For instance, imagine one of your students arranging a handful of 24 pennies into groups with the same number of pennies in each group. Conceptual understanding does not come from us telling a student what a concept is. More likely a student can be led to memorize this wonderful explanation, but not really understand what it means or how to use it. Most humans need to explore messy examples for a while before any explanation makes any sense. One way to think about this process is that people create their own representations of what they are dealing with.
Some representations are very concrete, like using objects to help see what is going on.
In the example of the 24 pennies, some students would need to actually have the pennies and move them around on the table to see the relationships among the groups. Others could imagine having the pennies and draw pictures of the possible groups. The picture or drawing would be a different and somewhat more abstract representation. Similarly, others could write a list of equations that represented the possibilities. The NCTM Standards encourage teachers to help students create their own representations to build conceptual understanding.