who are low achieving or underserved.71 The U.S. eighth-grade curriculum is not as advanced as those of other countries. In the Third International Mathematics and Science (TIMSS) Video Study, for example, whereas 40% of U.S. eighth-grade lessons included topics from arithmetic, German and Japanese eighth-grade lessons were more likely to cover algebra and geometry.72 Over the past decade, however, more and more U.S. schools have started to offer first-year algebra in the eighth grade. According to data collected by NAEP, 25% of eighth-grade students were enrolled in algebra in 1996 compared with 16% in 1990.73 Further, all but 3% of the twelfth-grade students reported that they had taken first-year algebra, the majority in grade 9. Although the goal of “algebra for all” has essentially been achieved by the time students reach the end of high school, many of these students experience difficulties in their first course in algebra.
The study of algebra need not begin with a formal course in the subject. Recent research and development efforts have been encouraging. By focusing on ways to use the elementary and middle school curriculum 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.74 From the earliest grades of elementary school, students can be acquiring the rudiments of algebra, particularly its representational aspects. They can observe that over time and across different circumstances, numerical quantities may vary in principled ways—the essence of the concept of variable. They can learn about functions by studying how a change in one variable is reflected in the behavior of another. As students encounter algebraic ideas, they discover the value of precise language and of working with clear definitions.
Once students are familiar with the laws of arithmetic, they can learn to see them as a convenient summary of arithmetic practice and as a valuable guide to methods that work. Students can learn to express the laws algebraically and can use them to support their reasoning and to justify their claims about numbers. It is important that they become aware of the role played by general statements expressed in algebraic symbols when justifying numerical arguments or discussing classes of situations. Little is known, however, about the relative effectiveness of strategies for helping students learn to justify their claims. With the development of new approaches to algebra and the infusion of the rudiments of algebra in the elementary and middle grades, an algebra-proficient population might become a reality.