1. Supporting Rigorous Mathematics Teaching and Learning
The Instructional Tasks Matter:
Analyzing the Demand of Instructional Tasks
Tennessee Department of Education
Elementary School Mathematics
Grade 2
Overview of the Module:
This module is foundational to the work of understanding the new Common Core State Standards and the design of the performance-based assessments.
Without a cognitively demanding task, students will not have an opportunity to learn to think, reason, problem solve, and communicate mathematically. We know that ALL students can learn challenging mathematics and that our job is to help teachers accomplish this goal. The QUASAR project, an NSF-funded study done with five middle schools in very diverse communities in the United States, found that regular use of high-level tasks made a difference in student performance in mathematics.
Materials:
Slides with note pages
Mathematics Common Core State Standards (the Standards for Mathematical Practice and the grade-specific Standards for Mathematical Content)
Participant handout
Chart paper and markers

2. Rationale Comparing Two Mathematical Tasks
Tasks form the basis for students’ opportunities to learn what mathematics is and how one does it, yet not all tasks afford the same levels and opportunities for student thinking. [They] are central to students’ learning, shaping not only their opportunity to learn but also their view of the subject matter.
Adding It Up, National Research Council, 2001, p. 335
By analyzing two tasks that are mathematically similar, teachers will begin to differentiate between tasks that require thinking and reasoning and those that require the application of previously learned rules and procedures.
Directions:
Give participants a minute to read the rationale slide. or
Paraphrase the rationale if desired.

3. Learning Goals and Activities
Participants will:
compare mathematical tasks to determine the demand of the tasks; and
identify the Common Core State Standards (CCSS) for Mathematical Content and Mathematical Practice addressed by each of the tasks.
Directions:
Read the goals and activities.

4. Comparing the Cognitive Demand of Mathematical Tasks
What are the similarities and differences between the two tasks?
The Strings Task
The Apples Task
(SAY) Analyze the two tasks, the Strings Task and the Apples Task. You will find the tasks in your participant packet. Do not solve the tasks, but instead compare the way the tasks are written and what the task is asking the student to think about and do. In your participant packet you will find a table where you can make a list of similarities and a list of differences between the two tasks.
Individual Work:
Work individually. It is important to allow for individual think time before the group share, so that all participants will have an opportunity to formulate their own ideas. The individual time not only provides broader access to the activity, but also allows opportunity for a range of ideas to be developed and discussed.
Small Group Work:
Discuss your responses with others at your table.
Group Discussion:
We will discuss our responses. What are some similarities and differences between the two tasks? Chart the similarities and differences. (See possible responses on the next two notes pages, the notes page for each task.)

5. The Strings Task
Solve the set of addition expressions. Each time you solve a problem, try to use the previous equation to solve the problem = ___ = ___ = ___ = ___ = ___ Solve each problem two different ways. Make a drawing or show your work on a number line. What pattern do you notice? If the pattern continues, what would the next three equations be?
(SAY) What are the similarities and differences among the tasks?
Possible Responses:
Similarities:
Both tasks deal with addition.
Both tasks deal with the addition of a two-digit number and a one-digit number.
Differences:
One task focuses on apples and the other works on computation without a context.
The Strings Task asks students to solve the task more than one way.
The Strings Task directs students to represent the problem with a number line or a drawing.
The Strings Task requires students to notice patterns and relationships.
The String Task ask students to notice a pattern. 5

6. The Apples Task
One basket has 27 green apples and 3 fell out of the basket. How many green apples do we have? Another basket has 37 red apples and 3 fell out of the basket. How many red apples do we have?
Probing Facilitator Questions and Possible Responses (Differences):
How do the Strings Task and the Apples Task differ from each other?
The Strings Task provides opportunities for students to make their thinking public and for reasoning when they model their solution to each expression. Whereas in the Apples Task, students are just asked for the total number of apples.
The Strings ask asks students to make connections between representations (to make a diagram and then write an equation that represents the diagram).
In the Strings Task, students must identify the pattern in the string of expressions/equations.

7. The Common Core State Standards (CCSS)
Examine the CCSS:
for Mathematical Content
for Mathematical Practice
Will second grade students have opportunities to use the standards within the domain of Operations and Algebraic Thinking and Number Operations in Base Ten?
What kind of student engagement will be possible with each task?
Which Standards for Mathematical Practice will students have opportunities to use with each task?
Directions: Take 10 minutes for these questions.
Probing Facilitator Questions and Possible Responses:
Which standards for mathematical content do students have opportunities to make sense of when solving the Strings Task?
Students have opportunities to work on the Number and Operations in Base Ten standards when solving the Strings Task.
Which standards for mathematical content do they have opportunities to work on when solving the Apples Task?
The Apples Task is a story problem so 2.OA.2A.
Which standards for mathematical practice does each task require students to use?
The Apples Task does not require a mathematical model, whereas the Strings Task does.
The Apples Task requires students to make sense of the story problem.
The String tasks ask students to notice the structure of the tens and ones place. Students have to notice a pattern.

8. The CCSS for Mathematical Content: Grade 2
Operations and Algebraic Thinking OA
Represent and solve problems involving addition and subtraction.
2.OA.A.1
Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.
Add and subtract within 20.
2.OA.B2
Fluently add and subtract within 20 using mental strategies. By end of Grade 2, know from memory all sums of two one-digit numbers.
Common Core State Standards, 2010

9. The CCSS for Mathematical Content: Grade 2
Operations and Algebraic Thinking OA
Work with equal groups of objects to gain foundations for multiplication.
2.OA.C.3
Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counting them by 2s; write an equation to express an even number as a sum of two equal addends.
2.OA.C.4
Use addition to find the total number of objects arranged in
rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.
Common Core State Standards, 2010

10. The CCSS for Mathematics: Grade 2
Number and Operations in Base Ten NBT
Understand place value.
2.NBT.A.1
Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases:
100 can be thought of as a bundle of ten tens—called a “hundred.”
The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds (and 0 tens and 0 ones).
Directions:
Read the standards. Which student work demonstrates that the student has met the standards?
2.NBT.B.6 Did we work on this standard? Yes, we worked with three-digit numbers. We used place value and compensation to make sense of the amounts we added.
2.NBT.B.7 Did we meet this standard? Yes, because we used concrete models. We also worked with three-digit numbers. We decomposed and recomposed quantities.
2.NBT.B.8 Why aren’t we working on this standard? What questions would we have to ask in order to be working on this standard? We would need to say = 58, = 68, = 78, etc.
2.NBT.B.9 Did we work on this standard? Explain why addition and subtraction strategies work, using place value and the properties of operations.
Common Core State Standards, 2010, p. 19, NGA Center/CCSSO

11. The CCSS for Mathematics: Grade 2
Number and Operations in Base Ten NBT
Understand place value.
2.NBT.A.2
Count within 1000; skip-count by 5s, 10s, and 100s.
2.NBT.A.3
Read and write numbers to 1000 using base-ten numerals, number names, and expanded form.
2.NBT.A.4
Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the results of comparisons.
(SAY) Did we work on these standards?
Some students might skip count from 48 by ten to 100.
Common Core State Standards, 2010, p. 19, NGA Center/CCSSO

12. The CCSS for Mathematics: Grade 2
Number and Operations in Base Ten NBT
Use place value understanding and properties of operations to add and subtract.
2.NBT.B.5
Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.
2.NBT.B.6
Add up to four two-digit numbers using strategies based on place value and properties of operations.
(SAY) How might working with quantities subtracted from 100 or adding from a two- digit amount to 100 build students’ fluidity with solving problems eventually?
Common Core State Standards, 2010, p. 19, NGA Center/CCSSO

13. The CCSS for Mathematics: Grade 2
Number and Operations in Base Ten NBT
Use place value understanding and properties of operations to add and subtract.
2.NBT.B.7
Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds.
2.NBT.B.8
Mentally add 10 or 100 to a given number 100–900, and mentally subtract 10 or 100 from a given number 100–900.
2.NBT.B.9
Explain why addition and subtraction strategies work, using place value and the properties of operations.
(SAY) Which of these students might students have opportunities to work to make sense of when solving the task?
2.NBT.B.7 Students might decompose 100 into and then subtract 48 from it. They might also add on tens to 48 till they arrive at 100.
2.NBT.B.9 Students are asked to use addition and subtraction when solving the problem and to explain why either operation can be used to solve the task.
Common Core State Standards, 2010, p. 19, NGA Center/CCSSO

14. Table 1: Common Addition and Subtraction Situations
Facilitator Notes:
After solving the task, we will consider the kind of situational word problem that we solved.
Common Core State Standards, 2010

15. The CCSS for Mathematical Practice
Make sense of problems and persevere in solving them.
Reason abstractly and quantitatively.
Construct viable arguments and critique the reasoning of others.
Model with mathematics.
Use appropriate tools strategically.
Attend to precision.
Look for and make use of structure.
Look for and express regularity in repeated reasoning.
Directions:
Read the directions on the slide.
Probing Facilitator Questions and Possible Responses:
What about the design of the task requires students to construct a viable argument?
Students must analyze the situation for the Strings Task, whereas there is not much to figure out when doing the Apples Task.
Does the Strings Task require students to reason abstractly and quantitatively?
Students do not use this practice for the Strings Task, but they might use it for the Apples Task.
Will students have to look for and make use of structure?
Students have to show that they know the evenness of two even numbers or two odd numbers because they have to write a generalization or draw conclusions. The Apples Task does not require that students explain what an even number is.
Common Core State Standards, 2010 15

16. Comparing Two Mathematical Tasks
How do the differences between the Strings Task and the Apples Task impact students’ opportunity to learn the Standards for Mathematical Content and to use the Standards for Mathematical Practice?
Facilitator Note:
After the whole group discussion of the differences and similarities between the two tasks, you should have a list of characteristics of high-level and low-level tasks similar to those on the Mathematical Task Analysis Guide (TAG). Participants will analyze the TAG later in the session and ideally they should recognize the similarity between their list and the one produced by researchers from the QUASAR project.
Directions:
Ask participants the question on the slide. Participants may need time to Turn and Talk with a partner before responding to the question.
Solicit responses from the whole group (5 minutes).
Possible Responses:
The differences do matter because students need the kinds of opportunities that they get from doing the Strings Task. This task develops students’ ability to problem-solve. They must figure out what to do, implement the plan, and then draw conclusions about how the two equations can be equivalent. If students never get this opportunity, they don’t know what to do when specific numbers aren’t given.
The Strings Task also provides opportunities for students to make public their thinking and reasoning.

17. Linking to Research/Literature: The QUASAR Project
…Not all tasks are not created equal - different tasks require different levels and kinds of student thinking.
Stein M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A. (2000). Implementing standards-based mathematics instruction: A casebook for professional development, p. 3. New York: Teachers College Press
Directions:
Give participants a minute to read the slide. or
Paraphrase the quote.

18. Linking to Research/Literature
There is no decision that teachers make that has a greater impact on students’ opportunities to learn and on their perceptions about what mathematics is than the selection or creation of the tasks with which the teacher engages students in studying mathematics.
Lappan & Briars, 1995
Directions:
Give participants a minute to read the slide. or
Paraphrase the quote.

19. Instructional Tasks: The Cognitive Demand of Tasks Matters
(SAY) Let’s look at the cognitive demands of a few more tasks and see if we make similar observations related to the characteristics of the demands of tasks. Are some more challenging than others? What makes them more challenging? Are some easier?

20. Linking to Research/Literature: The QUASAR Project
The Mathematical Tasks Framework
TASKS
as they appear in curricular/ instructional materials
TASKS
as set up by the teachers
TASKS as
implemented
by students
Student Learning
(SAY) We are now focusing on the first phase in the Mathematical Task Framework. The Framework was developed by the QUASAR Project. The study recognized that math tasks pass through phases during lessons. The most important phase is the first, the selection of a high-level task; without a high-level task, it is not possible to engage students in thinking and reasoning. In addition to the selection of high-level tasks, the QUASAR project learned that it was also important for teachers to think about how a task plays out as a teacher sets it up in the classroom and as students explore and discuss the task. 67% of high-level tasks are NOT carried out the way they are intended to be carried out. Therefore, it is important that teachers have opportunities to consider ways of maintaining the cognitive demand of tasks during implementation.
Stein, Smith, Henningsen, & Silver, 2000, p. 4

21. Linking to Research/Literature: The QUASAR Project (continued)
Low-Level Tasks
The Apples Task
High-Level
The Strings Task
(SAY) The QUASAR Project identifies two types of high-level tasks—“Doing Mathematics” and “Procedures With Connections.” The “Doing Mathematics” task is a more open-ended task, whereas in the “Procedures With Connections” task procedures are given to the student and they do not have to figure out how to solve the task. Instead, they must make connections or discuss mathematical relationships when solving the task. A “Doing Mathematics” task requires that students engage in formulating and carrying out a plan when solving a problem. The characteristics of these tasks are on the Mathematical Task Analysis Guide (TAG ) in your handout. Both appear on the right-hand side of the document.
Small Group:
Allow time for participants to review the characteristics in the TAG and compare them to the list of characteristics that they generated when comparing the Strings Task and the Apples Task.

22. Linking to Research/Literature: The QUASAR Project (continued)
Low-Level Tasks
Memorization
Procedures Without Connections (e.g., The Apples Task)
High-Level Tasks
Doing Mathematics (e.g., The Strings Task)
Procedures With Connections
Whole Group Discussion:
Ask participants to be specific about how the task aligns with the features on the TAG.
(SAY) Do any of the characteristics on the left-hand side of the TAG describe the Strings Task? How do the characteristics that we identified for the Strings Task compare with those on the TAG?
Probing Facilitator Questions and Possible Responses:
Do you think the Strings Task is a “Procedures With Connections” task or a “Doing Mathematics” task?
We call the Strings Task a “Doing Mathematics” task. This task is the highest level task. It requires the most thinking and reasoning from students.
What kind of task is the Apples Task?
It is a “Procedures Without Connections” task because students do not have to demonstrate any connections.

23. The Mathematical Task Analysis Guide
Research has identified characteristics related to each of the categories on the Mathematical Task Analysis Guide (TAG). How do the characteristics that we identified when discussing the Strings Task relate to those on the TAG? Which characteristics describe the Apples Task?
(SAY) Take a minute to read the characteristics in the “Doing Mathematics” category.
Probing Facilitator Questions and Possible Responses:
How do the characteristics that we identified when discussing the Strings Task relate to those on the TAG?
The task focuses on working with finding the addition equations two different ways.
Students must notice patterns in the list of equations. They may notice that is always 10, therefore the tens place always goes up by one.

24. The Cognitive Demand of Tasks (Small Group Work)
Working individually, use the TAG to determine if tasks A – L are high- or low-level tasks.
Identify and record the characteristics on the TAG that best describe the cognitive demand of each task.
Identify the CCSS for Mathematical Practice that the written task requires students to use.
Share your categorization in pairs or trios. Be prepared to justify your conclusions using the TAG and the Mathematical Practice Standards.
Facilitator Notes:
Three sets of tasks are available. Choose either A-D, E-H, or I-L.
Directions:
Read the directions on the slide. Give participants time to determine the differences among the four tasks that you have selected for teacher review. Responses can be written on the recording sheet in the participant handouts.
Tasks A-D and Possible Responses:
Task A - Low-Level Task (Memorization Task): The numbers are friendly numbers and ones that students often learn or memorize. Most of us can immediately arrive at the sum of the two numbers.
Task B - High–Level Task (Procedures with Connections): Students are shown two sides of a scale and asked to make the expressions balance one another without writing the same expression on each side. This task requires that students understand the meaning of equivalence.
Task C - High–Level Task (Procedures with Connections): Students are shown an algorithm and then the expectation is that they will just follow the procedure. Students must know the meaning of the amounts in each place, the ones and the tens place, in order to decompose and combine the amounts.
Task D - High–Level Task (Doing Mathematics): The prompt requires that students write a story problem that matches the addition expression. In doing so, students will have to determine what words to use to convey the meaning of addition or subtraction.
Tasks E-H and Possible Responses:
Task E - High–Level Task (Doing Mathematics): Students must show two different ways of solving the two-digit subtraction problem. In order to know the ways are different from each other, the student must be flexible when problem solving.
Task F - High–Level Task (Doing Mathematics): Students must solve each task and then compare and contrast the tasks to determine how they are similar or how they differ from each other. When comparing the situations, students will have to write about situations in which the whole is known and the part being subtracted is unknown. Students may also indicate that some situations involve an action and others involve work with static sets (the vanilla cookies and the chocolate cookies).

25. Identifying High-level Tasks (Whole Group Discussion)
Compare and contrast the four tasks. Which of the four tasks are considered to have a high-level of cognitive demand and why?
Directions:
Task E-H and Possible Responses::
Task G - Low-Level Task (Procedures without Connections): The student can use mental math to solve the problems. The student can also count back by ones or tens easily. The student has to know the meaning of the ones and tens; however, once the student knows an algorithm, this task does not have a high level of cognitive demand.
Task H - High-Level Task (Procedures without Connections): Students are shown an algorithm for a subtraction problem and then they are asked to repeat the problem solving process. Although they are asked to repeat the process, the student must be aware of the place value and have knowledge that a three-digit number can be decomposed. The student must also keep track of what gets combined in the end.
Task I - L and Possible Responses:
Task I High-Level Task (Low Mathematics): Once students know a rule for determining the largest number and ordering the numbers, this task can be done with very little effort. If students are not taught a rule and this task is used early in the study and ordering of sets, then this task will require students to know the value of the amounts in each place.
Task J High–Level Task (Doing Mathematics): Students are not given a way of thinking about this problem. There are 23 tens in 236 and 36 tens in 368. There is only one solution path; however, if the student was not taught a procedure initially, this would be a “Doing Mathematics” task. It would not be long before the level of this task would not be considered a high–level task.
Task K High-Level Task (Procedures With Connections): The student can use mental math to solve the problems. The request for students to study the patterns in the set of problems and to write an explanation of why one place changes continuously when other numbers in the three-digit number do not change makes this a high-level task.
Task L Low–Level Task (Memorization): Students are shown an algorithm for a subtraction problem and then they are asked to repeat the problem solving process. Although they are asked to repeat the process, the student must be aware of the place value and have knowledge that a three-digit number can be decomposed. The student must also keep track of what gets combined in the end.

26. Relating the Cognitive Demand of Tasks to the CCSS for Mathematical Practice
What relationships do you notice between the cognitive demand of the written tasks and the Standards for Mathematical Practice?
Possible Responses :
I notice that the high-level tasks include work on Standards for Mathematical Content and the Standards for Mathematical Practice, whereas the low-level tasks do not engage students in using the Standards for Mathematical Practice

27. The CCSS for Mathematical Practice
Make sense of problems and persevere in solving them.
Reason abstractly and quantitatively.
Construct viable arguments and critique the reasoning of others.
Model with mathematics.
Use appropriate tools strategically.
Attend to precision.
Look for and make use of structure.
Look for and express regularity in repeated reasoning.
Directions:
Read the directions on the slide.
Probing Facilitator Questions and Possible Responses:
Let’s look at some of the tasks we called “Doing Mathematics” tasks. Did they require that students use the Standards for Mathematical Practice to do the tasks?
Let’s look at Task K.
Yes, students must write about the structure of mathematics. They must know that the 10s place changes when a 10 is added to it and that the 100s place changes when 100 is added to the 100s place.
Let’s look at Task F.
Yes, students must know that each of these problems starts with a whole amount. They must recognize that two other parts exist. In some cases, the part being taken away is known and in other situations, it is not known. Students must be able to model the problems with equations and manipulatives. We will also get a sense of whether or not students refer to the context once they write an equation (MP2). They also must know about the structure of these kinds of problems and know that it is okay to know the part being subtracted or to not know the part being subtracted. Students must construct an argument for why the amounts are similar or different.
Let’s look at Task E.
In order to create more than one solution path to this task, students must know the meaning of the quantities and that they can decompose and recompose the amounts. Students will model with mathematics. We know if they understand the structure of subtraction and addition as the student decomposes and recomposes the amounts.
Let’s look at Task B.
The student must understand the meaning of equivalence (MP7). The students’ equations serve as an argument about equivalence. Students must create an argument for determining if the expressions are equivalent without solving each expression (MP3). In order to do so, students must know that it is acceptable to “juggle” the amount in one set to the other set and that this will not change the sum.
Common Core State Standards, 2010

28. Linking to Research/Literature: The QUASAR Project
If we want students to develop the capacity to think, reason, and problem-solve, then we need to start with high-level, cognitively complex tasks.
Stein, M. K. & Lane, S. (1996). Instructional tasks and the development of student capacity to think and reason: An analysis of the relationship between teaching and learning in a reform mathematics project. Educational Research and Evaluation, 2 (4),
Directions:
Give participants a minute to read the quote or
Paraphrase the quote if desired.

29. Linking to Research/Literature
Tasks are central to students’ learning, shaping not only their opportunity to learn but also their view of the subject matter.
Adding It Up, National Research Council, p. 335, 2001
Directions:
Give participants a minute to read the quote or
Paraphrase the quote if desired.

30. Gallery Walk Procedure
Circulate and analyze the modified tasks of the other groups.
On a yellow sticky-note, comment about the ways in which the task was modified to increase the cognitive demand of the task.
On a pink sticky-note, write wonderings if you can think of other ways the demand of the task can be increased.
Directions:
Ask participants to identify aspects of the textbook pages that have a high level of cognitive demand. If the textbook pages do not have a high level of cognitive demand, then what questions might you ask to increase the cognitive demand of the task?
Textbook Page 1a and 1b
The last question on page 240 is a high level of demand because it requires that students consider the relationship between the tens and ones when considering when regrouping is needed.
Other questions to consider asking:
What does the 1 mean when you write it above the tens place?
Look at Problem 1 ( ). Can we just write 11 as the sum and then add and write 50 in the sum? Will this give us the same sum? If so, why?
Textbook Page 2a and 2b
Can you do Problem 1 two different ways? Why can you add and then subtract, or why can you show the two sets that you have (the 16 and the 11, and take 14 from the 16, this results in 2 and then you combine this with 11).
Tell me about the two problems in box one. Are they put together or take apart type problems?
What tells you that the amounts are being put together? What tells you when something is being taken away?
Can you draw the part-part whole to show which of these you know in the situation?
Why does one situation increase the size of the set and the other problem results in a smaller set?

31. References
Smith, M. S., Stein, M. K., Arbaugh, F., Brown, C. A., & Mossgrove, J. (2004). Characterizing the cognitive demands of mathematical tasks: A task-sorting activity. In G. W. Bright and R. N. Rubenstein (Eds.), Professional development guidebook for perspectives on the teaching of mathematics: Companion to the sixty-sixth yearbook (pp ). Reston, VA: National Council of Teachers of Mathematics. Smith, M. S. & Stein, M. K. (1998). Selecting and creating mathematical tasks: From research to practice. Mathematics Teaching in the Middle School, 3 (5),