Reform mathematics is an approach to mathematics education, particularly in North America. It is based on principles explained in 1989 by the National Council of Teachers of Mathematics (NCTM). The NCTM document, Curriculum and Evaluation Standards for School Mathematics, attempted to set forth a vision for K-12 (ages 5-18) mathematics education in the United States and Canada. Their recommendations were adopted by many education agencies, from local to federal levels through the 1990s. In 2000, NCTM revised its standards with the publication of Principles and Standards for School Mathematics (PSSM). Like the first publication, these updated standards have continued to serve as the basis for many states' mathematics standards, and for many federally funded textbook projects. The first standards gave a strong call for a de-emphasis on manual arithmetic in favor of students' discovering their own knowledge and conceptual thinking. The PSSM has taken a more balanced view, but still emphasizes conceptual thinking and problem solving.
Mathematics instruction in this style has been called standards-based mathematics or reform mathematics.
Main article: Principles and Standards for School Mathematics
The momentum for reform in mathematics education began in the early 1980s, as educators reacted to the "new math" of the 1960s and 1970s. The work of Piaget and other developmental psychologists was shifting the focus of mathematics educators from mathematics content to how children best learn mathematics.  The National Council of Teachers of Mathematics summarized the state of current research with the publication of Curriculum and Evaluation Standards in 1989 and Principles and Standards for School Mathematics in 2000, bringing definition to the reform movement in North America.
Reform mathematics curricula challenge students to make sense of new mathematical ideas through explorations and projects, often in real contexts.  Reform texts emphasize written and verbal communication, working in cooperative groups, making connections between concepts, and connections between representations. By contrast, "traditional" textbooks emphasize procedural mathematics and provide step-by-step examples with skill exercises.
Traditional mathematics focuses on teaching algorithms that will lead to the correct answer. Because of this focus on application of algorithms, the traditional math student must always use the specific method that is being taught. This kind of algorithmic dependence is de-emphasized in reform mathematics. Reformers do not oppose correct answers, but prefer to focus students' attention on the process leading to the answer, rather than the answer itself. The presence of occasional errors is deemed less important than the overall thought process. Research has shown that children make fewer mistakes with calculations and remember algorithms longer when they understand the concepts underlying the methods they use. In general, children in reform classes perform at least as well as children in traditional classes on tests of calculation skill, and considerably better on tests of problem solving.
Main article: Math wars
"Principles and Standards for School Mathematics" was championed by educators, administrators and some mathematicians as raising standards for all students, but it was criticized by some for valuing understanding processes more than learning standard procedures. Parents, educators and some mathematicians opposing reform mathematics complained about students becoming confused and frustrated, claiming that it was an inefficient style of instruction characterized by frequent false starts. Proponents of reform mathematics countered that research showed that, when done correctly, students in reform math curricula learned basic math skills at least as well as those in traditional programs, and additionally understood the underlying concepts much better. Communities that adopted reform curricula generally saw increased math scores by their students.  However, one study has found that first-grade students with a below-average aptitude in math responded best to teacher-directed instruction.
During the 1990s, the development and large-scale adoption of curricula such as Mathland was criticized for partially or entirely abandoning teaching of standard arithmetic methods such as regrouping or common denominators. Protests from groups such as Mathematically Correct led to many districts and states abandoning such textbooks. Some states such as California revised their mathematics standards to partially or largely repudiate the basic beliefs of reform mathematics, and re-emphasize mastery of standard mathematics facts and methods.
The American Institutes for Research (AIR) reported in 2005 that the NCTM proposals "risk exposing students to unrealistically advanced mathematics content in the early grades." This is in reference to NCTM's recommendation that algebraic concepts, such as understanding patterns and properties like commutativity (2+3=3+2), should be taught as early as first grade.
Some, such as the 2008 National Mathematics Advisory Panel, called for a balance between reform and traditional mathematics teaching styles rather than a "war" between the two styles. In 2006 NCTM published its Curriculum Focal Points, which made clear that standard algorithms were to be included in all elementary school curricula, as well as activities aiming at conceptual understanding.
A common misconception was that reform educators did not want children to learn the standard methods of arithmetic. As the NCTM Focal Points made clear, such methods were still the ultimate goal, but reformers believed that conceptual understanding should come first. Reform educators believed that such understanding is best pursued by allowing children at first to solve problems using their own understanding and methods. Under guidance from the teacher, students eventually arrive at an understanding of standard methods. Even the controversial NCTM Standards of 1989 did not call for abandoning standard algorithms, but instead recommended a decreased emphasis on complex paper-and-pencil computation drills and greater attention to mental computation, estimation skills, thinking strategies for mastering basic facts and conceptual understanding of arithmetic operations.
During the peak of the controversy in the 1990s, unfavorable terminology for reform mathematics appeared in press and web articles, including Where's the math?, anti-math, math for dummies, rainforest algebra, math for women and minorities, and new new math. Most of these critical terms refer to the 1989 standards rather than the PSSM.
Beginning in 2011, most states adopted the Common Core Standards, which attempted to incorporate reform ideas, rigor (introducing ideas at a younger age) and a leaner math curriculum.