Rigor vs. Cognitive Demand

Between work on creating my school’s instructional vision and attendance at the Standards Institute this week I have been doing a lot of thinking around what makes good instruction at different grain size—lesson, unit, curriculum, sequencing, and overall educational experience. Two concepts, which I would once have used interchangeably, have come up frequently.

Cognitive Demand:

Stein, Smith, Henningsen, and Silver define cognitive demand as “the kind and level of thinking required of students in order to successfully engage with and solve the task.” I have analyzed tasks with their four levels of cognitive demand in many settings. The biggest question that comes up for me is what is the ideal placement and balance between low- and high-cognitive demand tasks? One of the professors in my teacher prep program always suggested  “a steady diet of high cognitive demand tasks” which I found both wise and frustratingly vague.

Rigor:

The idea of rigor as a concept separate from cognitive demand is a newer one for me. In the past nine months, though, it has come up for me in two places—as part of my district’s curriculum review tools and at the Standards Institute. At corestandards.org, they define rigor as “deep, authentic command of mathematical concepts” and clarify that it is “not making math harder or introducing topics at earlier grades.”

Here are the elements of rigor. I’ve also included the definitions from the five strands of mathematical proficiency, because I think they better describe the three elements.

	Corestandards.org	Adding it Up Chapt. 4
Conceptual Understanding	The standards call for conceptual understanding of key concepts, such as place value and ratios. Students must be able to access concepts from a number of perspectives in order to see math as more than a set of mnemonics or discrete procedures.	Comprehension of mathematical concepts, operations, and relations.
Procedural skills and fluency	The standards call for speed and accuracy in calculation. Students must practice core functions, such as single-digit multiplication, in order to have access to more complex concepts and procedures. Fluency must be addressed in the classroom or through supporting materials, as some students might require more practice than others.	Skill in carrying out procedures flexibly, accurately, efficiently, and appropriately.
Application	The standards call for students to use math in situations that require mathematical knowledge. Correctly applying mathematical knowledge depends on students having a solid conceptual understanding and procedural fluency.	Ability to formulate, represent, and solve mathematical problems. (This is called strategic competence, not application, here)

The language of the shift calls for students to pursue the three elements with “equal intensity.” However, this leaves me with questions about how the three elements interact.  One of the tools that my district used in evaluating curriculum starts to further illustrate exactly what balance of rigor looks like. 

The authors of Principles to Actions state “Effective teaching of mathematics builds fluency with procedures on a foundation of conceptual understanding so that students, over time, become skillful in using procedures flexibly as they solve contextual and mathematical problems.” With this in mind, we were really looking to see whether and how procedural fluency was explicitly built off of conceptual understanding.

And then there’s the question of applications. In my experiences with both the Standards Institute and on the curriculum review team, applications were generally defined as any problem with a “real world” context. I find this interpretation problematic for three reasons.

1. Instead of applications acting as a third leg in the three-legged stool of rigor, all conceptual understanding and procedural fluency problems can be categorized as applications or not
2. There wasn’t a larger conversation around how well the real-world context was executed (See Dan Meyer here on one of my biggest pet peeves).
3. There’s a reason that the phrase “real-world and mathematical problems” occurs in the common core 31 times rather than just “real-world problems.” I think we need to better define what it means to have an application that is in a purely mathematical context, but I don't think we should just drop it.

I am not yet ready to fully develop my ideas around how to better define application, but I think that the strategic competence description in Adding it Up would be a good place to start.

Relationship

Based on these definitions, it is clear to me that I can’t use the terms “rigor” and “cognitive demand” interchangeably. But these are both words that I hear thrown around a lot when trying to evaluate teaching and curricular materials. So what relationship do they have to each other?

In looking at the text of a problem and the execution by both students and teacher, I think it makes sense to identify the foremost element of rigor and the level of cognitive demand separately. However, I would imagine that conceptual and application problems have a higher likelihood of being high cognitive demand, and that procedural fluency problems have a higher likelihood of being low cognitive demand.

I also think that it’s worth trying to push a little bit beyond the concept of “a steady diet”—which I think can be applied both to high cognitive demand tasks as well as tasks of each type of rigor.  In the standards institute we started to look at arcs of the elements of rigor over a particular topic or unit. In general, we seemed to see the sequence of conceptual understanding --> procedural fluency --> application, though of course it was not quite that clear cut. I would also question the placement of application primarily at the end because I think it underestimates students and impedes their learning to say that procedural fluency is required before being able to apply a concept.

This makes me wonder if there is a similar general flow that makes sense for cognitive demand. I would certainly argue against low cognitive demand being a prerequisite for high cognitive demand. I think that there’s a lot of value in starting with high cognitive demand tasks because they are an ideal place to surface student reasoning and build upon the foundation that students already have. But I wonder if it is oversimplified to suggest high cognitive demand --> low cognitive demand --> high cognitive demand as an arc for a unit. It might make more sense to have a unit that is composed of a series of shorter cognitive demand arcs of that nature.