Valuing Intuition

In any classroom and any learning situation it is important to have an idea of what students are bringing to the learning.  Having a sense for what they already solidly grasp, what they find difficult, and what their misconceptions are all help you to better target your teaching.  
In looking at what students bring to the learning it is often the diagnostic testing or a pre-test on the particular content that may form the basis of knowing what they bring. However I think that this only looks at a very narrow band of understanding, what formal mathematical tools they have or remember.  This is approaching it from a basis of procedural fluency rather than conceptual understanding.  In looking at what they bring to the learning it is very important to me to know if they have a feel for the concept, do they have a sense of what it is before I start to apply too much formal mathematics to it.  Can they intuitively determine a way to do the problem without knowing the formal mathematics and if they can, can I use this intuitive understanding as a base for building further understanding.

An activity that we did in class other day really brought that into clear focus for me as it does most years.  The problem and the specifications for the task are shown below.

Obviously with this task the food helps, there is some competition here to help maximise the number.  It is  clear that this becomes a problem on volume, but this was not something students were told, a few came to that conclusion as they completed the task and it was good for me to know that some had a sense of volume, but it was not necessary to complete the task.  

The intention behind the task was to see if they could connect their idea of area to that of volume of shapes with uniform cross sections.  How that intention was set up was by limiting the number of cheese balls they were given.  By limiting the materials you force them into thinking about the problem in more depth they need to think flexibly about how they can use the resources they have to solve the problem they had been given.  The most common approach to solving this problem was the one shown below

Using this method students created a layer of cheese balls on the bottom and multiplied it by how many of those layers could fit in the container, they determined this by measuring the side in cheese balls.  In this case the student would have found that they could fill it with 80 balls (10x8).  What is interesting about this approach is that it is the exact method they would be taught to calculate the volume of these shapes, multiplying the area of the end by the length of the shape and they had got to the point where they had generalised the solution.  Many had started with rectangular prisms and determined the area of the base by multiplying the length in cheese balls by the width in cheese balls but then ran into difficulty when looking at a cylinder, they had to come up with a way to approach that and realised that area was the link that generalised for any shape. 

 It was a situation where they did not need to be taught it, they had a sense for it already, my role was then to formalise the thinking at the end of the task or