Deep Dive into the Math Shifts Understanding Focus, Coherence, and Rigor in the Common Core State Standards for Mathematics This module is designed for the participant who has experienced the Introduction to the Math Shifts module.

The Three Shifts in Mathematics Focus: Strongly where the standards focus Coherence: Think across grades and link to major topics within grades Rigor: Require conceptual understanding, fluency, and application This is a reminder of the three shifts that are required by the Common Core State Standards for Mathematics. Read slide

Focus on the Major Work of the Grade Two levels of focus: What’s in/What’s out The shape of the content that is in There are two levels of focus. The first level is the focus of what is in versus what is out; what is being taught at each grade level compared to what is not. It is because of this level of focus that teachers will have the time to go deeper with the math that is most important. Compared to the typical state standards of the past (which in some cases were literally volumes of standards that would have taken years to “cover,” even one grade’s worth of math), the Common Core State Standards for Mathematics have fewer standards which are manageable and it is clear what is expected of the teachers and students at each grade level. That is the 1st level of focus. The other level of focus is the shape of the content that is in each grade or course. What that means is that if you look at the “focused” list, say for Kindergarten, you can see the list in terms of shades. There are things that are really sharp and focused in the middle, that are the major content for that grade. The other topics are there in a supporting way and help to support that major work. So, even within the list that exists, there is focus. That is the 2nd level of focus.

NOTE: Depending on the audience, the presenter should choose to do the following slides for grade band K-2 and 3-5 or 3-5 and 6-8. This is the Kindergarten overview chart that is found in the Standards document. It is important to note that these are NOT the standards, but simply an overview of the domains and cluster headings for each grade. The pages that follow this overview will have the actual standards and will tell us what really is going on in each of these areas, but it can be useful to look at a grade level from the overview, where we only see the domain and cluster headings. We can see that there are 5 domains in Kindergarten. They are not the same five domains in all of the other grades. In kindergarten the focus is on counting and cardinality, operations and algebraic thinking, number and operations in base ten, measurement and data, and geometry. The clusters give us a general idea of what is happening within each domain. The slide is small and may be hard to read, but in your Standards booklets you have one of these pages for each of the grade levels. This is one way of looking at Focus in the first sense. You can see what is notably absent from Kindergarten that used to be a part of that grade.   We will need to turn the page and look at the actual Standards to see what really is happening at each of these levels, but looking at this page is one way of getting a general picture of what is going on in terms of what is in and what is out.

This next slide shows the second level of focus: the shape of the content that is in. This is the same table, it shows the domains and cluster headings for Kindergarten. But if a mathematician or a math education expert, with the knowledge of how things develop over time, looked at the content for this grade, she would see the main emphases for that grade rising to the surface. Here we see the items marked with green squares are the major clusters. These are the things that are the clear focus of the grade. The other content listed here, the supporting clusters (blue squares) and the additional clusters (yellow circles), are also things that are important for that grade. Everything in the Standards is important and all needs to be covered. What this slide shows is the relationship, the connections among the Standards, and how things support one another. These green squares are concepts that the student must leave Kindergarten knowing. Without a solid foundation of these ideas, they will be set up for failure as they continue through the years. They really have to leave Kindergarten with a secure knowledge of these things.   The other things are not to be left out. It is important to understand how the additional and supporting clusters interact with the major focuses when thinking about time and making decisions.

Briefly acknowledge the same slides for 1st and 2nd grade.

Engaging with the K-2 Content How would you summarize the major work of K-2? What would you have expected to be a part of the major work that is not? Give an example of how you would approach something differently in your teaching if you thought of it as supporting the major work, instead of being a separate, discrete topic. For this activity, we are going to spend a few minutes diving into K-2 and reflecting on what we just saw. We will then spend some time doing a similar activity for the 3-5 grade band. You will need the emphases charts and the worksheet “Engaging with the Content.” Remember that the first page with the Domains and Cluster Headings for each grade - the overview - can be found in the Standards document. Working with a few colleagues at your table, walk through the emphases for Kindergarten, First Grade, and Second Grade and reflect on the questions that are here. How would you summarize the major work of the grade band? After you look at the details in the tables for K, 1, and 2, how would you summarize it?  

NOTE: Only use these notes if you skipped K-2 and instead are going over 3-5 and 6-8. If you’ve already completed K-2, simply preview these charts with participants and move on to slide 17 for the activity. This is the 3rd grade overview chart that is found in the Standards document. It is important to note that these are NOT the standards, but simply an overview of the domains and cluster headings for each grade. The pages that follow this overview will have the actual standards and will tell us what really is going on in each of these areas, but it can be useful to look at a grade level from the overview, where we only see the domain and cluster headings. We can see that there are 5 domains in 3rd grade. They are not the same five domains in all of the other grades. In 3rd grade the focus is on operations and algebraic thinking, number and operations in base ten, number and operations – fractions, measurement and data, and geometry. The clusters give us a general idea of what is happening within each domain. The slide is small and may be hard to read, but in your Standards booklets you have one of these pages for each of the grade levels. This is one way of looking at Focus in the first sense. You can see what is notably absent from 3rd grade that used to be a part of that grade.   We will need to turn the page and look at the actual Standards to see what really is happening at each of these levels, but looking at this page is one way of getting a general picture of what is going on in terms of what is in and what is out.

NOTE: Again, only use these notes if you skipped K-2. This next page shows the second level of focus: the shape of the content that is in. This is the same table, which shows the domains and cluster headings for 3rd grade. But if a mathematician or a math education expert, with the knowledge of how things develop over time, looked at the content for this grade, she would see the main emphases for that grade rising to the surface. Here we see the items marked with green squares are the major clusters. These are the things that are the clear focus of the grade. The other content listed here, the supporting clusters (blue squares) and the additional clusters (yellow circles), are also things that are important for that grade. Everything in the Standards is important and all needs to be covered. What this slide shows is the relationship, the connections among the Standards, and how things support one another. These green squares are concepts that the student must leave Kindergarten knowing. Without a solid foundation of these ideas, they will be set up for failure as they continue through the years. They really have to leave Kindergarten with a secure knowledge of these things.   The other things are not to be left out. It is important to understand how the additional and supporting clusters interact with the major focuses when thinking about time and making decisions.

Summarize or briefly discuss grades 4 and 5.

Engaging with the 3-5 Content How would you summarize the major work of 3-5? What would you have expected to be a part of the major work that is not? Give an example of how you would approach something differently in your teaching if you thought of it as supporting the major work, instead of being a separate discrete topic. NOTE: Only use these notes if you skipped K-2. If you’ve already done the K-2 activity, simply ask participants to repeat the activity with grades 3-5. For this activity, we are going to spend a few minutes diving into 3-5 and reflecting on what we just saw. We will then spend some time doing a similar activity for the 6-8 grade band. You will need the emphasis charts and the worksheet Engaging with the Content. Remember the first page with the Domains and Cluster Headings for each grade can be found in the Standards. Working with a few colleagues at your table, walk through the emphases for 3rd grade, 4th grade, and 5th grade and reflect on the questions that are here. How would you summarize the major work of the grade band? After you look at the details in the tables for grades 3, 4, and 5, how would you summarize it?

(It is recommended that middle school teachers first explore 3-5 and then 6-8. If the facilitator chooses to only go over 6-8, please use the outline of notes on slides #11 and #12 to assist with this and the next page).

Engaging with the 6-8 Content How would you summarize the major work of 6-8? What would you have expected to be a part of the major work that is not? Give an example of how you would approach something differently in your teaching if you thought of it as supporting the major work, instead of being a separate, discrete topic. Let’s take some time to closely review 6-8 grade band. After reviewing, please discuss with your groups the questions on this page.

Coherence Across and Within Grades It’s about math making sense. The power and elegance of math comes out through carefully laid progressions and connections within grades. Coherence is about math making sense. Just like there are two levels of focus, there are two types of coherence found in the math standards. One is the coherence of topics across grades and the other is the coherence within a grade. You will see coherence across grades as the Standards direct us to have students apply learning from a previous grade to learn a new topic. You will see coherence across grades as you see the thoughtfully laid out progressions of mathematics that are meaningful and make sense. You will see coherence within a grade in the Standards as they direct us to have students reinforce a major topic in a grade by utilizing a supporting topic. You will see coherence within a grade as you see the meaningful introduction to topics in the same grade that complement each other.

Looking For Coherence Within Grades Examples: 1st grade – 5th grade: Represent and Interpret Data 3rd grade & 5th grade: “Relate area (volume) to multiplication and to addition.” 6th grade: Solve problems by graphing in all 4 quadrants. (1st year of rational numbers) 8th grade: “Understand the connections between proportional relationships, lines and linear equations.” Here are some examples of where coherence within a grade can be found. In every case in 1st – 5th grade, Represent and Interpret Data is a supporting cluster. It is clear that the intent for 1st – 5th is that representing and interpreting data needs to be done in a way that supports and deepens the understanding of the major work of the grade. Teachers do not have to search for the coherence in the Standards - in many cases it is very clear such as in 3rd and 5th grade when teachers are asked to relate area and volume to multiplication and addition. Here the teacher is shown that area and volume are not to be taught in isolation but as a topic that relates to something else. Again, this is about math making sense. In 6th grade, students are graphing in all 4 quadrants for the first time. It shouldn’t be a surprise that that is the first year students are introduced to rational numbers. In 8th grade, cluster 8.EE.B demands that students actually understand the connections between proportional relationships, lines, and linear equations. This is not left up to chance, but is written in the standards.

Coherence Within A Grade Use addition and subtraction within 100 to solve word problems involving lengths that are given in the same units, e.g., by using drawings (such as drawings of rulers) and equations with a symbol for the unknown number to represent the problem. 2.MD.5 In the 2nd grade standard 2.NBT.5, students are required to add and subtract within 100. 2.MD.5 strengthens this for students by asking them to solve word problems involving length as a context for adding and subtracting within 100. Here we see standards that support each other so math makes sense to students and that the math they are doing is related to other math that they are doing, rather than a endless list of discrete topics to learn.

Coherence Within A Grade Make a line plot to display a data set of measurements in fractions of a unit ( ½, ¼, 1/8). Solve problems involving addition and subtraction of fractions by using information presented in line plots. For example, from a line plot find and interpret the difference in length between the longest and shortest specimens in an insect collection. 4.MD.4 In the 4th Grade Cluster of Represent and Interpret Data, students use line plots to reinforce the major work of adding and subtracting fractions with like denominators. Line plots are not a separate unrelated topic in 4th grade, but a vehicle for students to practice their newly learned skill of adding and subtracting fractions, as well as a vehicle for making sense of the sums and differences. Again, not a new discrete topic to learn, but something that relates to the major work of the grade.

Looking for Coherence Across Grades Coherence is an important design element of the standards. “The Standards are not so much built from topics as they are woven out of progressions.” Structure is the Standards, Publishers’ Criteria for Mathematics, Appendix There are many meaningful progressions of mathematics in the CCSSM. We are going to look at some ways the coherence across grades is evident in the Standards. In the activity we are going to do next, you will see how themes, topics, and language build across grades in the CCSSM.

Coherence Card Activity Activity: Place the standards of each color under the appropriate grade (K-8). Determine a “theme” for each color. No grade has two of the same color card. Some “themes” that have only a few cards might represent consecutive grades and some may not. Read each card in its entirety to help determine placement. Do not check your Standards until you and your colleagues agree on the final product. Discuss horizontal and vertical observations with your partners. For the coherence activity, we really want you to dig into multiple standards. There are so many examples of coherence in the standards, just looking deeply at one of them is not enough. The key is to read the standards on the cards. This isn’t an activity to just see if you know the standards for each grade or can reason the order they should go in. Really read the standards. See the coherence of the writing. See how the topics progress from one grade to another. See why it is important for teachers to know the standards for their grade as well as for the grades before and after theirs. This activity is most beneficial when not using the standards as a guide. We want you to have to mull over these standards and hash out tough decisions with your colleagues. When you are finished and comfortable with your choices, use the Standards to check your work. (Note: Read the complete directions for this activity under the coherence card activity folder in the module.)

Rigor: Illustrations of Conceptual Understanding, Fluency, and Application Here rigor does not mean “hard problems.” It’s a balance of three fundamental components that result in deep mathematical understanding. There must be variety in what students are asked to produce. Rigor, as defined here, does not mean hard problems. It doesn’t mean more difficult. Rigor, here, means something very specific. We are talking about the balance of these components of conceptual understanding, fluency, and application. We are going to look at a set of problems; some assess fluency, some require conceptual understanding, and some are examples of application. By working through these problems, we can start seeing what this looks like.

Frequently Asked Questions How can we assess fluency other than giving a timed test? Is it really possible to assess conceptual understanding? What does it look like? Aren’t the Common Core State Standards for Math all about application and meaningful tasks? Here are a few frequently asked questions about the idea of rigor and balance. How can we assess fluency other than giving a timed test? When you talk about assessment items that get at fluency, it is not always only going to be weekly timed tests. That may be an instructional practice that works for certain people, but it is not a mandate that everyone across the country will be doing timed tests. There are lots of different ways to get to this idea of fluency and there are lots of different ways to know if a student is fluent in something. On a side note, it is important to remember that assessing fluency is not necessarily the same as practicing fluency. Often schools will do timed tests 3 days a week and think they are fulfilling the “fluency requirement.” Students need opportunities to practice fluency, as well.   Is it really possible to assess conceptual understanding? What does it look like? So, how do you take the math that we want students to know out of context and how do we ask them something that just by doing the problem will tell us that they have conceptual understanding? We will be doing a set of sample problems in a moment. Spend some time really thinking about the problems under conceptual understanding and see what is different. Asking students to show work and explain can be informative, but it isn’t the only way to assess conceptual understanding and can become tiring for students. Lastly, are the Common Core Standards in Math all about application and meaningful tasks?  It is a common misconception that CCSSM is only about rich application tasks. That is only one piece of the puzzle. You cannot just do these tasks and think that everything else is going to happen. This is not a criticism of the tasks that are out there. The tasks are often good. Performance tasks are great, but where in those performance tasks are you getting at procedural skill and fluency? Where are you getting at conceptual understanding? We cannot just expect those things to happen if we only do those tasks. Remember that rigor in the Standards is a balance of time spent on conceptual understanding, procedural skill and fluency, and application. The Standards themselves will typically guide the reader to what aspect of rigor is expected.

Rigor Conceptual Understanding: Procedural Skill and Fluency: 3.NF.1 Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b. Procedural Skill and Fluency: 5.NBT.5 Fluently multiply multi-digit whole numbers using the standard algorithm. Application: 7.NS.3 Solve real-world and mathematical problems involving the four operations with rational numbers. Here are some of the key words to look for when determining if a standard should be assessed for conceptual understanding, procedural skill and fluency, or through an application type problem. What are some other words that would highlight a “conceptual understanding” standard? (Interpret, recognize, describe, explain…) Of course, these three aspects of rigor are not always addressed separately from each other, but what is clear is that the wording of each standard can help the teacher determine what the main goal of this standard is.

Sample Problems Activity Work on a few problems from each aspect of rigor. Be prepared to discuss something you observed from one of the problems you tried. How can assessing (with tests, HW problems, exit tickets) all 3 aspects of rigor affect student learning? What does it look like when we are asking students to work on procedural skill and fluency, conceptual understanding, or application? Work on Sample Problems. After taking some time to work through the sample problems, discuss the following question with those around you: What does it look like when we are asking students to work on procedural skill and fluency, conceptual understanding, or application?  

The Three Shifts in Mathematics Focus: strongly where the standards focus Coherence: Think across grades and link to major topics within grades Rigor: Require conceptual understanding, fluency, and application