24. Mathematics Education Research
Paper
"Exploring Kindergartners’ Thinking in Division: A Case Study"
Galatia Kontakki, Iliada Elia
University of Cyprus, Cyprus
Presenting Author: Kontakki, Galatia
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).
Methodology, Methods, Research Instruments or Sources UsedThe 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).
Conclusions, Expected Outcomes or FindingsBoth 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.
ReferencesBoyatzis, 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.
24. Mathematics Education Research
Paper
Exploring the Possibilities of the Use of Picture Books for Inducing Mathematical Thinking in Early Childhood
Lucy Alambriti, Iliada Elia
University of Cyprus, Cyprus
Presenting Author: Elia, Iliada
In recent years, there is a growing interest in early childhood mathematics education research at an international level (Elia et al., 2023). This interest is attributed to a large extent to the increasing emphasis given on preschool education in many countries (e.g., Kagan & Roth, 2017) and to the findings of various studies which provide evidence for the significant role of young children’s early mathematical competences in their mathematics learning and performance later at school (Watts et al., 2014). Based on the above, the need of high-quality mathematics learning experiences from the beginning of children’s education is stressed.
A major pedagogical tool that is systematically used in early childhood education is picture books. Picture books are books that convey information either through a combination of images - text, or only through a series of images (Kümmerling – Meibauer et al., 2015). Picture books are used to nurture children’s emotional, social, and intellectual development as well as to develop children in content areas such as mathematics (Cooper et al., 2020). Particularly, picture books can provide a meaningful framework for learning mathematics and provide an informal base of experience with mathematical ideas that can be a starting point for more formal levels of understanding (Van den Heuvel-Panhuizen et al., 2009). Based on the findings of van den Heuvel-Panhuizen et al.’s study (2016), reading picture books should have an important place in the kindergarten curriculum to support children’s mathematical development.
Picture book reading in preschool can be done as an informal and spontaneous activity in which children are involved during free play and also as an activity that is organized and guided by the teacher (Van den Heuvel-Panhuizen & Elia, 2013). Considering the latter case, picture books can be used in all phases of the learning process, such as introducing new mathematical concepts, assessing children’s prior knowledge, deepening understanding and revising topics (Van den Heuvel-Panhuizen & Elia, 2012). Educators can make use of picture books by asking questions, posing problems to children, offering opportunities to discuss mathematical ideas and by adding relevant activities to provoke further exploration of the mathematics included in picture books.
In previous studies, different types of picture books were used to stimulate children’s mathematical development. With respect to the mathematical content included in the picture books, based on Marston’s (2014) work, a distinction can be made between (a) picture books with explicit mathematical content, which are written with the purpose to teach children mathematics, (b) picture books with embedded mathematical content, which are written primarily to entertain but the mathematics is intentional, and (c) picture books with perceived mathematical content, which tell an appealing story and in which mathematics is unintentional and implicit in the story.
According to the recent review on picture book reading in early years mathematics by Op ‘t Eynde et al. (2023), research studies that investigate the interplay between the picture books characteristics and the quality of picture book reading in early mathematics, based on the children’s and/or readers’ utterances, are rare. The present study could be considered as a step towards this research dimension, as it aims to explore the potential of the use of picture books with different characteristics in prompting children’s mathematical thinking.
Considering that, even if picture books are not written to teach mathematics, they may offer many opportunities for the exploration of mathematical ideas by young children (e.g., Dunphy, 2020), our study addresses the following research question: What are the possibilities offered by the use of picture books with embedded mathematical content and picture books with perceived mathematical content for inducing mathematical thinking in early childhood?
Methodology, Methods, Research Instruments or Sources UsedTo provide a deeper insight into the possibilities of using different types of picture books to stimulate mathematical thinking in the early years, we conducted a case study in which a 4-year-old girl participated. The girl has attended nursery and then kindergarten since the age of 4 months. She has not received formal instruction in mathematics or reading.
Two picture books were used in the study: “The Very Hungry Caterpillar” (Carle, 2017) and “How to hide a Lion from Grandma” (Stephens, 2014). These picture books are high quality books, which tell appealing stories and are not written to teach children mathematical concepts or skills. However, the book “The Very Hungry Caterpillar” (Book 1) includes mathematical content that is intentional, while in the book “How to hide a Lion from Grandma” (Book 2) the mathematics is unintentional and incidental. Therefore, based on Marston’s (2014) proposed distinction, in Book 1 the mathematical content is embedded, while in Book 2 the mathematics is perceived.
The story of Book 1 is about a small caterpillar that comes out of its egg very hungry. So, every day of the week, she eats a different amount of fruit or sweets, starting with one fruit on Monday, two fruits on Tuesday, etc., until it is full and makes her cocoon where she falls asleep. After two weeks it comes out, and from a small caterpillar, it turns into a beautiful butterfly.
The story of Book 2 is about a little girl named Elli, who has a secret: she lives with a lion. Elli has to hide the lion so that her grandmother, who will stay with her on the weekend, does not find it. In the end, however, it seems that Elli’s grandmother is also hiding something she brought from home in her bedroom.
For the data collection, the researcher (first author of this paper) read each picture book to the child in a separate session. A book reading scenario was used during each session. The reading scenarios were developed for the two books separately, prior to the reading sessions, and included questions and activities related to the mathematical content of the books, aiming at inducing the child’s mathematical thinking during the picture book reading. Both sessions took place in a quiet place in the school and were recorded. Each session lasted 20-30 minutes. The child’s mathematical thinking was examined by analyzing her utterances and her productions.
Conclusions, Expected Outcomes or FindingsThe findings of the study show that using the picture books had the power to elicit the child’s mathematical thinking and activate her cognitively. Based on the child’s utterances, the use of both the book with the embedded mathematical content (Book 1, Caterpillar) and the book with the perceived mathematical content (Book 2, Lion) elicited mathematical thinking related to different mathematical concepts. Specifically, although the embedded mathematical content of Book 1 focuses on numbers and counting, its use evoked thinking not only in numbers, but also in measurement and algebra. The use of Book 2 elicited the child’s spatial reasoning and thinking in measurement and numbers. These possibilities for engaging the child in mathematics were offered by the picture books through their rich environment, but also by the discussions and interactions with the reader/researcher and the additional activities that accompanied the narrative. This finding provides evidence for the important role of the reader in evoking the child’s mathematical thinking. For example, in our study more specific questions were asked to the child by the reader to trigger her mathematical thinking in the pages of the picture books in which mathematical content is not explicit. This occurred to a larger extent with Book 2 in which mathematical concepts are incidental and unintentional. Based on our findings, this variation in how the reader used the picture books during reading seemed to be effective, but additional research is needed to provide further insight into this issue.
Finally, based on our findings the pictures of both picture books had a crucial role in stimulating the child’s mathematical thinking, since most of the child’s mathematical utterances were focused on the pictures of the books irrespectively of the way the picture books were used (e.g., dialogic reading or accompanying mathematical activities related to the book).
ReferencesCarle, E. (2017). Μια κάμπια πολύ πεινασμένη [The very hungry caterpillar]. Kalidoskopio.
Cooper, S., Rogers, R. M., Purdum-Cassidy, B., & Nesmith, S. M. (2020). Selecting quality picture books for mathematics instruction: What do preservice teachers look for? Children’s Literature in Education, 51(1), 110-124.
Dunphy, L. (2020). A picture book pedagogy for early childhood mathematics education. In A. MacDonald, L. Danaia, & S. Murphy (Eds.), STEM Education across the learning continuum (pp. 67-85). Singapore: Springer.
Elia, I., Baccaglini-Frank, A., Levenson, E., Matsuo, N., Feza, N., & Lisarelli, G. (2023). Early childhood mathematics education research: Overview of latest developments and looking ahead. Annales de Didactique et de Sciences Cognitives, 28, 75-129.
Kagan, S. L., & Roth, J. L. (2017). Transforming early childhood systems for future generations: Obligations and opportunities. International Journal of Early Childhood, 49, 137-154.
Kümmerling-Meibauer, B., Meibauer, J., Nachatigäller, K., & Rohlfing, J. K. (2015). Understanding learning from picturebooks. In B. Kümmerling-Meibauer, J. Meibauer, K. Nachatigäller, & J. K. Rohlfing (Eds.), Learning from Picturebooks: Perspectives from child development and literacy studies (pp. 1-10). New York: Routledge.
Marston, J. (2014). Identifying and Using Picture Books with Quality Mathematical Content: Moving beyond" Counting on Frank" and" The Very Hungry Caterpillar". Australian Primary Mathematics Classroom, 19(1), 14-23.
Op ‘t Eynde, E., Depaepe, F., Verschaffel, L., & Torbeyns, J. (2023). Shared picture book reading in early mathematics: A systematic literature review. Journal für Mathematik-Didaktik, 44(2), 505-531.
Stephens, H. (2014). Πώς να κρύψεις ένα λιοντάρι από τη γιαγιά [How to hide a lion from grandma]. Athens: Ikaros.
Van den Heuvel-Panhuizen, M., & Elia, I. (2012). Developing a framework for the evaluation of picturebooks that support kindergartners’ learning of mathematics. Research in Mathematics Education, 14(1), 17-47.
Van den Heuvel-Panhuizen, M., & Elia, I. (2013). The role of picture books in young children’s mathematical learning. In L. English & J. Mulligan (Eds.), Advances in Mathematics Education: Reconceptualizing Early Mathematics Learning (pp. 227-252). New York: Springer.
Van den Heuvel-Panhuizen, M., Elia, I., & Robitzsch, A. (2016). Effects of reading picture books on kindergartners’ mathematics performance. Educational Psychology, 36(2), 323-346.
Van den Heuvel-Panhuizen, M., van den Boogaard, S., & Doig, B. (2009). Picture books stimulate the learning of mathematics. Australian Journal of Early childhood, 34(3), 30-39.
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.
24. Mathematics Education Research
Paper
Primary School Students and Prospective Teachers' Perspective Drawing Abilities in Geometry
Elif Tuğçe Karaca
KIRIKKALE UNIVERSITY, Turkiye
Presenting Author: Karaca, Elif Tuğçe
In primary education, geometrical drawing abilities hold pivotal importance. The ability to visually represent geometric shapes is a foundational skill that not only introduces students to the world of mathematics but also serves as a precursor to advanced spatial reasoning capabilities (Clements & Battista, 1992). This research aims to assess primary school students' geometrical drawing abilities comprehensively. By employing paper-pencil tests utilizing grid and isometric paper, the objective is to gauge the student's proficiency in visually representing two- and three-dimensional geometric shapes. This endeavor seeks to understand primary school students' current geometrical drawing skills.
The second objective involves an evaluation of the geometrical drawing abilities of pre-service primary school teachers. Through similar paper-pencil tests, the aim is to gauge the aptitude of prospective teachers to represent geometric shapes in various dimensions. This assessment is crucial for identifying potential areas of improvement in teacher training programs to ensure that future educators are equipped to impart geometric concepts effectively. The third objective
involves a comparative analysis of primary school students and pre-service teachers' performance in two- and three-dimensional geometric drawing tasks. By discerning potential differences in their abilities, this research seeks
to contribute valuable insights into the relationship between educational background and geometrical drawing proficiency. For these purposes, the three research questions are as follows: (a) What are the geometrical drawing abilities of primary school students? (b) What are the geometrical drawings? What are the abilities of pre-service primary school teachers? (c) What are the similarities and differences in drawing abilities for 4th graders and prospective teachers?
Methodology, Methods, Research Instruments or Sources UsedThe participants will consist of primary school students in 4th grade from a public school and pre-service primary school teachers from a primary school teacher education program in Kırıkkale province in Türkiye. The data will be collected in the 2024 spring semester by the researcher. The data will be analyzed qualitatively.
Paper-pencil tests will be designed for primary school students and pre-service teachers to assess geometrical drawing abilities. Grid paper and isometric paper will be utilized to facilitate the representation of two- and three-dimensional geometric shapes, respectively. The tests will encompass a range of shapes, including squares, rectangles, cubes, and prisms, ensuring a comprehensive evaluation of participants' abilities. Scoring rubrics will be developed to measure accuracy, precision, and creativity in geometric representation, providing a multifaceted assessment of geometrical drawing proficiency.
Conclusions, Expected Outcomes or FindingsThe data will be collected in the 2024 spring semester, and the findings will be reported according to the data. Understanding the participant's proficiency levels and identifying the strengths and weaknesses in their abilities are the expected outcomes of this research.
ReferencesClements, D. H., & Battista, M. T. (1992). Geometry and spatial reasoning. Handbook of research on mathematics teaching and learning, 420, 464
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