1. Leadership Pathway: Coherence in Grades K–5 July 2016
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Leadership Pathway:
Coherence in Grades K–5
119 min as designed (including 5 min for survey at the end)
Materials:
Vertical Articulation Challenge slips
Content Coaching Unbound handout (from earlier session)
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
July 2016

2. 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 among 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.

3. 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:
First, reflect on how you would explain this to a third grader who is just beginning the study of fractions. (1 min independent)
Share responses such as multiplying numerator and denominator by same number, using visuals, and 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 minute)
Let’s look at some standards

4. 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
4 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
TAKE AWAY – 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, developing ideas from existing ones. Emphasize logical progression.

5. 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:
Let them read it on their own first.
Highlight the 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:

6. 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?
4 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

7. 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. 
1 min
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:

8. 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.

9. 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 I 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.

10. 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 fraction 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 I 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.

11. Vertical Coherence Challenge
COHERENCE IN GRADES K–5
Vertical Coherence Challenge
In your groups, you have eleven standards on pieces of paper. Most standards come from the Number and Operations-Fractions domains in Grades 3–5.
The standards are not labeled!
Determine which standards are prerequisites for other standards.
Note: There is more than one vertical strand.
Bonus: Can you determine which standards belong in which grade?
25 min
Speaker Notes:
2 min setup
We just talked about the fact that 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.
We are going to “see” the vertical Coherence of the standards.
Work in pairs
Participants will use the vertical Coherence challenge cards.
See the answer key below.
Let participants know that there is more than one branch.
Tell them they should not use resources for the first ten minutes, just what they know about skills progression. Focus on reading the words in the standards and discuss the grades and the transitions and the way the verbs change. String together standards coherently.
8 min – work without resources
15 min - Now we are going to let you use resources to continue this challenge. (They can use their standards app or the SAP Focus document from the Focus session earlier.)
Link to standards doc participants will use:

12. COHERENCE IN GRADES K–5 A Picture of CoherenceB J
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?
Vertical Coherence Challenge Answer Key
A. 2.G.A.3
B. 4.NF.B.3
C. 5.NF.A.1
D. 3.NFA.1
E. 4.NF.A.2
F. 3.NF.A.2
G. 3.NF.A.3
H. 5.NF.A.2
I. 4.NF.A.1
J. 4.NF.C.5
K. 2.MD.B.6 K F

13. COHERENCE IN GRADES K–5 1 min 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.

14. 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

15. Table Talk
What are the implications of Coherence for students who are struggling mathematically in your schools?
5 min
Speaker Notes:
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 how 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.
IMAGE CREDITS

16. 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. 
1 min
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.
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17. 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
5 min
Speaker Notes:
Try this one
<Click> 4.OA.A.3, <Click> 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.

18. 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.

19. 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.

20. 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 not getting it, is the teacher leading students to make connections to previous learning?
Are the nonmajor work standards being taught actually supporting priority content?
Post-observation
Ask: what prerequisite knowledge is a student lacking that he or she needs to be able to make those connections?
Consider: share time studying the wiring diagram, and studying linking standards, with the next step being to dig 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. It’s 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.

21. 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 not getting it, is the teacher leading students to make connections to previous learning?
Are the nonmajor work standards being taught actually supporting priority content?
16 min (5 min set-up; 11 min video)
Speaker Notes:
5 min: Look up the standard. Use the Coherence Map to check out the prerequisites for this standard and any within grade connections.
Play video
Note: If they map the standard in the Coherence Map, they find that prerequisite standards include:
2.MD.A.1 Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes.
1.G.A.2 Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape.

22. 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 not getting it, is the teacher leading students to make connections to previous learning?
Are the nonmajor work standards being taught actually supporting priority content?
10 min
Speaker Notes:
5 min – First, you will deconstruct what you just saw in terms of COHERENCE. Turn & Talk w/ partner
5 min – Whole group share:
MAIN POINTS:
Using the ruler to measure the sides of the rectangle is connecting to the prerequisite: 2.MD.A.1 Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes → so yes, this teacher is helping the entire class of students make connections to prior learning
For the children who struggle, the teacher doesn’t really make a special effort to connect them to prior learning since she is more focused on SMP6, Attend to Precision.
Yes, the nonmajor work standards (2.MD.A.1) are supporting priority content.
Note: the video refers to linking multiplication with area as an example of coherence, but that is not really evident in this video. You can see how the teacher will build up to that, but jumping prematurely to the procedure of LxW=A undermines development of the conceptual understanding needed for this standard. Consider coming back to this point in the next session on Rigor

23. 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.

24. 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 not getting it, is the teacher leading students to make connections to previous learning?
Are the nonmajor work standards being taught actually supporting priority content?
If students are still not making connections:
Ask: what prerequisite knowledge is a student lacking that he or she needs to to be able to make those connections?
Consider: share time studying the wiring diagram, and studying linking standards, with the next step being to dig 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:
These questions are in their Content Coaching Unbound handout from the earlier session.
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 in order to learn the mathematics and to demonstrate this standard. It also leads the discussion into the “what’s next” for these students.
How are these questions the same or different from the way you usually debrief classroom observations? How would these questions help develop your teachers?

25. Reflection: Coherence
COHERENCE IN GRADES K–5
Reflection: Coherence
What is the state of Coherence in curriculum, planning, and professional development in your school or the schools you support?
Curriculum
Does your curriculum identify the prerequisite skills needed for a unit or lesson?
Is across grade Coherence evident, helping students deepen understanding by making links between domains and clusters?
Planning
Is Coherence a regular consideration in teacher and team lesson planning? How and when are prerequisite skills addressed when students are below grade level?
Professional Development
What professional development has occurred for teachers? Do they know Coherence well enough to effectively include it in planning and instruction?
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 among 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 your 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. You 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?

26. 1 min
Speaker Notes:
Before we adjourn for the day, let’s take one moment to appreciate the teachers we have observed today. All week, we will watch videos of teachers who took risks and submitted their instructional videos to various open education resource sites. We are able to learn because of them and from them.
IMAGE CREDITS:

27. Feedback
Please fill out the survey located here:
-Click “Summer 2016” on the top right
-Click “Details” on the center of the page
5 min
Speaker Notes:
Please fill out the survey to help us improve!

28. References COHERENCE IN GRADES K–5 Slide Source 5 13 17
21-23
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?

29. Image Credits COHERENCE IN GRADES K–5
Slide 5:
Slide 7, 15:
Slide 16:
Slide 26:
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?