Proficiency is much more likely to develop when a mathematics classroom is a community of learners rather than a collection of isolated individuals. In such a classroom, students are encouraged to generate and share solution methods, mistakes are valued as opportunities for everyone to learn, and correctness is determined by the logic and structure of the problem, rather than by the teacher. Questioning and discussion that elicit students’ thinking and solution strategies and build on those strategies lead to greater clarity and precision. A significant amount of class time is spent developing mathematical ideas, not just practicing skills.13
In addition to working as members of the classroom community, students also need to become independent learners, both inside and outside the classroom. Students should have opportunities to work independently of the teacher, individually and in pairs or groups. When homework is assigned for the purpose of developing skills, students should be sufficiently familiar with the skills so that they do not practice incorrect procedures. Only by becoming independent learners can students come to see mathematics as doable and useful.
One of the strongest findings from research is that time and opportunity to learn are essential for successful learning. For all students to develop mathematical proficiency, schools should devote a substantial and regular amount of time to mathematics instruction. As an overall guideline, an hour each school day should be devoted to mathematics instruction from kindergarten through eighth grade. This time should be apportioned so that all the strands of mathematical proficiency receive adequate attention.