**Using the Mathematical Tasks Framework to Analyze Subtraction Tasks**

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**

Low Cognitive Demand Tasks

:

1) Memorization

2) Procedures without connections

(These tasks depend on prior knowledge)

High Cognitive Demand Tasks

:

3) Procedures with connections

4) Doing Mathematics

(These tasks BUILD onto prior knowledge)

Using the Framework to Classify Subtraction Tasks

Mathematical Task Analysis Guide

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 Mathematics Task Framework is made up of 3 phases through which mathematical tasks go through:

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.

The 2nd phase the task goes through is the 'set-up' phase, in which the teacher makes neccessary adjustments to the task so that it matches the goals for student learning. The 3rd and final phase is the implementation phase. In this phase, the task is both implemented by the teacher and the student. This is a very crucial stage in the framework, because there are many variables teachers must consider when trying to maintain a high cognitive deman task, such as: allowing sufficient time for students to complete the task and then give feedback after completion, the type of 'help' the teachers give the students (not giving away clues on how to complete the task), classroom management allowing for students to stay fully engaged when working on the task, etc.- " This task is also focused on how the students go about completeing the task. This depends on whether or not the students are fully engaged while completing the task as well as how they go about completing the task.

“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)

I made a few changes to my concept map. I added 3 bubbles that connect straight to the main bubble of subtraction. The first bubble I added was Tasks - connected to this bubble is a bubble called "Mathematical Tasks Framework" and "Cognitive Demand of Tasks". Connected to the Cognitive Demand of tasks are 2 bubbles -The 1st bubble is Low Cognitive demand - with 2 bubbles stemming from it named Memorization and Procedures without connections. The 2nd bubble is High Cognitive Demand - with 2 bubbles stemming from it named Procedures with connections and Doing Mathematics. I changed my bubble named Methods and now it is named procedures. I added 2 procedures to this bubble - Austrian Algorithm and Equal Additions. The Second major bubble I added stemming from the main Subtraction bubble was Problem Structures which I also directly connected to the new Tasks bubble I added. Stemming from the Problem Structures are 3 bubbles which are part-part-whole, number sentence, and separate. The last thing I changed on my concept map was that I added a bubble named Skills to the main bubble of subtraction and I attatched this bubble to the Unitizing, Place Value Understanding, and regrouping bubbles that are stemming from the related vocabulary bubble.

**Task Sort Recording Sheet**

**Teaching Implications**

How I See each Level of Cognitive Demand

Task E: Memorization.

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.

Task F: Procedures without connections

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.

c) Explain how you could check your answer for correctness and why checking your answer that way works.

"

**LCD**

**HCD**

By re-working the problem, I have changed it from an LCD task to an HCD task. This task now requires the students to provide multiple representations by using pictures to represent how they would solve the problem (students could solve with a student-invented strategy). The students must also be able to understand the mathematical concepts of unitizing and place value by modeling and solving the problem with cubes/base ten blocks. This problem now also requires the student to make connections between addition and subtraction by requiring an explanation of how they could check their answer.

(Stein & Smith, 1998)