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**Math Common Core Standards**

Jennie Winters Lake County ROE

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**Focus for Today 3 types of change Standards for Mathematical Practice**

Focus, Coherence & Rigor Assessment Curriculum: Quality Units/Lessons Q & A

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**Common Core Implementation**

Instructional Change Curricular Change Assessment Change Common Core Implementation

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**1. Make sense of problems and persevere in solving them.**

Practice Standards 1. Make sense of problems and persevere in solving them. Make each standard individual slide. Add image to each slide – STUDENTS WORKING TOGETHER Add to handout

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**2. Reason abstractly and quantitatively.**

Practice Standards Make each standard individual slide. Add image to each slide Add to handout 2. Reason abstractly and quantitatively.

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**3. Construct viable arguments and critique the reasoning of others.**

Practice Standards 3. Construct viable arguments and critique the reasoning of others. Make each standard individual slide. Add image to each slide Add to handout

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**4. Model with mathematics.**

Practice Standards Make each standard individual slide. Add image to each slide – YOUNG STUDENTS WORKING WITH MATH MANIPULATIVES Add to handout 4. Model with mathematics.

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**5. Use appropriate tools strategically.**

Practice Standards 5. Use appropriate tools strategically. Make each standard individual slide. Add image to each slide Add to handout

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**6. Attend to precision. Practice Standards**

Make each standard individual slide. Add image to each slide Add to handout 6. Attend to precision.

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**7. Look for and make use of structure.**

Practice Standards 7. Look for and make use of structure. Make each standard individual slide. Add image to each slide – 100’s CHART – JENNIE WILL FIND Add to handout

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**8. Look for and express regularity in repeated reasoning.**

Practice Standards 8. Look for and express regularity in repeated reasoning. Make each standard individual slide. Add image to each slide Add to handout

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**Modes of Representation (Lesh, Post, & Behr, 1987)**

Manipulative Models Real-world Situations Pictures Oral/Written Language Written Symbols

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**Modes of Representation**

Manipulative/ Tools Real-Life Situations Picture/Graph Oral & Written Language Table/Chart The 5 (6) modalities are a helpful frame of reference for teachers, administrators, students and parents. Since the CCSSM are expectations of what students will know, do AND UNDERSTAND, it is key that students can transfer concepts between various modalities. Symbols (Equations, etc.)

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**The CCSS Requires Three Shifts in Mathematics**

Focus: Focus strongly where the standards focus. Coherence: Think across grades, and link to major topics 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.

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**Rigor The CCSSM require a balance of: Solid conceptual understanding**

Procedural skill and fluency Application of skills in problem solving situations Pursuit of all threes requires equal intensity in time, activities, and resources. {read slide}

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**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.

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**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.

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**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 as meaning the same thing as when we say that somebody is fluent 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 and others with pencil and paper. But for each of them, there should be no hesitation about how to proceed in getting the answer with accuracy.

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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.

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Assessment Conceptual Assessment Includes Observational Tools

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2nd Grade Example Students each create their own paper airplane and take turns flying them. They use a tape measure to find the distance each has flown to the nearest foot. What concepts are being assessed?

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**Assessment Conceptual Assessment Procedural Skill & Fluency Assessment**

Includes Observational Tools Procedural Skill & Fluency Assessment Includes Extended Response

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**Assessment Conceptual Assessment Procedural Skill & Fluency Assessment**

Includes Observational Tools Procedural Skill & Fluency Assessment Includes Extended Response Application Includes Rich Tasks

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**Curriculum Sequence of units (Coherence) Prioritization**

Sequence within units: Conceptual before procedural Application throughout

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**The Tri-State Quality Review Rubric Criteria**

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**4 Dimensions Alignment to the Rigor of the CCSS**

Key Areas of Focus in the CCSS Instructional Supports Assessment

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**1 – Alignment to the Rigor of the CCSS**

The unit aligns with the letter and spirit of the CCSS: Targets a set of grade level mathematics standard(s) at the level of rigor in the CCSS for teaching & learning

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**1 – Alignment to the Rigor of the CCSS**

The unit aligns with the letter and spirit of the CCSS: Standards for Mathematical Practice that are central to the unit are identified, handled in a grade-appropriate way, and well connected to the content being addressed.

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**1 – Alignment to the Rigor of the CCSS**

The unit aligns with the letter and spirit of the CCSS: Presents a balance of mathematical procedures and deeper conceptual understanding inherent in the CCSS

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**2-Key Areas of Focus in the CCSS**

The unit reflects evidence of key shifts that are reflected in the CCSS. Focus Centers on the concepts, foundational knowledge and level of rigor that are prioritized in the standards.

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**2-Key Areas of Focus in the CCSS**

The unit reflects evidence of key shifts that are reflected in the CCSS. Coherence Makes connections and provides opportunities for students to transfer knowledge and skills within and across domains and learning progressions.

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**2-Key Areas of Focus in the CCSS**

The unit reflects evidence of key shifts that are reflected in the CCSS. Rigor Requires students to engage with an demonstrate challenging mathematics in the following ways:

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**2-Key Areas of Focus in the CCSS**

Conceptual Understanding Requires students to demonstrate conceptual understanding through complex problem solving, in addition to writing and speaking about their understanding.

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**2-Key Areas of Focus in the CCSS**

Procedural Skill & Fluency Expects, supports and provides guidelines for procedural skill and fluency with core calculations, mathematical procedures and strategies (when called for in the standards for the grade) to be performed quickly and accurately.

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**2-Key Areas of Focus in the CCSS**

Application Provides opportunities for students to independently apply mathematical concepts in real-world situations and problem solve with persistence, choosing and applying an appropriate model or strategy to new situations.

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**3 – Instructional Supports**

The unit is responsive to varied student needs: Includes clear and sufficient guidance to support teaching and learning of the targeted standards, including, when appropriate, the use of technology and media.

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**3 – Instructional Supports**

The unit is responsive to varied student needs: Uses and encourages precise and accurate mathematics, academic language, terminology, and concrete or abstract representations (e.g. pictures, symbols, expressions, equations, graphics, models) in the discipline.

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**3 – Instructional Supports**

The unit is responsive to varied student needs: Engages students in productive struggle through relevant, thought-provoking questions, problems, and tasks that stimulate interest and elicit mathematical thinking.

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**3 – Instructional Supports**

The unit is responsive to varied student needs: Addresses instructional expectations and is easy to understand and use.

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**3 – Instructional Supports**

Provides appropriate level and type of scaffolding, differentiation, intervention, and support for a broad range of learners: Supports diverse cultural and linguistic backgrounds, interests and styles.

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**3 – Instructional Supports**

Provides appropriate level and type of scaffolding, differentiation, intervention, and support for a broad range of learners: Provides extra supports for students working below grade level.

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**3 – Instructional Supports**

Provides appropriate level and type of scaffolding, differentiation, intervention, and support for a broad range of learners: Provides extensions for students with high interest or working above grade level.

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**3 – Instructional Supports**

Recommend and facilitate a mix of instructional approaches for a variety of learners such as using multiple representations, (including models) using a range of questions, checking for understanding, flexible grouping, pair-share, etc.

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**3 – Instructional Supports**

Gradually remove supports, requiring students to demonstrate their mathematical understanding independently.

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**3 – Instructional Supports**

Demonstrate an effective sequence and a progression of learning where the concepts or skills advance and deepen over time.

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**3 – Instructional Supports**

Expects, supports, and provides guidelines for procedural skill and fluency with core calculations and mathematical procedures (when called for in the standards for the grade) to be performed quickly and accurately.

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4 – Assessment The lesson/unit regularly assesses whether students are mastering standards-based content and skills: Is designed to elicit direct, observable evidence of the degree to which a student can independently demonstrate the targeted CCSS.**

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4 – Assessment The lesson/unit regularly assesses whether students are mastering standards-based content and skills: Assesses student proficiency using methods that are accessible and unbiased, including the use of grade level language in student prompts.**

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4 – Assessment The lesson/unit regularly assesses whether students are mastering standards-based content and skills: Includes aligned rubrics, answer keys, and scoring guidelines that provide sufficient guidance for interpreting student performance.

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4 – Assessment The lesson/unit regularly assesses whether students are mastering standards-based content and skills: Use varied modes of curriculum embedded assessments that may include pre-, formative, summative and self-assessment measures.

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**Questions and (Hopefully) Answers**

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