The intent of the year-long focus on representation is to help teachers, school leaders, and teacher educators understand what representation is and to expand their thinking about the role of representation in teaching and learning mathematics. Representation, as articulated in Principles and Standards for School Mathematics, describes how the understanding of mathematics is advanced when concepts are explored in a variety of forms including symbols, graphs, tables, and physical models, as well as spoken and written words.
The term representation refers both to process and to product. It refers to the process of expressing a mathematical concept or relationship in some form and to the form itself. For example, a student who draws pictures of apples to determine the total number of apples he has when he combines two apples and three apples is using the picture of apples both to find and represent the answer. Representation may be reflected in a variety of forms such as using manipulatives to model a mathematical idea, expressing an idea verbally, using symbols to rewrite an idea as a mathematical expression, creating tables to show different values an expression may take, and making a graph to give a picture of what is happening in a relationship.
Examining students’ representations gives teachers valuable insight into the students’ thinking, illuminating how students comprehend mathematical ideas, solve problems, and model mathematical situations. Since students’ external representations reflect their internal thinking, having insight into this internal dialogue can be a powerful instructional tool. Students’ representations shed light on their thought process, and how they organized and approached the problem. Therefore, it is essential that teachers encourage students to engage with the mathematics and represent their mathematical ideas in ways that make sense to them. Teachers should also encourage the use of multiple representations and technology-assisted representations.
The focus of the year gives teachers, school leaders, and teacher educators the opportunity to expand their understanding of representation so that they are able to make or support instructional decisions that increase the mathematical learning and achievement of all students. The focus of the year is intended to highlight the value and power of mathematical representation and encourage its integration in all aspects of mathematics teaching and learning.
Learn Reflect Strands
The three 2006 Regional Conferences and Expositions, as well as the 2007 Annual Meeting and Exposition, featured Professional Development Focus of the Year Learn Reflect Strands.
The strands culminated with reflection sessions that allowed participants to discuss the following questions:
- How has your understanding of representation changed or focused?
- What new ideas about representation do you plan to incorporate into your classroom? How? Why?
- What role can representation play in developing students’ insight and understanding of mathematics?
- What role can representation play in assessing mathematics learning?
Recent Focus of the Year Topics