While we often think responding to student understanding means adjusting instruction and changing methods to address unfinished teaching, we also adjust instruction to extend and deepen student understanding. That is, when we determine a representative sample of students demonstrate the correct mathematical reasoning and we have the opportunity to improve students’ reasoning skills and enrich content knowledge, we should take it. Below are two high-leverage ways to do so that advance students’ reasoning and sense making about important mathematical ideas and relationships.
| Question Type | Description | Examples |
| Encourage reflection and justification (NCTM 2014, p. 37) | Students reveal deeper understanding of their reasoning and actions, including making an argument for the validity of their work. |
How might you prove that December 22nd is the solution to this system of equations? How do you know that the difference between two odd numbers will always be even?
Why would this scale factor be constant for all these similar blueprints? |
| Summarize | After students have engaged in the mathematical thinking of a task, they should extract and summarize (i.e. generalize) the learning goal’s most important mathematical ideas. |
What patterns do we notice? What is a rule we can follow when we see a pattern like this? What would the nth stage be, then? What would happen with a different set of values? |
Additionally, consider that extending and deepening student understanding requires teachers not just to ask questions yet also demonstrate actions that promote student engagement in mathematical reasoning. As you continue to ask the questions in response to correct mathematical reasoning, use the following table from the National Council of Teachers of Mathematics (2014, p. 41) to help respond to these opportunities.
