The Importance of Informal Language in Representing Mathematical Ideas
Cramer, Kathleen, Karnowski, Lee, Teaching Children Mathematics
Mathematics as Problem Solving, Mathematics as Communication, Mathematics as Reasoning, and Mathematical Connections - these four Standards, which open the NCTM's Curriculum and Evaluation Standards for School Mathematics (1989), can be considered the pedagogical standards. Together they paint a picture of how the nine content standards for grades K-4 and 5-8 can be taught. The K-6 curriculum should afford students opportunities for exploring problems, conjecturing, and generating hypotheses. The content learned should be rich in connections, both within a specific topic and across mathematics topics. Instruction should include opportunities for students and teachers to talk about mathematics.
This article presents a model for implementing this vision of instruction with particular emphasis on the importance of communication in learning mathematics.
The different roles that students' language plays in learning mathematics are articulated in the K-4 Mathematics as Communication standard (NCTM 1989, 26):
* Language helps children construct links between their informal mathematics experience and abstract symbols used in mathematics.
* Language facilitates connections among different representations of mathematical ideas.
* Writing about mathematics helps students clarify their thinking and deepen their understandings.
The communication standard also describes how representing mathematical ideas in multiple ways is a form of communication. This process involves taking a mathematical idea represented one way and translating it to another way. For example, a student may represent the symbol 3/4 by drawing a picture of a circle divided into four equal parts with three parts shaded. The student is translating from written symbols to pictures. A student may translate a story problem for division into actions with manipulative material. This activity would represent translation from a real-life context to a manipulative one. Children are communicating about mathematics when they are given opportunities to represent concepts in different ways and to discuss how the different representations reflect the same concept.
Understanding in mathematics can be defined as the ability to represent a mathematical idea in multiple ways and to make connections among different representations. Students' informal mathematical language mediates the translations among different representations and can support students' abstraction of mathematical ideas. Consider the conversation with a fourth grader shown in figure 1. This student had been using colored pieces of circles to study fractions. Before a lesson on fraction addition, she was asked to solve a fraction-addition problem.
Her language is imprecise and exemplifies the type of informal language that children use as they explain their actions with manipulatives. She translated her actions with manipulatives into her informal language. This translation mediated the translation to written symbols. She recorded what she verbalized about her actions with manipulatives.
The Lesh translation model (Lesh 1979) shown in figure 2 can help teachers organize instruction (a) to emphasize children's representation of mathematical ideas in different ways and (b) to use children's informal mathematics language to build links between different representations. The model suggests that mathematical ideas can be represented in five different ways, or modes: manipulatives, pictures, real-life contexts, verbal symbols, and written symbols. The arrows connecting the different ovals represent translations between modes; the arrows within each oval represent translation within modes. To develop a deep understanding of mathematical ideas, children should have experience in all five modes and be able to see how the different modes are related.
The use of manipulatives is important for young children's learning. The manner in which children use manipulatives may aid teachers in understanding those children's developmental levels and may also serve as a basis for students' discussion of mathematical ideas. …