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# Mathematical Tasks: High Cognitive Demand and Low Cognitive

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## Haley Throneburg

on 18 November 2013

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#### Transcript of Mathematical Tasks: High Cognitive Demand and Low Cognitive

RESOURCES
Doing Mathematics:
This task requires students to explore and understand relationships between the concept of subtraction and real-life situations. So the student must be able access relevant knowledge (real-world events) and apply that knowledge to be able to make a connection by putting this problem into a real world-context. This cannot be solved by using a learned algorithm and there are no suggested pathways given on how to solve the problem. There is a great deal of cognitive demand required for the student to completely work through this task.

Procedures With Connections
In this taskStudents must be able to make real world connections by being able to relate a real-life situation to a number sentence. This task also requires the student model each problem using manipulatives. The student then must go even further and explain their thinking and reasoning either with words or by drawing pictures. These aspects of the problem require students to make connections among multiple representations and develop a depper understanding of subtraction. This task suggests different pathways to follow but students cannot work through each part of the problem by using specific procedures. The students must be able to apply their prior knowledge of using manipulatives (must understand the concept of unitizing and place value) and then explain their thinking and reasoning using words or pictures. (again, multiple representations)
4 Levels of Cognitive Demand
:
1) Memorization
2) Procedures without connections
(These tasks depend on prior knowledge)

:
3) Procedures with connections
4) Doing Mathematics
(These tasks BUILD onto prior knowledge)
Using the Framework to Classify Subtraction Tasks
LCD:
Low Cognitive Demand
HCD:
High Cognitive Demand
Examples of Each Type of Task
HCD
HCD
HCD
LCD
LCD
HCD
HCD
LCD
Memorization:
This task only requires the student to reproduce previously learned rules, subtract 38 from the sum to find the missing number. This problem has no connection to the concepts or meanings that are associated with completing this problem.
LCD
" The cognitive demand is the kind and level of thinking required of students in order to successfully engage with and solve the task."

(
Stein, M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A., 2000.
Stein, M.K. & Smith, M.S. (1998). Mathematical tasks as a frameworkfor reflection: From research to practice. Mathematics Teaching in the Middle School, 3, 268-275.

Smith,M.S. & Stein, M.K. (1998). Selecting and creating mathematical tasks. Mathematics Teaching in the Middle School,3, 344-350.
The 1st phase is selecting a task in its original form/ as it appears in textbooks, instructional materials, etc; here the teacher must select a task that is aligned with the target knowledge they want students to obtain.
“Not all tasks are created equal, and different tasks will provoke different levels and kinds of student thinking.”

(Stein, et al., 2000)

Stein & Lane (1996) emphasize the importance of beginning with high-level tasks that are cognitively demanding and complex so that our students can gain essential mathematical skills such as reasoning and problem solving.

Doing Math
This task is complex and requires non-algorithmic thinking, in that there is no procedure for the student to follow. This task requires the student to analyze and explore the concept of regrouping and be able top explain their thinking. Students must access their own relevant knowledge of place value and regrouping and be able to use this knowledge to explain the significance of the "1" in each of the examples.
Procedure without Connections
This task is algorithmic, in that 'the use of procedure is evident from prior instruction/experience. It require limited cognitive demand in that the student can use their prior experience to procedurally solve the problem 2 different ways. It is not connected to the concepts or meanings that underlie the procedure being used. It is only focused on finding the correct answer 2 different ways. This task does not require the student to give an explanation describing how the procedure was used.
Procedure with Connections
This task requires the student to be able to relate the problem to a real-world context. It requires the student to know how time works in terms of years and be able to relate that to someones age. This task also requires the student to be able to relate the change in time to subtraction. There are multiple entry points to this problem and it cannot be completed in just one step.
Doing Mathematics
This task requires complex, non-algorithmic thinking by requiring that the students solve the problem using a strategy they understand (could be student invented which requires a high level of cognitive demand and also requires the student to be fully engaged). The students must also be able to explain the problem simply, because the last step of the problem requires that they be able to explain their solution to a child much younger then them, so the student must be able to easily display their knowledge in a simplified way as well as make connections among multiple representations using manipulatives. This task requires students to access relevant knowledge and make real-world connections (know how age is calculated) and then apply that knowledge to successfully work through each step of the problem. Students must be able to make real world connections, (the ages of people) and then connect the ages to the use of sutraction. This explore and understand the nature of mathematical concepts, processes, and relationships. This task requires a considerable amount of cognitive effort due to it's multiple entry and exit points which are not explicitly suggest by the task.
Procedure without connections
There are many ways in which this word problem can be solved, but all of the ways in which it can be solved are given to the student, so the use of a procedure is specifically called for. Also since the procedures to solve are given, there is no ambiguity therefore requiring very little cognitive demand from the student. To answer this problem, the student is only asked to choose a given strategy.
Memorization

Reproduction of Facts
Using Formulas/Rules
No explanations are required

I know have much more insight on the importance of selecting and implementing rich tasks for my future students. The Mathematical Tasks Framework is an amazing tool that all educators should use when planning for instruction. The framework can be used as a tool for implementation as well as refelction. I now have a much better understanding of how different tasks yield different levels of cognitive demand. Low level tasks only require students to memorize procedures and facts and then solve mathematical problems by only using memorized standard procedures and rules. These types of tasks do not require students to 'think for themselves' rather they are just following rules and guidelines that have been given to them. Students can only become problem solvers when they are required to think for themselves and are able make connections that are meaningful in mathematics. I have learned that in order for students to be able to make connections with important mathematical concepts they must be given tasks that require them to gain a conceptual understanding knowledge of mathematics and be able to relate these concepts to real-world contexts. I now also understand how important it is that we as teachers act as facilitators in the classroom so that we do not lower the levels of cognitive demand for the tasks we are implementing. choose, design, and implement tasks in our classrooms that have a high level of cognitive demand and require students to 'work through' the tasks so that they are pushed to think conceptually and make connections with important mathematical concepts.
Procedures without Connections

Require Algorithmic Thinking
Focus is on producing the right answer
No explanations are requires

Procedures with Connections

Require Algorithmic Thinking
Tasks are focused on gaining an understanding of mathematical concepts
Require some degree of cognitive effort
Involves multiple representations
Doing Mathematics

Requires students to 'work-through' a problem rather than just follow a procedure and solve for an answer.
Requires Non-Algorithmic Thinking
Requires an understanding of mathematical concepts and ability to apply these concepts
Explanationas are required

The Mathematical Tasks Framework (Stein et al., 2000)
Teaching Implications
How I See each Level of Cognitive Demand
This problem involves reproducing previously learned facts . Both fractions have a common denominator already, so all the student has to do is subtract, which only requires the student to use basic subtraction rules and get the answer.

This task is only focused on obtaining the correct answer. The use of the basic procedure for subtracting fractions is evident (finding a common denominator and then using the basic subtraction rules to find the correct answer.)

For both tasks, it is evident what procedures needs to be done to solve the problems. Neither of these tasks require any explanations

HCD
LCD
Procedure without Connections

This task is very simple. It requires very little cognitive demand and there is virtually no ambiguity about what needs to be done to solve this problem. All that is required of the student is to find the difference between the 2 groups of people. This task does not require any explanations nor does it connect to any meaningful mathematical concepts.
Procedure with Connections
This problem definitely focuses the student's attention on the use of a procedures (either 'adding up' or subtracting) for the purpose of developing a deeper understanding of mathematical concepts and ideas. This problem suggests 2 explicit pathways to follow that involve using general procedures (adding/subtracitng) that have close connections to underlying conceptual ideas (being able to explaing why/how both of these procedures are acceptable ways to solve this problem. This task requires some degree of cognitive effort and requires students to understand the full concept of how adding and subtracting are interrelated in mathematics.
Doing Mathematics
Memorization
Procedures without Connections
Procedures with connections
Changing a task from LCD to HCD

There are 30 people in the music room. There are 74 people in the cafeteria. How many more people are in the cafeteria than the music room?

a) Draw a picture to show how you would solve this problem.

b) Use words or pictures to explain how you could use cubes to model and solve this problem.