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Mathematical Tasks with High Cognitive Demands
This page will address the following questions:
What you already know about cognitive complexity
On this page, you will find materials and resources to help you understand the concepts of cognitive complexity and DepthofKnowledge levels and think about how they can impact student learning in your classroom.
Before we get to those materials, please consider the following two points:
 Even if you have not heard the terminology before, you will most likely find that you already have some sense of both of these concepts.
Here are some examples of math problems that differ in their cognitive complexity. Reading through them will begin to give you an idea of what cognitive complexity means. Try solving them all and answer the reflection questions below:
Column A

Column B

Sarah is recarpeting her bedroom which is 15 feet long and 10 feet wide. How many square feet of carpeting will she need to purchase?

Sarah is planning to build a garden in her backyard. She needs to erect a strong fence around the garden to protect her vegetables from local animals. She purchases 24 feet of this fencing.
Sarah wants a rectangularshaped garden. Given the amount of fencing she has, several different rectangle shapes are possible.
What garden dimensions will give Sarah the largest amount of area to plant her vegetables? Explain your reasoning, including any diagrams you used.

Calculate the mean, median and mode for the following set of numbers:
2, 34, 5, 16, 37, 45, 5

Construct a set of ten numbers, with a mean of 7, a median of 6 and a mode of 3.
Is there more than one way to do it?

Calculate the following:
34 ÷ 8
3 ¼ ÷ ½
2/3 x ¾

Create a real world situation for the following problems:
34 ÷ 8
3 ¼ ÷ ½
2/3 x ¾

What is 60% of 510?

Show as many ways as you can to find 60% of 510.

How are the problems in Column A similar?
How are the problems in Column B different from the problems in Column A?
Why might students who get the problems in Column A right, get the problems in Column B wrong?
 The importance of using more cognitively complex mathematical tasks in instruction is not new to adult literacy.
Below are three problems from the old GED Official Practice Test. They represent the kinds of problems that our students have been facing on the GED exam between 20022013. As you can see from these three examples, students already need time working on problems that go beyond the use of a simple procedure or formula. The problems all require students to understand concepts, so that they can be flexible and adapt their reasoning. These problems also each test a student's ability to work through a problem that is different from how they are likely to have seen it before.
One major contributing factor to students getting these kinds of problems wrong is they don't fit with the ideas that students have about what it means to do math. Students are used to problems like, "Kelly's sales for January were $40,000. Her sales in February were $20,000. Her sales in March were $15,000. What was her average sales for the three months?" Those kinds of problems have set procedures that students can perform by rote. The problem on the left below is more cognitively complex. When student see problems like that they often give up, because they assume there is some rote procedure, and that they just don't know it  the reality is that here are several ways to solve each of them and students have to figure out their own way to solve it.





What are the four DepthofKnowledge Levels?
The four (4) DepthofKnowledge levels were developed by Norman Webb to provide a way of thinking about our students and how they are engaging with the content in our classes. The idea behind the DOK framework is to evaluate a task in terms of how deeply a student has to understand the content to be able to successfully complete the task. DOK is not about difficulty, it is about complexity, or depth.
DepthofKnowledge Levels

Math Descriptors
(These are just a few examples and characteristics of what task at each DOK level might require students to do.)
adapted from Karin Hess

When tasks at each
DOK Level
are Expected on the TASC

DOK Level 1:
Recall & Reproduction

 There is usually a right answer
 Recall or recognize a fact, definition, term or property
 Apply/commute a wellknown algorithm or formula (i.e. sum, quotient, etc.)
 Perform a specified or routine procedure
 Solve a onestep word problem
 Retrieve information from a graph or table
 Make conversions between and among representations or numbers (fractions, decimals, percents) or within and between customary metric measures
 Locate points on a coordinate grid
 Determine the area or perimeter of rectangles or triangles given a drawing a labels
 Identify shapes and figures

Starting in 2014

DOK Level 2:
Basic Skills & Concepts

 There is usually a right answer
 Classify shapes and figures
 Interpret information from a simple graph
 Solve a routine problem that require multiple steps/decision points or the application of multiple concepts
 Provide justification for steps in a solution process
 Use models or diagrams to represent and explain mathematical concepts
 Make and explain estimates
 Make basic inferences or logical predictions from data/observations
 Organize or order data
 Choose an appropriate graph type and organize and display data
 Extend/continue a pattern
 Retrieve information from a table, graph, or figure and use it to solve a problem requiring multiple steps
 Specify and explain relationships between facts, terms, properties, or operations

DOK Level 3:
Strategic Thinking/Reasoning

 There may be more than one right answer and/or more than one way to get there
 Use concepts to solve nonroutine problems
 Explain your thinking when more than one response is possible
 Having to plan a strategy and decide how to approach a math task when more than one approach is possible
 Generalize a pattern
 Write your own problem, given a situation
 Describe, compare, contrast different solution methods
 Use evidence to develop logical arguments for a concept
 Draw conclusions from observations and data, citing evidence
 Interpreting information from a complex graph
 Make and/or justify conjectures
 Perform procedure with multiple steps and multiple decision points
 Solve a multistep problem and provide support with a mathematical explanation that justifies the answer
 Interpret data from a complex graph
 Verify the reasonableness of results

Starting in 2015

DOK Level 4:
Extended Thinking

 Relate math concepts to other content areas
 Relate math concepts to realworld applications in new situations
 Conduct a project that specifies a problem, identifies solution paths, solves the problem and reports results
 Design a mathematical model to inform and solve a practical or abstract situation
 Apply understanding in a novel way, providing an argument/justification for the application

Starting in 2016

Depth of Knowledge: A very brief overview by Karin Hess
This clip is an excerpt from a longer presentation.
If you'd like to watch the entire video (23 min), click here.
Another helpful video (11 min) by Karin Hess can be viewed here.


Are there other helpful ways for teachers to think about the cognitive demands of math tasks?
Webb's DepthofKnowledge Levels look at different depths of engaging with content. The D.O.K. levels are generalized and work across all content levels. Another useful place for math teachers to look is the work of Mary Kay Stein.
Mary Kay Stein, along side others, has developed a four category scale for specifically categorizing mathematical tasks. The four categories are (1) "Memorization", (2) "Procedures without Connections", (3) "Procedures with Connections" and (4) "Doing Mathematics"
Here are some examples of problems at each of the four levels:
Memorization
What is the rule for multiplying fractions?

Procedures with Connections
Find 1/6 of 1/2. Draw your answer and explain your solution.

Procedures Without Connections
Multiply:
2/3 x 3/4
5/6 x 7/8
4/9 x 3/5

Doing Mathematics
Create a realworld situation for the following problem:
2/3 x 3/4
Solve the problem you have created without using a rule, and explain your answer

 A few brief, readable and helpful articles by Mary Kay Stein and Margaret Schwan Smith exploring the use of cognitively demanding mathematical tasks with our students
 Here's an activity to do with fellow teachers at your program. It offers some definitions four categories of cognitive demands ("memorization", "procedure without connections", "procedures with connections", and "doing mathematics")and then asks teachers to categorize a variety of problems and discuss their process. This is a good way to begin to develop our abilities to choose and create problems that are appropriately demanding for our students.

Why are we talking about cognitive complexity and DepthofKnowledge levels anyway?
One of the ways the TASC is going to change over the next three years (20142016) is in the kinds of questions they ask. The questions are going to get more challenging, not only because content will be harder, but because the questions will require students to demonstrate a deep understanding. You will hear the terms "Depth of Knowledge" of "DOK levels" when CTB/McGraw Hill talks about the rigor of the problems on the TASC. Moving into higher levels of DOK is one of the big ways that the TASC will get more challenging each year.
 In 2014, the questions will be at Depth Of Knowledge Levels 1 and 2.
 In 2015, the focus will shift to more cognitively demanding math tasks, with problems at DOK Levels 2 and 3 (with some DOK 1)
 In 2016, the focus will shift to even more complex math tasks  the exam will be mostly DOK Levels 2 and 3 (with some DOK 4 and very little DOK 1)

How can I create my own, more rigorous mathematical tasks?
This article looks at the practice of using openended math problems to work with a mixedlevel math group and to allow students at all levels to engage in rich and thoughtful mathematics. Many of you wrote in your applications that mixedlevels classrooms are a huge challenge in instruction. One response to a mixedlevel class is to give advanced students extra work or to give them harder problems. These strategies can sometimes lead to different challenges. One of the strategies suggested by this article is that there are benefits to having all students work on the same problem  if it’s the right problem.

A great resource for finding good, challenging tasks
Mathematical Tasks with High Cognitive Demands

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