# Deep Dive into the Math Shifts

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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.

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: 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

(It is recommended that middle school teachers first explore 3-5 and then 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.

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 MD.5 strengthens this for students by asking them to solve word problems involving length as a context for adding and subtracting within 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.

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.

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

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.

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

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