ACTIVITY: INTRODUCTION TO DEPTH
MATHEMATICS (50 MINUTES)
OF
KNOWLEDGE
FOR
OVERVIEW:
In this activity, participants will learn about Norman Webb’s Depth of Knowledge (DOK) levels and Karin
Hess’s Cognitive Rigor Matrix through participating in a text-based protocol. They will then practice
using the Cognitive Rigor Matrix to evaluate the rigor of questions and adjust the level of rigor when
appropriate.
TARGET AUDIENCE:
This activity is appropriate for pre-K-12 educators who have some familiarity with the Common Core
Learning Standards for Mathematics.
OUTCOME:
Participants will be able to explain how Webb’s DOK levels and Hess’s Cognitive Rigor Matrix can help
educators evaluate the rigor of their tasks and questions. They will also be able to use these tools to
evaluate and adjust the rigor of questions.
GUIDING QUESTIONS:


How are Webb’s DOK levels and Hess’s Cognitive Rigor Matrix used to evaluate the rigor of
questions and tasks?
How do these tools relate to the transition to the Common Core Learning Standards (CCLS)?
MATERIALS:


Handouts for participants:
o Depth of Knowledge Levels for Four Content Areas (Norman Webb)
o Cognitive Rigor Matrix for Math/Science (Karin Hess)
o Note-Taking Guide
o Common Core Learning Standards for Mathematics
o 2012-2013 Citywide Instructional Expectations
Facilitator’s Resource:
o What exactly do “fewer, clearer, and higher standards” really look like in the classroom?
(Hess et al.)
o Introduction to Depth of Knowledge – Appendix
PREPARING TO FACILITATE:

Read the article What exactly do ‘fewer, clearer, and higher standards’ really look like in the
classroom? by Karin Hess et al. for background information about DOK.

Read Hess’s Cognitive Rigor Matrix for Math/Science to familiarize yourself with the different DOK
levels.
Facilitation Guide

Read Introduction to Depth of Knowledge - Appendix for further information about Depth
Knowledge, including its relationship to Bloom’s Taxonomy.
FACILITATION NOTES:
1. Introduction. Facilitator briefly explains the connection between DOK and the CCLS and introduces
the goals for the activity:
 A key component of the transition to the Common Core Learning Standards (CCLS) is
strengthening student work by examining and refining curriculum, assessment, and classroom
instruction. Depth of Knowledge is a framework that can help educators examine the level of rigor
in these areas to ensure that students are being pushed to meet the demands of the new
standards.
 In this activity we will become familiar with the DOK framework through a text-based protocol.
We will then practice applying this framework to questions.
2. Surface Current Thinking about Rigor. Facilitator asks participants to record their definition of
cognitive rigor in their Note-Taking Guide. Participants share their definitions. Facilitator asks
participants to keep these definitions in mind during the activity and to note any shifts in their
understanding throughout the activity.
Facilitator asks the group to find the DOE-Selected Domain of Focus in the 2012-2013 Citywide
Instructional Expectations. Facilitator then instructs participants to each write two types of questions
for one of the standards in the Domain of Focus: a basic comprehension question and a question that
the participant considers rigorous. Facilitator tells participants that they will come back to these
questions later in the activity and examine them using the Depth of Knowledge framework.
As an example, facilitator models the process for a fifth grade mathematics standard from the CCLS
for mathematics in the “Number and Operations – Fractions” Domain. Standard 5.NF.1 asks students
to
“add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with
equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators.”
A basic question for this standard might be “What is 2/3 plus 1/6?” and a “rigorous question for this
standard might be “explain the steps you would take to subtract 3/8 from 4/5. Why would you take
each step?” (10 minutes)
3. Overview of Depth of Knowledge. Facilitator directs participants to the Depth of Knowledge outline
in their Note-Taking Guide and has them silently read it. Facilitator then introduces the framework
using the following talking points: (5 minutes)
 These four levels indicate how much students must know about a concept or topic to complete an
assignment or answer a question. The Webb levels do not necessarily indicate degree of
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Facilitation Guide




“difficulty.” A Level 1 question can ask a student to recall or state both a simple and a much more
complex concept. For example, doing 100 long division problems might be difficult, but not
complex.
Greater depth of understanding is required to be able to explain how or why a concept works or
use understanding to complete a routine assignment (Level 2), to apply it to real-world
phenomena with supporting evidence (Level 3), or to integrate one concept with other concepts
or other perspectives (Level 4).
What distinguishes Level 3 from Levels 1 and 2 is that it is often more abstract and less routine,
with more than one possible answer and/or more than one way to arrive at that answer.
Level 4 will probably be an extended activity, with extended time provided. Level 4 tasks often
involve multiple sources or content areas.
The point is not to eliminate DOK Level 1 and 2 questions and tasks– those levels are necessary
but not sufficient. They provide the background necessary to reach DOK Levels 3 and 4. However,
if you are only assessing at levels 1 and 2, chances are the instruction is only at that level. It is
important to incorporate a range of DOK levels into curriculum, instruction and assessment
throughout the course of the school year.
4. Text-based Discussion about DOK. Facilitator explains and then leads participants through the
process of reading and discussing the mathematics article in Depth of Knowledge Levels for Four
Content Areas by Norman Webb. (15 minutes)
Step 1: Participants silently read the portion of the article related to mathematics, thinking about the
following questions as a guide: (5 minutes)
 What does “cognitive rigor” look like in mathematics?
 How does the Depth of Knowledge framework relate to your daily practice?
 Where do you see Depth of Knowledge fitting into the lesson/unit planning process?
Step 2: Each participant shares their responses to the guiding questions. (5 minutes)
Step 3: Facilitator leads a short discussion around these reflections using a question such as:
 Based on the overview of DOK and this text and, how might DOK be a useful lens for thinking
about rigor in student work as we transition to the Common Core Learning Standards? (5
minutes)
5. Apply DOK Levels and the Cognitive Rigor Matrix to Questions. Facilitator hands out and explains
the Cognitive Rigor Matrix – Math-Science:
o Bloom’s Taxonomy provides a way to think about what students are being asked to do when
completing an assignment in terms of the skill they must have to complete a task or question.
Bloom’s does not examine how deeply students must engage with the content to complete the
assignment, which is where DOK comes in.
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Facilitation Guide
o The Cognitive Rigor Matrix allows us to look at two dimensions simultaneously so that we can
consider not only what students are asked to do with content, but also the depth with which
they need to engage with content to complete an assessment task. This tool combines Bloom’s
Taxonomy with Webb’s Depth of Knowledge, and was developed by Karin Hess.
o This matrix can help us to assess the level of cognitive rigor of question or task, think about how
to increase cognitive rigor, and track the amount of exposure students are getting to various
levels of cognitive rigor.
Facilitator leads participants through the process of evaluating the rigor of the questions they
developed earlier using the Cognitive Rigor Matrix. Facilitator explains that this is not about locating
questions on a rubric or assigning a value; this is about using the descriptors within the matrix to
make an informed assessment of the level of rigor required by a particular question or task. (15
minutes)
Step 1: Participants look back at the questions they created. Using their understanding of DOK
and the Cognitive Rigor Matrix – Math-Science, participants evaluate the DOK level(s) that their
questions incorporate. (5 minutes)
Step 2: Each participant shares his/her evaluations. The group provides feedback on the
participant’s thinking when appropriate and cites language in the matrix to support their ideas. (5
minutes)
Step 3: Participants use the Cognitive Rigor Matrix to revise their original questions to increase
the level of cognitive rigor where appropriate. Facilitator encourages participants to be as specific
and clear as possible in rewriting their questions to ensure that they will produce the type of
thinking they want from students. Participants share out their revised questions and gather
feedback from the group. (5 minutes)
6. Debrief. Facilitator asks participants to revisit their original definitions of the term cognitive rigor
and share any changes or revisions that resulted from this activity. (5 minutes)
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