- Identify the standards you are targeting. This is pretty much the starting point if you are designing any kind of math or science curriculum for a public school district today.
- Identify the tasks that a student should be able to do when he or she meets the standard. These are your performance assessments.
- Break down the tasks into sub-tasks. Each sub-task should either be intuitive or build on a sub-task that students have already mastered.
- Evaluate the list of sub-tasks to determine how many students should be able to perform the task. You should expect at least 95% of the students to be able to do these sub-tasks. If that is not the case, then return to step 3 and break down the task some more.
- Design learning experiences in the curriculum so that each sub-task can be mastered. Provide formative assessments and interventions.
- Evaluate the learning experiences in the curriculum to determine how many students should be able to perform the sub-tasks once they have gone through the curriculum. You should expect at least 95% of the students to be able to do the sub-tasks. If that is not the case, then return to step 5 and improve the learning experiences.
- Implement the curriculum.
- Compare actual student performance to your expectations, including retention and transfer. If student performance does not match expectations, then re-evaluate your assumptions and return to step 2.
In this post, we are going to talk a little bit more about step 2. When designing tasks and performance assessments, you want to include everything that a student should be able to do when they have met a standard. This means pushing tasks as far as you can go.
If a student has learned the rules for order of operations (e.g., PEMDAS) and can apply them, then he or she should be able to evaluate expressions like this one:
5 + 4(27 − 3(9 − 7)^3)^2×(2(8 − 120÷(4 + 10(3^3 − 5^2))) − 1) − (3 + 4)^2
During an open house one year, a parent walked up to me and commented that he had never learned the rules for order operations needed to evaluate expressions like these and that he was super impressed that his daughter was learning them in 6th-grade. This made me smile because there are no additional rules needed to evaluate expressions like these; if you know PEMDAS, then you can evaluate them. In fact, I'd go so far as to say that if you can't evaluate expressions like these, then you haven't met the standard.
According to Richard Elmore, a researcher at the Harvard Graduate School of Education, student learning is bounded by the tasks we ask students to do. If we ask them to evaluate expressions with three operations or less, then that is what students are going to learn how to do, and no more. They won't learn to do more until we ask them to do more.
For expressions with three operations or less, the set of possible permutations is very small. This means that most students use pattern recognition to evaluate these expressions instead of PEMDAS. If they see a single parenthesis in an expression, they know to evaluate it first. If they see an addition operation next to a multiplication operation, they know to evaluate the multiplication first. These students may know PEMDAS well enough to recite it back to you and even explain how it works, but they aren't using it to evaluate these simple expressions. They don't have to.
But give them an expression with over a dozen operations in it and their pattern recognition suddenly breaks down. Now they need to read the entire expression and run through PEMDAS one step at a time. And then they need to do it all over again after evaluating each operation. Over time, they will identify more sophisticated patterns to increase their speed and efficiency, bypassing PEMDAS once again (e.g., going straight for the "inside" parentheses first), but they will have mastered the rules for order of operations by then and they will know what to do when they encounter something they haven't seen before. Over 95% of my 6th-graders are able to evaluate expressions like these less than two lessons after being introduced to PEMDAS for the first time.
6th-grade textbooks avoid asking students to evaluate expressions with more than three operations because the textbook authors don't expect students to really meet this standard. They view this approach as a bridge to eventual mastery. Maybe they mistake pattern recognition for application. The writers of standardized tests make the same mistake and don't assess students against the actual standard. Don't do that. Come up with the most challenging tasks that you can that fit within the standard(s). Just make sure that you are doing your task analysis. When you have nested parentheses, one of the sub-tasks that students will need to be able to perform is to match up opening and closing parentheses. Push your tasks and do your task analyses, and you'll be surprised what students are able to do.