In some countries by the end of eighth grade, all students have been studying algebra for several years, although not ordinarily in a separate course. “Algebra for all” is a worthwhile and attainable goal for middle school students. In the United States, however, some efforts to promote algebra for all have involved simply offering a standard first-year algebra course (algebra through quadratics) to everyone. We believe such efforts are virtually guaranteed to result in many students failing to develop proficiency in algebra, in part because the transition to algebra is so abrupt. Instead, a different curriculum is needed for algebra in middle school:

• Teachers, researchers, and curriculum developers should explore ways to offer a middle school curriculum in which algebraic ideas are devel oped in a robust way and connected to the rest of mathematics.

Research has shown that instruction that makes productive use of computer and calculator technology has beneficial effects on understanding and learning algebraic representation. It is not clear, however, what role the newer symbol manipulation technologies might play in developing proficiency with the transformational aspects of algebra. We recommend the following:

• Research should be conducted on the effects on students’ learning of using the symbol-manipulating capacities of calculators and computers to study algebraic concepts and to transform algebraic expressions and equa tions.

An important part of our conception of mathematical proficiency involves the ability to formulate and solve problems coming from daily life or other domains, including mathematics itself. That ability is not being developed well in U.S. pre-K to grade 8 classrooms. Studies in almost every domain of mathematics have demonstrated that problem solving provides an important context in which students can learn about number and other mathematical topics.

Problem-solving ability is enhanced when students have opportunities to solve problems themselves and to see problems being solved. Further, problem solving can provide the site for learning new concepts and for prac-