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
highcognitive 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.

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 singledigit 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.
 Instead of applications acting as a third leg in the threelegged stool of rigor, all conceptual understanding and procedural fluency problems can be categorized as applications or not
 There wasn’t a larger conversation around how well the realworld context was executed (See Dan Meyer here on one of my biggest pet peeves).
 There’s a reason that the phrase “realworld and mathematical problems” occurs in the common core 31 times rather than just “realworld 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.
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