Mathematical proficiency is key in developing a strong mathematical community. Kilpatrick, Swafford and Findell (2001) define mathematical proficiency as having five intertwining strands:
- conceptual understanding—an understanding of concepts, operations and relations. This frequently results in students comprehending connections and similarities between interrelated facts.
- procedural fluency—flexibility, accuracy and efficiency in implementing appropriate procedures. Skill in proficiency includes the knowledge of when and how to use procedures. This includes efficiency and accuracy in basic computations.
- strategic competence—the ability to formulate, represent and solve mathematical problems. This is similar to problem solving. Strategic competence is mutually supportive with conceptual understanding and procedural fluency.
- adaptive reasoning—the capacity to think logically about concepts and conceptual relationships. Reasoning is needed to navigate through the various procedures, facts and concepts to arrive at solutions.
- productive disposition—positive perceptions about mathematics. This develops as students gain more mathematical understanding and become capable of learning and doing mathematics.
The development of mathematical proficiency takes time. In each grade, students need to make progress along every strand mentioned above. Each strand is important and interwoven with the others.
Adapted with permission from Jeremy Kilpatrick, Jane Swafford and Bradford Findell (eds.), Adding It Up: Helping Children Learn Mathematics (Washington, DC: National Academy Press, 2001), pp. 116, 118, 121, 124, 127, 129, 131.