This is put into play first with a quiz. For most instances, I'll call a quiz a formative assessment mid-way through the unit to check for understanding before going on to more complex applications. My hang-up with this is that:
- Quizzes are generally lower level rungs on the ladder of understanding. My quizzes have questions that are accessible and easily understood because I don't want to confuse students early on. For example, a quiz question on area and perimeter might be something like: "Here is a rectangle with a side of 7 meters and a side of 8 meters. What is the area and perimeter?" This feels to me like a low level task of recall and not so much application. Because a student can demonstrate this does not mean that they could apply this to a higher order question like "Joe wants to make a garden with an area of 20 square meters but use the least amount of fencing to build around his garden (perimeter). Using whole numbers, what are the dimensions of the garden that would have an area of 20 square meters but also the least amount of fencing to encircle around the garden? Label the dimensions of the garden and show any mathematical notation to explain your reasoning" Using mathematics in reality happens more like the second question and such understanding is much richer and deeper.
- I have had countless math students that are whizzes with plugging numbers into formulas but fail to apply what they've learned on projects or summative assessments. Some of these students fail to find the underlying relationship between these variables that we want them to discover. For instance, in the problem above, they might learn that with the changing dimensions of a rectangle, a rectangle can have the same area but changing perimeter. For students who are merely number crunchers, they can show a surface level competency, but I can't get my head around on how they would apply such skills to real world situations until they were asked to do it again.