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How Do Kindergarteners Express Their Mathematics Understanding? Universal Journal of Educational Research 3(12): 1015-1023, 2015 http://www.hrpub.org DOI: 10.13189/ujer.2015.031210 How Do Kindergarteners Express Their Mathematics Understanding? Kyoko Johns Department of Curriculum and Instruction, Jacksonville State University, USA Copyright © 2015 by authors, all rights reserved. Authors agree that this article remains permanently open access under the terms of the Creative Commons Attribution License 4.0 International License Abstract This article describes how kindergarten Studies have indicated the importance on focusing on students represent their understanding of mathematical students’ thought process rather than end products [4, 11, 17, knowledge. The study examines the students’ use of oral 18], but a very few studies have focused on young students expressions, drawings, written language, and gestures when and their use of representations. There is a great need to communicating mathematically with their classmates and investigate the use of representation in the primary grade teacher. settings. The research question “How do kindergarten students express their mathematical understanding?” guided Keywords Mathematics, Representations, Kindergarten this inquiry. The following section describes literature relevant to the use of mathematical representation in primary classrooms. It is organized into three sections: (1) the use of representation by young children, (2) the role of teachers in 1. Introduction young children’s mathematics classrooms, and (3) the function of representation in mathematics lessons. Issues Young children are capable of solving mathematical regarding future research and the use of representation in problems even before they enter formal school settings. They primary classroom and its benefits are discussed. count objects in their environment and share things with their families and peers daily. There are many occasions where young children are challenged to solve problems such as 2. Review of Literature sharing their snack and dividing toys and food with their siblings and friends. By interacting with others in their 2.1. The Use of Representation by Young Children community, children learn to communicate and express their mathematical thoughts and knowledge through a variety of Representation is a part of the process of construction of ways. knowledge, which can be performed either by sharing the According to the Standards produced by the National ideas with others or individually [2, 14]. According to Council of Teachers of Mathematics (NCTM) [14], the previous studies based on Piaget’s constructivist theory, world's largest organization with more than 80,000 members there are three kinds of knowledge: physical knowledge, that are concerned about mathematics education across the social knowledge, and logico-mathematical knowledge. US and Canada, and internationally and the Common Core Physical knowledge is the “knowledge of objects in external State Standards (CCSS) [2] which are an US initiative reality;” social knowledge is the “knowledge of developed by the National Governors Association (NGA) conventions;” and logico-mathematical knowledge consists and the Council of Chief State School Officers (CCSSO), of “mental relationship, and the ultimate source of these representation is regarded as an important topic of the relationships is the human mind” [9-11]. Constructivists mathematics curriculum and an essential aspect for emphasize that children construct knowledge by interacting mathematical competence. Both NCTM and CCSS seek to with their environment and creating meaningful connections provide mathematics standards that are consistent across the between new information and their surroundings. Their US based on best practices and research [2, 14]. In order to knowledge is built upon their prior knowledge and become mathematically literate and to be able to function in experiences. Representation is the way in which young the 21st century as a productive citizen, students must be children express their interpretation and understanding of able to represent and interpret mathematical ideas and mathematics concepts using their oral and written language concepts. Much research has been conducted on the [9 - 11]. The use of multiple representations in a classroom functions of representation for older students in the past. is recommended by the CCSS [2] and NCTM standards 1016 How Do Kindergarteners Express Their Mathematics Understanding? [14]. representations. After involving the children in several Kamii [9-11] concentrated on working with young counting activities using children’s literature, they were children based on constructivist theory. Her work produced asked to create visual representations of those experiences. information concerning children’s social and intellectual The results indicated that children use a variety of strategies development. Kamii was critical of traditional practices of to explain their thinking process and the talk that surrounds teaching mathematics in primary classrooms and raised the this experience enriches their mathematical understanding. question of teacher-centered activities and the use of The study has suggested that if children are to appreciate worksheets. Characteristically, her studies examined how the traditional mathematical symbols, they need an young children develop their mathematical knowledge and opportunity to create such symbols themselves preceding how teachers support and facilitate their learning. Kamii the formal introduction to them [17]. presented a strong argument that teachers train children to Earlier studies have affirmed the fact that young children become passive learners by using worksheets and do and can use representation when communicating demanding “one right answer” rather than encouraging mathematically. These research findings also support them to actively participate in their learning. The studies Piaget’s theory on how young children acquire their also pointed out that children respect the rules they create knowledge. It is only recently that researchers have begun and increase their mathematical achievement when teachers to systematically examine the effect of the role teachers expect more from them by challenging their intellect. play in teaching mathematics and the function of The ideas expressed by Kamii lead to a broader representation in mathematics classrooms. One of the conceptualization of children’s abilities to utilize and central questions of the current studies is the role of master mathematical concepts. The study of 60 Japanese teachers in teaching mathematics concepts and skills in children ages three to seven years revealed the relationship young children’s mathematics classroom. between their levels of abstraction and their levels of representation [12]. The researchers interviewed children to 2.2. The Role of Teachers in Young Children’s investigate their abilities to represent a group of objects. Mathematics Classrooms The results suggested that children’s construction of number concepts were closely related to their levels of The CCSS [2] and NCTM standards [14] emphasize the abstraction. For example, if a child sees a group of five importance of the use of concrete, pictorial, and symbolic objects and does not mentally connect them with a number representation to facilitate student mathematics learning. five, that child is not ready to make one-to-one Previous studies have revealed the roles teachers play in correspondence and can only describe the objects as teaching representation in mathematics classroom in recent “bunch”, which is considered a lower level of years. Flevares and Perry [6] conducted a study on how representation. As children develop their one-to-one three first-grade teachers incorporated nonspoken correspondence skills and number concepts, they can representations such as gestures, pictures, objects, and represent these abstract concepts using shapes and symbols writing during mathematics lessons in their classrooms over accurately. 3 years. This study indicated that teachers used nonspoken Young children use representations in their mathematics representations along with verbal instructions to aid student classroom in so many ways. Woleck [18] studied the ways learning. The use of nonspoken representations “supplies an first graders make use of their drawings to convey their essential but incomplete depiction of classroom mathematical understanding. The study found children’s communication” (p.343). The study revealed that those drawings served as many learning tools for them to explore, teachers used gestures the most during mathematical investigate, debate, refine, and confirm one’s mathematical lessons as a spontaneous communication tool. Researchers understanding. There was a strong link between children’s conclude that the modeling of the use of representation by oral language development and the use of the pictorial classroom teachers encourages children to acquire effective representations. According to Kamii [9, 10] and Woleck representation skills. [18], children are able to engage in abstraction when they According to McClain and Cobb [13], teachers need to move from using pictures to symbols in order to become proactive in leading students in a right direction communicate their mathematical ideas and knowledge. when teaching mathematics. They conducted an experiment Piaget emphasizes that children construct knowledge from to analyze one teacher’s attempt to reorganize her first within, not by internalizing from the environment
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