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Lesh, R., Cramer, K., Doerr, H., Post, T., Zawojewski, J., (2003) Using a translation model for curriculum development and classroom instruction. In Lesh, R., Doerr, H. (Eds.) Beyond Constructivism. Models and Modeling Perspectives on Mathematics Problem Solving, Learning, and Teaching. Lawrence Erlbaum Associates, Mahwah, New Jersey.


Chapter 24

Using a Translation Model for Curriculum
Development and Classroom Instruction

Kathleen Cramer
University of Minnesota

Please note - At this time, only the introduction and summary of this chapter are online. We are seeking copyright permission to include the entire chapter.

Other chapters in this book emphasize a variety of ways that representational fluency is an important component of students' models and modeling abilities - and an important part of what it means to understand basic mathematical costructs in topic areas ranging from early number concepts, to rational number concepts, to concepts in algebra, geometry, probability, statistics, or calculus.  The purpose of this chapter is to report some ways that the National Science Foundation supported Rational Number Project (RNP) has used the following translation model (Lesh, 1979) to develop curriculum materials and classroom activities that are aimed at helping both students and teachers develop deeper and higher order understandings of some of the most important ideas in the school mathematics curriculum.
FIG 24.1. The Lesh Translation Model

The Lesh translation model suggests that elementary mathematical ideas can be represented in five different modes: manipulatives, pictures, real-life contexts, verbal symbols, and written symbols. It stresses that understanding is reflected in the ability to represent mathematical ideas in multiple ways, plus the ability to make connections among the different embodiments; and, it emphasizes that translations within and between various modes of representation make ideas meaningful for students. Thus, the Lesh translation model extends Bruner's theory by adding to his three modes of representation real-life context and verbal symbols.  The Lesh Model emphasizes interactions within and among representations.  For example, the arrows connecting the different modes depict translations between modes; and, the internal arrows depict translations within modes.  The model suggests that the development of deep understanding of mathematical ideas requires experience in different modes, and eperience making connections between and within these modes of representation.  A translation requires a reinterpretation of an idea from one mode of representation to another.  This movement and its associated intellectual activity reflect a dynamic view of instruction and learning.


The purpose of this chapter was to present a case for considering the Lesh translation model as a guide for curriculum development. Building on the theories of Piaget, Bruner and Dienes, the Lesh model suggests that a deep undestanding of mathematical ideas can be developed by involving students in activities that embed the mathematical ideas to be learned in five different modes of representation with an emphasis on translations within and between modes.  The Lesh model was used in developing the RNP fraction curriculum. A large-scale study with fourth- and fifth- graders showed the effectiveness of this curriculum in developing fraction understanding over traditional curricula. 

Mathematics content courses developed with this model have been successfully used with elementary and middle-school teachers.  These courses, with Minneapolis teachers, have proved so popular that the district's new Urban Systemic Initiative aimed at improving mathematics instruction in Grades K through 5 will be using these courses as a foundation for its teacher enhancement activities.

Classroom teachers can use the Lesh translation model to guide how they implement district-adopted curriculum within their classrooms.  The model can help teachers determine the types of activities to supplement a curriculum so they can effectively meet students' instructional needs.  Generally, most curriculums can benefit from more manipulative models and more problem contexts with translations from these modes to other modes of representation.

Although it has not been addressed in this chapter, the Lesh translation model can be an effective tool for developing assessment items (Cramer & Bezuk, 1991).   Instruction and assessment should be aligned.  Assessment tasks can be constructed around the translations within and between modes of representation. This allows teachers to assess understanding beyond procedural skill.

The power of the Lesh Translation Model can be seen in its multiple uses: a model for curriculum development, a model for classroom curriculum decisions, and a model for assessment.