MATH
Depth Of Knowledge
Depth Of Knowledge
for mathematics
Norman Webb explains why the teaching of math should be
at Level 3 are more abstract than at Levels
1 or 2. The complexity does not result from
aligned with the complexity of the subject
the fact that there are multiple answers,
a possibility for both Levels 1 and 2, but
TEACHERS ARE FACED with a vast array of equations with two unknowns generally because the task requires more demanding
guidance regarding what students should requires a set procedure for eliminating one reasoning. Other Level 3 activities include
know and should be able to do. Teachers variable and solving for the second variable. drawing conclusions from observations;
use textbooks and planning guides, national Because students are expected to apply a citing evidence and developing a logical
organizations make recommendations, and standard procedure, fi nding the values of argument for concepts; explaining
researchers disclose greater insights into the two variables is a DOK level 1. phenomena in terms of concepts; using
the learning sequence and process. Teachers Level 2 (Skill/ Concept) includes the concepts to solve problems; and critiquing
need to make sense of how their own engagement of some mental processing experimental designs.
methods fi t, and align their teaching with
these expectations, so that students learn
Frequently, content complexity is interpreted as content
what they are expected to know and do. But
how can they do this?
diffi culty. Diffi culty can be related to complexity, but what
makes a mathematical activity hard for a student depends on
Content complexity
“Content complexity” is a theory that has
more factors than just how complex the activity is
been discussed by academics since the late
1940s and is one technique that teachers can beyond a habitual response. A Level 2 Level 4 (Extended Thinking) requires
use to ensure that their teaching is aligned assessment task requires students to make deep reasoning, planning, developing, and
with learning expectations and assessments. some decisions as to how to approach the thinking activities over an extended period
Content complexity differentiates learning problem or activity. Level 2 expectations and of time. At Level 4, the cognitive demands
expectations and outcomes by considering activities imply more than one step. Action of the task should be high and the work
the mental processing of concepts and verbs, such as “explain,” “describe,” or should require drawing upon multiple
skills in particular prior knowledge and the “interpret” could be classifi ed at different resources or analyses. Students should
number of steps that need to be considered levels depending on the object of the action. be required to make several connections
to complete a task. In mathematics, content For example, interpreting information – relate ideas within the content area or
complexity is related to a student performing from a simple graph – reading information among content areas – and have to select
a set procedure or recalling information, from the graph by considering the units one approach among many alternatives
applying a multiple step process or on the axes and other attributes – is a on how the situation should be solved, in
conceptual understanding, or solving a non- Level 2. Level 2 activities are not limited order to be at this level. Level 4 activities
routine problem where several approaches to just number skills, but can involve include developing and proving conjectures,
are possible. visualization skills and probability skills. designing and conducting experiments,
Other Level 2 activities include: extending making connections between a fi nding and
Depth Of Knowledge non-trivial patterns, explaining the purpose related concepts and other phenomena,
Depth Of Knowledge (DOK) is a language and use of experimental procedures; and combining and synthesizing ideas
system used to describe different levels of carrying out experimental procedures; into new concepts. Conducting a research
complexity. Four levels specify the degree making observations and collecting data; project including developing the questions,
of complexity of mathematical content, as classifying, organizing, and comparing data; creating the design, collecting and
it relates to typical students at a given age. and organizing and displaying data in tables, analyzing data, drawing conclusions, and
These are: graphs, and charts. reporting the results would be a typical
Level 1 (Recall) includes the recall of Level 3 (Strategic Thinking) requires Level 4 activity.
information such as a fact, defi nition, term, reasoning, planning, using evidence, and a Frequently, content complexity is
or a simple procedure, as well as performing higher level of thinking than the previous interpreted as content diffi culty. Diffi culty
a simple algorithm or applying a formula. two levels. In most instances, requiring can be related to complexity, but what
Generally in mathematics a one-step, well- students to explain and justify their thinking makes a mathematical activity hard for a
defi ned, and straight algorithmic procedure mathematically is a Level 3 task. Activities student depends on more factors than just
should be included at this basic level. At that require students to make conjectures how complex the activity is. If a student
secondary level, solving a system of two are also at this level. The cognitive demands has not had the opportunity to learn a
14 Better: Evidence-based Education fall 2009
Better(US) Fall09.indb 14 14/10/09 13:05:10
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