1. Leadership Pathway: Coherence in Grades K–5 February 2016
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Leadership Pathway:
Coherence in Grades K–5
114 min as designed
Materials:
Vertical Articulation Challenge slips
1 video - Be sure to have loaded the video prior to participants’ arrival by opening the link and letting it begin to run. This should then have the video ready to view without buffering.
11 min play video
February 2016

2. COHERENCE IN GRADES K–5 At a Glance
Day
Ideas
Wednesday
8:30-4:00
The Foundation
Components of a Comprehensive Literacy Program
Thursday
Building Knowledge and Vocabulary
The Juicy Language of Text
Organizational Systems and Structures
Friday
Focus, Coherence and Rigor
Saturday
8:30-2:30
Observing the Standards And Shifts
Adaptations for Struggling Learners
L E A R N I N G C O N T E N T
1 min
Speaker Notes:

3. COHERENCE IN GRADES K–5 Coherence Session
Objectives:
Participants will be able to describe the coherence shift both as a logical sequencing of content across grades and as important connections between standards, clusters, and domains within the grade
Participants will be able to identify prerequisite standards for grade level standards
Participants will be able to observe and coach the coherence shift in teacher practice
Agenda:
Opening and Activator
Coherence: What and Why
Coherence Across Grades
Coherence Within the Grade
Observing for Coherence
Summary and Reflection
1 min
Speaker Notes:
We’ll start by looking carefully at the shift of coherence in math, with an emphasis on why it’s important and what it looks like in grades K-5.

4. Norms that Support Our Learning
COHERENCE IN GRADES K–5
Norms that Support Our Learning
Take responsibility for yourself as a learner
Honor timeframes (start, end, activity)
Be an active and hands-on learner
Use technology to enhance learning
Strive for equity of voice
Contribute to a learning environment where it is “safe to not know”
1 min
Speaker Notes:
Choose to remind them of a norm if you think it has been slipping:
Keep an open mind (esp. about what don't know or thought you knew);
Stay in learning orientation vs. performance orientation—growth mindset;
Be active during video observation by capturing evidence in writing
Appreciate everyone's perspective and journey;
Share ideas and ask questions, one person at a time (airtime);
Be okay with discomfort and focus on growth;
Be present (monitor multi-tasking, technology, honoring timeframes)

5. Coherence: What and Why?
COHERENCE IN GRADES K–5
Coherence: What and Why?
How would you explain to a student why 2/3 is equal to 4/6?
3 min
Speaker Notes:
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. COHERENCE IN GRADES K–5
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
3 minutes
Speaker Notes:
Ask participants to try to order these standards sequentially – work individually
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. 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.”
1 min
Speaker Notes:
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);
IMAGE CREDITS:

8. Picture of Coherence, Part 1
COHERENCE IN GRADES K–5
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?
5 min
Speaker Notes:
Have participants do these problems individually the way they normally would do them.
Independent activity (1 min) including reflection question.
Table Share Out (2 min) – with a partner, discuss your reflection on what your procedures had in common.
Whole Group Share Out (1 min) - Can I have one or two people in this room share out what your procedures have in common?
1 min (including providing the answers) - 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

9. Coherence ACROSS Grades
LEADERSHIP PATHWAY: COHERENCE IN HIGH SCHOOL
Coherence ACROSS Grades
Learning is carefully connected across grades so that students can build new understanding onto foundations built in previous years. 
Across-Grade Coherence
Learning is carefully
connected across grades
so that students can build new understanding onto foundations built in previous years. 
30 seconds
Speaker Notes:
One of our objectives for you today is to be able to describe BOTH aspects of coherence. Coherence will have a couple of meanings for us, but one important one is about the progression of content ACROSS grades.
IMAGE CREDITS:

10. Truths About Numbers . . . Numbers have organizers
COHERENCE IN GRADES K–5
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
1 min
Speaker Notes:
Share this information. What you were doing in the previous math included composing and decomposing numbers.

11. Picture of Coherence, Part 2a
COHERENCE IN GRADES K–5
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
2 min
Speaker Notes:
By applying these two truths, these two problems can be attacked in the same manner. Look at how important it is to learn this deeply in kindergarten so that you can apply it to more complex versions in 2nd grade.
Standards: K.NBT.A.1 and 2.NBT.B5
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.

12. Picture of Coherence, Part 2b
COHERENCE IN GRADES K–5
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
2 min
Speaker Notes:
These two fractions problems can also be attacked in the same manner as the whole numbers. And look at how important these same truths are for fractions in 4th grade. 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.
Standard: 4.NF.B3
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
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.

13. Vertical Coherence Challenge
COHERENCE IN GRADES K–5
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.
30 min
Speaker Notes:
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.
Hand out activity slips
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:

14. COHERENCE IN GRADES K–5 30 secs Speaker Notes:
Our friends at Student Achievement Partners have created a tool to help us determine Coherence (both across and within grades). In addition to coherence, this tool also identifies whether the standards are major, additional or supporting clusters, and it also includes sample tasks for some of the standards. We’re going to show you how to use the tool now, and then you will check your Vertical Progressions Challenge accuracy using the tool.

15. Coherence Map Go to http://achievethecore.org/coherence-map/ Click on
COHERENCE IN GRADES K–5
Coherence Map
Go to
Click on
Choose a grade level to map-out the Operations and Algebraic Thinking (OA) Domain. Use the map to check the accuracy of your Vertical Progression Challenge.
Revise as needed. Note which standards belong in which grade.
10 min
Speaker Notes:
Do this with your partner

16. A Picture of Coherence D I F K P C M A R B O H J E Q L N G
COHERENCE IN GRADES K–5
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
5 min
Speaker Notes:
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

17. COHERENCE IN GRADES K–5
Table Talk
What are the implications of coherence for students who are struggling mathematically in your schools?
5 min
Table talk – 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.

18. Within-Grade Coherence
Coherence is also built into the standards in how they reinforce a major topic in a grade by utilizing supporting, complementary topics. 
15 sec
Speaker Notes:
You may have seen examples of this in the Coherence Map, and we want to focus on it a bit. Coherence has another meaning-- coherence within the grade.
Instead of a list of unrelated standards, content is connected across domains. Let’s see what it looks like in a student task.
IMAGE CREDITS:

19. COHERENCE IN GRADES K–5
Do the Math—Grade 4
For the task below, name (at least) two different standards, clusters, or domains that are present.
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
2 min
Speaker Notes:
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.

20. Observing for Coherence
1 min
Speaker Notes:
Transition - We will now apply what we have learned to the classroom setting. We want to be proficient at identifying the grade level standards, determining whether the standards being taught are the major work of the grade, and know the questions to ask that will deepen teachers’ understanding of the math coherence. We’re going to observe another math lesson – this time for Coherence (across and within grade), the math knowledge and skills required, and the math above and below the grade level.

21. Key Supervision Questions for Coherence
COHERENCE IN GRADES K–5
Key Supervision Questions for Coherence
Across Grade Coherence
Does the instruction carefully connect learning across grades so that students can build new understanding onto foundations built in previous years?
Within Grade Coherence
Is the instruction leveraging how the standards within a grade were built to reinforce a major topic by utilizing supporting, complementary topics?
1 min
Speaker Notes:
These are the key questions leaders and coaches ask when engaging classroom observations from a coherence lens. These questions will help you collect the right evidence while in the classroom; they are terrific guiding questions around teacher development as well.

22. Observing for Coherence
COHERENCE IN GRADES K–5
Observing for Coherence
Leader Tasks
Across Grade Coherence
Within Grade Coherence
Preparation
Determine domain of focus. Review standards, progressions document.
Use standards app and Coherence Map tools.
Look-fors
Are the students who get it making connections to previous learning?
For students who are are not getting it, is the teacher leading students to make connections to previous learning?
Are the non-major work standards being taught supporting priority content?
Post-observation
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 be emphasized in this grade?
4 min
Speaker Notes:
Before you observe for Coherence, prepare by knowing what you are walking in to see. If you can, get the information ahead of time. If you cannot, figure out the the domain of focus and the intended standard(s). Use your app!
During your time in the classroom, keep looking for student learning, the teacher’s role in providing connections to prior learning, the teacher’s response to struggling learners, etc.
Post-observation If you haven’t done it before the observation, use the progressions documents and/or the Coherence Map to prepare for the discussion with the teacher. Even better if you use them during the discussion. Use these questions to discuss what you saw, develop the teacher’s knowledge and skill around coherence and math.

23. Observing for Coherence
COHERENCE IN GRADES K–5
Observing for Coherence
Standard: 3.MD.5
Prepare:
Determine the domain of focus for the
unit or module
Look up the standard
Preview the corresponding progressions
document or Coherence Map
Capture Evidence:
Are the students who get it making connections to previous learning?
For students who are are not getting it, is the teacher leading students to make connections to previous learning?
Are the non-major work standards being taught supporting priority content?
14 min (3 min set-up; 11 min video)
Speaker Notes:
3 min: Look up the standard. Use the Coherence Map to check out the prerequisites for this standard and any within grade connections.
Play video

24. After the Observation Is this part of the major work of that grade?
COHERENCE IN GRADES K–5
After the Observation
Is this part of the major work of that grade?
Are the students who get it making connections to previous learning?
For students who are are not getting it, is the teacher leading students to make connections to previous learning?
Are the non-major work standards being taught supporting priority content?
8 min
Speaker Notes:
5 min – First, you will deconstruct what you just saw in terms of COHERENCE. Turn & Talk w/ partner
3 min – Whole group share:
MAIN POINTS:

25. COHERENCE IN GRADES K–5
After the Observation
What questions would you now want to ask Ms. Rasso?
2 min
Speaker Notes:
Solicit 1-3 answers (don’t spend much time here). We have a couple to suggest that can help unpack her thinking and provide opportunities for her development.

26. Questions that Develop
COHERENCE IN GRADES K–5
Questions that Develop
Leading the Conversation
Are the students who get it making connections to previous learning?
For students who are are not getting it, is the teacher leading students to make connections to previous learning?
Are the non-major work standards being taught supporting priority content?
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?
2 min
Speaker Notes:
As we did with the video observation of Focus, we’d like to offer these additional post-observation questions that can help develop a teacher’s knowledge and skill around Coherence. We’ve already spent time with this first set of questions in grey. This second set of questions will provide a window into the teacher’s knowledge and skill with what children need to learn the mathematics and demonstrate this standard. It also leads the discussion into the “what’s next” for these students.
How are these questions the same or different than the way you usually debrief classroom observations? How would these questions help develop your teachers?

27. Reflection: Coherence
COHERENCE IN GRADES K–5
Reflection: Coherence
Think about Common Core implementation in your school.
What is the state of curriculum, planning and professional development around Coherence in your building?
What are some preliminary steps that are needed to improve the state of Coherence in your building?
What systems can you access, adjust or initiate?
Professional development
Planning
Observation and supervision
5 min
Speaker Notes:
1 min set-up; 4 min silent write
Restate session objectives: this session was designed for you to be able to describe the coherence shift both as a logical sequencing of content across grades and as important connections between standards, clusters, and domains within the grade. We also want you to be able to observe and coach the coherence shift in teacher practice.
In whatever format you are using to track your reflections and action steps, take a few minutes to individually, or in district/school teams, reflect on how COHERENCE is currently going in our context and what next steps for improvement you could take.
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 instruction. Charter vs district - less control over curriculum. How is coherence relevant to me? If you're stuck with something it's still your responsibility. Work with what you have but ensure shifts and standards are being taught. Moral obligation to speak to decision-makers about curriculum and show why not aligned?

28. Take a break… IMAGE CREDITS:

29. References COHERENCE IN GRADES K–5 Slide Source 7 15 19
23-25
5 min
Speaker Notes:
1 min set-up; 4 min silent write
Restate session objectives: this session was designed for you to be able to describe the coherence shift both as a logical sequencing of content across grades and as important connections between standards, clusters, and domains within the grade. We also want you to be able to observe and coach the coherence shift in teacher practice.
In whatever format you are using to track your reflections and action steps, take a few minutes to individually, or in district/school teams, reflect on how COHERENCE is currently going in our context and what next steps for improvement you could take.
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 instruction. Charter vs district - less control over curriculum. How is coherence relevant to me? If you're stuck with something it's still your responsibility. Work with what you have but ensure shifts and standards are being taught. Moral obligation to speak to decision-makers about curriculum and show why not aligned?

30. Image Credits COHERENCE IN GRADES K–5
Slide 7:
Slide 9, 17:
Slide 18:
Slide 28:
5 min
Speaker Notes:
1 min set-up; 4 min silent write
Restate session objectives: this session was designed for you to be able to describe the coherence shift both as a logical sequencing of content across grades and as important connections between standards, clusters, and domains within the grade. We also want you to be able to observe and coach the coherence shift in teacher practice.
In whatever format you are using to track your reflections and action steps, take a few minutes to individually, or in district/school teams, reflect on how COHERENCE is currently going in our context and what next steps for improvement you could take.
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 instruction. Charter vs district - less control over curriculum. How is coherence relevant to me? If you're stuck with something it's still your responsibility. Work with what you have but ensure shifts and standards are being taught. Moral obligation to speak to decision-makers about curriculum and show why not aligned?