The purpose of this study is to suggest the direction for development of function materials integrating mathematics and science. First, we must investigate curricular scope and sequence of mathematics and science. Second, the science contexts selected need to support the mathematics concepts, not overwhelm them. The mathematics can easily get lost if science becomes too complicated. We may be tempted, which can result in misconceptions that are hard to correct later. Third, Many different examples of mathematics-science integration exist, therefore, it is important to find rich science contexts to connect with mathematics.

In this paper, firstly examined various proofs types that cover informal empirical justifications by Balacheff, Miyazaki, and Harel ＆ Sowder and Tall. Using these theoretical frameworks, justification activities by 5th graders were analyzed and several conclusions were drawn as follow: 1) Children in 5th grade could justify using various proofs types and method ranged from external proofs schemes by Harel & Sowder to thought experiment by Balacheff This implies that children in elementary school can justify various mathematical statements of ideas for themselves. To improve children's proving abilities, rich experience for justifying should be provided. 2) Activities that make conjectures from cases then justify should be given to students in order to develop a sense of necessity of formal proof. 3) Children have to understand the meaning and usage of mathematical symbol to advance to formal deductive proofs. 4) New theoretical framework is needed to be established to provide a framework for research on elementary school children's justification activities. Research on proof mainly focused on the type of proof in terms of reasoning and activities involved. But proof types are also influenced by the tasks given. In elementary school, tasks that require physical activities or examples are provided. To develop students'various proof types, tasks that require various justification methods should be provided. 5) Children's justification type were influenced not only by development level but also by the concept they had. 6) Justification activities provide useful situation that assess students'mathematical understanding. 7) Teachers understanding toward role of proof(verification, explanation, communication, discovery, systematization) should be the starting point of proof activities.

In recent days, it is stressed that problem solving ability and inference ability to get a higer accomplishment are very important. The purpose of this research is to explore the affective factors related the problem solving ability and reasoning ability. Also, we explored the difference between the two affective factors focusing on 6th graders in primary school.

The purpose of this study is to analyze the interaction patterns and the commonly accepted norms of reaching a consensus among small-group children when solving open-ended problems. In conclusion, open-ended problems have various strategies or different acceptable answers, so they give children learning opportunities to compare the answers and to participate in communication. And more valuable interaction patterns come from 'measuring','classifying' problems and open-ended problems with implicit solution. Therefore, teachers might as well consider the relation between problems and interaction patterns when they pose open-ended problems in a small-group study setting. They are expected to empower children to have sociomathematical norms of reaching a consensus un der indirect and supportive guidance.