1. Coherence in High School
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Leadership Pathway:
Coherence in High School
120 min
Materials:
Sticky notes on table, chart paper, markers
Handouts:
Vertical Coherence Challenge slips
Content Coaching Unbound handout from the 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 have the video ready to view without buffering.
11 min. – Play video:
July 2016

2. Coherence: What and Why Coherence across Grades
COHERENCE IN HIGH SCHOOL
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 high school.

3. You’ve Got Some Explaining to Do!
COHERENCE IN HIGH SCHOOL
You’ve Got Some Explaining to Do!
What does a student need to know about functions in high school mathematics?
3 min.
Speaker Notes:
What does a student need to know about functions?
Elicit responses: the definition, domain and range (vocabulary), use in modeling contexts, graphing, connections to algebra
This is a lot of content! Where would you start? Luckily, this logical progression is baked into the standards.
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.
Let’s look at some standards.

4. What’s the Right Order? COHERENCE IN HIGH SCHOOL Standard
A. Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If (f) is a function and (x) is an element of its domain, then f(x) denotes the output of (f) corresponding to the input (x).
B. Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.
C. Solve an equation of the form f(x) = c for a simple function (f) that has an inverse and write an expression for the inverse.
D. Represent transformations in the plane using transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).
Algebra I
Grade 8
Algebra II
5 min.
Speaker Notes:
Let’s focus just on the concept of a function—what it is.
What is the correct order of these standards?
Give participants some time to try to order these, then <Click> to display the answers.
The standards are organized to develop ideas gradually over time. Ask participants to try to order these standards sequentially—work individually.
Note that both the Algebra I and Geometry standards follow from Grade 8 (Algebra I not necessary for understanding Geometry std). Algebra I std is a prerequisite for Algebra II.
Emphasize logical progression and the connection between geometry and algebra in thinking about transformations.
Geometry

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 it 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).
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6. 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.
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7. Vertical Coherence Challenge
COHERENCE IN HIGH SCHOOL
Vertical Coherence Challenge
In your groups, you have 16 standards on pieces of paper. Most standards come from the high school standards—a few come earlier!
The standards are not labeled.
Determine which standards are prerequisites for other standards.
First 10 minutes—work without resources.
Note: There is more than one vertical strand.
Bonus: Can you determine which standards belong in which grade?
26 min.
Speaker Notes:
2 min. – Set-up:
We talked about the fact that there are fundamental truths in mathematics that connect across grade levels. Now you are going to explore 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 10 minutes, just what they know about skills progression. Focus on reading the words in the standards and discuss the grades, 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 them use resources to continue this challenge. They can use their standards app or the PARCC Model Content Frameworks document from the Focus session earlier.
Vertical Coherence Challenge Answer Key (accompanies diagram on next slide)
A. 8.F.3
B. 8.EE.6
C. G-CO.7
D. A-REI.7
E. 7.RP.2C
F. 8.F.1
G. 5.G.2
H. F-BF.1C
I. 8.G.5
J. 6.EE.5
K. F-LE.2
L. 6.G.3
M. 8.EE.8A
N. F-IF.2
O. 8.G.3
P. A-REI.6

8. COHERENCE IN HIGH SCHOOL A Picture of Coherence
Grades 5–7
Grade 8
Algebra I
Geometry
Algebra II O
8.G.3 C
G-CO.7 L
6.G.3 G
5.G.2 N
F-IF.2 F
8.F.1 E
7.RP.2C H
F-BF.1C I
8.G.5 B
8.EE.6 A
8.F.3 K
F-LE.2
5 min.
Speaker Notes:
Here is a picture of how these standards are organized by grade, as well as some logical connections between them. These are not the only logical connections, of course, but some worth highlighting.
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 progression of content along standards G, L, O, and C shows students learning coordinates, then the basic transformations (dilations, translations, rotations, reflections), followed by using a subset of these (the rigid motions) to reason about congruence in triangles.
The progression of content along standards E, F, N, and H shows how understanding of functions starts as early as Grade 7, with proportional relationships, then moves on to the definition of a function in Grade 8, followed by use of function notation in Algebra I, and then further study with composing functions.
The progression of content along standards I, B, A, and K shows the development of understanding the graph of a linear function, based on similarity in Grade 8, then eventually constructing functions (both linear and exponential), in Algebra I.
The progression of content along J, M, P, and D shows the development of understanding of systems of equations, from first understanding the meaning of “solution” in Grade 6 all the way up to solving a linear-quadratic system in Algebra II. P
A-REI.6 D
A-REI.7 M
8.EE.8A J
6.EE.5

9. COHERENCE IN HIGH SCHOOL
1 min.
Speaker Notes:
We’d also like to share a second tool to help you identify prerequisites from Grade 8 and earlier. 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 give you a chance to use the tool now.

10. Coherence Map Go to http://achievethecore.org/Coherence-map/ Click on
COHERENCE IN HIGH SCHOOL
Coherence Map
Go to
Click on
Beginning with 8th grade,
map out the Geometry
prerequisites.
10 min.
Speaker Notes:
Do this with your partner.
Open the Coherence Map.
And open the 8th grade geometry domain.
Click on “Map Standard” to explore how this tool can help you find the prerequisites for high school geometry.
Explore for 10 minutes.
MAIN POINTS:
We want participants to see how they can use this tool. If participants go to the 8th grade Geometry square and click on it, the standards open up for them. Then, if they click on "map this standard," it opens up the across- and within-grade Coherence. High school leaders should have the experience of using the tool to see what kids should be walking into 10th grade geometry with AND to notice things like how 8.G.C.9 connects to 8.EE.A.2. 

11. 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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12. COHERENCE IN HIGH SCHOOL Do the Math—Geometry
G-CO.6
G-GPE.4
G-GPE.5
G-SRT.5
Is the quadrilateral with vertices (−6, 2), (−3, 6), (9, −3), (6, −7) a rectangle? Explain.
10 min.
Speaker Notes:
5 min. – Take a look at the following task.
(a) Facilitator Model the math for participants and (b) explain which 3 domains are evident in the task.
What standards and/or clusters are evident in the task?
G­CO.6 (Major), G­GPE.4 (Major), G­GPE.5 (Major), G­SRT.5 (Major)
5 min. – Discussion of standard/cluster alignment/within-grade Coherence:
KEY POINT: [Discussion of standard/cluster alignment.] There can be more than one domain/cluster within a grade level for any given tasks. During our first session, we discussed major work of the grade. This is an example of how different domains help build links in several major work standards.

13. 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 grades), the math knowledge and skills required, and the math above and below the grade level.

14. Key Supervision Questions for Coherence
COHERENCE IN HIGH SCHOOL
Key Supervision Questions for Coherence
Across-Grade Coherence
Does the instruction carefully connect learning across grades so 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.

15. Observing for Coherence
COHERENCE IN HIGH SCHOOL
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 for
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 non-major work standard(s) that are 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 and 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.

16. Observing for Coherence
COHERENCE IN HIGH SCHOOL
Observing for Coherence
Standard: G.CO.A.4, G.CO.A.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 non-major work standard(s) being taught 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
G.CO.A.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.
G.CO.A.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using (e.g., graph paper, tracing paper, or geometry software). Specify a sequence of transformations that will carry a given figure onto another.
Note: If they map the standard in the Coherence Map, they find that prerequisite standards include:
8.G.A.1 Verify experimentally the properties of rotations, reflections, and translations
8.G.A.1.a Lines are taken to lines, and line segments to line segments of the same length.
8.G.A.1.b Angles are taken to angles of the same measure.
8.G.A.1.c Parallel lines are taken to parallel lines.
8.G.A.2 Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.
8.G.A.3 Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.
8.G.A.4 Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two- dimensional figures, describe a sequence that exhibits the similarity between them.
8.G.A.5 Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.

17. After the Observation Is this part of the major work of that grade?
COHERENCE IN HIGH SCHOOL
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 non-major work standard(s) being taught supporting priority content?
10 min.
Speaker Notes:
5 min. – First, you will deconstruct what you just saw in terms of Coherence. Turn & Talk with partner.
5 min. – Whole group share:
MAIN POINT:
These two standards are not major work; they are supporting clusters.
The collaborative analysis of the formative assessment could be much stronger if they were also looking at the prerequisites for these standards.
Ask how analyzing where students are struggling from a “misconceptions” lens is different from looking at their struggles from the lens of missing prerequisites.
Misconceptions indicate they misunderstand something and that you must un-do some learning. Missing prerequisites mean there are gaps to be filled of either skill or knowledge. Each of these deductions would require a different response by the teacher.
There is some Coherence, but 4 is not really addressed. Kids are not developing definitions, so much as they are applying them as in 5 (which could be OK).

18. COHERENCE IN HIGH SCHOOL
After the Observation
What questions would you now want to ask Ms. Mekita?
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.

19. Questions that Develop
COHERENCE IN HIGH SCHOOL
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 non-major work standard(s) that are 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 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 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 gray. 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?

20. Reflection: Coherence
COHERENCE IN HIGH SCHOOL
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
This session’s reflection once again focuses on implementation and systems.
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 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.
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.

21. 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 video of teachers who took risks and submitted their instructional videos to various open educational resource sites. We are able to learn because of them and from them.
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5 min
Speaker Notes:
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23. References COHERENCE IN HIGH SCHOOL Slide Source 5 9 12
16–18
Speaker Notes:

24. Image Credits COHERENCE IN HIGH SCHOOL
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Slide 21:
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