1. SESSION 3: Rigor Stop 3 of your instructional rounds
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SESSION 3: Rigor Stop 3 of your instructional rounds

2. We will be looking to see if the next lesson...
Session 3: Rigor
We will be looking to see if the next lesson...
Includes opportunities for students to become more fluent (efficient and accurate) with procedures.
Includes opportunities for students to build their conceptual understanding.
Includes opportunities for students to apply their thinking, particularly in real world situations.

3. Session 3: Rigor Participants will be able to:
Goals and Purpose
Session 3: Rigor
Participants will be able to:
Participants will be able to describe the three aspects of rigor (procedural fluency, conceptual understanding, and application) generally and specifically at this grade/course-level.
We’ll take that jargon-y word, “rigor” and see what it really looks like at this level.

4. Agenda
Session 3: Rigor
Rigor: Leading the implementation of a rigorous curriculum
What is rigor?
Why rigor?
Rigor in the standards
Rigor in balance
Rigor at this grade level
How to prepare to look for rigorous instruction
What to look for in the lesson
Guiding conversations post-observation
Preparing feedback for growth
Here is an overview of the approach today. We’ll start by digging into the meaning of rigor at this grade level, evaluate some lessons for rigor, and then look at assessment materials.

5. Why Rigor?

6. Session 3: Rigor
Why Rigor?
“Rigor refers to deep, authentic command of mathematical concepts, not making math harder or introducing topics at earlier grades.”
Why is “rigor” emphasized in the standards?
A common misconception is that rigor just means “hard.” It doesn’t. “Rigor” has a specialized meaning in the context of Common Core math. From CoreStandards.org: “Rigor refers to deep, authentic command of mathematical concepts, not making math harder or introducing topics at earlier grades.”
This morning we talked about what mathematical content is important, and what connections exist between standards.
Rigor has everything to do with how students engage with mathematical content: it implies a balance of conceptual understanding, procedural skill and fluency, and application. But why do we care about these aspects of mathematical understanding?

7. From “Adding It Up” Session 3: Rigor
Researchers in the late 90s identified 5 components (or strands) of mathematical proficiency.
Note particularly these three: (conceptual understanding, procedural fluency, and strategic competence). These form the basis of what we call “rigor” in the standards.
We will revisit the other strands tomorrow when discussing the mathematical practices.

8. From the National Mathematics Advisory Panel
Session 3: Rigor
From the National Mathematics Advisory Panel
“To prepare students for Algebra, the curriculum must simultaneously develop conceptual understanding, computational fluency, and problem solving skills.”
The National Mathematics Advisory Panel noted the importance of pursuing the aspects of rigor in balance.
From the report: “Debates regarding the relative importance of these aspects of mathematical knowledge are misguided. These capabilities are mutually supportive, each facilitating learning of the others. Teachers should emphasize these interrelations; taken together, conceptual understanding of mathematical operations, fluent execution of procedures, and fast access to number combinations jointly support effective and efficient problem solving.”

9. From TIMSS Video Study Session 3: Rigor
TIMSS video study of the late 1990s compared eighth grade math instruction in a variety of countries.
Despite the recognition that a balance of the different strands of mathematical proficiency is necessary (Adding It Up), on average, 75% of “private work time” is spent repeating procedures in the United States.
NOTE: Repeating procedures in this context means following modeled algorithms/procedures rather than independently applying strategies (without heavy guidance) or time spent demonstrating conceptual understanding by explaining thinking, right?
QUESTION PARTICIPANT MAY ASK:
Might have a leader talk about how our students/culture is different....need to emphasize what the research says what students need, ... and what we control.

10. From TIMSS Video Study Session 3: Rigor
Additionally, the kind of non-routine problem solving implied by the application shift does not seem to be happening either.
Compared to some other high-performing nations, we spend a smaller amount of time on problems that take longer than 45 seconds. (Only 61% of problems are solved in longer than 45 seconds.)
QUESTIONS PARTICIPANTS MAY ASK:
I think a lot of times educators talk about how technology prevents students from spending extended focus time on problem solving...this graphic completely debunks that. Looking at developed nations, have large access to technology, so excuse that students don't have stamina isn't valid.

11. Session 3: Rigor
Shift 3: Rigor
Procedural Skill and Fluency: The standards call for speed and accuracy in calculation.
Conceptual Understanding: The standards call for conceptual understanding of key concepts, such as place value and ratios. 
Application: The standards call for students to use math in situations that require mathematical knowledge.
The Common Core names and emphasizes three aspects of rigor that are a direct reflection of this research on mathematical understanding.
Procedural shift refers both to “skill” and “fluency”. In some cases, speed with calculations is particularly emphasized.
Fluency should be built after conceptual understanding has been achieved.
Students can still do the work if they aren’t fluent - they just do it slowly.
From the front matter: “But what does mathematical understanding look like? One way for teachers to do that is to ask the student to justify, in a way that is appropriate to the student’s mathematical maturity, why a particular mathematical statement is true or where a mathematical rule comes from.”
Application is not “just doing a bunch of real-world problems” but should genuinely require that students know which ideas to apply when.

12. Rigor in the Standards

13. Rigor in the Standards Protocol: Do the math.
Session 3: Rigor
Rigor in the Standards
Protocol:
Do the math.
For each, what aspects of rigor are emphasized and how do you know?
Bonus: What are the grade level standards associated with each one?
Rigor is baked into the standards. We’ll start to see what rigor looks like and how it is embodied in the standards.
For each task, “do the math” and discuss which aspects of rigor are present.
If you can, determine the grade level standard associated with each one.

14. Session 3: Rigor
Task #1
Juanita spent $9 on each of her 6 grandchildren at the fair. How much money did she spend?
Nita bought some games for her grandchildren for $8 each. If she spent a total of $48, how many games did Nita buy?
Helen spent an equal amount of money on each of her 7 grandchildren at the fair. If she spent a total of $42, how much did each grandchild get?
Take a moment to “do” this task from Illustrative Mathematics.
Answers:
a) Juanita spent $54.
b) Nita bought 6 games.
c) Each grandchild gets $6.
The highlighted aspect of rigor is application because students must use multiplication and division to solve real-world problems. Note the mix of multiplication and division within the task--students must choose different strategies for each problem. This is standard 3.OA.A.3.
Source:

15. Session 3: Rigor
Task #2
The teacher gives the equation 4 × ___ = 12. Charlie finds the answer by writing and solving 12 ÷ 4 = ___.
Explain why Charlie’s method works.
Take a moment to “do” this task from EngageNY.
Answer: Participants may come up with a variety of responses including verbal descriptions and drawings that relate division to multiplication.
The aspect of rigor is conceptual understanding -- students must have a deep understanding of the meaning of each equation and the relationship between multiplication and division. This is standard 3.OA.B.6.
Source:

16. Task #3 1. Facts for speed and accuracy: a. ____ x 5 = 15
Session 3: Rigor
Task #3
1. Facts for speed and accuracy:
a. ____ x 5 = 15
b. 10 ÷ 1 = ___
c. ___ = 6 x 10
d. ___ =20 ÷ 5
e. ___ = 7 x 10
f. 1 x 6 = ___
g. 9 x 2 = ___
h. 0 x 5 = ___
Take a moment to “do” this task from Student Achievement Partners.
Answers:
a) 3, b) 10, c) 0, d) 4, e) 70, f) 6, g) 18, h) 0
The highlighted aspect of rigor is procedural fluency because students must perform computations quickly and accurately.
It’s important to note that there are standards that emphasize procedures, but may not include the word “fluently”--we may emphasize performing procedures accurately with less emphasis on speed for these standards.
Source:

17. Where is the Rigor? - 3.O.A 3.OA.A.3
Session 3: Rigor
Where is the Rigor? - 3.O.A
3.OA.A.3
Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem. 1.
3.OA.B.6
Understand division as an unknown-factor problem. For example, find 32 ÷ 8 by finding the number that makes 32 when multiplied by 8. 2.
Note the rigor as it appears in the standards. Some standards strongly imply procedural skill and fluency, conceptual understanding, and application.
Task 1 was aligned to 3.OA.A.3; the emphasis on solving word problems in the standard implies application.
Task 2 was aligned to 3.OA.B.6; the emphasis on “understanding” in the standard implies conceptual understanding.
Task 3 was aligned to 3.OA.C.7; the emphasis on “fluently multiplying and dividing” in the standard implies procedural fluency.
Note that many standards may imply more than one aspect of rigor.
3.OA.C.7
Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers. 3.

18. Identifying Rigor in the Standards
Session 3: Rigor
Identifying Rigor in the Standards
Procedural Fluency
Conceptual Understanding
3.OA.C.7
Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers.

19. Looking at Grade 3, Module 1
Session 3: Rigor
Looking at Grade 3, Module 1
Examine the standards for Module 1 . What are the aspects of rigor associated with each one? (There may be more than one!)
Bonus: Describe the kinds of problems you’d expect to see associated with each standard.
Examine the standards for Module 1. What are the aspects of rigor associated with each one?
[Facilitators may choose to have participants actually do highlighting (markers will be provided at tables) of the standards for this activity. This will give a visual picture of the rigor in the standards.]

20. Share Out Session 3: Rigor
Ask participants to share with a neighbor, then highlight a few responses with the whole group.
[Facilitators generate some examples of standards and associated rigor for Module 1 from this grade band]
Standards w/ Associated Rigor for Module 1
Grade 3:
3. OA.A.1 - Interpret products (CONCEPTUAL UNDERSTANDING) of whole numbers, e.g., interpret 5x7 as the total number of objects in 5 groups of 7 objects each.
3.OA.A.3 - Use multiplication and division within 100 to solve (APPLICATION) word problems in situations involving equal groups, arrays, and measurement quantities.
3.OA.C.7 - Fluently multiply and divide (PROCEDURAL FLUENCY) within 100, using strategies such as the relationship between multiplication and division or properties of operations.
Grade 4:
What are the aspects of rigor associated with each standard in Module 1?

21. Rigor in the Balance
Let’s now consider the charge put forth by research -- how do we pursue a “balance of rigor”?

22. From the Publisher’s Criteria
Session 3: Rigor
From the Publisher’s Criteria
The Standards…set high expectations for all three components of rigor in the major work of each grade.
NOTE: Criteria were developed to help states, districts, schools faithfully implement the common core standards through evaluation of curricular materials." Crosswalking not good enough, good curricular must be able to speak to the spirit of the standards.
The Publishers’ Criteria makes clear this balance is important.
“Of course, that makes it necessary that we first follow through on the focus in the Standards” (From Publishers’ Criteria) - We must have a focused curriculum in order for teachers to be able to develop fluencies, conceptual understanding, and application (it’s an issue of time, essentially).
(May need to intro what the publisher’s criteria is and does)

23. From the Publisher’s Criteria
Session 3: Rigor
From the Publisher’s Criteria
(1) The three aspects of rigor are not always separate in materials.
(2) Nor are the three aspects of rigor always together in materials.
The aspects of rigor are not always neatly separated.
But the aspects of rigor can be separated.

24. Session 3: Rigor
Example
Task
Decide if the equations are TRUE or FALSE. Explain your answer.
a. 4 x5 = 20
b. 34 = 7 x 5
c. 3 x 6 = 9 x 2
d. 5 x 8 = 10 x 4
e. 6 x 9 = 5 x 10
f. 2 x (3 x 4) = 8 x 3
g. 8 x 6 = 7 x 6 + 6
h. 4 x (10 + 2) =
Take a moment to “do” this task. [Give participants time to try the task.]
What aspects of rigor do you see in this example?
In this example, we see two aspects of rigor highlighted -- conceptual understanding and procedural fluency. The task can be completed by performing a large number of computations, but also highlights the concept of equivalence (see language in the stem, “Decide the equations are true or false) and reasoning about the properties/finding patterns. For example, in part b, the equation is not true because the value of 7 x 5 is 35. Students may also recognize that all multiples of 5 end in 0 or 5 and conclude that this equation is not true without evaluating 7 x 5.
From IM commentary: “On the surface, both tasks can be completed with sound procedural fluency in addition and multiplication. However, these tasks present the opportunity to delve much more deeply into equivalence and strategic use of mathematical properties. These tasks add clarity to the often misunderstood or neglected concept of equivalence. Students often understand the equal sign as the precursor to writing the answer. Class discussion should be carefully guided to ensure that students come to the understanding that the equal sign indicates equivalence between two expressions. Though these tasks can be completed by evaluating each expression on either side of the equal sign, they present deliberate next levels of reasoning that invite students to look for different approaches. Anyone facilitating a conversation about this task should constantly ask, "Is there another way to know whether this equation is true?" “

25. Balance of Rigor in Grade 3, Module 1
Session 3: Rigor
Balance of Rigor in Grade 3, Module 1
What will a “balance of rigor” look like?
What will it look like to have tasks or activities where rigor is not “separated”?
NOTE: high quality fluency exercise--- one that reinforces conceptual understanding while also reinforcing fluency
For Module 1 in your grade, what will a “balance of rigor” look like?
[Participants should come to see here that a “balance” does not necessarily imply 33% - 33% - 33%. The standards represent a balance. Lessons and units should reflect the rigor in the standards. Some activities, lessons, assessments, and units may emphasize one or more aspects, depending on the targeted standards. (i.e. a fluency activity that is focused on one skill or standard, or a sequence of lessons that are focused on a standard about word problems) We do not need “all three every day.”]
What will it look like to have tasks or activities where rigor is not “separated”?
Possible responses:
Grade 3: The standards for Module 1 imply a pretty even balance among procedural fluency, conceptual understanding, and application. An example of an activity in which rigor is “not separated” is finding quotients using a division procedure, while explaining the quotient as an unknown factor (procedural/conceptual).
Grade 4: The standards for module 1 primarily involve application and procedural skill. An example of an activity in which rigor is “not separated” is solving a word problem that involves fluently adding or subtracting multi-digit numbers.
Grade 5: The standards for Grade 5 module 1 mostly imply procedural skill/fluency and conceptual understanding. An example of an activity in which rigor is “not separated” is performing multiplication by powers of ten while also explaining patterns in the number of zeros.

26. Rigor at this Grade Level

27. Rigor at this Grade Level
Session 3: Rigor
Rigor at this Grade Level
Examine the tasks and activities in the lessons and problem sets within Grade 3 Module 1.
Find at least one task or activity that emphasizes procedural fluency.
What evidence do you have?
Find at least one task or activity that emphasizes conceptual understanding.
Find at least one task or activity that emphasizes application.
Let’s have a look at what rigor actually looks like for your grade level in Module 1.
What is the evidence of each aspect of rigor you can find?
Before you dive in, let’s take a quick look at the features of the ENY lessons.

28. Fluency Practice Session 3: Rigor
Each lesson has a fluency activity associated with it.

29. Application Problem Session 3: Rigor
Lessons typically include an application problem at the beginning, as well.

30. Concept Development Session 3: Rigor
The concept development is the heart of the lesson where new content is introduced.

31. Problem Set Session 3: Rigor
The problem set is a mix of exercises for student practice.

32. Student Debrief Session 3: Rigor
The student debrief are closing questions the teacher uses after the problem set.

33. Exit Ticket Session 3: Rigor
The lesson assessment is an exit ticket, tied to the outcomes of the lesson.

34. Homework Session 3: Rigor
Each lesson also comes with a homework assignment.

35. Rigor at this Grade Level
Session 3: Rigor
Rigor at this Grade Level
Examine the tasks and activities in the lessons and problem sets within Module 1.
Find at least one task or activity that emphasizes procedural fluency.
What evidence do you have?
Find at least one task or activity that emphasizes conceptual understanding.
Find at least one task or activity that emphasizes application.
Okay, now go into your first module and find evidence of rigor.
[Optional] Record each task or activity on a separate piece of chart paper, so you can present it to the group.

36. Session 3: Rigor
Share Out
What are the characteristics of activities that show evidence of rigor at your grade level?
Share out. [Optional] Participants hang their chart papers on the wall and everyone walks around and reads the tasks/activities and description of evidence.
[Facilitators create a list of examples of activities/tasks from Module 1 that show evidence of rigor.]

37. Share Out Session 3: Rigor
Share out. [Optional] Participants hang their chart papers on the wall and everyone walks around and reads the tasks/activities and description of evidence.
[Facilitators create a list of examples of activities/tasks from Module 1 that show evidence of rigor.]
Standards w/ Associated Rigor for Module 1
Grade 3:
3. OA.A.1 - Interpret products (CONCEPTUAL UNDERSTANDING) of whole numbers, e.g., interpret 5x7 as the total number of objects in 5 groups of 7 objects each.
3.OA.A.3 - Use multiplication and division within 100 to solve (APPLICATION) word problems in situations involving equal groups, arrays, and measurement quantities.
3.OA.C.7 - Fluently multiply and divide (PROCEDURAL FLUENCY) within 100, using strategies such as the relationship between multiplication and division or properties of operations.
What are the characteristics of activities that show evidence of rigor at your grade level?

38. Leading Rigorous Implementation
Session 3: Rigor
Leading Rigorous Implementation
Is the instruction appropriately rigorous while remaining aligned to the standards?
How to prepare to look for rigorous instruction as intended by the standards:
Prework:
Read the standard closely, looking for explicit language that calls out expectations around:
Fluency
Conceptual understanding
Application
Complete instructional rounds with standards app

39. Leading Rigorous Implementation
Session 3: Rigor
Leading Rigorous Implementation
Is the instruction appropriately rigorous while remaining aligned to the standards?
What to look for:
Opportunities for students to become more fluent (efficient and accurate) with procedures.
Students practice, many times “on the clock”, with facts and procedures
Fluency activities build upon conceptual understandings students already have
Opportunities for students to build their conceptual understanding
Students being asked “why” and to rationalize their thinking
Students working with models to process their thinking
Opportunities for students to apply their thinking, particularly in real world situations
Students working on rich math problems
Students use math absent of external prompts (e.g. “use addition to solve this problem”)

40. Leading Rigorous Implementation
Session 3: Rigor
Leading Rigorous Implementation
Is the instruction appropriately rigorous while remaining aligned to the standards?
Guiding conversations after the walk through - Fluency
If fluency opportunities are not present:
Ask where fluency practice is/will be built in upcoming lessons
students show fluency as a limiter in their math work (evidence: can still do the work but are slower or inaccurate with facts, procedures):
Ask how students’ lack of fluency will be addressed
Consider curriculum: fluency activities from study high quality lessons for the area that is limiting students
slow

41. Leading Rigorous Implementation
Session 3: Rigor
Leading Rigorous Implementation
Is the instruction appropriately rigorous while remaining aligned to the standards?
Guiding conversations after the walk through – Conceptual Understanding
If conceptual understanding opportunities are not present:
Ask how more opportunities may be worked in to dig into what the students are thinking when working with math concepts
If students show conceptual understanding as a limiter in their math work:
Consider gaps: Re-ask questions in coherence activities
Consider curriculum: study high quality lessons aligned to the standard of focus

42. Leading Rigorous Implementation
Session 3: Rigor
Leading Rigorous Implementation
Is the instruction appropriately rigorous while remaining aligned to the standards?
Guiding conversations after the walk through – Application
If application opportunities are not present:
Ask how more application opportunities can be folded into the student math experience.
If students are provided external prompts to complete application problems:
Ask how the teacher can adapt opportunities so that students can apply math without the prompting.
Consider curriculum: study high quality tasks aligned to the standard of focus

43. Leading Rigorous Implementation
Session 3: Rigor
Leading Rigorous Implementation
Is the instruction appropriately rigorous while remaining aligned to the standards? Prepare feedback for growth.

44. Stop 3 of your instructional rounds
Session 3: Rigor
Stop 3 of your instructional rounds
5th Grade Measurement and Fractions
Standards: 5.MD.1, 5.NF.3
Prepare:
Read the standard closely, looking for explicit language that calls out expectations around:
Fluency
Conceptual understanding
Application
Complete instructional rounds with standards app

45. Stop 3 of your instructional rounds
Session 3: Rigor
Stop 3 of your instructional rounds
Watch the 11min video: 5th Grade Measurement and Fractions
What to look for:
Fluency:
Students practice, many times “on the clock”, with facts and procedures.
Fluency activities build upon conceptual understandings students already have.
Conceptual understanding:
Students being asked “why” and to rationalize their thinking.
Students working with models to process their thinking.
Application:
Students working on rich math problems.
Students use math absent of external prompts (e.g. “use addition to solve this problem”)
NOTES
Fluency:
Students practice, many times “on the clock”, with facts and procedures
Fluency activities build upon conceptual understandings students already have; no evidence of "on the clock" fluency
Conceptual understanding:
Students being asked “why” and to rationalize their thinking; Students working with models to process their thinking; Do now discussion and kids explaination of their math (sub being split among 4 people); the girl in the pink sweater explained a lot - but her thinking is incorrect conceptually; In response to are they getting more than or less than a whole sub: "Less than because a decimal is less than a whole number." There was not explicit evidence that the whole class was engaged in deep conceptual learning. The conversations (whole group/small group) seemed to be dominated by 1 or 2 students. When a student did not understand, there did not seem to be an opportunity to check back to see if they got it.
Application:
Students working on rich math problems; Students use math absent of external prompts (e.g. “use addition to solve this problem”); The sub DO Now activty is a rich math problem; the Do NOw seemed to be designed for them to "use math absent of external prompts" because it didn't tell them what math to use to solve it;

46. After the Walk Through - Questions to Ask
Session 3: Rigor
After the Walk Through - Questions to Ask
Leading the Conversation – Fluency
If fluency opportunities are not present:
Ask where fluency practice is/will be built in upcoming lessons.
If students show fluency as a limiter in their math work:
Ask how students’ lack of fluency will be addressed.
Consider curriculum: fluency activities from study high quality lessons for the area that is limiting students

47. After the Walk Through - Questions to Ask
Session 3: Rigor
After the Walk Through - Questions to Ask
Leading the Conversation – Conceptual Understanding
If conceptual understanding opportunities are not present:
Ask how more opportunities may be worked in to dig into what the students are thinking when working with math concepts.
If students show conceptual understanding as a limiter in their math work:
Consider gaps: Re-ask questions in coherence activities
Consider curriculum: study high quality lessons aligned to the standard of focus

48. After the Walk Through - Questions to Ask
Session 3: Rigor
After the Walk Through - Questions to Ask
Leading the Conversation – Application
If application opportunities are not present:
Ask how more application opportunities can be folded into the student math experience.
If students are provided external prompts to complete application problems:
Ask how the teacher can adapt opportunities so that students can apply math without the prompting.
Consider curriculum: study high quality tasks aligned to the standard of focus

49. After the Walk Through - Feedback for Growth
Session 3: Rigor
After the Walk Through - Feedback for Growth

50. Reflection Think about Common Core implementation in your school:
Session 3: Rigor
Reflection
Think about Common Core implementation in your school:
What is the state of curriculum/planning around rigor in your building?
What are some preliminary steps that are needed to improve the state of rigor in your building?
REFLECTION:
Highlight - Research showing amount of time students around the world spend time on the actual work of doing math. Think about how this might impact the design/planning of math time in the classroom.
QUESTIONS PARTICIPANTS MAY ASK:
How do I get teachers to engage in this level of work when students won't sit in the chair? Real issue faced by teachers and administrators. Have administrators think about how/why students behave the way they do in the classroom? What might this level of deep scaffolded engagement building to fluency support a more productive learning environment? What structures would need to be in place to engage students in the work
So should my teachers be spending each day with 1/3, 1/3, 1/3 of class period on each component of rigor? NO! It varies

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Day1: Setting Teachers Up For Success: Math Standards Implementation
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53. Day1: Setting Teachers Up For Success: Math Standards Implementation
Reference List
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54. Day1: Setting Teachers Up For Success: Math Standards Implementation
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