We address common criticisms of the Common Core State Standards--Mathematics, evaluating them based on comprehensive reviews of existing documentation and research to better ground future debates and to ameliorate negative effects of possible misconceptions or misinterpretations. The four main criticisms follow. (1) No one who helped develop the standards had any expertise in the education of young children. (2.) The CCSSM dictates scripted curricula and didactic instruction rigidly applied to all children at the same pace. (3.) The standards emphasize academic skills and leave no time for play, exploratory approaches, or social-emotional development. (4.) The standards are too early and therefore developmentally inappropriate for children in the early grades. We conclude that these criticisms are not valid, and that, given the importance of mathematics to academic success in all subjects, all children need and deserve to build a robust knowledge of mathematics in their earliest years and can do so if we use the research knowledge and research-based standards and programs presently available. We summarize and exemplify the research-based balanced approach to teaching based on learning trajectories that can provide guidance for engaging and developmentally appropriate mathematical experiences that have been demonstrated to help all children learn to high standards. (author abstract)