Having taught in several schools I have worked under several different definitions of rigor. Rigor was defined as “higher up Bloom’s taxonomy”, “meeting the standards”, “higher-level thinking”, “going faster”, “giving harder problems”, and “making math real life”. Not only were these vague and frustrating, they really didn’t address all of rigor.
The CCSS definition of rigor is “Pursue conceptual understanding, procedural skills and fluency, and application with equal intensity.” (Key Shifts in Mathematics, http://www.corestandards.org/other-resources/key-shifts-in-mathematics/, Accessed Sept. 30, 2019). This is definitely a more complete definition of rigor. Not only does it cover all of Bloom’s, it includes all the other definitions except “go faster”.
I have seen many students just plug numbers into a calculator and decide if the number might work for the answer. Then the student entered the numbers in a different way and decided if the new number might work for the answer. Eventually, the student would come up with the correct answer but couldn’t explain how they got that answer. The student solved a hard problem and justified the answer without having to think about meaning or process. So “harder problems” and “going up Bloom’s taxonomy” does not meet rigor.
I was one of those students that could calculate effortlessly. It was fun and easy to plug numbers into formulas. I loved learning new formulas and crunching more numbers. I calculated fluently but had very little idea what the formulas meant or where they came from. Fluency is great but lack of application became very frustrating for me as a teacher because students would ask, “When will we use this?” I could give broad examples but nothing specific enough to show the power of math. All of those “higher level” formulas became meaningless. And “going faster” isn’t a good option by itself because it just means more algorithms and formulas with even less understanding.
Without a combination of all the pieces that have been used to describe it, we do not reach rigor. Rigor has three parts: conceptual understanding, procedural skills and fluency, and application.
With conceptual understanding, the student gains a collection of tools and concepts to pull out as needed. Conceptual understanding means being able to show or explain the math in front of you. Part of it is definitions, part of it is formulas, and part is modeling and describing.
The student needs to not only know what to do but also how to do it. Fluency is more than just memorizing facts. It means being able to do the process without having to think about all the parts. A student that struggles to remember 6×7=42 will struggle to solve 432/6=72 because of the energy it takes to go through possible facts. But a student that knows the facts yet doesn’t remember the steps for long-division will not succeed either. It takes practice to gain skill and competence. Thankfully, practice on one skill can increase the fluency of another.
When a student has conceptual understanding and procedural skills and fluency, the application makes more sense. Because the student knows the concepts the students can “see the math”. The problem has pieces that the student is familiar with. The student can now pull out a known formula or mathematical model (expression, equation, graph, chart, diagram, table…) to start to solve the problem.
Rigor does not have to be a straight line – concept, fluency, then application. I love it when I can give the students examples of an application or procedure and they discover the concept themselves. It is inspiring when a student discovers “multiply by the reciprocal” for dividing by a fraction by looking for patterns in several completed division by fraction equations. When they can say “in 2/3÷1/5=10/3 they just multiplied the first numerator by the second denominator” I can tell that the student is on the way to develop the algorithm. (Of course they usually just say multiply the top by the bottom but they do get better at “talking math” with practice.) But the student must also be able to explain why the algorithm works. The student can show understanding with a drawing or using manipulatives or explaining that “we need to know how many 5ths there are so we have to create 5ths by multiplying by the denominator.”
Teaching with rigor is complex. You need to juggle time and lessons to allow students to develop conceptual understanding, gain skill and fluency, and apply what they understand and can do. There isn’t a one-size-fits-all version of teaching with rigor. Teachers know their students and content and should be given the freedom to teach with rigor.