Preschool children’s views on unsolvable math problems

Olgan, Refika
Cengizoğlu, Seçil
Solmaz, Gizem
Introduction Early childhood mathematics education immerses young children into many kinds of reasoning through seeing, listening and talking, touching and moving that reveal how mathematics associates with the real world. In this array, mathematics education in the early years provides many opportunities for children to handle everyday tasks and inspires them about a way of thinking with interesting and challenging problems (Sperry-Smith, 2006). One of the aims of mathematics education in the early years is to bring children the skill of composition and decomposition of numbers. In traditional view, composition and decomposition of numbers can be learned simply through rote learning and there is no need for children to experience any cognitively embedded process. It is also believed that composition of numbers is a kind of mechanic procedure in which young children are not capable to do it (Baroody, Lai and Mix, 2005). On the other hand, more contemporary approaches in mathematics education argue that the relation among the numbers and composition and decomposition skills can be accomplished by young children through easily realizing the patterns and the relations among the numbers. According to INMPC (2008), children ages between 4 to 6 years begin to develop composition and decomposition skills. In this point, decomposition and composition process make children realize that there could be more than one strategy to solve a problem (Tirosh, Tsamir, Levenson, Tabach and Barkai, 2015). The National Council for the teachers of Mathematics (NCTM, 1989) proposed that developing the skills of composition and decomposition of numbers is important step for problem solving that should be in the center of learning activities in a daily routines of preschools and daily activities should be designed specifically for promoting problem solving skills (Baroody, 2006). In the related literature, there are studies that aimed to reveal young children’s mathematical problem solving skills. In these studies, problems set in a real-life context and also have a solution. In the current study, however, the children received a decomposition problem that has no mathematical solution. Therefore, the aim was to reveal strategies the children use for problem solving when they engage with unsolvable mathematical problems. Methodology The current research is a kind of replication study which was originally conducted by Tirosh, Tsamir, Levenson, Tabach, and Barkai (2015). The participants were 60-72 month-old preschool children (N=50) attending three different preschools in Ankara, Turkey. The convenient sampling method was utilized. Before conducting the study, required ethical measures and consents were gained from University Ethical Council, administrators of the school, and parents of the participant children, as well. The verbatim transcripts were analyzed through multiple coders in order to reach an agreement on codes and categories to ensure the reliability. In the data collection procedure, the children were proposed a birthday party task which is very common a daily life experience for children. For this task, four empty plates and eight cards (on which different amount of candies were seen) placed on a table. One of the eight cards was a blank card that has no candy picture on it. The problem that has no solution since twenty-seven is not divisible by four and seven candies cannot be placed on each of the four plates was introduced to each child as “You have a birthday party and four children are coming. You want to give each child seven candies on their plate. 26. Uluslararası Eğitim Bilimleri Kongresi, 20-23 Nisan 2017 Özetler Kitabı 2488 Can you arrange it so that there are seven candies on each plate? You can arrange the candies however you wish, but there have to be seven candies on each plate” While each child was working out on a task, the researcher sometimes puts remarks if needed to remind children that they can place more than one card on a plate. When the child finished the task, the researcher asked “So, does every child get seven candies?” The researchers followed the same procedure with each child on the same task. Findings There were two salient findings of the current study. The first one is that counting was the most common strategy that the children employed to decompose the candies into the plates equally. The children tried to count the number of candies through touching them and tried to place seven candies into each plates. While decomposition of the 27 candies, children tried to recall the numbers in each plate, thus they made simply subitizing. Moreover, most of the children did not recognize that this problem is an unsolvable but they tried to put forward their own strategies. One of the interesting sides of the problem is the blank card. Even though the blank card represents the zero, children who were 60 month-old (n=18) preferred to place the blank card in one of the plate. They also proposed some solutions including drawing a candy on the blank card. On the other hand, the children who were 72 month-old (n=15) did not use the blank card, but only two of them asserted that this problem is an unsolvable problem. There is only one child in the group of 72 month-old children tried out several strategies and then he decided that blank card is meaningless and the problem is an unsolvable problem. In conclusion, counting the number of the candies was the most frequently used strategy. Moreover, the older children (72 month-old) were more successful than the younger ones (60 month-old) in estimating and subitizing the candies. All of the children did not give up and tried to develop their own strategies to solve the problem that has no solution.
International Conference on Educational Sciences, (20 - 23 Nisan 2017)


Seventh grade students’ attitudes toward mathematics in terms of cognitive, affective and behavioral components: a modeling study
Gün, Özge; Bulut, Safure; Department of Secondary Science and Mathematics Education (2011)
The purpose of the study is threefold: (1) to examine students’ attitudes toward mathematics in terms of cognitive, affective and behavioral components, (2) to examine the relationships among students’ perceptions of their mathematics teacher’s teaching profession, their mathematics teacher’s and parents’ attitudes toward and expectations from them as learners of mathematics, some affective variables students have and the time they spent on mathematics at home and (3) to examine the relationships between st...
Secondary mathematics teachers’ mathematics related beliefs throughout a professional development program based on mathematical modelling
Ören Vural, Duygu; Çetinkaya, Bülent; Department of Secondary Science and Mathematics Education (2015)
The purpose of this qualitative study is to investigate in-service secondary school mathematics teachers’ mathematics related beliefs, to explore changes in their beliefs after they participated in a one-year professional development program (PDP) based on mathematical modelling as well as teachers’ perception about the effects of PDP on their belief change. Semi-structured interviews and an open-ended analogy questionnaire were the main data sources. Results showed that most of the teachers held more than ...
Pre-service elementary mathematics teachers' knowledge about definitions of integers and their knowledge about elementary students' possible misconceptions and errors in describing integers
Kubar, Ayşenur; Çakıroğlu, Erdinç; Department of Elementary Science and Mathematics Education (2012)
The aim of the study is to examine the nature of pre-service elementary mathematics teachers’ subject matter knowledge about definition of integers and the nature of pre-service elementary mathematics teachers’ pedagogical content knowledge about definitions of integers. For this purpose, pre-service mathematics teachers’ knowledge about definitions of integers, their interpretations of quoted definitions of integers, their knowledge about elementary students’ possible misconceptions and errors regarding de...
Preschool children's perceptions on human-environment relationship: follow-up research
Cengizoğlu, Seçil; Olgan, Refika; Teksöz, Gaye (Informa UK Limited, 2020-05-01)
In the current study, we focus on how the early childhood education for sustainability (ECEfS) program develops the perceptions of preschool children about the human-environment relationship. The sample of the study consisted of preschoolers aged 60-66 months in Ankara, Turkey. The data were collected through the drawings of children on the topic of 'human-environment' and interviews on drawings obtained before and after the program. The findings revealed that children perceive their environment as a peacef...
Middle school mathematics teachers' pedagogical content knowledge in relation to statistical reasoning
Gokce, Rukiye; Kazak, Sibel (2017-01-01)
To investigate middle school mathematics teachers' pedagogical content knowledge (PCK) with regard to statistical reasoning, an interview protocol was developed and used with nine teachers. This paper focuses on one of the problems in this interview protocol (Basketball problem) to illustrate teachers PCK in relation to four components: big ideas, student responses, student difficulties, and instructional intervention. Our analyses showed that levels of teachers' PCK varied in each component. Teachers had d...
Citation Formats
R. Olgan, S. Cengizoğlu, and G. Solmaz, “Preschool children’s views on unsolvable math problems,” 2017, p. 2487, Accessed: 00, 2021. [Online]. Available: