Content Knowledge for Teaching Math

Highly effective math teachers have specialized content knowledge beyond the ability to carry out procedures.
The chart below gives examples of this specialized content knowledge in the area of Operational Sense.

Teachers with deep content knowledge in mathematics can articulate, for a given content area: Examples of this Content Knowledge in the area of Operational Sense
A developmental sequence/trajectory of concepts and skills

 A partial sequence of early multiplication skills can be seen by clicking the image below.  Effective math teachers use their knowledge to sequence skills, but also help them understand why a student may be struggling.

Learning goals and success criteria
(in “student-friendly’ language)
 Common misconceptions, the thinking they reveal, and student and teacher next steps

 A  student uses a tree diagram to partition both numbers when subtracting.  What are the next steps?  Is this a suitable tool and strategy?

 A range and variety of possible student solutions to a given problem and the connections between them
 Conceptual understanding of the relevant curriculum (research base, big ideas, etc.)

 The operations of addition and subtraction can occur in 12 situations, but we traditionally give students questions in just a couple of them.  When we broaden our scope,  students will understand the operations more deeply and apply their knowledge successfully in more situations.

 Tools and representations that will advance student thinking  Common tools to support multi-digit addition and subtraction are base 10 materials, hundreds charts, tree diagrams and open number lines.  Each of these tools have pros and cons and are used to illustrate student thinking for different strategies.
Connections between grades, topics, and strands  Addition and subtraction occurs naturally across the strands.  The comparison meaning of addition and subtraction have a particular connection to data management.  The Part-Whole meaning follows from Part-Part-Whole relationships and conceptual subitizing in Kindergarten and grade 1, and should influence the materials and approaches used to introduce these operations.