Asking someone if they understand can be unreliable. One reason is that students may give the answer that they think the teacher wants to hear rather than say the through. Another reason is because students may not be able to self-evaluate on the spot the level of their understanding. Rather than ask if they understand, it might be more useful to ask students to complete another task that would allow them to demonstrate their level of understanding.
Of course asking "Do you understand what the homework assignment is?" is different than "Do you understand convergence?" The point here is that if you want know if students understand a mathematical concept, then it is best to ask them to demonstrate it to you rather than just ask if they get it.
- "Now that Joe solved problem N, let's see if we can take it another step. Here's a related problem... (write problem on the board)."
- "I would like to know if you understand this concept. Here is a question that I'd like you to work on and answer in the next few minutes..."
- "Can you come up with conjectures that extend the problem just presented? What else might be true (or false)?"
- "We have just started a new section, and there have been a couple of presentations on problems related to definition 25. Here's a question about this definition I'd like you all to answer..."
- "Janis just solved problem 35. Using a similar strategy to Janis, how would you solve this type of problem...?"