Mastery of Mathematics - What does it look like?

Recently, I read Grant Wiggins' article "Getting Students to Mastery: How Good is Good Enough?"  The title alone for this article as made me think about several conversations with teachers over the past few weeks.  Often, in mathematics, we refer to mastery at the elementary school level as quick recall of basic facts.  But mathematics is so much more.....

While Common Core and earlier sets of standards have given educators an idea of "what" needs to be taught, the 2001 National Research Council report Adding It Up provides a clear description for mathematical proficiency.  In their definition, mathematical proficiency is comprised of five strands: conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive disposition.

As shown in the picture above, the five strands are intertwined.  No particular strand takes precedence over another strand - and the strands are interdependent.  No single strand can exist in isolation.  In their description of mathematical proficiency, Kilpatrick, Swafford, and Findell write, "How learners represent and connect pieces of knowledge is a key factor in whether they will understand it deeply and can use it in problem solving" (p. 117) and "Learning with understanding is more powerful than simply memorizing because the organization improves retention, promotes fluency, and facilitates learning related material" (p. 118).

This brings me back to the article by Grant Wiggins.  In the article, Wiggins proposes the following definition for mastery:
Mastery is effective transfer of learning in authentic and worthy performance.  Students have mastered a subject when they are fluent, even creative, in using their knowledge, skills, and understanding in key performance challenges and contexts at the heart of that subject, as measured against valid and high standards.

The Common Core State Standards for Mathematics provide the standards for us to use as the measurement.  Our job as educators is to provide a balance of learning opportunities that will engage students and require them to use all five strands of mathematical proficiency in their learning.  Today's learners need authentic learning tasks to master the mathematical content.  Imagine how many people might say "Math was my favorite subject in school!" if we had received different instruction when we were students!