**Norman Webb** explains why the teaching of math should be aligned with the complexity of the subject

**TEACHERS ARE FACED **with a vast array of guidance regarding what students should know and should be able to do. Teachers use textbooks and planning guides, national organizations make recommendations, and researchers disclose greater insights into the learning sequence and process. Teachers need to make sense of how their own methods fit, and align their teaching with these expectations, so that students learn what they are expected to know and do. But how can they do this?

What we know |
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● Paying attention to content complexity is important. Depth Of Knowledge is one way of defining content complexity. There are four levels: Level 1 – Recall Level 2 – Skill/Concept Level 3 – Strategic Thinking Level 4 – Extended Thinking |

**Content complexity **

“Content complexity” is a theory that has been discussed by academics since the late 1940s and is one technique that teachers can use to ensure that their teaching is aligned with learning expectations and assessments. Content complexity differentiates learning expectations and outcomes by considering the mental processing of concepts and skills in particular prior knowledge and the number of steps that need to be considered to complete a task. In mathematics, content complexity is related to a student performing a set procedure or recalling information, applying a multiple step process or conceptual understanding, or solving a non-routine problem where several approaches are possible.

**Depth Of Knowledge **

Depth Of Knowledge (DOK) is a language system used to describe different levels of complexity. Four levels specify the degree of complexity of mathematical content, as it relates to typical students at a given age. These are:

**Level 1 ****(***Recall***)** includes the recall of information such as a fact, definition, term, or a simple procedure, as well as performing a simple algorithm or applying a formula. Generally in mathematics a one-step, well-defined, and straight algorithmic procedure should be included at this basic level. At secondary level, solving a system of two equations with two unknowns generally requires a set procedure for eliminating one variable and solving for the second variable. Because students are expected to apply a standard procedure, finding the values of the two variables is a DOK level 1.

**Level 2 ****(***Skill/Concept***)** includes the engagement of some mental processing beyond a habitual response. A Level 2 assessment task requires students to make some decisions as to how to approach the problem or activity. Level 2 expectations and activities imply more than one step. Action verbs, such as “explain,” “describe,” or “interpret” could be classified at different levels depending on the object of the action. For example, interpreting information from a simple graph – reading information from the graph by considering the units on the axes and other attributes – is a Level 2. Level 2 activities are not limited to just number skills, but can involve visualization skills and probability skills. Other Level 2 activities include: extending non-trivial patterns, explaining the purpose and use of experimental procedures; carrying out experimental procedures; making observations and collecting data; classifying, organizing, and comparing data; and organizing and displaying data in tables, graphs, and charts.

**Level 3 ****(***Strategic Thinking***)** requires reasoning, planning, using evidence, and a higher level of thinking than the previous two levels. In most instances, requiring students to explain and justify their thinking mathematically is a Level 3 task. Activities that require students to make conjectures are also at this level. The cognitive demands at Level 3 are more abstract than at Levels 1 or 2. The complexity does not result from the fact that there are multiple answers, a possibility for both Levels 1 and 2, but because the task requires more demanding reasoning. Other Level 3 activities include drawing conclusions from observations; citing evidence and developing a logical argument for concepts; explaining phenomena in terms of concepts; using concepts to solve problems; and critiquing experimental designs.

**Level 4 ****(***Extended Thinking***)** requires deep reasoning, planning, developing, and thinking activities over an extended period of time. At Level 4, the cognitive demands of the task should be high and the work should require drawing upon multiple resources or analyses. Students should be required to make several connections – relate ideas within the content area or among content areas – and have to select one approach among many alternatives on how the situation should be solved, in order to be at this level. Level 4 activities include developing and proving conjectures, designing and conducting experiments, making connections between a finding and related concepts and other phenomena, and combining and synthesizing ideas into new concepts. Conducting a research project including developing the questions, creating the design, collecting and analyzing data, drawing conclusions, and reporting the results would be a typical Level 4 activity.

Frequently, content complexity is interpreted as content difficulty. Difficulty can be related to complexity, but what makes a mathematical activity hard for a student depends on more factors than just how complex the activity is. If a student has not had the opportunity to learn a concept or skill, applying the concept or skill will probably be difficult. Also, applying a repetitive action, such as memorizing and recalling a large number of digits of π, can be difficult to achieve, but is still just recall of information and is therefore a DOK Level 1 task.

**Case study **

In one school district in the U.S., a mathematics coordinator used DOK to help teachers understand the inconsistency between students’ grades and their scores on the state assessment. Most students in the districts were receiving high grades in mathematics. However, their scores on the state assessment were below proficiency. Teachers used the DOK levels to analyze the complexity of the state standards and assessments, and compared this to the complexity of the teaching methods they used. Teachers found that most of their techniques focused on DOK Level 1 activities (recall of information) whereas the state standards expected students to have a conceptual understanding of the main ideas (a DOK Level 2) and some solving of non-routine problems (a DOK level 3).

**Conclusion **

Attention to content complexity is important for ensuring that instruction is aligned with expectations. Depth Of Knowledge is one means for defining content complexity. A number of considerations are necessary to assign a DOK level to instructional activities, expectations, or assessment activities. Among these are the actions, the subject of the actions, prior experience, and mathematical sophistication. Awareness of content complexity through the use of DOK levels helps to ensure that students will learn mathematics as fully expressed in high expectations and assessments.

**About the author **

**Norman L Webb** is an emeritus research scientist at the Wisconsin Center for Education Research at the University of Wisconsin–Madison. He is currently a visiting research scientist for the National Science Foundation.