Responding to Student Understanding Guide: Secondary Math

We adjust instruction when we encounter misconceptions worthy of adjustment. As the term suggests, misconceptions are conceptual misunderstandings. When students do not fully understand a concept they continue to construct a framework for a concept that is inaccurate; it is through that incorrect framework that they solve tasks and, inevitably, incorrectly. Students also do so consciously, i.e. believe their incorrect framework is correct. These frameworks have a variety of origins: unfinished teaching; the entanglement of prior conceptual misunderstandings; inaccurate teaching.

When diagnosing misconceptions, we search for underlying patterns of incorrect mathematical reasoning to signal the conceptual framework students consciously rely on.  Research points to three ways we can anticipate patterns of incorrect reasoning to occur.  After asking students questions in response to students’ work and/or thinking, we may notice misconceptions are the result of one or more of the following:

When we find evidence of a misconception and if it sources from vocabulary, computation and/or false beliefs, we must decide if it actually is worthy of instructional adjustment at that moment and for the entire class (as opposed to one or few students).  We should adjust instruction if the misconception: 

• Is closely aligned with the mathematical goals of the lesson.
• Provides powerful insight into students' conceptual understanding, fluency with procedures, or competence in selecting strategies for problem solving. 
• Aligns well with the class's general progress toward solving the problem. 
• Offers a viable answer that may be contrary to the class’s accepted solution or solution strategies because of hidden assumptions about the problem (e.g., the student interpreted the problem differently than intended).