When do you and when don't you give the answer to your learners?

Brooke, you share a very common question of when to provide an answer and when to continue to try to string the learner to discovering the answer. I find that I very often listen to a student request for an answer and start thinking of questions or models I can offer that will help the student remember any concepts or procedures he or she might feel are relevant.

When offering questions,I adjust my questions to hit at general steps that might need to be accomplished. These steps are often in English language (Algorithms) rather than math language. Once we are on the right track, I bring back in the math lingo as the student starts finding success. A simple example might be that a student shares, "We need to get those bottom numbers the same somehow." I may respond, "Why would we want those bottom numbers the same? What does that even mean?". If the student's response indicates that he or she understand what the denominator really means, then I might restate their statement with the math language, "OK, so we want to get both fractions to have the same denominators?". If the student was not able to conceptually share what those bottom numbers are or why we might want them to be the same, we have identified some critical information that we need to solidify and we back up accordingly. Effectively, my questions act as a sort of formative assessment sequence to better help me understand what concepts or procedures are poorly understood so I have a foundation on which to offer questions that build up from that foundation. 

In some cases, especially with very visual learners, I ask questions about what things may look like. Often learners may be so focused on the abstract procedures that the student fails to have any visual of what is going. This is a case where I may offer information that helps the learner view the problem from another angle. For example: When trying to figure out how to solve a square root problem, especially when the a variable is in the square root symbol, I may first draw a square with sides measuring 5 and ask about the area. Assuming they know about area I may ask them what the square root of 25 is and how it is related to what they just did. Very often, this visual relationship between squaring a number and finding the square root as it relates to areas of squares establishes some conceptual understanding that was missing. I can then bring the student back to the troublesome problem and ask if they can sketch that problem out similar to our area situation. The student often identifies how the given information fits the simple model. That usually brings them to some successes with that type of problem. 

When questioning does not work and demonstrating similar models does not work, the student may remain frustrated with no understanding still. At this point, I may ask if the student would like to see a demonstration. I fabricate a similar example and do the entire thing out myself being sure to talk through what I am thinking and doing and why. Then I come up with another and ask the student to lead me through it with me offering corrections if needed. Then with some success we jump back into the original problem usually with little difficulty. Note, I still did not just offer an answer. I demonstrated, then supported the students efforts until success with similar problems helped the student answer their question. 

I listed the above strategies in a specific order. Question first, Similar Model next, then Demonstration.

The first strategy offers the least direct help typically and often pushes the learner to struggle to make some connections that currently may not be well established. This will help create new synapses in the learner's brain as the student works through the productive struggle. These new connections help with longer retention and more flexibility in thinking over time.

The demonstration strategy is often reported to be the easiest process from the learners. It often gets the student successfully processing the quickest and frustration is typically fairly low. I tend to save this as a last resort because it does not appear that the "assistance" I offer helps things stick in the middle to long term. That day or even the next day things may be great, but just one or two weeks out a similar example will confound the student who will now be more frustrated because, "This was so easy just the other week! Why is this so hard again?"

The modeling strategy seems to be as effective as the questioning when dealing with students that are strong visual or spacial learners. This can also be effective for hands-on learners if manipulatives or actual models can quickly be fabricated with paper or goods easily in reach. The model helps learners conceptualize what the abstract formulas might look like and what is happening in the math. 

I try to not give an answer unless the student has reached the point of anxiety where positive functioning seems to be in danger. There have been a few students that consistently wished me to be an answer machine so they could "do the work" for some remedial college course. Sadly, when you can't divert the learner from that need of "Just give me the answer!", the student often does not experience much success until the shift from completing work to understanding what is going on has started. 

How about others? When or how do you answer questions, especially when you notice the student is frustrated or anxious?