Moving beyond procedures
To help clarify this idea of moving students beyond only knowing how to perform procedures and towards deeper conceptual understanding, let me give you an example.
Example
If we look at the addition problem, 23 + 18, the correct answer is 41. If we teach students the standard algorithm or traditional algorithm for doing this problem it looks like this:
Procedural Fluency
To determine if the student understands the procedure, we would look at their answer and we might look at how they showed their work. For example, did they write a small 1 above the 2. If the student was able to carry out that procedure for multiple problems and arrive at the correct answers, we would say they are developing procedural fluency.
Conceptual Understanding
If we were trying to develop conceptual understanding of that same math concept, we would want students to understand why they wrote the 1 above the 2. To develop this understanding we would probably teach the concept differently to begin with. For example, we might use manipulatives like base ten blocks to visually and physically show them what regrouping means. As we assessed their understanding we would engage them in Math Talk asking them to explain why they carried the 1. We would still be looking for the correct answer, but in addition we would be looking for students to be able explain that carrying the 1 really means regrouping ten ones into a new group of ten. The exact wording from students might vary, but the goal would be to have students understand why they were doing what they were doing rather than just knowing which step to execute.
