examined. Qualitative case study was performed where / data was collected from the pre-service teachers at the end of the spring semester of 2004-2005. Pre-service teachers were senior students enrolled in a teacher education program at a public university. Results revealed that pre-service teachers could easily symbolize and solve the basic questions on multiplication and division of fractions. However, in terms of pre-service teachers&amp / #8217 / interpretation and reasoning of key facts and principles on multiplication and division of fractions, their subject matter knowledge could not be regarded as conceptually deep. Furthermore, although pre-service teachers have strong belief that they should teach multiplication and division of fractions conceptually, where the logical background of the operations is explained, they do not have sufficient knowledge to represent and explain these topics and relationships conceptually.

The purpose of this study is to examine the effects of a mathematics teaching methods course on pre-service elementary mathematics teachers&rsquo / content knowledge for teaching mathematics (CKTM). In order to accomplish this purpose, pre-service mathematics teachers&rsquo / understanding of basic concepts and procedures in school mathematics, use of mathematical definitions, presentation of mathematical content to students, identification of common errors, misconceptions and solution strategies and evaluation of unusual solution methods were examined with the help of a multiple choice test. The data were collected from 43 senior pre-service mathematics teachers from a teacher education program at a large public university in Ankara. The participants were given an 83-item test to measure their content knowledge for mathematics teaching at the beginning and after the methods course. The purpose of the pre- and post-test assessment was to measure the amount of change in the participants&#039 / knowledge for mathematics teaching. The test was developed and piloted at the University of Michigan in the USA for Learning Mathematics for Teaching (LMT) Project. Quantitative data analysis techniques were used to answer the research questions. The results indicated that there was a significant effect of the mathematics teaching methods course on pre-service teachers&rsquo / content knowledge for teaching mathematics. Moreover, the findings showed that there is no significant mean difference between male and female pre-service teachers, and between the pre-service teachers who have taken at least one mathematics teaching elective course and the ones who have not taken any elective course related to mathematics teaching in terms of their CKTM. Also, the study showed that there is a significant positive relationship between pre-service teachers&rsquo / CKTM and their academic achievement on undergraduate mathematics content courses. The study is expected to make important contributions to the literature by providing information about whether the methods courses significantly contribute to pre-service teachers&rsquo / understanding of knowledge for mathematics teaching. Moreover, the findings of the study is hoped to inform teacher educators and policy makers about the needs and improvements in teacher preparation programs.

The purpose of this study was to investigate the nature of changes in preservice elementary mathematics teachers&rsquo / views and reflections about elementary mathematics classes based on their observations in School Experience I and School Experience II courses. Specifically, this study investigated pre-service elementary mathematics teachers&rsquo / views and reflections about instruction, assessment, and classroom management issues in elementary mathematics classes based on their observations in School Experience I and School Experience II courses. The sample consisted of 19 pre-service elementary mathematics teachers from a teacher education program at Middle East Technical University (METU). Pre-service elementary mathematics teachers involved in this study made their observations in their cooperating schools for 3 months during the spring semester of 2004-2005 academic year and the fall semester of 2007-2008 academic year. The data were collected by means of their School Experience I and School Experience II course reports. The results indicated that although there was commonality in the views and reflections of pre-service elementary mathematics teachers&rsquo / about instruction, assessment, and classroom management issues in elementary mathematics classes, there were differences in their knowledge in the way they perceive instruction, assessment, and classroom management as they progressed through their education. In other words, when School Experience I course reports were compared with School Experience II course reports, pre-service teachers perceived the mathematics instruction as more student-centered and they gave more importance to the alternative assessment strategies in their School Experience II course reports. Furthermore, they defended more positive classroom management methods in their School Experience II course reports.

The main purpose of the study was to investigate pre-service elementary mathematics teachers&rsquo / personal efficacy beliefs and outcome expectancies about using concrete models in teaching mathematics. Data were collected from the pre-service teachers in the middle of the spring semester of 2008-2009. Pre-service teachers were junior students enrolled in elementary mathematics teaching program at a public university. Six instructional sessions based on using concrete models in teaching mathematics were carried out during a three week period. In this study, the researcher was also the teacher of the instruction at the same time. A survey on pre-service mathematics teachers&rsquo / efficacy beliefs about using concrete models was administered to the students before and after the instruction to evaluate the contribution of the instruction on pre-service teachers&rsquo / efficacies. After the instruction, semi-structured interviews were conducted. The present study demonstrated that the instruction based on using concrete models had positive contributions on the pre-service elementary mathematics teachers&rsquo / self-efficacy beliefs and outcome expectancies about using concrete models in teaching mathematics. In addition, results revealed that pre-service elementary mathematics teachers had confidence in themselves about using concrete models both as learners and as teachers. Moreover, they believed that using concrete models in teaching mathematics would have positive consequences in teaching process and students&rsquo / learning. However, the interview data indicated that, pre-service teachers had relatively low personal efficacies and outcome expectancies about classroom management, when the concrete models were involved in the instruction.

This study was conducted in an attempt to integrate a number of cognitive, motivational and behavioral factors in elementary mathematics education, and to develop a theoretical model that explains the direct and indirect relationships among these concepts and their underlying dimensions. In particular, it was intended to examine the interrelationships among students&rsquo / achievement goal orientations, perception of classroom goal structure, self-efficacy, use of self-regulatory strategies, and academic achievement in mathematics. Participants were 1019 seventh grade students, enrolled in public elementary schools, located in four different urban and rural districts in Ankara. A self-report questionnaire and a mathematics achievement test were administered to the participants during their regular class periods. A pilot study was carried out with 250 seventh grade students, for conducting exploratory factor analysis. Structural equation modeling technique was used for data analysis. First, confirmatory factor analyses were conducted for each factor in the questionnaire. Then, a structural equation model was developed for the whole sample. Results revealed that students&rsquo / perceptions of classroom goal structure were directly linked to their adoption of achievement goal orientations. Among these goal orientations, only mastery goal orientation was associated with students&rsquo / use of learning strategies, which, in turn, related to their mathematics achievement. Among the learning strategies, only elaboration was significantly related to students&rsquo / mathematics achievement. Besides, self-efficacy was both directly and indirectly related to students&rsquo / adoption of achievement goals, use of learning strategies, and mathematics achievement.

The purpose of this study was to investigate the prospective teachers&rsquo / knowledge of mistakes held by elementary students in fractions and their proposed strategies to overcome those mistakes. The data were collected from 149 prospective elementary mathematics teachers enrolled in the elementary mathematics education programs from a public university in Central Anatolian Region. Fraction Knowledge Questionnaire was used to accomplish the purpose of the study. The data collection tool included nine open ended questions, and each question had two sub-tasks. In this study, the items in the &ldquo / Fraction Knowledge Questionnaire&rdquo / were analyzed in-depth in order to reach a detailed description of prospective teachers&rsquo / knowledge about students&rsquo / mistakes on fractions. The results of this study revealed that prospective elementary mathematics teachers mostly could identify the students&rsquo / mistakes. However, although prospective teachers could notice the students&rsquo / mistakes, they could give superficial reasons for these mistakes. Furthermore, verbal explanations, using area representation, using real life model, reviewing prior knowledge, teaching standard algorithm, asking guided questions, using simple examples, using counter examples, using drill and practice, making students aware of their mistakes, and increasing students&rsquo / motivation were the suggested strategies by prospective teachers in order to overcome students&rsquo / mistakes in fractions.

The purpose of this study was to investigate the views of upper elementary mathematics teachers and students about the use of manipulatives in teaching and learning mathematics. This study is a qualitative case study. The participants of this study were four elementary mathematics teachers in a private school and their 6th, 7th, and 8th grade students. The data were collected through one-to-one interviews, observations and analyzing documents consisting of annual plan, daily plan, notebooks of students, and the field note that the researcher kept throughout the study. This study revealed that although all the teachers advocate the use of manipulatives in teaching mathematics, they use traditional teaching techniques in their classes. They mentioned different factors affecting their use of manipulatives in teaching mathematics such as not knowing how to use them, grade level, availability of materials, time constraints, students&rsquo / reactions (seeing them as a toy or not being accustomed to them), school administration, classroom management, not finding materials appropriate for the subject being taught and classroom size. In fact, these are the factors that are seen on the surface level. This study indicated that even when teachers are provided with training about the use of manipulatives, supported by the school administration, and provided with manipulatives, the use of manipulatives is largely determined by their views / beliefs about the nature of mathematics, how students can learn mathematics, the effect of manipulatives and their knowledge in using them. Students seemed to like learning by using manipulatives. When conditions were arranged for learning, they were willing to learn through manipulatives.

Many countries are reporting of difficulties in exact education at schools: mathematics, informatics, physics etc. Various methods are proposed to awaken and preserve students’ interest in these disciplines. Among them, the simplification, accent on applications, avoiding of argumentation (especially in mathematics) etc. must be mentioned. As one of reasons for these approaches the growing amount of knowledge/skills to be acquired at school is often mentioned. In this paper we consider one of the possibilities to integrate partially teaching of important chapters of discrete mathematics and informatics not reducing the high educational standards. The approach is based on the identification and mastering general combinatorial principles underlying many topics in both disciplines. A special attention in the paper is given to the so-called “pigeonhole principle” and its generalizations. In folklore, this principle is usually formulated in the following way: “if there are n + 1 rabbits in n cages, you can find a cage with at least two rabbits in it“. Examples of appearances of this principle both in mathematics and in computer science are considered.

Many countries are reporting of difficulties in exact education at schools: mathematics, informatics, physics etc. Various methods are proposed to awaken and preserve students’ interest in these disciplines. Among them, the simplification, accent on applications, avoiding of argumentation (especially in mathematics) etc. must be mentioned. As one of reasons for these approaches the growing amount of knowledge/skills to be acquired at school is often mentioned. In this paper we consider one of the possibilities to integrate partially teaching of important chapters of discrete mathematics and informatics not reducing the high educational standards. The approach is based on the identification and mastering general combinatorial principles underlying many topics in both disciplines. A special attention in the paper is given to the so-called “pigeonhole principle” and its generalizations. In folklore, this principle is usually formulated in the following way: “if there are n + 1 rabbits in n cages, you can find a cage with at least two rabbits in it“. Examples of appearances of this principle both in mathematics and in computer science are considered.