Our analyses of the mathematics to be learned, our reading of the research in cognitive psychology and mathematics education, our experience as learners and teachers of mathematics, and our professional judgment have led us to adopt a composite view of successful mathematics learning. Recognizing that no term completely captures all aspects of expertise, competence, knowledge, and facility in mathematics, we have chosen mathematical proficiency to express what we think it means for anyone to learn mathematics successfully.
Mathematical proficiency has five strands:3
Understanding: Comprehending mathematical concepts, operations, and relations—knowing what mathematical symbols, diagrams, and procedures mean.
Computing: Carrying out mathematical procedures, such as adding, subtracting, multiplying, and dividing numbers flexibly, accurately, efficiently, and appropriately.
Applying: Being able to formulate problems mathematically and to devise strategies for solving them using concepts and procedures appropriately.
Reasoning: Using logic to explain and justify a solution to a problem or to extend from something known to something not yet known.
Engaging: Seeing mathematics as sensible, useful, and doable—if you work at it—and being willing to do the work.
The most important feature of mathematical proficiency is that these five strands are interwoven and interdependent. Other views of mathematics learning