Language, Arithmetic and Young Children's Interpretations
Warren, Elizabeth, Focus on Learning Problems in Mathematics
Throughout the last two decades the language of mathematics has been acknowledged as being important to the development of mathematics understanding. It is believed that one of the main reasons children have difficulties with mathematics is the learning of the vocabulary. It seems that children are required to attach new meaning and learn new noun or verb forms to words that already exist in the learner's everyday speech. Halliday (1979) referred to this phenomenon as dealing with the mathematical register, a separate international language system with its own symbols, vocabulary, syntax, grammar and semantics. Children have to become aware that the meaning of words in the mathematics classroom is more constrained or precise as compared with words used in everyday speech (Gal, 1999).
The goal of this paper is to contribute to a reflection on young children's understanding of words commonly used in addition and subtraction, the meaning of the vocabulary, the relationships they see between the words, and the conceptualization they use in delineating this connectedness. In times where the focus is on reading and writing, few studies have explored how young children comprehend the language of mathematics. The literature often refers to this as children's metalinguistic awareness of mathematical words (MacGregor & Stacey, 1999), that is, the linguistic ability that allows the language user to reflect on and analyze the spoken or written word. Metalinguistic awareness is important for understanding algebraic notation (MacGregor & Price, 1999). Past research has tended to examine language proficiency with English as a second language (Adler, 1995; Bums, 1992; Lopez Real, 1997) or investigated children's knowledge of the meanings of words, their familiarity with patterns of discourse used in the classroom, and their ability to comprehend word problems.
As our attention turns to a social constructivist world and the learning of algebra in the early years, the role of language increases in importance. A vast amount of recent research has focused on the processes of communication in classrooms and the emergence of shared meaning. The underlying belief is that when people communicate through all sorts of signs (both idiosyncratic and conventional) knowledge emerges, with the individual continually interpreting and reinterpreting these signs (Peirce, 1960). However, MacGregor (1993) suggests research has shown that group activities involving sharing ideas, discussing procedures and writing reports may not necessarily be helpful to all students. These discussions themselves involve group language that participants may not feel comfortable engaging in as they may not understand the explanations of others and may not be able to communicate their ideas verbally. Nevertheless Arzarello (1998) believes that natural language is crucial to developing an algebraic way of thinking. When solving problems the verbal code activates an inner language that serves to aid the communication of knowledge and solutions to problems. Signs and tools are used by the learner to mediate the learning activity (Vygotsky, 1978, 1986), and words and dialogue play an important role in both external and internal processes of learning. From this premise the meanings of words used by young children become of great interest.
The epistemological stance taken in this analysis is the science of semiotics, a means of addressing signs, their connections and meanings. In this instance signs refer to external representations. Presmeg (1997) suggests that when one recognizes the structure of the system he or she engages in, explains this structure to others by such means as encoding it in a diagram or applying some overarching framework then mathematics exists. So while semiotics is commonly used to construct links between cultural and historical practices and mathematics (Presmeg, 1997; Radford, 1997) it also assists us to understand classroom discourse in mathematics (Saenz-Ludlow, 2001), and children's understanding of the relationship between words. …