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.

Abstract Students in grades 3 and 4 attending a traditional public elementary school in a northeastern state did not meet proficiency levels in mathematics as measured by the state's assessment system. Published reports indicated that students attending the Montessori programs were more proficient in solving math problems compared to students in traditional schools. However, researchers had not compared Montessori and traditional teachers' perceptions of teaching elementary mathematics. The purpose of this qualitative case study was to explore the perceptions of traditional and Montessori teachers regarding teaching basic problem solving skills in mathematics. Koehler and Grouws' model provided the theoretical framework. Data collection included semistructured interviews with 6 traditional and 4 Montessori elementary teachers, field notes, and journaling. Data were analyzed using a coding scheme that incorporated the theoretical model's categories. Findings indicated that both groups of teachers reported that concrete (manipulatives) to abstract (pen and paper) learning was an effective approach to teaching basic math concepts and problem-solving skills. Social change will be realized when struggling elementary students in both Montessori and traditional settings begin to meet proficiency levels in mathematics and benefit from instruction that balances concrete and abstract learning skills. As such, students will be able to explore, develop, and become more actively engaged in learning math and problem solving in all elementary grades. The project deliverable, a position paper supporting the principal theme of concrete to abstract learning, may be used to promote effective instructional practices in mathematics, hence, positive social change.