mathematical language

The article substantiates the potential and pedagogical conditions for using loose parts to develop mathematical competence in preschool and primary school children. Loose parts are defined as natural, household, manufactured, or recycled materials with which children can engage in a variety of practical activities, including those involving mathematical content. These may include stones, sand, water, branches, leaves, seeds, shells, pieces of fabric, threads, cardboard boxes, bottle caps, containers, sticks, and other materials. It has been established that, at preschool age, the use of loose parts provides a sensory basis for developing concepts of quantity, number, shape, size, space, and patterns. In primary school, in addition to deepening this knowledge, the same materials can be used to support modeling, solving problems in several ways, testing assumptions, justifying strategies, and working with data. This enables continuity between preschool and primary education in the development of children’s mathematical competence. The potential of loose parts for developing mathematical competence in preschool and primary school children lies in the fact that they enable the practical acquisition of concepts related to number, shape, size, spatial relationships, and patterns, while stimulating the development of skills in counting, comparing, classifying, measuring, modeling, and problem-solving. Their use encourages learners’ independence, investigative activity, and the development of mathematical language. The following pedagogical conditions for using loose parts to develop mathematical competence in preschool and primary school children have been identified: ensuring the purposeful selection and diversification of materials; combining support for children’s initiative with the implementation of pedagogical intentions; and organizing mathematical reflection among preschool and primary school learners.

Proving is a fundamental skill that pre-service mathematics teachers are expected to develop and to support in their future students. This study aimed to examine the proving processes of pre-service middle school mathematics teachers. To this end, a total of 30 participants were selected through maximum variation sampling from among pre-service teachers who had just been admitted to a middle school mathematics teacher education program and pre-service mathematics teachers at all year levels of the program, with six participants from each level, and the data of this qualitative study were collected through clinical interviews. The data were collected in the context of the concepts of function and number. According to the results of the study, participants at the higher year levels generally showed greater success in constructing the formal-rhetorical part of a proof and in solving its problem-centered part, indicating a cross-year pattern rather than individual longitudinal change. It was also concluded that even when participants in the upper years possessed knowledge of proof methods, either their lack of sufficient conceptual knowledge to solve the problem-centered part or their deficiencies in the use of mathematical language hindered the proving process. Furthermore, it was observed that the participants were unable to use the full range of proof methods; whereas proof by counterexample showed the highest success in the proving process at every level, participants experienced difficulties in the direct proof and proof by contraposition processes. The findings suggest that the formal-rhetorical and problem-centered parts of proof should be addressed explicitly and systematically across all year levels of mathematics teacher education programs.