1. SESSION 2: Coherence Stop 2 of your instructional rounds
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Marc
Welcome to the 2015 Standards Institute hosted by NewOrg!
SESSION 2: Coherence Stop 2 of your instructional rounds

2. We will be looking to see...
Session 2: Coherence
We will be looking to see...
carefully connects learning across grades so that students can build new understanding onto foundations built in previous years.
leverages how the standards were built in how they reinforce a major topic in a grade by utilizing supporting, complementary topics.
30 secs
Marc

3. Session 2: Coherence Participants will be able to:
Goals and Purpose
Session 2: Coherence
Participants will be able to:
Describe the coherence shift both as a logical sequencing of content across grades and important connections between standards, clusters, and domains within the grade.
Marc
Here’s our objective. (30 sec)

4. Agenda
Session 2: Coherence
Coherence: Leading the implementation of a coherent curriculum
What is coherence and what does it look like?
What does coherence look like during implementation and how do we guide growth?
How to prepare to lead coherent instruction
What to look for in the lesson
Guiding conversations post-observation
Preparing feedback for growth
Marc
Here’s what we’ll do today during this session.

5. Coherence: What and Why?
Session 2: Coherence
Coherence: What and Why?
How would you explain to a student why 2/3 is equal to 4/6?
Marc
(3 min)
Ask participants how they would explain this to a 3rd grader that is just beginning the study of fractions. (1 min independent)
Share responses such as multiplying numerator and denominator by same number, using visuals, using number line. (1 minute)
(Click to show multiplication.) Explaining fraction equivalence is challenging when students do not yet understand how to multiply fractions; most ways we think of to explain equivalence rely on fraction multiplication. (1 min)
The study of mathematics rests on the idea of developing ideas from existing ones. Similarly this is how students learn; they learn mathematics based on what they already understand.
TAKE AWAY – Students learn based on what they already understand, developing ideas from existing ones.

6. Session 2: Coherence
What’s the Right Order?
a. Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line.
b. Interpret multiplication as scaling (resizing) by…relating the principle of fraction equivalence a/b = (n × a)/(n × b) to the effect of multiplying a/b by 1.
c. Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size.
Grade 3
Grade 5
Grade 4
Marc
(2 minutes)
Do individually – acknowledge technical and mathy
Ask participants to try to order these standards sequentially.
Click to Share correct answer:
A = Grade 3
B = Grade 5
C = Grade 4
Emphasize logical progression. It is baked into the standards for us.

7. Session 2: Coherence
Coherence is Key
“A focused, coherent progression of mathematics learning, with an emphasis on proficiency with key topics, should become the norm in elementary and middle school mathematics curricula. Any approach that continually revisits topics year after year without closure is to be avoided. By the term focused, the Panel means that curriculum must include (and engage with adequate depth) the most important topics underlying success in school algebra. By the term coherent, the Panel means that the curriculum is marked by effective, logical progressions from earlier, less sophisticated topics into later, more sophisticated ones. Improvements like those suggested in this report promise immediate positive results with minimal additional cost.”
Marc
(1 min)
KEY POINT: The idea of logical progressions of learning is an important one. This is what mathematics is: developing new ideas from existing ones.
In the Final Report of the National Mathematics Advisory Panel, this idea was highlighted.
Anticipated Questions: how do we know this is important? Can connect to what lead writers of CCSS heard in testimony, noticed in college students: seeing math as branching out of isolated skills and procedures that get more complicated without seeing underlying principles and how all math is unified by them (McCallum video on coherence);

8. Session 2: Coherence
The Progressions
“The Common Core State Standards in mathematics were built on progressions: narrative documents describing the progression of a topic across a number of grade levels, informed both by research on children's cognitive development and by the logical structure of mathematics…”
Katie
(15 seconds)
In fact logical progressions undergird the standards. These documents describe the logical, gradual progression of content across grades.
These are “research-based learning progressions detailing what is known today about how students’ mathematical knowledge, skill, and understanding develop over time.”

9. The Progressions Session 2: Coherence Katie (1 min)
Sample of one of the progressions documents
Rhetorical question – Think about how powerful this document could be for the development of your teachers?

10. Coherence Across Grades

11. Session 2: Coherence
Definition #1
Learning is carefully connected across grades so that students can build new understanding onto foundations built in previous years. 
Katie
(30 seconds)
Coherence will have a couple of meanings for us, but one important one is about the progression of content across grades.

12. Picture of Coherence, Part 1
Session 2: Coherence
Picture of Coherence, Part 1
Complete the following problems without a calculator. How did you compute them?
8 + 5
3/7 + 6/7
2 5/ /9
What do your procedures have in common?
Katie
Independent activity (1 min 30 secs)
Have participants do these problems the way they normally would do them.
Table Share Out (2 min) – discuss at your table
Whole Group Share Out (1 min) - Can I have two or three people in this room share out what your procedures have in common?
This is about grouping and redistributing (aka composing and decomposing) but don't give too much away here since going over in detail in next several slides
Answers 13 85
1 2/7
8 4/9

13. Truths About Numbers... Numbers have organizers
Session 2: Coherence
Truths About Numbers...
Numbers have organizers
10 is the organizer for whole numbers
1 is the organizer for fractions
Numbers can be composed and decomposed, and by doing so, the value of those numbers does not change
Katie
Share this information. (30 secs)

14. Picture of Coherence, Part 2a
Session 2: Coherence
Picture of Coherence, Part 2a
These are whole numbers, so we will apply the organizer of 10.
We will also apply the fact that we can compose and decompose numbers to reorganize to complete these problems by applying the same principles.
8 + 5
Kindergarten
Grade 2
Katie
(1 min 30 sec)
By applying these two truths, these two problems can be attacked in the same manner.
Problem 1:
I look at these numbers and know the answer will be greater than 10.
I see I can make a ten with the 8 if I had a 2.
Then I strategically “decompose” the 5 into a 2 and 3. This does not change the value of 5.
Then I compose a 10 with the 8 and 2.
This new organization gives me a much simpler problem of , which equals 13.
Problem 2:
I see immediately that have groups of 10s and groups of 1s, so I start by decomposing the groups of tens from the groups of ones (20, 7, 50, 8.)
Then I see the numbers in the ones place will be greater than 10.
I see I can make a ten with the 7 if I had a 3 (I could have also done this with the 8 and a 2, but I am going to work with the 7, just because I can!)
Then I strategically “decompose” the 8 into a 3 and 5. This does not change the value of 8.
Then I compose a 10 with the 7 and 3.
This new organization gives me a much simpler problem of , which equals 85.

15. Picture of Coherence, Part 2b
Session 2: Coherence
Picture of Coherence, Part 2b
These are fractions, so we will apply the organizer of 1.
We will also apply the fact that we can compose and decompose numbers to reorganize to complete these problems by applying the same principles.
3/7 + 6/7
2 5/ /9
Grade 4
Katie
(1 min 30 sec)
Better yet, these two fractions problems can also be attacked in the same manner as the whole numbers.
Problem 3:
I look at these numbers and know the answer will be greater than 1.
I see I can make a one with the 6/7 if I had a 1/7.
Then I strategically “decompose” the 3/7 into a 1/7 and 2/7. This does not change the value of 3/7.
Then I compose a 1 with the 6/7 and 1/7.
This new organization gives me a much simpler problem of 1 + 2/7, which equals 1 2/7.
Problem 4:
I see immediately that have groups of 1s and groups of parts of 1 (fractions), so I start by decomposing the groups of ones from the groups of fractions (2, 5/9, 5, 8/9.)
Then I see that when I add the fractions I will get a number greater than 1.
I see I can make a one with the 8/9 if I had a 1/9 (I could have also done this with the 5/9 and a 4/9, but I am going to work with the 8/9, just because I can!)
Then I strategically “decompose” the 5/9 into a 1/9 and 4/9. This does not change the value of 5/9.
Then I compose a 1 with the 8/9 and 1/9.
This new organization gives me a much simpler problem of /9, which equals 8 4/9.
Do you remember how we used to add mixed numbers? Stack them vertically, bring down the denominator, add the numerators, add the whole numbers, go to the side and divide the numerator by the denominator, and then remember which is the whole number, which is the numerator, which is the denominator, and then add this new whole number to the previous whole number sum.
This other way - using the truths about numbers, is much simpler, makes sense, and allows students to see the connection across different types of numbers and it sets the stage for combining like terms, a foundational skill in algebra which is learned in middle school, and then is amply applied in calculus many years later.
KEY Don't forget to sum up with last point: these problems are tied together by fundamental truths about numbers that connect to math from kindergarten addition through calculus

16. Vertical Coherence Challenge
Session 2: Coherence
Vertical Coherence Challenge
In your groups, you have 17 standards on pieces of paper; most come from the OA domain in grades 1-5.
They are not labeled!
Determine which standards are prerequisites for other standards.
Note: They will not all string together in a single branch.
Katie - We just talked the fact there are fundamental truths in mathematics that connect across grade levels. Now you are going to have time to explore the how the standards span across grades.
Heather Takes Over
(5 min)
Can we see the vertical coherence of the standards?
Work in pairs
Let participants know that there is more than one branch.
Tell them they should not use resources, just what they know about skills progression. Read the words and discuss the grades and the transitions and the way the verbs change.
String together standards coherently.
(2 min)
Independent - Take a minute to reflect on how you are feeling about this activity. Share with your table.
Now we are going to let you use resources and to continue this challenge.
Link to standards doc participants will use:
See bottom of facilitator guide for “answer key.

17. Progression Intervention
Session 2: Coherence
Progression Intervention
Read OA progression.
Using standards app, annotate where the standards lie.
Revise: Use the Progression on Operations & Algebraic Thinking to help revise your organization.
Bonus: Can you determine which standards belong in which grade?
Grades 1 and 2
Grade 3
Grade 4
Grade 5
Grades 6 and 7
Heather
( 8 minutes)
Take 5 minutes to read the OA progression for Grades Think about might change the order of you sequence based on this document.
Take 3 minutes to work with your partner to revise your sequence.
Now take a moment to try to revise your ordering.

18. A Picture of Coherence D I F K P C M A R B O H J E Q L N G
Session 2: Coherence
A Picture of Coherence
Grades 1 and 2
Grade 3
Grade 4
Grade 5
Grades 6 and 7 D
4.OA.3 I
7.NS.3 F
3.OA.8 K
2.OA.1 P
5.OA.1 C
1.OA.1 M
4.OA.1 A
5.OA.2 R
6.EE.2.B B
2.OA.4 O
3.OA.1 H
4.OA.2 J
1.OA.7
Heather
(2 min)
Here is a picture of how these standards are organized by grade, as well as some logical connections between them. How does this compare to what you had? Review the standards where you had differences. Recall there is more than one traceable set of connections.
What is something you noticed about the coherence across grades?
Ideas to highlight:
The C, K, F, D, and I standards trace the progression of word problem solving from Grade 1 to Grade 7. (Grades 5 and 6 problem solving focuses on fractions.)
The J, B, O, M, and H standards trace the progression of multiplication from Grade 1 to Grade 4.
The J, G, L, Q, N, and E standards trace the progression of understanding patterns through two variable equation, Grade 1 to Grade 6.
The P, A, and R standards trace the development from numerical expressions to algebraic ones in Grades 5 and 6. E
6.EE.9 Q
4.OA.5 L
3.OA.9 N
5.OA.3 G
2.OA.3

19. CCSSM Wiring Diagram Session 2: Coherence Heather (30 sec)
An extremely useful resource for understanding coherence is SAP’s “wiring diagram” that maps the coherence of the standards.
this document is helpful to show the different lead-ins to a particular standard; if a student is struggling with “getting it”, this document can help identify all the standards that lead into the one he’s struggling with. we're showing it now to let you know it exists and what it can help with.
We will use the notion of coherence heavily as we move throughout the week.

20. Session 2: Coherence
Now what…so what
Discuss the implications of these resources (i.e. progressions documents and wiring diagram) in your role as the school leader.
Heather
(2 min)
Table take – limited share out
KEY TAKEAWAYS - Stress usefulness as resource for thinking through scaffolding for students struggling with grade level standards - can trace connections backwards to previous grades. Also think we talk about the importance about professional development/conversations with teachers across grade levels. I actually think this part is the leadership development: helping leaders think about how they create systems/routines that are rooted in growing and improving the implementation of the shifts. Speaks to the importance of vertical mapping and teachers to engage in this work together across grade levels.

21. Coherence Within the Grade
Let’s consider “horizontal coherence” as well.

22. Session 2: Coherence
Definition #2
Coherence is also built into the standards in how they reinforce a major topic in a grade by utilizing supporting, complementary topics. 
Katie
(15 sec)
has another meaning-- coherence within the grade.
Instead of a list of unrelated standards, content is connected across domains.

23. Quick Review Session 2: Coherence Domain Grade Level Cluster Standard
Katie
(30 sec)
Quickly discuss structure of the standards.
Lessons, activities and tasks, and assessment items should roll up to the domain and cluster level.

24. Session 2: Coherence
Coherent Tasks
For the next task, name (at least) two different standards, clusters, or domains that are present.
Katie
(15 sec)
Look at the next task
(a) do the math and (b) explain which two domains are evident in the task.
What standards and/or clusters are evident in the task?

25. Session 2: Coherence
Grade 4
Task
Karl’s rectangular vegetable garden is 20 feet by 45 feet, and Makenna’s is 25 feet by 40 feet. Whose garden is larger in area?
4.OA.A.3
4.MD.A.3
Katie
(2 min)
Try this one
4.OA.A.3, 4.MD.A.3
KEY POINT - [Discussion of standard/cluster alignment.] There can be more than one domain/cluster within a grade level for any give tasks. During our first session, we discussed major work of the grade. This is an example of how supporting work connects to the major work and help build links.

26. Leading Coherent Implementation
Session 2: Coherence
Leading Coherent Implementation
Does the instruction carefully connect learning across grades so that students can build new understanding onto foundations built in previous years?
Prework:
Determine the domain of focus for the unit/module
Read the corresponding progressions document
Complete instructional rounds with standards app
Heather
We have been looking at coherence within curriculum now we will shift to look at coherent implementation.

27. Leading Coherent Implementation
Session 2: Coherence
Leading Coherent Implementation
Does the instruction carefully connect learning across grades so that students can build new understanding onto foundations built in previous years?
What to look for:
Are the students who are getting it making connections to previous learnings?
For students who are are not getting it, is the teacher leading students to make connections to previous learnings?
Heather

28. Leading Coherent Implementation
Session 2: Coherence
Leading Coherent Implementation
Does the instruction carefully connect learning across grades so that students can build new understanding onto foundations built in previous years?
Guiding conversations after the walk through:
If students are still not making connections:
Ask: what prerequisite knowledge is a student lacking to be able to make those connections?
Consider: Share time studying the wiring diagram, studying linking standards, with next steps being digging into curriculum for additional lessons on knowledge gaps.
Heather

29. Leading Coherent Implementation
Session 2: Coherence
Leading Coherent Implementation
Does the instruction carefully connect learning across grades so that students can build new understanding onto foundations built in previous years? Prepare feedback for growth.
Heather
Teacher exercise: read the progressions document for the domain.

30. Leading Coherent Implementation
Session 2: Coherence
Leading Coherent Implementation
Is the instruction leveraging how the standards within a grade were built to reinforce a major topic by utilizing supporting, complementary topics?
How to prepare to look for within-grade coherent instruction
Complete instructional rounds with:
standards app
content emphasis in-hand
What to look for:
Are the non-major work standard(s) that are being taught supporting priority content?
Heather

31. Leading Coherent Implementation
Session 2: Coherence
Leading Coherent Implementation
Is the instruction leveraging how the standards within a grade were built to reinforce a major topic by utilizing supporting, complementary topics?
Guiding conversations after the walk through:
If supporting standards are not linking to major work of the grade:
What do the standards say?
Same question as before: How can this chosen standard authentically lead students back to working with math content that is to be emphasized in this grade?
Prepare feedback for growth.
Heather

32. Stop 2 of your instructional rounds
Session 2: Coherence
Stop 2 of your instructional rounds
Watch the 11 min video
Standard: 3.MD.5
Prepare:
determine the domain of focus for the unit/module
Preview the corresponding progressions document (
Complete instructional rounds with standards app and content emphasis
(Marc)
Put the video up on the screen.
Place prework/what to look for on chart paper

33. Stop 2 of your instructional rounds
Session 2: Coherence
Stop 2 of your instructional rounds
Watch the 11 min video
Standard: 3.MD.5
What to look for:
Are the students who are getting it making connections to previous learnings?
For students who are are not getting it, is the teacher leading students to make connections to previous learnings?
Are the non-major work standard(s) that are being taught supporting priority content?
16 min
11 min
Watch video
Marc
(5 min)
Table - Talk at the tables about what you saw.
Whole Group Share Out

34. After the Walk Through - Questions to Ask
Session 2: Coherence
After the Walk Through - Questions to Ask
Leading the Conversation
If students are still not making connections:
Ask: what prerequisite knowledge is a student lacking to be able to make those connections?
Consider: Share time studying the wiring diagram, studying linking standards, with next steps being digging into curriculum for additional lessons on knowledge gaps.
If supporting standards are not linking to major work of the grade:
What do the standards say?
Same question as before: How can this chosen standard authentically lead students back to working with math content that is to emphasized in this grade?
Marc
(2 min)
Follow up questions for the teachers

35. After the Walk Through - Feedback for Growth
Session 2: Coherence
After the Walk Through - Feedback for Growth
Marc
(5 min)
Table – What feedback centered on coherence would you give this teacher?

36. Reflection: Coherence
Session 2: Coherence
Reflection: Coherence
Think about Common Core implementation in your school:
What is the state of curriculum/planning around coherence in your building?
What are some preliminary steps that are needed to improve the state of coherence in your building?
How can you use some of the resources we shared with you today (i.e. progressions document, wiring diagrams) support you with this?
Marc
Anticipated Questions
A leader or two might suggest that the district has aligned the curriculum to the standards. And it has a scope and sequence in place. Might want to highlight the work that the teachers are doing around aligning teaching to curriculum. Emphasize Engage as an aligned curriculum that takes some of this pressure off teachers to ensure that the curriculum is both focused and coherent. Allow teachers and leaders to focus on implementation during istruction. Charter vs district - less control over curriculum. How is coherence relevant to me? If you're stuck with something it's still your responsiblie. Work with what you have but ensure shifts and stadnards are being taught. Moral obligation to speak to decision-makers about curriculum and show why not aligned?

37. Take a break…