By Juli Dixon, posted September 28, 2015 –

Teaching mathematics based on rigorous, focused, and coherent standards requires teachers to know mathematics in ways that are likely different from how they were taught. Such teaching requires an understanding of the mathematics taught but also the mathematics that comes before and after that content so that appropriate connections can be established. Teaching for depth often necessitates a shift in how we teach, what we teach, and what we look for in student work.

This is less likely. This error is the type that teachers with a deep understanding of mathematics for teaching might anticipate. When mathematics is taught for depth of understanding, so that concepts are addressed before procedures are introduced, even student errors will be different. Teaching for depth does not necessarily eradicate errors. However, it does change them.

How might this problem be addressed when teaching for understanding? It would likely be introduced either in context or with an expectation that students develop a context to bring meaning to the problem. Think of a context before reading further. How does your word problem compare to the one provided here?

Once this error is identified, it is easily resolved. However, unless teachers have a deep understanding of mathematics for teaching, errors such as this will go unnoticed as common errors. What are other “new common errors” teachers need to anticipate? Discussion of common errors should be part of the process of planning for instruction so that the errors—both new and old—can be identified and addressed during instruction. In this way, errors become springboards for learning rather than long-term issues in building understanding.

Anticipating common errors has always been an important part of planning, and teaching, mathematics. Now, knowing mathematics for teaching is more important than ever, including how misconceptions may appear as students persevere to make sense of mathematics. How are we as teachers and teacher leaders addressing the need to know elementary school mathematics with depth? What are we doing in teacher preparation, in school-based collaborative teams, and in professional development to address the need for deeper content knowledge for teaching elementary school mathematics? In what ways are we planning for a new set of common errors so that they can be addressed during instruction and through a formative assessment process? Until all teachers have this deep understanding of the mathematics they teach, all students will not have access to mathematics based on rigorous, focused, coherent standards.

Juli K. Dixon, Juli.dixon@ucf.edu, a professor of mathematics education at the University of Central Florida in Orlando, is interested in mathematics content knowledge for teaching as well as communicating and justifying mathematical ideas.