Math and the Common Core: Layers of Math Knowledge

Questioning That Develops Layers of Math Knowledge
As a math educator, how often do we hear students stating, “I know how to do this”? With the advent of the Common Core, the duality of mathematics knowledge is highlighted and provides an avenue to explore this claim. The Common Core State Standards not only delineate the content of the discipline of mathematics, but they also describe the content of a student’s mathematical character through the mathematical practices. Both parts are integral to substantiating a student’s claim to know and helping students reach the expectation of a deep understanding of the content.

Change Just One Word
Questioning that targets the math practices can be used to examine and extend a student’s claim of knowing a particular concept. This examination can occur by changing one word in the statement I know how to do this. Changing one word assesses multiple knowledge domains (i.e. conceptual, procedural, metacognition) and targets the development of the standards for mathematical practice.  If we focus on the function words of how, why, where, and when, an interesting cognitive shift occurs that unveils a student’s comprehensive understanding of a concept.

HOW: Do you know how to do this?
Students that know how to do a particular concept will be able to represent that concept in multiple ways. They will be able to demonstrate concrete verbal and pictorial representations, numeric representations, and abstract symbolic representations. This demonstrates a student’s application of mathematical practice (MP) 2.  Students will be able to reason abstractly and quantitatively.

Students that know how to do a particular concept will know what tools are needed to negotiate the task or scenario.  Math practice 5 notes that students will be able to strategically choose tools that will aid in problem-solving.

WHY: Do you know why to do this?
When students begin to understand the value of a math concept, the likelihood that they will continue to use and retain understanding of that concept beyond the classroom increases. Students that know why are able to provide a rationale for the selection of an algorithm or law used to negotiate a problem and the importance of the concept. An explanation of why triggers the development of multiple math practices including MP3 and MP1. Students will be able to make meaning of a problem and construct a viable argument as a result.

WHERE: Do you know where to do this?
Mathematics is a discipline that describes patterns, relationships, and changes in everyday life.  Math practice 4, modeling, captures the essence of why we study mathematics – to be able to interpret the world we live in. Students that know how to apply a math concept should be able to explain the context in life that calls for where that particular application is needed.

WHEN: Do you know when to do this?
Students that know will be able to explain when a concept relates to other concepts and when that concept may be called upon in a logical progression of thought.  One of the instructional shifts introduced by the Common Core is that of coherency.  Knowing when a concept is needed and how it relates to other concepts being employed demonstrates a coherent understanding of the discipline. The ability to look for and make use of structure, generalize, and to know when to use these generalizations in new situations (Math Practices 7 and 8) demonstrates a comprehensive understanding of the math concept.

Changing the Paradigm
As we continue to change the paradigm in mathematics education of what it means to be a student of mathematics, we will continue to hear the response of “I know how to do this.”  Redefining this idea of knowledge calls upon us as educators to ask the questions of not only Do you know how to do this?, but also Do you know why to do this?, Do you know where to do this?, and Do you know when to do this?  The precise (Math Practice 6) answers to these questions using the language of the discipline will help assess a student’s claim of knowing and provide support, resulting in students developing as mathematicians.

Next topic: Making the math practices explicit through prompting

Your Turn!
What do you think may be the biggest obstacle in shifting students from an “I know how to do this” perspective?