Setting Teachers Up For Success: Math Standards Implementation

Setting Teachers Up For Success: Math Standards Implementation
Global Neutral a Global Warm Neutral d3d1c8 Global Accent On Dark ffbf00 Global Accent on Light ff9800 Global Accent Alt 97c410 ELA - Coral ff5147 Math 009f93 Leadership 7872bf Setting Teachers Up For Success: Math Standards Implementation Welcome to the 2015 Standards Institute hosted by NewOrg! Day 1 | Session 1-3 Grades 6-8

Introduction: Who We Are
Setting Teachers Up For Success Introduction: Who We Are We are a team of former classroom teachers, curriculum writers, school leaders, and education experts who have worked in the private, public, and non-profit sectors. We are dedicated to teacher learning and teacher growth. We know that teaching is hard work and requires excellent training, high quality materials, and meaningful support for practitioners who are continuously striving to better serve their students. We provide educators with high-quality materials and hands-on professional development to help their students achieve the learning goals set by higher standards. We empower educators to make strong instructional decisions through immersive training and access to free standards-aligned resources to adapt for their classrooms, schools, and districts.

Tiffany Hardrick and Liam Honigsberg
Facilitator Introduction Tiffany Hardrick and Liam Honigsberg Needs name, photo, and description

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 in which it is “safe to not know”

The Learning Arc these Two Days
Setting Teachers Up For Success: Math Standards Implementation The Learning Arc these Two Days Learn how to: determine if instruction is aligned to the standards determine if instruction intentionally builds on mathematical foundations determine if there is a balance of rigor in instruction identify if instructional practices support mathematical practices we want all students to use evaluate a lesson for quality and standards alignment

The Main Focus of These First Two Days
Setting Teachers Up For Success: Math Standards Implementation The Main Focus of These First Two Days Upgrading our instructional rounds practices to be better aligned to the spirit of the math standards, including: How to best prepare before going out on rounds What to look for when in a classroom Questions to ask when following up with a teacher Plus, using lots of video to practice new skills!

Goals and Purpose We will... Go deeper on standards and shifts but also see how they play out in school. Examine and evaluate many resources. Do a lot of math problems. Think about our students and how we can adapt resources for them. Think about our teachers needs as they relate to the common core shifts and how we can support them. The standards and shifts have been around for a long time and some of us know them well. That’s awesome! We will go deeper in our study and see how the standards and shifts look instructionally at the highest level. We’ll do a lot of math. And we’ll never forget our students, the ones who need us to get this right.

Leadership Sessions Day 1 Session 1: Focus Session 2: Coherence
Math Days Leadership Sessions Day 1 Session 1: Focus Session 2: Coherence Session 3: Rigor Day 2 Session 1: Mathematical Practices Session 2: Instructional Rounds Session 3: “Monday Morning Plan”: Good Curriculum Here’s what we’re looking over the next two days.

Session 1: Focus Time for a Walkthrough Prepare as you normally would: What do you check in advance? What are the go to tools you use for instructional rounds? Watch the video: Treat the video as instruction in one of your classrooms. Consider feedback: What points of feedback would help this teacher grow in terms of instruction?

Stop 1 of your instructional rounds
Session 1: Focus Stop 1 of your instructional rounds Watch the 10min video Content Developer: Need video/Video Link

Did this lesson... Address grade level standards?
Session 1: Focus Did this lesson... Address grade level standards? focus on grade-level cluster(s), grade-level content standard(s) or part(s) thereof? reflect the full intent of the grade-level cluster(s), grade-level content standard(s) or part(s) thereof being addressed? intentionally relate new concepts to students’ prior skills and knowledge? intentionally targets the aspect(s) of rigor (conceptual understanding, procedural skill and fluency, application) called for by the standard(s) being addressed?

Goals and Purpose Session 1: Focus Participants will be able to: Describe the focus shift generally and specifically describe the major work of the grade. We’ll start by looking carefully at the shift of focus in math, with an emphasis on why it’s important, and what it looks like in Grades 3-5.

Agenda Session 1: Focus Focus: Leading the implementation of a focused curriculum What is focus and what does it look like? What does focus look like during implementation and how do we guide growth? How to prepare to look for focused instruction What to look for in the lesson Guiding conversations post-observation Preparing feedback for growth Here’s what we’ll do today during this session.

Before We Start… Orienting to the Organization of the Math Standards
Session 1: Focus Before We Start… Orienting to the Organization of the Math Standards 6.NS.A.1 6: grade level NS: domain: domains are larger groups of related standards. Standards from different domains may sometimes be closely related. Domains generally go across multiple grade levels. A: cluster: clusters summarize groups of related standards. Note that standards from different clusters may sometimes be closely related, because mathematics is a connected subject. 1: standard number: standards define what students should understand and be able to do.

Before We Start… Orienting to the Organization of the Math Standards
Session 1: Focus Before We Start… Orienting to the Organization of the Math Standards Domain Grade Level Cluster title Standard Standard Cluster Quickly discuss structure of the standards. Lessons, activities and tasks, and assessment items should roll up to the domain and cluster level.

Before We Start… Using the App
Session 1: Focus Before We Start… Using the App Download the above app. Apple: Android: Windows: Amazon:

Session 1: Focus Before We Start… Using the App To find standards: Choose “Standards” Choose “Math-Traditional.” (Traditional refers to the high school course organization of “Algebra 1”, “Geometry”, and Algebra 2”) Choose a grade.

Session 1: Focus Before We Start… Using the App Swipe back and forth… By swiping, find standard 3.NBT.A.2 What does NBT stand for? What is is the cluster title for this standard? To find standards - continued: Scroll to find a standard. (language is included after the code to give some indication of what the standard is about.) Once a standard is chosen: Swipe left and right to see the standards before and after the one you chose.

Quick Task: Using the App
Session 1: Focus Quick Task: Using the App You are in a 7th grade classroom and students are subtracting rational numbers. Navigate the app to find the standard the students are likely addressing. possible answers: 7.NS.A.1, 2, or 3 Share these ideas for getting to the standard: Possibility 1: select grade 7 select domain “Number System” select the first standard in the domain and swipe until you read the standard about subtracting rational numbers. Possibility 2: type “rational numbers” in the search bar scroll until you see 7th grade standards Ask participants if they had any other ways of finding the standard. of finding the standard.

Session 1: Focus “The Common Core calls for greater focus in mathematics. Rather than racing to cover many topics in a mile-wide, inch-deep curriculum, the standards ask math teachers to significantly narrow and deepen the way time and energy are spent in the classroom.” The first shift is indeed called focus, and requires a deep focus on a smaller number of topics.

Session 1: Focus Content Emphasis Guidance from SAP organizes content in terms of major, supporting, and additional content: Major clusters are the highest priority. Supporting clusters are designed to support and strengthen areas of major emphasis. Additional clusters may not connect tightly or explicitly to the major work. Student Achievement Partners (SAP) is a nonprofit founded by lead writers of the Common Core and works to provide resources for understanding the Common Core. Major content is the most essential for future work in mathematics. Major content indicates what the majority of time should be spent teaching this content. The materials should devote at least 65% and up to approximately 85% of the class time to the major work of the grade with Grades K–2 nearer the upper end of that range, i.e., 85%. These are guiding principles of the PARCC and Smarter Balanced assessments.

Focus in Grade 6 Session 1: Focus
What is the emphasis in 6th grade? (operations with fractions, ratios and proportions, and beginning work with expressions and equations)

What is the emphasis in 7th grade? (expressions and equations, ratios and proportions, and rational number arithmetic)

What is the emphasis in 8th grade? (expressions and equations, functions, and congruence/similarity)

Session 1: Focus Summary What are the common threads in the Major Work content for grades 6-8? Multiplication and division of whole numbers Work with fractions

Task Analysis (Whole Group)
Session 1: Focus Task Analysis (Whole Group) What does the major work look like at each grade level? Let’s dive in and do some math. Our purpose for this section is to understand what it means for tasks to align to content standards and to understand better what constitutes the major work at each grade level. Illustrative Mathematics is an important resource we will look at throughout the week.

Protocol Do The Math Discussion Questions:
Session 1: Focus Protocol Do The Math Discussion Questions: What are the knowledge and skills required to be successful on this task? What grade and standard is the task aligned to? Is this part of the major work of that grade? How does this task connect to the major work in the grades above and below? Here is the protocol we’ll use. Before we do the math - some notes about “math phobia”: Principals will need to deal w/ math a bit and have to have the audacity to supervise it from a content place. even in high school. This training will require us to do some math, but more importantly, will give you some look-fors and some questions to ask when you’re completing those instructional rounds - even if you don’t have a math background.

Session 1: Focus Task 1 8th grade task Take a look at this example task and do the math. What are the knowledge and skills required to be successful on this task? What grade and standard is this aligned to? Is it part of the major work? How does this connect to major work in the grades above and below? [Students have to be able to solve a ratio problem, this is 7.RP.1. (It would be a more challenging 6th grade problem because of the work with fractions.) This is part of the major work and connects to multiplication of fractions in Grade 5, ratio work in Grade 6, and functions in Grade 8.]

Session 1: Focus Task 2 The number of siblings for a group of sixth grade students is shown below: 1, 0, 2, 1, 6, 0, 2, 0, 1, 10 Make a dot plot of the data. Find the mean and the median of the data. What does the mean tell you about the data? Which measure of average (mean or median) do you think best describes the data? Why? 6th grade task Take a look at this example task and do the math. What are the knowledge and skills required to be successful on this task? What grade and standard is this aligned to? Is it part of the major work? How does this connect to major work in the grades above and below? [Students have to be able to make a dot plot and answer questions about the mean and median, including interpreting them. The item signals 6.SP.A.2, 6.SP.B.4, and 6.SP.B.5.c. These are NOT part of the major work in Grade 6. It is the beginning of statistical work that is continued in Grade 7, where students compare populations using statistics.]

Session 1: Focus Task 3 To the right is a picture of two congruent rectangles: Show that the rectangles are congruent by finding a translation followed by a rotation which maps one of the rectangles to the other. Explain why the congruence of the two rectangles can not be shown by translating Rectangle 1 to Rectangle 2. Can the congruence of the two rectangles be shown with a single reflection? Explain. 7th grade task Take a look at this example task and do the math. What are the knowledge and skills required to be successful on this task? What grade and standard is this aligned to? Is it part of the major work? How does this connect to major work in the grades above and below? [The task requires using transformations to reason about the congruence of figures. The task aligns to 8.G.A.2 which is part of the major work of the grade. This relates to further work in Geometry in high school with congruence (G-CO) and has its roots in 7th grade work with drawing figures with given conditions (7.G.2).]

Task Analysis - Small Group
Session 1: Focus Task Analysis - Small Group Do The Math Discussion Questions: What are the knowledge and skills required to be successful on this task? What grade and standard is the task aligned to? Is this part of the major work of that grade? How does this task connect to the major work in the grades above and below? Share out Results At your tables, take a look at some more examples. Let’s look at some more. [Facilitators prepare responses.] A: Task requires an understanding of multi-digit multiplication and division and the long division algorithm; 6th grade, 6.NS.B.2:Fluently divide multi-digit numbers using the standard algorithm.; This is not a part of the major work - is part of the additional clusters; connects to prior major work from 5th grade of understanding operations. Move to fluency with operations with rational numbers in 7th grade B: Task requires understanding of ratios and percents (specifically discounts). Aligned to 6.RP.A.3.c: Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.This is a part of the major work on ratios in 6th grade. connects to the major work with fractions in 5th grade & ratios and proportions in 7th grade & equations in 8th grade. C: task requires an understanding of rates, setting up inequalities, graphing solutions to inequalities, and determining reasonableness of an answer in context. Aligned to 7.EE.B.4b: Solve word problems leading to inequalities of the form px+q>r or px+q<r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem. For example: As a salesperson, you are paid $50 per week plus $3 per sale. This week you want your pay to be at least $100. Write an inequality for the number of sales you need to make, and describe the solutions.This is a part of the major work for the grade level. Connects to work with expressions/equations in 6th & 8th grade. Extends to work with linear equations/functions in 8th grade. D: Task requires an understanding of the definition of a function (specifically that every input has exactly one output), and interpreting information from a table.Aligned to 8.F.A.1: 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; this is part of the major work of the grade - functions. Connects to high school work with functions (in fact, there is an extension of this task for high school that asks students to create a trigonometric function to represent the number of foxes and rabbits with respect to time). Connects to earlier work with coordinate points and data analysis. Participants will look at these tasks, without the standards labeled: A. B. C. D.

Share Out What do you notice about the major work of your grade?
For example, seventh grade participants should notice the emphasis on a very small number of key areas: ratio and proportions, expressions and equations, and the rational number system. 6th grade: Does not include probability. Focus on ratios, rational numbers, and expressions and simple equations/inequalities 8th grade: focus on linear relationships, congruence, and Pythagorean Thm.

Leading Focused Implementation
Session 1: Focus Leading Focused Implementation Is the instruction explicitly teaching grade level standards? How to prepare to look for focused instruction: Complete instructional rounds with standards app What to look for: What standard(s) are being taught? Is the instruction addressing the intended standard?

Focused Implementation
Session 1: Focus Focused Implementation Is the instruction explicitly teaching grade level standards? Guiding conversations after the walk through: If not grade level standards: Why was instruction not addressing grade level standards? What data or other work supports the decision to teach non-grade level standards? Prepare feedback for growth.

Session 1: Focus Focused Implementation Is the instruction explicitly teaching grade level standards? How to prepare to look for focused instruction: Complete instructional rounds with content emphases in-hand standards app What to look for: Are the standard(s) that are being taught major work of the grade?

Session 1: Focus Focused Implementation Is the instruction explicitly teaching grade level standards? Guiding conversations after the walk through: If not major work of the grade: How will this chosen standard authentically lead students back to working with math content that is to emphasized in this grade? Prepare feedback for growth.

Session 1: Focus Stop 1 of your instructional rounds Watch the 10min video again Prepare: standards app content emphasis Look fors: What standard(s) are being taught? Is the instruction addressing the intended standard? Are the standard(s) that are being taught major work of the grade?

After the Walk Through - Questions to Ask
Session 1: Focus After the Walk Through - Questions to Ask Leading the Conversation: If not grade level standards: Why was instruction not addressing grade level standards? What data or other work supports the decision to teach non-grade level standards? If not major work of the grade: How will this chosen standard authentically lead students back to working with math content that is to emphasized in this grade?

After the Walk Through - Feedback for Growth
Session 1: Focus After the Walk Through - Feedback for Growth

Session 1: Focus Reflection: Focus Think about Common Core implementation in your school: What is the state of curriculum/planning around focus in your building? What are some preliminary steps that are needed to improve the state of focus in your building?

Take a lunch…

SESSION 2: Coherence Stop 2 of your instructional rounds
Global Neutral a Global Warm Neutral d3d1c8 Global Accent On Dark ffbf00 Global Accent on Light ff9800 Global Accent Alt 97c410 ELA - Coral ff5147 Math 009f93 Leadership 7872bf Welcome to the 2015 Standards Institute hosted by NewOrg! SESSION 2: Coherence Stop 2 of your instructional rounds

We will be looking to see if the next lesson...
Session 2: Coherence We will be looking to see if the next lesson... 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.

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. Here’s our objective.

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 Here’s what we’ll do today during this session.

Coherence: What and Why?
Session 2: Coherence Coherence: What and Why? How would you teach students to graph the proportional relationship between the number of pounds of coffee and the total cost? Lena paid $18.96 for 3 pounds of coffee. Ask participants how they would explain this to an 8th grader who is just beginning to relate proportional reasoning to graphing. Responses include: teaching unit rate, teaching slope, teaching how to construct an appropriately scaled graph Ask which one would be appropriate to teach first. (unit rate comes in 6th grade, must be a secure foundation before moving on; lines cannot be graphed before students know how to construct an appropriate scale)

Session 2: Coherence What’s the Right Order? Use ratio and rate reasoning to solve real-world and mathematical problems by making tables of equivalent ratios, finding missing values in tables, and plotting the pairs of values on the coordinate plane. Graph proportional relationships, interpreting the unit rate as the slope of the graph; compare different proportional relationships in different ways. Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate. Grade 6 Grade 8 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. Ask participants to try to order these standards sequentially. Share correct answer.: A = Grade 6 B = Grade 8 C = Grade 7 Emphasize logical progression. Grade 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.” 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.

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. These documents were spliced together and then sliced into grade level standards. From that point on the work focused on refining and revising the grade level standards. The early drafts of the progressions documents no longer correspond to the current state of the standards.” 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.”

The Progressions Session 2: Coherence This grade 7 piece taken from:
A screen shot This grade 7 piece taken from:

Coherence Across Grades

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

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? have participants do these problems the way they normally would do them.

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 Share this information.

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 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.

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 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.

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 3-5. They 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? Can we see the vertical coherence of the standards? Let participants know that there is more than one vertical strand. String together standards coherently. Share out some “pieces of the puzzle” that participants have come up with. Link to standards doc participants will use: See bottom of facilitator guide for “answer key.”

Progression Intervention
Session 2: Coherence Progression Intervention Use the progression on Ratios & Proportions to help revise your organization. Using standards app, annotate where the standards lie. Revise: Use the Progression on Operations & Algebraic Thinking to help revise your organization. Take a moment to read and annotate the RP progression for grades 6-7. Participants may also want to consult the Functions progression for grades 8-HS. Now take a moment to try to revise your ordering.

A Picture of Coherence Session 2: Coherence Grades 4 and 5 Grade 6
High School Content Developer: Is this table supposed to be blank? A way to organize the task

A Picture of Coherence I D F K P C M A E B O H J Q L N G
Session 2: Coherence A Picture of Coherence Grades 4 and 5 Grade 6 Grade 7 Grade 8 High School I 8.F.5 D 7.RP.1 F 6.RP.1 K 5.NF.5 P 8.F.2 C 4.NF.1 M 7.RP.2 A 8.EE.6 E HS-BF.1 B 5.NF.3 O 6.RP.2 H 7.G.1 J 4.MD.2 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: While this exercise was centered in the RP domain, it’s impossible to stay there without tapping into other domains. This is the richness of the standards, and also the difficulty of teaching the standards. Standards C, K, and F highlight how the NF domain is a foundation for proportional reasoning from grades 4-6 Standards O, M, H, A, and E illustrate how proportional reasoning (in grades 6 and 7) opens the door to geometry, expressions/equations, and functions in grades 8 and 9 Q 7.RP.3 L 6.RP.3a N 8.EE.8 G 5.G.2

CCSSM Wiring Diagram Session 2: Coherence
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.

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

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. Coherence has another meaning-- coherence within the grade. Instead of a list of unrelated standards, content is connected across domains.

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

Session 2: Coherence Coherent Tasks For each task, name (at least) two different standards, clusters, or domains that are present. Content Developer: Should there be any tasks on this slide, or is the task on next slide being referenced here? Take a look at the following 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?

Grade 7 7.EE.B.3 7.EE.B.4 7.RP.A.3 Session 2: Coherence Try this one.
Try this one. This problem can be solved with a ratio table or by finding a unit rate, or by using expressions and /or equations. This problem fosters proportional thinking and lends itself to a discussion of algebraic reasoning, especially if the numbers are substituted for “less clean” ones. The problem highlights using expressions and equations (7.EE.B.3 and 7.EE.B.4) as well as proportional reasoning (7.RP.A.3). 7.EE.B.3 7.EE.B.4 7.RP.A.3

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

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?

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.

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. Teacher exercise: read the progressions document for the domain.

Session 2: Coherence Leading Coherent Implementation Is the instruction leveraging how the standards were built in how they reinforce a major topic in a grade 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?

Session 2: Coherence Leading Coherent Implementation Is the instruction leveraging how the standards were built in how they reinforce a major topic in a grade 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.

Session 2: Coherence Stop 2 of your instructional rounds Watch the 10min video Prepare: Prework determine the domain of focus for the unit/module read the corresponding progressions document Complete instructional rounds with standards app content emphasis

Session 2: Coherence Stop 2 of your instructional rounds Watch the 10min video 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?

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?

Session 2: Coherence After the Walk Through - Feedback for Growth

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?

Take a break…

SESSION 3: Rigor Stop 3 of your instructional rounds
Global Neutral a Global Warm Neutral d3d1c8 Global Accent On Dark ffbf00 Global Accent on Light ff9800 Global Accent Alt 97c410 ELA - Coral ff5147 Math 009f93 Leadership 7872bf SESSION 3: Rigor Stop 3 of your instructional rounds

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

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

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

Why Rigor?

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

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

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

From TIMSS Video Study Session 3: Rigor
TIMSS video study of the late 1990s compared eighth grade math instruction in a variety of countries. Despite the recognition that a balance of the different strands of mathematical proficiency is necessary (Adding It Up), on average, 75% of “private work time” is spent repeating procedures in the United States.

From TIMSS Video Study Session 3: Rigor
Additionally, the kind of non-routine problem solving implied by the application shift does not seem to be happening either. Compared to some other high-performing nations, we spend a smaller amount of time on problems that take longer than 45 seconds. (Only 61% of problems are solved in longer than 45 seconds.)

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

Rigor in the Standards

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

Session 3: Rigor Task #1 A mixture of concrete is made up of sand and cement in a ratio of 5 : 3. How many cubic feet of each are needed to make 160 cubic feet of concrete mix? Take a moment to “do” this task from Illustrative Mathematics. Answer: 100 cubic feet of sand and 60 cubic feet of cement The highlighted aspect of rigor is application because students must use an understanding of ratios to solve real-world problems. This is standard 6.RP.3.

Task #2 Session 3: Rigor Take a moment to solve this problem. Answer:
Take a moment to solve this problem. Answer: The emphasized aspect of rigor is procedural fluency -- students must fluently be able to divide fractions. This is standards 6.NS.1.

Task #3 Hippos sometimes get to eat pumpkins as a special treat.
Session 3: Rigor Task #3 Hippos sometimes get to eat pumpkins as a special treat. If 3 hippos eat 5 pumpkins, how many pumpkins per hippo is that? Lindy made 24 jelly-bread sandwiches with a 16-ounce jar of jelly. How many ounces of jelly per sandwich is that? Purslane bought 350 rolls of toilet paper for the whole year. How many rolls of toilet paper per month is that? In the world's longest running experiment, scientists have tried to capture tar pitch dripping on camera. In the past 86 years, 9 drops have formed. How many years per drop is that? Imagine that 12 goats got into a dumpster behind a pizza parlor and ate 3 pizzas. How many goats per pizza would that be? Take a moment to “do” this task from IM. Answers: 5/3 pumpkins per hippo 2/3 ounces of jelly per sandwich 350/12 or 29 ⅙ rolls per month. 86/9 or 9 5/9 years per drop; this is about 9 years and 7 months per drop. 12/3 or 4 goats per pizza. The highlighted aspect of rigor is conceptual understanding because students must use the concept of ratio to explain their answer to this problem. Note that computation is not necessary for this example. In this problem, students may easily get confused with ⅗ and 5/3 -- this is where the conceptual understanding comes in. Students must deeply understand the concept of a unit rate. The emphasis is not on computation here. This is standard 6.RP.2

Session 3: Rigor Where is the Rigor? 6.RP.3 Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations. 1. 6.NS.1 Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. 2. #1 – application #2 – procedural fluency #3 – conceptual understanding Note the rigor as it appears in the standards. Some standards strongly imply procedural skill and fluency, conceptual understanding, and application. Task 1 was aligned to 6.RP.3; the emphasis on solving real world problems in the standard implies application. Task 2 was aligned to 6.NS.1. The “compute” part of the standard implies procedural skill. We’ll talk in a moment about the other parts of this standard. Task 3 was aligned to 6.RP.2; the emphasis on “understanding the concept” in the standard implies conceptual understanding. 6.RP.2 Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship. 3.

Identifying Rigor in the Standards
Session 3: Rigor Identifying Rigor in the Standards Procedural Skill Application 6.NS.1 Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. Conceptual Understanding

Looking at Grade 6, Module 1
Session 3: Rigor Looking at Grade 6, Module 1 Examine the standards for Module 1 for your grade level. What are the aspects of rigor associated with each one? (There may be more than one!) Bonus: Describe the kinds of problems you’d expect to see associated with each standard. Examine the standards for Module 1. What are the aspects of rigor associated with each one? [Facilitators may choose to have participants actually do highlighting (markers will be provided at tables) of the standards for this activity. This will give a visual picture of the rigor in the standards.]

Share Out Session 3: Rigor
Ask participants to share with a neighbor, then highlight a few responses with the whole group. [Facilitators generate some examples of standards and associated rigor for Module 1 from this grade band] What are the aspects of rigor associated with each standard in Module 1?

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

From the Publisher’s Criteria
Session 3: Rigor From the Publisher’s Criteria The Standards…set high expectations for all three components of rigor in the major work of each grade. The Publishers’ Criteria makes clear this balance is important. “Of course, that makes it necessary that we first follow through on the focus in the Standards” (From Publishers’ Criteria) - We must have a focused curriculum in order for teachers to be able to develop fluencies, conceptual understanding, and application (it’s an issue of time, essentially). (May need to intro what the publisher’s criteria is and does)

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

Session 3: Rigor Example Can you find an inconsistency in the information on this box of staples? Explain. Take a moment to “do” this task. [Give participants time to try the task.] What aspects of rigor do you see in this example? In this example, we see two aspects of rigor highlighted -- procedural skill and application. The task requires performing a division computation in order to solve a real world problem. From IM commentary: The goal of this task is to perform long division with remainder in a context. The teacher will likely need to provide multiple levels of support on this question. First there is a lot of information on the box of staples and the relevant information for the task is the total quantity of staples and number of staples in each strip.

Balance of Rigor in Grade 3, Module 1
Session 3: Rigor Balance of Rigor in Grade 3, Module 1 What will a “balance of rigor” look like? What will it look like to have tasks or activities where rigor is not “separated”? For Module 1 in your grade, what will a “balance of rigor” look like? [Participants should come to see here that a “balance” does not necessarily imply 33% - 33% - 33%. The standards represent a balance. Lessons and units should reflect the rigor in the standards. Some activities, lessons, assessments, and units may emphasize one or more aspects, depending on the targeted standards. (i.e. a fluency activity that is focused on one skill or standard, or a sequence of lessons that are focused on a standard about word problems) We do not need “all three every day.”] What will it look like to have tasks or activities where rigor is not “separated”? Possible responses: Grade 6: The standards for Module 1 mostly imply application and conceptual understanding. An example of an activity for which rigor is “not separated” is using precise ratio or unit rate language in the context of solving a word problem.

Rigor at this Grade Level

Session 3: Rigor Rigor at this Grade Level Examine the tasks and activities in the lessons and problem sets within Grade 3 Module 1. Find at least one task or activity that emphasizes procedural fluency. What evidence do you have? Find at least one task or activity that emphasizes conceptual understanding. Find at least one task or activity that emphasizes application. Let’s have a look at what rigor actually looks like for grade 6 Module 1. What is the evidence of each aspect of rigor you can find? Before you dive in, let’s take a quick look at the features of the ENY lessons.

Teacher Version Session 3: Rigor
Every lesson has a teacher version and a student version. The teacher version is indicated with a “T” in the top right corner. The teacher version contains helpful information about lesson design and lesson reasoning.

Student Version Session 3: Rigor
The student version has all of the same work as the teacher version but without the narrative/answers.

Student Outcomes Session 3: Rigor
These are the learning objectives for the lesson.

Examples Session 3: Rigor
Each lesson usually begins with examples. These form the basis for a discussion, exploration, or other ways of developing new ideas.

Problem Set Session 3: Rigor
Each lesson has a problem set for independent work for the students. This can be used strategically in class or as a homework assignment.

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

Exploratory Challenge
Session 3: Rigor Exploratory Challenge One additional feature of the ENY lessons in the 6-8 grade band is the exploratory challenge. Building conceptual understanding, these challenges allow students to dive into (i.e., explore) a concept on their own for later classroom discussion.

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

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

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

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

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

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

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

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

Session 3: Rigor Stop 3 of your instructional rounds Watch the 10min video again Prepare: Prework Read the standard closely, looking for explicit language that calls out expectations around: Fluency Conceptual understanding Application Complete instructional rounds with standards app

Session 3: Rigor Stop 3 of your instructional rounds Watch the 10min video again What to look for: Fluency: Students practice, many times “on the clock”, with facts and procedures. Fluency activities build upon conceptual understandings students already have. Conceptual understanding Students being asked “why” and to rationalize their thinking. Students working with models to process their thinking. Application: Students working on rich math problems. Students use math absent of external prompts (e.g. “use addition to solve this problem”)

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

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

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

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

Reflection Think about Common Core implementation in your school:
Session 3: Rigor Reflection Think about Common Core implementation in your school: What is the state of curriculum/planning around rigor in your building? What are some preliminary steps that are needed to improve the state of rigor in your building?

Check your email for a feedback survey link or submit online via our website.

Day1: Setting Teachers Up For Success: Math Standards Implementation
Reference List Slide Source 20 22-24 Student Achievement Partners 26,28, 29, 30, 66 49 50 61 84 87 88 89 Content Developer: FYI This slide was added by ppt designer.

Day1: Setting Teachers Up For Success: Math Standards Implementation
Reference List Slide Source 92 93 94 100 101 102 Content Developer: FYI This slide was added by ppt designer. Image Credits: Slide 30, 128 (Dollar Photo Club), Slide 1, 78 (Unsplash.com)

Setting Teachers Up For Success: Math Standards Implementation

Similar presentations

Presentation on theme: "Setting Teachers Up For Success: Math Standards Implementation"— Presentation transcript:

Similar presentations

About project

Feedback

Log in

Auth with social network:

Setting Teachers Up For Success: Math Standards Implementation

Similar presentations

Presentation on theme: "Setting Teachers Up For Success: Math Standards Implementation"— Presentation transcript:

Similar presentations

About project

Feedback