Non-numerical and Numerical Understanding of the Part-Whole Concept of Children Aged 4 to 8 in Word Problems

Abstract

For the development of mathematical competencies from preschool children and children at primary school age, the acquisition of a comprehensive part-whole concept is essential. The principle of understanding numbers as compositions of other numbers as well as being able to decompose numbers and interpret quantitative relations, enables a flexible understanding of operations. It is known that the part-whole concept is based on various schemas and is subject to a longer learning process. The aim of the study was to examine the non-numerical and numerical understanding of the part-whole concept more closely. To achieve this, the availability of the complex part-whole concept in children aged 4 to 8 years was examined by means of a sample of 181 children on the basis of word problems with non-numerical and numerical tasks on different part-whole contents (compensation, covariation, final amount, initial amount, finding and evaluating decompositions of amounts). A Rasch analysis resulted in a two-dimensional model. According to this, two content-related competency dimensions must be differentiated: A non-numerical dimension with the understanding of solving part-whole tasks in word problems without number relation, and a numerical dimension with the understanding of exact quantification with numbers. The two dimensions are considered as two different domains of knowledge, though they are highly correlated. They develop partly parallel, while at the beginning the non-numerical part-whole understanding develops a little earlier. Two levels of competencies could be identified within each dimension. The findings are discussed regarding the distribution of the children on the respective levels of competencies depending on their age.