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Abstract
This article presents effective strategies for educators of young children to promote children’s mathematical communications, which include think-aloud during individual work time, utilizing reasoning and proof questions, and questioning back to children.
Mathematical communication is an important tool that allows children to demonstrate their mathematical thinking and understanding of mathematics. The National Council of Teachers of Mathematics (2000) emphasizes the critical importance of mathematical communication as a method to rationalize and to confirm children’s mathematical thinking process to others. According to the National Council of Teachers of Mathematics’ (2000) standards on “communication,” programs from pre-kindergarten through grade 12 should prepare all students to: (1) organize and consolidate their mathematical thinking through communication; (2) communicate their mathematical thinking coherently and clearly to peers, teachers, and others; (3) analyze and evaluate the mathematical thinking and strategies of others; and (4) use the language of mathematics to express mathematical ideas precisely. Previous empirical studies have found that mathematical communication promoted students’ conceptual understanding of mathematics (Hoyles, 1985), and mathematical thinking and problem-solving skills (Baxter et al., 2005; Kostos and Shin, 2010), and helped children correct misconceptions about mathematical concepts (Kinman, 2010). Mathematical communication also had positive effects on the mathematics performance of academically low-achieving children and children at risk (Baxter et al., 2001).
Although “communication” in mathematics has been emphasized in the field of mathematics education, there has been a deficiency of resources to promote and facilitate children’s communication in early childhood and elementary education settings. “Communication” has often been disregarded in mathematics classrooms. The mathematics class is considered to be one of the quietest classes, where children work on problems individually and quietly. In elementary schools, children have few opportunities to talk, draw, and write in order to communicate their mathematical thoughts in mathematics classrooms (Whitin and Whitin, 2002). Ongoing communication in mathematics is important, and it is a critical process in developing children’s mathematical thinking. Many mathematical tasks require children to demonstrate correct answers without rationalizing how they obtained the answers. In many cases, children are able to provide a correct answer without understanding how they solved the problem (Kostos and Shin, 2010; Lee, 2014). Communication helps children represent and clarify their thinking, as well as discuss new ways of solving problems (McLennan, 2014; Ririe and Redford, 2008). According to Hoyles (1985), encouraging children to communicate about their mathematical thinking not only helps them to express their thought processes, but also shifts practice in teaching mathematics from “teacher-directed” to a more constructivist “child-centered” approach.
When a teacher asks children reasoning and proof questions by asking them to explain what they found and how they solved the problem, common responses from children are: “Oh, I just had it in my head” or “That made sense to me in my head.” Children frequently show difficulty in presenting their mathematical thoughts to others, or their mathematical thinking is not reflected in words. This is because they lack experience in communicating and presenting their mathematical thoughts to others (Chapin et al., 2003). Mathematical communication is a tool for children to represent their thinking using their own logic via oral and/or written language, including drawings or other types of creations. Problem solving is an “integral part” of mathematics (National Council of Teachers of Mathematics, 2000). When solving a mathematical problem, it is critically important to help children “be able to monitor and reflect on the process of mathematical problem solving” (National Council of Teachers of Mathematics, 2000: 52). This article presents useful strategies for teachers of young children as they attempt to promote children’s mathematical communications in their teaching practice.
Think-aloud
The “think-aloud” strategy provides a way for children to stay on task and verbally express as well as hear their own voices when problem solving. Think-aloud also enables teachers to assess children’s mathematical thinking and problem-solving processes by observing and listening to children’s voices. During think-aloud, children organize and comprehend their thoughts in language while thinking aloud (Baumann et al., 1993). Individual work time is considered by most educators to be a quiet time for children to work alone and solve problems. However, it is necessary for children to speak their thoughts by utilizing a think-aloud strategy. They do not have to talk loudly, but they are allowed to think aloud and hear their own voices, in order to stay focused on their work and organize their mathematical thoughts without interrupting others. Think-aloud strategies in mathematics often actively engage children to be focused on their work and help them make sense of their problem-solving processes to reach the solution (Rhodes and Kaesshaefer, 2009; Silbey, 2002). English language learners also benefit from the think-aloud process, using their own native language. This think-aloud time allows English language learners to organize their thoughts using their own language and make sense of the problem (Lee et al., 2011).
Reasoning and proof questions
Questioning in a mathematics classroom is regarded as a “powerful tool” as a teacher attempts to build children’s mathematics knowledge and enhance their conceptual understanding of mathematics (Purdum-Cassidy et al., 2014). Reasoning and proof questions promote, in particular, children’s mathematical logic. It is important for teachers of young children to provide an emotionally safe mathematics environment, so that children can willingly share their mathematical thinking without worrying about whether their answers are correct. Such an environment allows children to learn and to think deeply about mathematics (De Garcia, n.d.; Lee et al., 2011). More importantly, being able to rationally and logically prove their thought processes is more critical than coming up with the right answers. The National Council of Teachers of Mathematics (2000) has emphasized that children should develop this skill by continuously using reasoning and proof processes in various contexts from the earliest grades.
Questioning back to children
Children in early childhood tend to ask numerous questions, but their teachers do not have to answer all of the questions being asked of them. Children often have their own answers and logical thoughts regarding their questions. Simply questioning them back is a strategy to hear children’s logic (e.g. “What do you think?” or “I wonder too why this happened”) by promoting their mathematical communication skills and to assess children’s misconceptions about mathematics. The strategy of questioning back to the child allows the child to think one more time about the question and the process (Ogu and Schmidt, 2009). It is also important to give children some thinking time. From the teacher’s perspective, this is “waiting time,” in order to allow children the necessary time to come up with their own logic.
Conclusions
According to Whitin and Whitin (2002), teachers of young children play a critical role in helping children develop and refine mathematical communication skills. It is necessary for teachers of young children to carefully plan questioning strategies and groupings in order to provide children with opportunities to practice mathematical communications as a part of the process of rationalizing and justifying their mathematical thoughts. The strategies found to be effective are individual think-aloud work time, utilizing reasoning and proof questions, and questioning back to children. As they use these strategies, children will grow as mathematical practitioners who are comfortable in expressing to others the results of their mathematical thinking.
Funding
This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.
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Author biography
Joohi Lee is Associate Professor of Early Childhood and Elementary Mathematics Education at the University of Texas. Her research centers on children’s mathematical performance associated with their demographic information (socio-economic status, gender, ethnicity, language, etc.) and high-quality mathematics education in early and elementary education, focusing on the attributes of pre- and in-service teachers.
