Teaching students to add appears to be a very linear, skill driven endeavor. Hidden in this is the concept of what it means to add and how to assess this conceptual understanding. Here is an approach to address and assess the concept of adding.
In the photo above a student is prompted to pull both groups into 1 pile (see photo below). The word, add, is not addressed. The symbol is absolutely not introduced yet.
Once the student has demonstrated a consistent performance of pulling the groups into 1 pile (addition) two other tasks are introduced, taking away and sorting. The student is presented each of these individually (field of 1).
After showing consistent performance in demonstration of these skills, the skills are then presented using a generalized mat (see below).
Then two skills as pairs. First “pulling together” and “taking away” are randomly prompted individually, e.g. “pull into a pile” using the generalized mat above. Then combine “pull together” and “sort” then “sort” and “take away.” Finally all 3 are randomly chosen (field of 3).