The reason why setting expectations is important is because it closes a few feedback loops. First, it lets you know if you have taken your task analysis to the appropriate level. If you review the sub-tasks that students need to complete a task and you feel that many students won't be able to master those sub-tasks, then you know that you need to drill down further or redesign the curriculum to bridge some of those gaps. Then, if you implement the curriculum and student performance doesn't match your expectations, you need to go back to re-examine your assumptions or take a closer look at how the curriculum is being delivered.
If you look at how we teach middle school students to use prime factorization to find greatest common factors and least common multiples, it is fairly obvious that neither the curriculum developers who wrote the curriculum nor the teachers who are delivering the curriculum expect students to do very well on these tasks. Students are memorizing sets of procedures that make no sense to them. No one is surprised when only half of the students perform adequately on these tasks for a chapter test or that less than 10% of the students can even recall the procedures a month later. Instead, we choose not to think about it until after it happens.
The same thing happens when we teach area. Typically, 10-20% of students entering 6th-grade are unable to find the area of a rectangle without some review. We consider that normal and wave it away with the explanation that students forget things over the summer or that, especially in the case of students with a learning disability, some students aren't yet developmentally ready to master a skill or concept. These judgments are based on our experiences with thousands of students over decades, but historical data is not what our expectations should be based on. If we expect past results and consider them normal, then how will we ever make the leap to better results?
If our area curriculum is aligned with the Common Core standards, we begin laying the foundation for finding the area of rectangles in 2nd-grade:
Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.
In 3rd-grade, students are formalizing their understanding:
A plane figure which can be covered without gaps or overlaps by n unit squares is said to have an area of n square units.
Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units).
Find the area of a rectangle with whole-number side lengths by tiling it, and show that the area is the same as would be found by multiplying the side lengths.
And by 4th-grade, the focus is on asking students to apply what they already know:
Apply the area and perimeter formulas for rectangles in real world and mathematical problems.
So when asked to find the area of a rectangle at the start of 6th-grade, a student should be able to complete the task using these sub-tasks:
- Recognize that the area of the rectangle can be found by tiling the rectangle with unit squares
- Recognize that the unit squares are arranged in an array and can be counted using multiplication
- Recall the area formula for a rectangle
The key step in this process is recognizing that the area of the rectangle can be found by tiling the rectangle with unit squares. This recognition is grounded in the concept of area. If the student makes that connection, it should trigger a mental image of a rectangle tiled by unit squares, which should lead into step 2. By now, students should be very familiar with arrays and area models for multiplication, so this should trigger the area formula for a rectangle. Notice that finding the area of the rectangle does not require the student to start by recalling the area formula for a rectangle.
If a student is unable to perform this task, we should break down the task and ask ourselves if the student is capable of performing these sub-tasks. If the student is capable of performing the sub-tasks, then he or she is capable of performing the task, and it is our job to find out why that isn't happening and what we can do to address it. Forgetting how to find the area of a rectangle over the summer isn't normal and it isn't the same as forgetting the area formula for a rectangle. It is feedback that we should be responding to.