In the past few years, the interest in the mathematical development of preschool children has increased. An important reason for this is the evidence provided by research that children’s competence levels in numeracy before or at the beginning of school are significant predictors of their achievement over the school years (e.g., Watts et al., 2014). Considering also that mathematical literacy is a key component of STEM education, which contributes to the knowledge and skills individuals need to develop to live and grow in our modern societies of information and technology, (early) mathematics education should be regarded as one of the most important constituents of the educational system. Early years mathematics education aims to offer children mathematical experiences and learning opportunities through which the children shall strengthen their mental abilities, to be able to structure mathematical concepts and develop mathematical skills both in the present and in the future.

In recent years several researchers have studied preschool children’s number sense and number-related abilities, including quantitative reasoning, that is, additive reasoning, which refers to addition and subtraction (e.g., Purpura & Lonigan, 2013) and multiplicative reasoning, which refers to multiplication and division (e.g., Nunes et al., 2015; Van den Heuvel-Panhuizen & Elia, 2020). Multiplicative reasoning, which is more complex than additive reasoning (Urlich, 2015), has received less research attention. 

The present study focuses on the mathematical concept of division. Specifically, the research objective of the study is to gain an in-depth insight into kindergartners’ thinking in division. The research questions that are addressed in the present study are the following: (a) How do kindergartners make sense of division?, (b) What strategies do kindergartners use to solve division problems?, (c) What difficulties do kindergartners encounter in division? A further concern of the study was to identify possible differences in making sense of division by kindergartners of different ages.   

Division is the process of dividing a quantity or a set into equal parts.  Partitive division and quotative division are two major types of division problems (Nunes et al., 2015). In partitive division a group of objects is divided into equal subgroups and the solver has to find the size of each subgroup. In the quotative division, the size of the whole group and the size of each equal subgroup are known and the solver must find out how many equivalent subgroups there are (Van de Walle et al., 2014). 

From the two types of division, partitive division is the type of division that children develop first (Clements et al., 2004). An informal strategy that is often used by children in partitive division with concrete objects is the distribution of the objects one by one (one-by-one strategy) or two by two (two-by-two strategy) to the recipients (subgroups). The difficulties encountered by the children in division are often caused by the increase of the quantity children are asked to divide among a certain number of recipients and also by the increase of the number of recipients to whom the certain quantity must be divided in partitive division or by the increase of the number of items of each equal subgroup in quotative division (Clements et al., 2004). 

The present study is a case study which explores the mathematical thinking of two kindergartners in the concept of division. Child 1 was six years old (6 years and 4 months) and Child 2 was almost five years old (4 years and 10 months) at the time of the interview. The children did not receive explicit instruction on division before the study. For the data collection clinical semi-structured interviews (Ginsburg, 1997) were used in order to better understand how children think about division and solve problems of division. Before the interviews, which were carried out individually for each child, a common question guide (protocol) was developed for both children, which included six division tasks and questions which aimed to reveal children’s ideas, conceptions and processes when solving each of the tasks. The six tasks involved either partitive or quotative division and were hierarchically ordered based on their difficulty level. During the interviews, for every task, each child had at his disposal relevant material (concrete objects or pictorial representations) which he was encouraged to use to solve the task and demonstrate his thinking. Two of the division problems that were used are the following: (1) John has some biscuits to give to his two dogs. He wants the two dogs to get the same number of biscuits. How can you help John to do this? Each child was asked to solve the task for different quantities of biscuits (n=2,4,6,10,14, or 20) (partitive division); (2) Mrs Rabbit has 7 carrots and she would like to put them into some baskets. She wants each basket to have 2 carrots. Draw the baskets that she will need (quotative division). Open-ended and more focused questions which prompted children to express their thinking were used at various moments throughout the interviews by the researcher, such as: “Can you explain to me how you got this answer”, “How did you do it?”, “Are there any carrots left? How many?”, “Can you draw the amount of carrots left?” The exact questions and their wording varied between the two children, depending on their responses. The interviews were conducted at a quiet place familiar to the children. The interview with Child 1 lasted 29 minutes, and with Child 2 37 minutes. Short breaks were taken when needed. The interviews were videotaped and after they were transcribed, the data analysis was carried out using the method of thematic analysis (Boyatzis, 1998).

Both children in the study demonstrated adequate awareness of various aspects of the concept of division. The use of concrete objects or pictures was a major part of both children’s processes of representing, making sense and solving most of the division problems. However, a few constraints were identified in the younger kindergartner’s thinking which were not found in the older kindergartner’s reasoning. Particularly, Child 1 (older) could solve both types of division problems which included quantities up to twenty items, while Child 2 (younger) could better solve partitive division problems with quantities of items up to ten and with up to two subgroups. Child 2 encountered difficulties in solving quotative division tasks mainly because he did not recognize that every group should have a specific size. Interestingly both children solved the incomplete division task successfully. This could be possibly due to the small quantity of the items included in the problem. Both children often used the one-by-one strategy to solve the partitive division problems. Grouping of the items of the whole set was mainly used for the solution of the quotative tasks. The older child was also found to use mental strategies for some partitive and quotative tasks. As this is a case study, these findings cannot be generalized, but they indicate that children can reason in division even prior to receiving any instruction on the specific concept, and this could be considered by teachers before starting the formal teaching of division. This intuitive thinking in division was found to differ between the younger kindergartner and the older one. Further quantitative and qualitative studies could be conducted to specify, to what extent and in what ways, age and other children-related characteristics (e.g., gender, language, home environment) influence children’s performance, their thinking and its development in division at a kindergarten level.

Boyatzis, R. E. (1998). Transforming qualitative information: Thematic analysis and code development. Sage. Clements, D.H., Sarama, J., & DiBiase, A.M. (Eds.) (2004). Engaging young children in mathematics. Standards for early childhood mathematics education. Mahwah, New Jersey: Lawrence Erlbaum Associates. Ginsburg, H. P. (1997). Entering the child's mind: The clinical interview in psychological research and practice. Cambridge University Press. Nunes, T., Bryant, P., Evans, D., & Barros, R. (2015). Assessing quan- titative reasoning in young children. Mathematical Thinking and Learning, 17(2–3), 178–196. Purpura, D. J., & Lonigan, C. J. (2013). Informal numeracy skills: The structure and relations among numbering, relations, and arith- metic operations in preschool. American Educational Research Journal, 50(1), 178–209. Ulrich, C. (2015). Stages in constructing and coordinating units additively and multiplicatively (Part 2). For the Learning of Math- ematics, 36(1), 34–39. Van den Heuvel-Panhuizen, M., & Elia, I. (2020). Mapping kindergartners’ quantitative competence. ZDM Mathematics Education, 52(4), 805-819. Van de Walle, J. A., Lovin, L. A. H., Karp, K. H., & Williams, J. M. B. (2014). Teaching Student-Centered Mathematics: Developmentally Appropriate Instruction for Grades Pre K-2 (Vol. 1). Pearson Higher Ed. Watts, T. W., Duncan, G. J., Siegler, R. S., & Davis-Kean, P. E. (2014). What’s past is prologue: Relations between early mathematics knowledge and high school achievement. Educational Researcher, 43(7), 352-360.