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Mathematical Tasks with High Cognitive Demands

Page history last edited by mark.trushkowsky@mail.cuny.edu 6 years, 9 months ago

 

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 Depth-of-Knowledge 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 re-carpeting 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 rectangular-shaped 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 2002-2013. 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 Depth-of-Knowledge Levels?

 

The four (4) Depth-of-Knowledge 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. 

 

 

Depth-of-Knowledge 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

 

 

D-O-K Level 1:

 

Recall & Reproduction

 

 

  • There is usually a right answer 
  • Recall or recognize a fact, definition, term or property 
  • Apply/commute a well-known algorithm or formula (i.e. sum, quotient, etc.) 
  • Perform a specified or routine procedure
  • Solve a one-step 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

 

 

D-O-K 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

 

 

 

D-O-K 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 non-routine 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 multi-step 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

 

D-O-K Level 4:

 

Extended Thinking 

 

  • Relate math concepts to other content areas
  • Relate math concepts to real-world 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 Depth-of-Knowledge 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 real-world situation for the following problem:

 

2/3 x 3/4

 

Solve the problem you have created without using a rule, and explain your answer

 

 

 

 

  • 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 Depth-of-Knowledge levels anyway?

 

One of the ways the TASC is going to change over the next three years (2014-2016) 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 open-ended math problems to work with a mixed-level math group and to allow students at all levels to engage in rich and thoughtful mathematics. Many of you wrote in your applications that mixed-levels classrooms are a huge challenge in instruction. One response to a mixed-level 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.

 

 

 

 

Check out the collection of rich Common Core math problems, tasks and lessons!

A great resource for finding good, challenging tasks

 

 

 

 

 

 

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