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#### Developing Algebraic Thinking

The formal study of algebra is both the gateway into advanced mathematics and a stumbling block for many students. The transition from arithmetic to algebra is often not an easy one. The difficulties associated with the transition from the activities typically associated with school arithmetic to those typically associated with school algebra (representational activities, transformational activities, and generalizing and justifying activities) have been extensively studied. Research has documented that the visual and numerical supports provided for symbolic expressions by computers and graphing calculators help students create meaning for expressions and equations. The research, however, has shed less light on the long-term acquisition and retention of transformational fluency. Although through generalizing and justifying, students can learn to use and appreciate algebraic expressions as general statements, more research is need on how students develop such awareness.

The formal study of algebra is both the gateway into advanced mathematics and a stumbling block for many students.

The study of algebra, however, does not have to begin with a formal course in the subject. New lines of research and development are focusing on ways that the elementary and middle school curriculum can be used to support the development of algebraic reasoning. These efforts attempt to avoid the difficulties many students now experience and to lay a better foundation for secondary school mathematics. We believe that from the earliest grades of elementary school, students can be acquiring the rudiments of algebra, particularly its representational aspects and the notion of variable and function. By emphasizing both the relationships among quantities and ways of representing these relationships, instruction can introduce students to the basic ideas of algebra as a generalization of arithmetic. They can come to value the roles of definitions and see how the laws of arithmetic can be expressed algebraically and be used to support their reasoning. We recommend that algebra be explicitly connected to number in grades pre-K-8:

• The basic ideas of algebra as generalized arithmetic should be anticipated by activities in the early elementary grades and learned by the end of middle school.

• Teachers and researchers should investigate the effectiveness of instructional strategies in grades pre-K-8 that would help students move from arithmetic to algebraic ways of thinking.

The National Academies of Sciences, Engineering, and Medicine
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