Catherine Puster, NPESC Curriculum

Ohio’s New Learning Standards in Mathematics: Supporting Depth of Understanding
Catherine Puster, NPESC Curriculum Karen Witt, Curriculum Director Genoa Andrea Smith, NPESC Assistant Superintendent Catherine

Testing Talk PARCC practice problems Activity (2 min)
Like/learn, what went well Need help/dislike, what did you not like Karen Practice problems were questions which were thrown out because it did not meet one of the seven criteria for validity.

Why the Common Core?: How these Standards are Different
Karen Instructional shifts are hard and an overnight change Process over time rethinking “Pull the weeds” Revise scope and sequence

Activity: What didn’t you have time for What could you get rid of What could you spend less time on How can you build on what you already do

Why are we doing this? We have had standards.
Before Common Core State Standards we had standards, but rarely did we have standards-based instruction. Long lists of broad, vague statements Mysterious assessments Coverage mentality Focused on teacher behaviors – “the inputs” Catherine In this time of implementing the Common Core State Standards, it is very easy to approach this as yet another round of “standards revision.” Many of us have lived through such revision processes, multiple times. Even though we have had standards across the country since the 1990s, we have not had systemic standards based instruction. There are plenty of reasons why previous versions of standards so rarely led to standards based instruction. Typical state standards which preceded the Common Core were excessively long and broad. Even if a teacher wanted to teach all of the standards included in a typical grade, there simply are not enough school days in a year – perhaps in several years – to teach what was listed. In addition, the standards themselves were often made of exceptionally broad statements. Many student learning objectives could be aligned to very broad descriptions of learning. One might think that the assessments designed to evaluate student learning of these standards would be a point of guidance to teachers working from a list of standards too long to teach. State assessments however were often built on vague blue prints that often surveyed or sampled the standards. Teachers are therefore often left with the “best worst option” of simply covering as many of the standards as possible in order to hedge their bets for what would appear on the assessments. This has created an intense pressure on teachers, who have limited time. The level of student learning is considered only after the pacing charts. For decades the long lists of standards coupled with accountability pressures have led to an unbalanced focus on what is being taught, rather than on what is being learned.

What are our expectations?
Based on the beliefs that A quality education is a key factor in providing all children with opportunities for their future It is not enough to simply complete school, or receive a credential – students need critical knowledge and skills This is not a 12th grade or high school issue. It is an education system issue. Quality implementation of the Common Core State Standards is a necessary condition for providing all students with the opportunities to be successful after high school. Catherine The Common Core State Standards are based on a belief that education can lead to improved opportunities for all children and that we need to do more than just give them a credential. We need to create a K-12 system of education in which we focus our energy, efforts, and resources on what really matters to support their success.

The Background of the Common Core
Initiated by the National Governors Association (NGA) and Council of Chief State School Officers (CCSSO) with the following design principles: Result in College and Career Readiness Based on solid research and practice evidence fewer, higher, and clearer Catherine To have more students meet the requirements for post-secondary success, we needed greater clarity in what was expected of students each year during K-12 education. Typical state standards that preceded the Common Core were too vague and too long to realistically inform instruction. So the Common Core was designed to be a set of standards that are fewer in number, clearer in describing outcomes, and higher. What is included is what is expected for ALL students. To help students achieve these standards, all practitioners must be honest about the time they require. Teaching less at a much deeper level really is the key to Common Core success. The decisions surrounding how to focus the standards had to be grounded in evidence regarding what students in fact need in order to have a solid base of education and to be well prepared for career or college demands. There had to be many decisions about what not to include. The focus was narrowed to what mattered most. You can look at the standards and probably agree that all of them are skills you would really want every student you know and care about to leave school able to exercise. So the CCSS represent that rarest of moments in education when an effort starts by communicating that we must stop doing certain things, rather than telling educators about one more thing they can add to their already overbooked agenda to help support their students. You can think of the standards and this focus on the shifts that really are at the center of what is different about them as the “power of the eraser” over the power of the pen.

https://vimeo.com/92784229 Catherine
David Coleman speaking on focus in the Common Core State Standards. This video, produced by Engage NY, can be seen in its entirety at

“These standards are not intended to be new names for old ways of doing business. They are a call to take the next step.” CCSSM, page 5 The Common Core State Standards clearly say that “These standards are not intended to be new names for old ways of doing business. They are a call to take the next step.” This is the message for everyone involved. In the past, state standards might have been revised and it didn’t necessarily have an impact on instruction or for curriculum —what happened in the classroom was business as usual. That can’t happen here—there are some real changes with these standards. There is no room for old ways of doing business.

Fewer - Clearer - Higher
Principles of the CCSS Fewer Clearer Higher Aligned to requirements for college and career readiness Based on evidence Honest about time So, where do the Common Core State Standards fit in with this conversation? In order to improve education, we need to have a set of standards that are powerful, meaningful, and achievable. During the development of the Standards, the design principles were often described as fewer, clearer, and higher. Although these are relative, and perhaps even subjective terms, it is worth understanding how these make the Common Core State Standards different in approach than typical state standards. In terms of fewer, the design principle is that these standards can be learned within a year. There is very little repetition from year to year of the same standards. The standards are clearer in that they more precisely describe outcome expectations, rather than vague or broad descriptions of learning. The standards are higher with respect to what is meant by higher – not harder – standards. Having higher standards means that what is included in the Common Core State Standards is actually intended for all students each year; there is congruence between what is stated and what is expected. The next issue then is how to get to fewer, clearer, higher. Unlike typical standards development or revision processes in which groups of stakeholders are gathered in committees to advocate for their individual positions, preferences, pet topics, etc., these standards relied on evidence for what students need in order to be prepared for college and careers. It turns out that a lot of what we spend time and energy on in school K-12, doesn’t buy students much after graduation. This of course wouldn’t be a problem if time was not such a finite resource. Because time is limited however, decisions had to be made. Rarely in education do we pay so much attention to the limited resource of time. We often, rather, keep adding and adding initiatives. It is always easy to add one more thing. These Standards were built with the awareness that each additional expectation came at the cost of time spent on what was already included. It is exceptionally important in understanding these Common Core State Standards that we acknowledge and accept the power of the eraser as well as, perhaps after, the power of the pen.

https://vimeo.com/92784226 {show video}
William McCallum and Jason Zimba (two lead writers of the Common Core State Standards for Mathematics) on the background of writing the Standards. This video, produced by The Hunt Institute, can be seen in its entirety at

Some Old Ways of Doing Business (1 of 2)
A different topic every day Every topic treated as equally important Elementary students dipping into advanced topics at the expense of mastering fundamentals Infinitesimal advance in each grade; endless review Incoherence and illogic – bizarre associations, or lacking a thread This list represents some of the old ways of doing business in textbooks that are simply not aligned with CCSS. Some books present a different topic every day and treat every topic as equally important. This goes against the focus and coherence of the Standards. There are cases in which textbooks just use a list of standards as their table of contents – math is not a flat list of topics, this is an old way of doing business. Some materials had elementary students dipping into advanced work at the expense of mastering fundamentals. This is back to the mentality of students doing a little bit of everything, every year. This leads to infinitesimal advance in each grade with endless review. Have you seen middle school books recently? They don’t list a grade anymore, they are just different colors or animals – red, blue and green for instance. It is almost impossible to know which grade is which in some middle school series even after opening the book, because the topics and level of expectations are minimally varied from each other. Infinitesimal advance in each grade with endless review. There are countless Algebra I books where ½ the year is spent covering middle school material with no indication to the teacher that this material is review. Books often contain incoherence and illogic. They put topics together that had no right to be together – one chapter of an Algebra book, chapter one, was actually titled “Angles and Order of Operations” This makes no sense. Math should make sense, and all too often the incoherence of textbooks sabotages that.

Some Old Ways of Doing Business (2 of 2)
Lack of rigor Reliance on rote learning at expense of concepts Aversion to repetitious practice Severe restriction to stereotyped problems lending themselves to mnemonics or tricks Lack of quality applied problems and real-world contexts Lack of variety in what students produce Materials contain a widespread lack of rigor. At one extreme, there is reliance of rote learning at the expense of conceptual understanding – all procedural skill and fluency. At another extreme, there is aversion to repetitious practice – books that dedicate so much time to applications that there is very little opportunity for students to practice the mathematical skills they need to develop. Both exist. Both are old ways of doing business. Content in a lot of books is severely restricted to stereotyped problems lending themselves to mnemonics or tricks. Assignments contain problems that all looked exactly the same. Who wouldn’t want to use tricks when there is no variety to what is being asked to do? For example, when a student is asked to write “I will not chew gum in class” 100 times, he/she soon realizes that it is easier to write” I” 100 times then “will” 100 times and so on… Students don’t internalize the message “I will not chew gum in class”, they just found a way to do it easier – almost mindless of what they were writing. When students are given math problems that all look the same – the same thing occurs – no internalization – no conceptual understanding – no ability to transfer what they are doing to something just slightly different. They just learn tricks. There can be a lack of quality applications and real-world contexts, and when they do appear they were often designated into the gutter - into the areas of the text that teachers rarely assigned problems from. There is also a lack of variety in what students are asked to produce. Overwhelmingly students are asked to produce only answers, and very seldom are they asked to produce arguments, diagrams, or models.

Cautions: Implementing the CCSS is...
Not about “gap analysis” Not about buying a text series Not a march through the standards Not about breaking apart each standard As decisions are made going forward, there are some cautions worth noting. A simple gap analysis between what topics we used to teach and where those topics exist in the Common Core is not an adequate approach to take. Such a topical gap analysis rarely addresses what we should not longer include in a curriculum. It does not address the conceptual understanding, fluency and rigor expectations, and most importantly it does not result in the signaling of focus on the major work of the grade. Another important note is that these standards do not dictate a particular order or scope and sequence. Students should not be marched through these standards one at a time, but rather many of these expectations develop over the course of the year. The architecture of the standards documents matters a lot. The domains and cluster headings are very purposefully organized to support teaching and learning of the standards. Take every opportunity to build on understanding and develop math proficiency rather than just going over or covering a list of math topics.

Evidence Centered Design can inform a deliberate and systematic approach to instruction that will help to ensure daily classroom work leads to all students meeting Ohio's New Learning Standards.

Evidence-Centered Design (ECD) in the Classroom - Start with the end in mind.
PARCC is using ECD to create the gr 3-11 assessments. Learning Targets/Objectives Design begins with the inferences (claims) we want to make about students—should be connected clearly to our new standards - What should students be able to DO or KNOW? Classroom Assessments Formative/Summative In order to support claims, we must gather evidence----what can teachers point to, underline or highlight to show that students are making progress toward doing what we claim they can do? Classroom Activities Classroom activities (tasks) are designed to elicit specific evidence from students in support of claims. This is an important slide - it helps teachers see how they can use evidence centered design in their own assessment planning - this is where you can make a connection between the process PARCC is using - and what they could do to write SLOs, common assessments, design units aligned to the New Learning standards, etc. Approach to assessment provides clear connection between the Standards and the inferences we want to draw about students (claims), the type of student work/demonstrations that allow us to draw those inferences, and the tasks or activities that allow us to elicit that student work. This same approach/concept applies in the classroom—with ideas like Understanding By Design—moving backwards from the Standards to Classroom Assessments. Confidential - Not for Distribution

Mathematical Practices
Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Don’t Bureaucratize The Common Core State Standards for Mathematics include standards for mathematical practices in addition to standards in content. The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important “processes and proficiencies” with longstanding importance in mathematics education. In integrating the practice standards, it is important that we do not bureaucratize these expectations – as in “problem solving Mondays” or “perseverance Tuesdays.” Rather, these practices should be integrated into the content instruction and practice.

The CCSS Requires Three Shifts in Mathematics
Focus: Focus strongly where the Standards focus. Coherence: Think across grades and link to major topics within grades. Rigor: In major topics, pursue conceptual understanding, procedural skill and fluency, and application. The Common Core State Standards for Mathematics were designed to address these issues. To learn more about the Standards we are going to talk about the three shifts, which represent the overarching messages in these new Standards. Here are the three shifts in mathematics. {Read the slide} These are not only things I’m (we’re) telling you, these are things I’m (we’re) asking you to tell other people. These are what you need to be fighting for. These are what you need to be thinking about when a speaker at a workshop or a publisher or even members of your district tell you about CCSS – you can test their message against these things. You can test anyone’s message against these touchstones. They are meant to be succinct, and easy to remember; we’ll discuss them each in turn.

Power of the Shifts Know them – both the what and the why
Internalize them Apply them to your decisions about Time Energy Resources Assessments Conversations with parents, students, colleagues Continue to engage with them: on Twitter The shifts are a very powerful entrance point to the standards. They should be well understood by all stakeholders. Once everyone understands these shifts, they can be used as “anchors” to guide the many decisions to be made about the use of time, energy, resources, assessments, etc. The CCSS and the larger mission of supporting students in college and career readiness should weave its way into all decisions and conversations in the education system.

Mathematics: 3 shifts Focus: Focus strongly where the Standards focus.
In the second shift, coherence, we take advantage of focus to actually pay attention to sense-making in math. Coherence speaks to the idea that math does not consist of a list of isolated topics. The Standards themselves, and therefore any resulting curriculum and instruction, should build on major concepts within a given school year as well as major concepts from previous school years.

Engaging with the shift: What do you think belongs in the major work of each grade?
Which two of the following represent areas of major focus for the indicated grade? K Compare numbers Use tally marks Understand meaning of addition and subtraction 1 Add and subtract within 20 Measure lengths indirectly and by iterating length units Create and extend patterns and sequences 2 Work with equal groups of objects to gain foundations for multiplication Understand place value Identify line of symmetry in two dimensional figures 3 Multiply and divide within 100 Identify the measures of central tendency and distribution Develop understanding of fractions as numbers 4 Examine transformations on the coordinate plane Generalize place value understanding for multi-digit whole numbers Extend understanding of fraction equivalence and ordering 5 Understand and calculate probability of single events Understand the place value system Apply and extend previous understandings of multiplication and division to multiply and divide fractions 6 Understand ratio concepts and use ratio reasoning to solve problems Identify and utilize rules of divisibility Apply and extend previous understandings of arithmetic to algebraic expressions 7 Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers Use properties of operations to generate equivalent expressions Generate the prime factorization of numbers to solve problems 8 Standard form of a linear equation Define, evaluate, and compare functions Understand and apply the Pythagorean Theorem Alg.1 Quadratic inequalities Linear and quadratic functions Creating equations to model situations Alg.2 Exponential and logarithmic functions Polar coordinates Using functions to model situations Without looking at your standards, please work with those around you to determine what two topics of each row are major work of that grade. You could circle the 2 topics that are major work, or cross out the one that is not. A hint to this is that the one item that is not the major work of the grade is actually not even part of the standards for that grade. {After a sufficient amount of time, have participants share their answers. An “answer key” can be found in the resources for this module.}

Move away from "mile wide, inch deep" curricula identified in TIMSS.
Focus Move away from "mile wide, inch deep" curricula identified in TIMSS. Learn from international comparisons. Teach less, learn more. “Less topic coverage can be associated with higher scores on those topics covered because students have more time to master the content that is taught.” The Third International Math and Science Study (TIMSS) not only ranks assessment performance of many industrialized nations, but also analyzes the education systems of those countries. The TIMSS study showed that the U.S. covers far more topics than those countries that significantly outperform us. In fact, the TIMSS study revealed that in grade 4, high scoring Hong Kong omitted 48 percent of the TIMSS items. The U.S., on average, omitted only 17%. In the U.S. we have been covering more topics with the net result of learning less about them. Interestingly, the slogan on the Singapore Ministry of Education’s website is “Teach less, learn more.” By focusing curriculum and instruction, teachers have the opportunity to support students in the development of strong foundations that pay off as the progress through more complex math concepts. {read quote from bottom of slide} – Ginsburg et al., 2005

Shift #1: Focus Strongly Where the Standards Focus
Significantly narrow the scope of content and deepen how time and energy is spent in the math classroom. Focus deeply on what is emphasized in the standards, so that students gain strong foundations. What does it mean to focus? {Read slide} So, focus in the Standards quite directly means that the scope of content is to be narrowed. This is that notion of “the power of the eraser”. After a decade of NCLB, we have come to see “narrowing” as a bad word – and when it means cutting arts programs and language programs, it is. But meanwhile, math has swelled in this country. It has become a mile wide and an inch deep. The CCSSM is telling us that math actually needs to lose a few pounds. Just as important is that we are deepening expectations as well. So rather than skating through a lot of topics – covering the curriculum – we are going to have fewer topics on our list, but the expectations in those topics are much deeper. Without focus, deep understanding of core math concepts for all students is just a fantasy. A study of the Standards reveals that there are areas of emphasis already engineered into the standards at each grade level.

How Are CCSS Assessments Different? An Overview
Shift 1: Focus strongly where the Standards focus From To Cover content that is a “mile-wide and an inch-deep” Assess fewer topics at each grade (as required by the Standards) Give equal importance to all content Dedicate large majority of score points to the major work of the grade

Traditional U.S. Approach
K Number and Operations Measurement and Geometry Algebra and Functions Statistics and Probability This slide presents a visualization of how U.S standards used to be arranged, giving equal importance to all four areas - like “shopping aisles.” Each grade goes up and down the aisles, tossing topics into the cart, losing focus. This visualization, and the curriculum which it represents, shows no priority. The CCSS domain structure communicates the changing emphases throughout the elementary years (e.g., Ratios and Proportional Relationships in grades 6 and 7).

The shape of math in A+ countries
Mathematics topics intended at each grade by at least two-thirds of A+ countries Mathematics topics intended at each grade by at least two-thirds of 21 U.S. states There are no detailed labels here because I want you to see visually that the overall shape of math topics in A+ countries and those typical in the US (pre-Common Core) is different. Each row is a math topic, like fractions, or congruence. In 2/3rds of the high-performing countries, the foundations are laid and then further knowledge is built on them. The design principle is focus and coherent progressions. In the U.S., the design principle is to teach everything every year that can possibly be taught… as well as many things that cannot. 1 Schmidt, Houang, & Cogan, “A Coherent Curriculum: The Case of Mathematics.” (2002).

Focusing attention within Number and Operations
Operations and Algebraic Thinking Expressions and Equations Algebra → Number and Operations— Base Ten The Number System Number and Operations—Fractions K 1 2 3 4 5 6 7 8 High School In contrast to the prior slides visualization, this picture shows a shape. Early emphasis on operations and algebraic thinking and number and operations – base ten build to more sophisticated concepts in middle school and then to authentic algebra, rather than the all too common “experience with algebra” for all.

https://vimeo.com/92784228 What is Focus?
“From the Page to the Classroom: Implementing the Common Core State Standards Mathematics. Produced by Council of the Great City Schools.” Math curriculum in the U.S. has often been described as a “mile wide, inch deep.” We cover a lot of topics, but with very little depth. Because of the pressure we feel to “get through it all,” there is often little time to meet expectations of mastery for all students. The consequences of this is students only learning a little bit of a lot of things and the students not gaining insight or depth of understanding of the important math topics that will allow them successfully build on that understanding each year.

Focus in K–8 Grade 2 Geometry Example
Traditional Approach (Grade 2 Geometry) CCSS-Aligned Approach (2.G.A.3) Shawn cut a rectangle along two lines of symmetry. How many equal shares will he have? Ms. Nim gave her students a picture of a rectangle. Then she asked them to shade in one half of the rectangle. Here are three pictures: Which ones show one half? Explain. Source: Illustrative Mathematics. /827 The item on the left is a topic that has long been covered in early grades; however, symmetry does not appear in the Standards until grade 4. As a result, it cannot be assessed before grade 4. The item on the right assesses standard 2.G.A.3 appropriately (Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.). Notice that partitioning is the key component of the item, not symmetry.

Focus in K–8 Measures of Center Example
Traditional Approach (Grade 3 Mean and Median) CCSS-Aligned Approach (6.SP.B.4, 5c)) Source: Illustrative Mathematics. Before, measures of center would have been assessed as early as grade 2 or 3 with an item like the one on the left. Now, measures of center is assessed starting in grade 6, with an item like the one on the right. The item on the left is a low-level "fake data" item and the item on the right is a richer more authentic item with full use of rational numbers (the mean is 5.35). The CCSS-aligned item shows the richness of the item when we wait for the appropriate grade for the content.

Focus in K–8 Supporting Work Reinforcing Major Work
Traditional Approach (Grade 1) CCSS-Aligned Approach (1.MD.C.4) The CCSS-aligned item on the right shows how supporting content enhances focus by engaging students in the major work of the grade. In grade 1, 1.MD.C.4 is a supporting cluster standard. That means it should support and reinforce the major work of the grade. In grade 1, focus areas are around addition and subtraction. Prior to CCSS, items like the one of the left just asked kids to read the information from the data display. As you can see, the questions on the left merely ask students to count. The item on the left does not require students to complete grade-level work and uses data displays as an end in themselves. Now with CCSS, students are not only asked to read data displays, but also to answer “how many more” and “how many less” questions, solving using addition and subtraction, part of the major work for grade 1. As you can see, the questions on the right have students interpret the data given and perform operations on the data. Note that the connections among content are explicitly written into the standards.

Focus in K–8 Supporting Work Reinforcing Major Work
Traditional Approach (Grade 7) CCSS-Aligned Approach (7.SP.C.8) A coin is flipped three times. Part A: Draw a tree diagram that shows all possible outcomes. Part B: Create an organized list that shows all possible outcomes. Source: EngageNY. hments/math-grade-7.pdf 𝟐 𝟖 = 𝒙 𝟐𝟒 𝒙=𝟔 students expected to get 3 heads or 3 tails 2 HHH HHT HTH HTT THH THT TTH TTT Prior to CCSS, items like the one of the left just asked kids to calculate simple and compound probabilities. As is, this item does align to 7.SP.C.8; however, it misses a prime opportunity to connect this learning back to major work of the grade (proportional relationships). Now with CCSS, work with probability links to major work of the grade (7.RP.A, Analyze proportional relationships and use them to solve real-world and mathematical problems). To answer part C, the student may do the following: 2 8 outcomes are either 3 heads or 3 tails, so for 24 students, 2 8 = 𝑥 24 x = 6 students.

Priorities in Mathematics
Grade Focus Areas in Support of Rich Instruction and Expectations of Fluency and Conceptual Understanding K–2 Addition and subtraction, measurement using whole number quantities 3–5 Multiplication and division of whole numbers and fractions 6 Ratios and proportional reasoning; early expressions and equations 7 Ratios and proportional reasoning; arithmetic of rational numbers 8 Linear algebra and linear functions This chart shows the major priority areas in K-8 math. These are concepts which demand the most time, attention, and energy throughout the school year. These are not topics to be checked off a list during an isolated unit of instruction. Rather, these priority areas will be present throughout the school year through rich instructional experiences.

Focus in K–8 Major Work of the Grade
The large majority of score points on any grade-level assessment system should be devoted to the major work of the grade. See achievethecore.org/focus for major work at other grade levels. In the CCSSM, the clusters of standards for K–8 are broken up into major, supporting, and additional clusters. On the left is the Grade 6 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 grade 6. They are not the same 5 domains in all other grades. In grade 6, the domains are ratios and proportional relationships, the number system, expressions and equations, geometry, and statistics and probability. The clusters give us a general idea of what is happening within each domain. On the right is the list of clusters with the emphasis of each cluster for grade 6. Looking at this, you can see the main emphases for the grade. 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 grade 6. The large majority of points in each grade K–8 should be devoted to the major work of the grade (the green clusters and standards within those clusters).

Content Emphases by Cluster: Grade Four
Key: Major Clusters; Supporting Clusters; Additional Clusters Here is an example of how focus and coherence works in a single grade. This is the emphases by cluster chart for fourth grade. The green boxes represent the major work of the grade, the blue are the clusters that support the major work of the grade, and the yellow are the additional clusters. The emphases charts make visible the intended focus and coherence of the subject matter by identifying the major work (focus) and the supporting and additional work (coherence within and across) For example, here are some connections between supporting and major. - Gain familiarity with factors and multiples: Work in this cluster supports students work with multi-digit arithmetic as well as their work with fraction equivalence. - Represent and interpret data: The standard in this cluster requires students to use a line plot to display measurements in fractions of a unit and to solve problems involving addition and subtraction of fractions, connecting it directly to the Number and Operations--Fractions clusters.

Shift #1: Focus strongly where the Standards focus.
Group Discussion Shift #1: Focus strongly where the Standards focus. In your groups, discuss ways to respond to the following question, “Why focus? There’s so much math that students could be learning, why limit them to just a few things?” {Pass out practice with shifts handout. Read the question from the top of handout – same as slide} Take a few minutes to discuss this question.

Coherence: Think across grades, and link to major topics with grades. In the second shift, coherence, we take advantage of focus to actually pay attention to sense-making in math. Coherence speaks to the idea that math does not consist of a list of isolated topics. The Standards themselves, and therefore any resulting curriculum and instruction, should build on major concepts within a given school year as well as major concepts from previous school years.

Shift #2: Coherence: Think Across Grades, and Link to Major Topics Within Grades
Carefully connect the learning within and across grades so that students can build new understanding on foundations built in previous years. Begin to count on solid conceptual understanding of core content and build on it. Each standard is not a new event, but an extension of previous learning. {Read slide} In the second shift, coherence, we take advantage of focus to actually pay attention to sense-making in math. Coherence speaks to the idea that math does not consist of a list of isolated topics. The Standards themselves, and therefore any resulting curriculum and instruction, should build on major concepts within a given school year as well as major concepts from previous school years. Typically, current math curriculum spends as much as 25% of the instructional school year on reviewing and re-teaching previous grade level expectations – not as an extension – but rather as a re-teaching because many students have very little command of critical concepts.

Shift 2: Coherence: think across grades, and link to major topics within grades From To Assessment as a checklist of individual standards Items that connect standards, clusters, and domains (as is natural in mathematics) as well as items that assess individual standards Each topic in each year is treated as an independent event Consistent representations are used for mathematics across the grades, and Content connects to and builds on previous knowledge The second Shift, coherence, requires: Thinking across grades: The Standards are designed around coherent progressions from grade to grade. Learning is carefully connected across grades so that students can build new understanding onto foundations built in previous years. Each standard is not a new event, but an extension of previous learning. In seeing math as a set of isolated topics, certain topics were taught year after year. Typically, recent math curricula would spend as much as 25% of the instructional school year on reviewing and re-teaching previous grade level expectations – not as an extension – but rather as a re-teaching because many students have very little command of critical concepts. If there's re-teaching, there's re-assessing and, as a result, the same topics were being assessed in every grade. Instead, the Standards require us to use the grade-level content and build on previous knowledge. Linking to major topics within a grade: Instead of allowing additional or supporting topics to detract from the focus of the grade, these concepts serve the grade level focus. For example, instead of data displays being an end in themselves, they are an opportunity to do grade-level word problems focused on operations.

https://vimeo.com/92784230 William McCallum speaking on coherence.
This video, produced by the Hunt Institute, can be seen in its entirety at

Coherence: Think Across Grades
Example: Fractions “The coherence and sequential nature of mathematics dictate the foundational skills that are necessary for the learning of algebra. The most important foundational skill not presently developed appears to be proficiency with fractions (including decimals, percents, and negative fractions). The teaching of fractions must be acknowledged as critically important and improved before an increase in student achievement in algebra can be expected.” Final Report of the National Mathematics Advisory Panel (2008, p. 18) {read slide regarding fraction example} As an example of coherence, the Common Core State Standards takes a different approach to preparing students for success in algebra. Previous approaches moved topics of pre-algebra and algebra earlier and earlier in the grade sequence rather than attend to careful progressions of conceptual development that in fact prepare students for algebra.

Coherence Across Grades Seeing the Structure
“The Standards were not so much assembled out of topics as woven out of progressions.” What It Means Aligning items to grade-level expectations requires understanding of all the standards at that grade (e.g., NBT standards often give bounds for OA items) and understanding how the standard fits into a progression with previous and future grades Why It Matters for Assessment The Standards were woven out of connected topics (the progressions) and so assessments should also rely on the connections between topics (coherence across grades). The progressions are narratives of how the standards play out over grades. The initial conception of these progression documents was to render the standards in narrative form so that you can readily see what is going on across the grades. The documents sometimes have useful sample questions in the margins as well. The progressions provide helpful information about alignment that can be used to ensure items are aligned to the expectations articulated in the Standards. It’s important to note that the distinction between “means” and “equals” is part of learning progressions. Do not think of the standards as assembled out of topics, but think of them as woven out of progressions, at least in the early grades.

Alignment in Context: Neighboring Grades and Progressions
One of several staircases to algebra designed in the OA domain. Algebra begins in 6.EE.3 in its cleanest sense. Here is a beautiful illustration of the design of the standards. {read slide} 43

4.NF.4. Apply and extend previous understandings of multiplication to multiply a fraction by a whole number. 5.NF.4. Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction. 5.NF.7. Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions. 6.NS. Apply and extend previous understandings of multiplication and division to divide fractions by fractions 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. Grade 4 Grade 5 Grade 6 CCSS Here you see some of the international data that informed the development of the CCSSM. On the left you see a coherent progression of multiplication and division of fractions over three grade levels. On the right you see excerpts from the Common Core that address fractions with purpose and direction. The term “apply and extend previous understandings…” is a sure sign of the expectation within a progression of concept development. Looking at the progression of multiplying and dividing fractions, you will see that over these 3 years the students are building on previous knowledge and not just repeating the same thing each year. You’ll see that in 4th grade students apply and extend previous understandings of multiplication to multiply a fraction by a whole number. In 5th grade that knowledge and skill is extended to multiply a fraction or whole number by a fraction. Also in 5th grade, students extend previous understandings of division to divide unit fractions by whole numbers and the reverse. In 6th grade, students will continue to build on the foundations laid and will divide fractions by fractions. These are clear and carefully laid progressions. Informing Grades 1-6 Mathematics Standards Development: What Can Be Learned from High-Performing Hong Kong, Singapore, and Korea? American Institutes for Research (2009, p. 13)

Coherence Across Grades Consistent Progressions
Traditional Progressions (Perimeter and Area) Grade 3: Grade 4: Grade 5: Grade 6: Write the area of the shape. Determine the area of the shape in square units. Find the perimeter of the figure. For the traditional progression illustrated by these items, notice how there is an infinitesimal advancement from year to year. However, there is no increase in the magnitude or type of numbers used: each example uses relatively small whole numbers. Even the grade 4 example, which shows halves via the corner sections, can be counted in pairs to represent a whole. There is also no progression in the type of questions asked. Select the rectangle with an area of 24 square units and a perimeter of 20 units.

Coherence Across Grades Consistent Progressions
CCSS-Aligned Progressions (Area and Surface Area) 3.MD.C.6: 4.MD.A.3: 5.NF.B.4b: 6.G.A, 6.RP.A.3: Find the area of each colored figure. Karl’s rectangular vegetable garden is 20 feet by 45 feet, and Makenna’s is 25 feet by 40 feet. Whose garden is larger in area? How much larger is that garden? An aerial photo of farmland shows the dimensions of a field in fractions of a mile. Create a model to show the area, in square miles, of a field that is 3/4 mile by 1/3 mile. In contrast to the traditional approach on the previous slide, the CCSS-aligned approach displays the following features as the items progress through the grades: The numbers increase in magnitude to reflect the grade-level expectations (from within 20, to within 1000, to fractions, to within 10,000) The types of numbers used changes to reflect the grade-level expectations (from whole numbers, to fractions, to general rational numbers) There is an increasing complexity in types of questions used to assess similar topics (e.g., the grade 4 item is a word problem that requires comparison AND broaches the commutative and associative properties). For reference: 3.MD.C.6: Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units). 4.MD.A.3: Apply the area and perimeter formulas for rectangles in real-world and mathematical problems. For example, find the width of a rectangular room given the area of the flooring and the length, by viewing the area formula as a multiplication equation with an unknown factor. 5.NF.B.4b: Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas. 6.G.A: Solve real-world and mathematical problems involving area, surface area, and volume. 6.RP.A.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. (Source for 5.NF.B.4b: adapted from (Source for other CCSS-aligned examples – Alexis needs to paint the four exterior walls of a large rectangular barn. The length of the barn is 80 feet, the width is 50 feet, and the height is 30 feet. The paint costs $28 per gallon, and each gallon covers 420 square feet. How much will it cost Alexis to paint the barn? Explain your work.

Coherence: Link to major topics within grades
Example: data representation Standard 3.MD.3 Instead of bar charts being “yet another thing to cover,” detracting from focus, this standard tells you how to link bar charts to the major work of the grade. This contrasts with typical practice in which it is not uncommon for instruction that focuses on data representation to miss the opportunity to reinforce work with operations by having students use calculators when they are not working on “calculation.”

Coherence: Link to Major Topics Within Grades
Example: Geometric Measurement 3.MD, third cluster Another example of coherence within a grade: Area is not just another topic to cover in Grade 3 . It is explicitly linked to addition and multiplication in the standards. This is clear in 5th grade as well when volume is introduced.

Group Discussion Shift #2: Coherence: Think across grades, link to major topics within grades In your groups, discuss what coherence in the math curriculum means to you. Be sure to address both elements—coherence within the grade and coherence across grades. Cite specific examples. {Page 2 of practice with shifts handout. Read the question from the top of handout – same as slide} Take a few minutes to discuss this question.

Progress to Algebra in Grades K-8
It’s important to know that an important subset of the Major Work of the grade is the progression that leads toward middle-school algebra. Materials should give especially careful treatment to these clusters and their interconnections. This document can be found in the Publishers’ Criteria as well as on the backside of each Focus by Grade Level document (K – 8).

Coherence: Think across grades, and link to major topics within grades. Rigor: In major topics, pursue conceptual understanding, procedural skill and fluency, and application. The third shift is Rigor. This word can mean many different things. For purposes of describing the shifts of the standards, it does not mean “more difficult.” For example, stating that “the standards are more rigorous” does not mean that “the standards are just harder.” Here rigor is about the depth of what is expected in the standards, and also about what one should expect to see happening in the classroom, in curricular materials, and so on. The Standards do not offer a choice between focus on conceptual understanding or fluency or application. They instead require equal intensity of all three. In practical terms, it is not enough to merely know your multiplication facts in third grade, but students must also understand the concept of multiplication and what it represents and be able to apply that understanding and fluency to solving real-world or unexpected application problems.

Shift 3: Rigor: in major topics pursue conceptual understanding, procedural skill and fluency, and application with equal intensity From To Unbalanced emphasis on procedure or application Assessment of all three aspects of rigor in balance A lack of items that require conceptual understanding Items that require students to demonstrate conceptual understanding of the mathematics, not just the procedures Fluency items that are only routine and ordinary Fluency items that are presented in new ways, as well as some that are routine and ordinary Application of mathematics to routine and contrived word problems Application of mathematics to authentic non-routine problems and real-world situations

Rigor Balanced Assessments
CCSSM-aligned assessments and sets of assessments will be balanced by distributing score points across the three aspects of rigor: Procedural skill and fluency Application Conceptual understanding With the third Shift, rigor, many educators appreciate the visual of the three legged stool. All three aspects of rigor (conceptual understanding, procedural skill and fluency and application) need to be addressed for students to be able to reach the depth of learning that is expected by the CCSS. That said, it is also noted in the Publishers’ Criteria that: 1. The three aspects of rigor are not always separate in assessment questions. Conceptual understanding and fluency go hand in hand; fluency can indirectly be assessed through applications; and applications can elicit information about students’ conceptual understanding 2. Nor are the three aspects of rigor always together in assessment questions. Fluency requires dedicated assessment questions. Rich applications cannot always be forced into any assessment question. And conceptual understanding cannot be assumed, but must be explicitly assessed. It is important to note that many educators have been attending to the various aspects of rigor for years. The difference that exists for all educators is the explicit attention to the balance of these three aspects of rigor to ensure the students have a firm grasp on the mathematics of their grade. The “Math Wars” of previous decades are over. We do not need understanding OR fluency for students to be college and career ready. College and career readiness requires flexible understanding as well as fast and efficient procedural skills. And, students must apply these mathematical skills in grade-appropriate applications.

The CCSSM require a balance of: Solid conceptual understanding
Rigor The CCSSM require a balance of: Solid conceptual understanding Procedural skill and fluency Application of skills in problem solving situations Pursuit of all three requires equal intensity in time, activities, and resources. {read slide}

Phil Daro (a lead writer of the CCSSM) discussing conceptual understanding versus “answer getting”.
This video can be seen in its entirety at

Solid Conceptual Understanding
Teach more than “how to get the answer” and instead support students’ ability to access concepts from a number of perspectives Students are able to see math as more than a set of mnemonics or discrete procedures Conceptual understanding supports the other aspects of rigor (fluency and application) One aspect of rigor is building solid conceptual understanding. Once we have a set of standards that are in fact focused, teachers and students have the time and space to develop solid conceptual understanding. {read the slide} There is no longer the pressure to quickly teach students how to superficially get to the answer, often relying on tricks or mnemonics. The standards instead require a real commitment to understanding mathematics, not just how to get the answer. As an example, it is not sufficient to simply know the procedure for finding equivalent fractions, but students also need to know what it means for numbers to be written in equivalent forms. Attention to conceptual understanding is one way that we can start counting on students building on prior knowledge. It is very difficult to build further math proficiency on a set mnemonics or discrete procedures.

Here is an example of a place value chart that you get when you search for “place value worksheets” online. It is also a non-example of work that would elicit conceptual understanding. As you can see, it would not be possible to assess whether your students had a conceptual understanding of place value by them completing this worksheet. It would be fairly obvious to a student who does not understand place value that the first number goes with hundreds, the 2nd number with tens and so on. Even on problem letter h, where it could have asked for deeper understanding, the worksheet places a 0 for tens to eliminate any need for thinking.

Here is a snapshot of a worksheet practicing place value understanding
Here is a snapshot of a worksheet practicing place value understanding. You can see how a teacher would be able to assess a student’s conceptual understanding of place value more clearly with the results of this worksheet. In problems 6-8, the base ten units in 106 are bundled in different ways. This is helpful when learning how to subtract in a problem like In #9, we see that if the order is always given “correctly,” then all we do is teach students rote strategies without thinking about the size of the units or how to encode them in positional notation.

The standards require speed and accuracy in calculation.
Fluency The standards require speed and accuracy in calculation. Teachers structure class time and/or homework time for students to practice core functions such as single-digit multiplication so that they are more able to understand and manipulate more complex concepts Another aspect of rigor is procedural skill and fluency. {read slide} Note that this is not memorization absent understanding. This is the outcome of a carefully laid out learning progression. At the same time, we can’t expect fluency to be a natural outcome without addressing it specifically in the classroom and in our materials. Some students might require more practice than others, and that should be attended to. Additionally, there is not one approach to get to speed and accuracy that will work for all students. All students, however, will need to develop a way to get there. It is important to note here that while teachers in grades K-5 may find creative ways to use calculators in the classroom, students are not meeting the standards when they use them--not just in the area of fluency, but in all other areas of the Standards as well.

Required Fluencies in K-6
Grade Standard Required Fluency K K.OA.5 Add/subtract within 5 1 1.OA.6 Add/subtract within 10 2 2.OA.2 2.NBT.5 Add/subtract within 20 (know single-digit sums from memory) Add/subtract within 100 3 3.OA.7 3.NBT.2 Multiply/divide within 100 (know single-digit products from memory) Add/subtract within 1000 4 4.NBT.4 Add/subtract within 1,000,000 5 5.NBT.5 Multi-digit multiplication 6 6.NS.2,3 Multi-digit division Multi-digit decimal operations This chart shows a breakdown of the required fluencies in grades K-6. Fluent in the particular Standards cited here means “fast and accurate.” It might also help to think of fluency in math as similar to fluency in a foreign language: when you’re fluent, you flow. Fluent isn’t halting, stumbling, or reversing oneself. The word fluency was used judiciously in the Standards to mark the endpoints of progressions of learning that begin with solid underpinnings and then pass upward through stages of growing maturity. Some of these fluency expectations are meant to be mental, others need pencil and paper. But for each of them, there should be no hesitation about how to proceed in getting the answer with accuracy.

Application Students can use appropriate concepts and procedures for application even when not prompted to do so. Teachers provide opportunities at all grade levels for students to apply math concepts in “real world” situations, recognizing this means different things in K-5, 6-8, and HS. Teachers in content areas outside of math, particularly science, ensure that students are using grade-level-appropriate math to make meaning of and access science content. Using mathematics in problem solving contexts is the third leg of the stool supporting the learning that is going on in the math classroom. This is the “why we learn math” piece, right? We learn it so we can apply it in situations that require mathematical knowledge. There are requirements for application all the way throughout the grades in the CCSS. {read slide} But again, we can’t just focus solely on application—we need also to give students opportunities to gain deep insight into the mathematical concepts they are using and also develop fluency with the procedures that will be applied in these situations. The problem-solving aspect of application is what’s at stake here—if we attempt this with a lack of conceptual knowledge and procedural fluency, the problem just becomes three times harder. At the same time, we don’t want to save all the application for the end of the learning progression. Application can be motivational and interesting, and there is a need for students at all levels to connect the mathematics they are learning to the world around them.

Engaging with the Shift: Making a True Statement
Rigor = ______ + ________ + _______ This shift requires a balance of three discrete components in math instruction. This is not a pedagogical option, but is required by the Standards. Using grade __ as a sample, find and copy the standards which specifically set expectations for each component. {note to presenter – you may fill in either grade 3 or grade 6 for this activity, or have them choose, depending on your audience} Continuing on page 3 of the practicing with the shifts handout, who can fill in the blanks of the three components of rigor in the Standards? Next we will find and copy the standards for grade 3 (or 6) which specifically set expectations for fluency, conceptual understanding or application. {You may want to remind the participants that the Standards are very clear about this and give the hint that the fluency standards say “fluent” the conceptual understanding standards often use the word “understand” or “relate” and the application standards often say “real world problem.”} You will have 10 minutes to locate the Standards – you may find that sometimes there is overlap between two components. You can just write the code of the standard instead of writing the entire standard. {Discuss the results – see sample answer in the module packet}

Group Discussion Shift #3: Rigor: Expect fluency, deep understanding, and application In your groups, discuss ways to respond to one of the following comments: “These standards expect that we just teach rote memorization. Seems like a step backwards to me.” Or “I’m not going to spend time on fluency—it should just be a natural outcome of conceptual understanding.”

Monica Sims (a 5th grade teacher) speaking about the Standards
Monica Sims (a 5th grade teacher) speaking about the Standards. This video, produced by America Achieves, can be seen in its entirety at

Rigor Activity Use your assessment and work through making an assessment item “Rigorous” Pick an assessment item from your group to put on chart paper Put the before and after item on chart paper Post on walls for a gallery walk

Activity: What didn’t you have time for What could you get rid of What could you spend less time on

Additional Resources:
For a growing bank of mini-assessments that can be used in the classroom, go to For a growing bank of tasks that can be used for instructional or assessment purposes, go to For more information about the Shifts, go to For Illustrative Math tasks, go to For PARCC sample items and practice tests, grades 3–11, go to For SBAC practice tests, grades 3–high school, go to For more about the progressions documents, go to Textbook Alignment, University of Michigan: Textbook Navigator/Journal – Michigan State University: Center for the Study of Curriculum an independent non-profit, has released free, web-based reviews of current K-8 math instructional materials.

Thank You! Catherine M Puster, cpuster@npesc.org
Karen Witt, Andrea Smith, Additional information from: Char Shyrock, iteachbay.blogspot.com

Catherine Puster, NPESC Curriculum

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