Common Core State Standards Math Workgroup Training

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  • Primary structure of the CCSS-M:Domains are large groups of related standards. Standards from different domains may sometimes be closely related. Look for the name with the code number on it for a Domain. All of the standards have domains that are the large groups of standards.These definitions are directly from the CCSSM. All parts of the CCSSM use these three partsClusters are groups of related standards.Standards from different clusters may sometimes be closely related, because mathematics is a connected subject. Clusters appear inside domains. This is the subgroup under the domain – standards here are more closely related.These definitions are directly from the CCSSM. All parts of the CCSSM use these three partsStandards define what students should be able to understand and be able to do – part of a cluster.More in-depth comparison and transitional documents are available on the OSPI CCSS web site http://k12.wa.us/Corestandards/default.aspx
  • Contained within the Common Core State Standards for ELA is a section on Literacy for Science, Social Studies, and other Technical subjects. Mathematics is considered a technical subject. Therefore, Mathematics teachers of grades 6-12 are responsible for the 10 reading and 10 writing literacy standards. Once again, these standards are found on pages 62-66 of the Common Core State Standards for ELA.
  • A few of the examples of what students should be seen doing that represent literacy skills being employed in the mathematics classroom are:Learning to read mathematical text including textbooks, articles, problems, problem explanationsCommunicating using correct mathematical terminology appropriate to the student’s mathematical developmentReading, discussing and applying the mathematics found in literature, including looking at the author’s purposeResearching mathematics topics or related problemsReading appropriate text providing explanations for mathematical concepts, reasoning or proceduresApplying readings as citing for mathematical reasoning – using information found in texts to support their reasoning; developing works cited documents for research done to solve a problemListening and critiquing peer explanations of their reasoningJustifying orally and in writing reasoningRepresenting and interpreting data with and without technologyUsing the Literacy standards as an adjunct to the Standards for Mathematical Practice will further students’ mathematical proficiency.
  • Brief overview of the format of the CCSS in the next few slides..
  • Each K-8 grade is made up of domains, clusters and standards.Domain names are in the shaded band; overarching ideas that continue across multiple grades; illustrates a progression of increasing complexity Clusters are underneath in bold with the standards numbered under each of the clusters; describes the big idea of a group of related standardsStandards are numbered and describe what students should know and be able to do at each grade level
  • (Each grade in K-8 begins with an Introduction.These introductions identify the critical areas, which cut across topics, in the gradeThe numbered description of each critical area illustrates the focus for the learning.
  • The second page of each grade is an Overview. It identifies the domains and clusters in that grade also reminding readers of the mathematical practices.
  • High school has an added level in the hierarchy – the Conceptual Category. These Conceptual Categories are: Number and Quantity, Algebra, Functions, Geometry, Statistics and Probability and Modeling. Modeling actually does not have Standards; it is embedded within the other Conceptual Categories. Standards connected to Modeling are indicated by asterisks. Conceptual Categories are not courses. Standards marked with a plus sign are standards that are necessary for advanced mathematics. This means that any student who intends to pursue advanced mathematics needs these experiences.
  • Within each Conceptual Category the structure parallels K-8. There are Domains or big ideas. The bolded statements are the Clusters. The Standards are numbered and further describe the Cluster. An added element is the (+) standards which identify standards that are needed for advanced study. They are not intended to be required for all students.
  • Like the K-8 grade level introduction, the Conceptual Category Introduction describes the mathematics within the Conceptual Category and places that mathematics within the K-16 perspective.
  • The Conceptual Category Overview serves as an outline of the conceptual category, listing the domains with their clusters, and reminding readers of the mathematical practices.
  • An Achieve committee organized the high school standards into course sequences.
  • Each Pathway prepares students for a menu of fourth-year courses such as:Pre-calculus (or AP Calculus)AP StatisticsDiscrete MathematicsAdvanced Quantitative ReasoningAnd courses designed for particular career technical pathways
  • Each of the Pathways has an overview that details the content addressed in each course. The columns represent each course with the related domains, clusters and standards.
  • The critical areas are fleshed out as the units within each course. The overview By the end of the third course in each pathway, all of the critical areas have been addressed.
  • Unit details are elaborated with the addition of unit overviews containing instructional notes. These notes are seen in italics and further clarify the related clusters and standards.
  • These critical areas in K-8 mathematics are intended to establish a firm foundation for high school algebraGreat implications on instructional practice – many instructional materials are surface and don’t provide the depth of practiceApproach mathematics in a different way – dive deep into a core area to develop habits of mind in those areas versus rushing through materials
  • This activity involves getting to know the HS Critical Areas or the K-8 Critical Areas of Focus.Choose a grade in K-8 or High School Algebra 1 or Math 1. The Critical Areas of Focus are on the Introduction page of each grade in K-8, or on the introduction page of the HS course in the Pathways. Locate your selected grade or course.You will be looking for the concepts, skills and procedures, and the relationships that students are to make.Before we move on. Let’s clarify the difference between a concept and a skill/procedure.
  • Concepts are shown in red, skills and procedures are in green. A concept represents a big idea in which relationships of key elements are developed. In this fourth grade example, the concept is how ten is the basis of our place value system. Comparing and rounding numbers is first accomplished by understanding how the place value system works. Ultimately, the understanding evolves into a systematic procedure. Reading and writing multi-digit numbers is a skill that also results from this understanding. The Common Core treats concepts, and skills and procedures equally. The expectation is that students develop understanding of concepts first. Over time and through usage, fluency with skills and procedures will develop. Teaching shortcuts, mnemonics, and rote procedures is premature if the underlying concepts have not been developed.
  • Concepts are shown in red, skills and procedures are in green. In this fourth grade example, the general concept is how ten is the basis of our place value system. In re, Comparing and rounding numbers is first accomplished by understanding how the place value system works. Ultimately, the understanding evolves into a systematic procedure. Reading and writing multi-digit numbers is a skill that also results from this understanding. The Common Core treats concepts, and skills and procedures equally. The expectation is that students develop understanding of concepts first. Over time and through usage, fluency with skills and procedures will develop. Teaching shortcuts, mnemonics, and rote procedures is premature if the underlying concepts have not been developed.
  • After your group finishes analyzing the critical area, look at the domains, clusters and standards that address that critical area, answer the question “How does the Critical Area of Focus inform their instruction?
  • Each grade level begins with this…integration of the learning that is to take place.
  • Handout: Domain illustration template from Step 1Highlighters (Extra for grade level facilitator to compile groups thoughts & share on document camera)Small group: Make group version for sharingWhole group: Share and discuss (doc cam)Whole group: Add language and examples from small groups(doc cam)Individual: Make revisions/edits for personal domain illustrationWork in small groups to repeat the process for next domain (based on 3-year implementation plan) or for other grade levels, depending on the makeup of the group.
  • (Read this slide.)(The next slides will describe these three types of progressions.)
  • This slide is taken from Adding It Up: Helping Children Learn Mathematics, NRC, 2001. It shows the typical learning progression from learning how to find a sum by first counting all, then starting from the last number said representing a set and counting on to using strategies such as making ten, using doubles, etc.
  • Bill McCallum is responsible for narratives describing the typical learning progression of a topic, informed both by research on children's cognitive development and by the logical structure of mathematics.In other words, they describe how ideas connect and grow across grades. A technical appendix, authored by Jason Zimba, highlights structural features that are not highly visible in the document.
  • Standards Progressions are the actual standards represented in a time progression.
  • This diagram illustrates how the domains are distributed across the Common Core State Standards. What is not easily seen is how a domain may impact multiple domains in future grades. An example is K-5 Measurement and Data, which splits into Statistics and Probability and Geometry in grade 6. Likewise, Operations and Algebraic Thinking in K-5 provides foundation Ratios and Proportional Relationships, The Number System, Expressions and Equations, and Functions in grades 6-8.
  • Nonetheless, to support some analysis of the progressions of standards across grades, we can place the text of the standards, for each domain, in a table as shown. Sometimes the clusters of standards support more fine-grained analysis.
  • This slide show the highlighted sections on place value from the previous slide. By reviewing standards progressions, one can see how one year builds upon another.
  • These pathways assume mathematics in each year of high school, describe 3 years of study, and lead directly to preparedness for college and career readiness.These are also located in the appendices of the document. The CCSS provide 3 years of instruction with a nod to students ready for the fourth year. The 4th year is denoted in the standards as a + and might not be applicable for all students.
  • Because of structural differences between the disciplines of mathematics and English Language Arts, the mathematics standards do not support such easy analysis of the progression of standards across grades. This diagram depicts some of the structural features of the mathematics standards, where several different domains from grades K-8 converge toward algebra in high school. This diagram does not include other “flows,” such as from Number and Operations—Fractions in grades 3-5, to Ratios and Proportional Relationships in grades 6 and 7, to Functions in grade 8 and high school, with connections to geometry and probability. An Algebra Idea Across K-12Compare and contrast: patterns, functions, and sequencesIn grades K-8, students study patternsIn grades 9-11, students study functionsIn grade 12, students might study sequencesA sequence is a patternA pattern suggests a functionA sequence is a function with a domain consisting of whole numbers
  • This is located in the front of the standards book in great detailThese are the habits of mind we want for all students regardless of the subject area. Student characteristics not teacher characteristicsThese should not be overlooked – how can we be intentional about building a strong foundation.
  • Not individually addressed….culture of the classroom
  • Teacher CommentaryThis lesson is a reengagement lesson designed for learners to revisit a problem-solving task they have already experienced. My colleague, Stacy Emory, best describes reengagement by comparing it to re-teaching. Re-teaching is a teacher directed activity where we plan a different lesson to address something that is perceived to be a misconception with our students. Reengagement is a learner-centered activity wherein the original task is posed in such a way that we may expose learners to different strategies, alternate solutions, or even misconceptions. Think of the original task as a formative assessment that helps you shape the lessons that follow.All of my 6th grade learners were able to successfully draw the 4th stage of this pattern and mostly all of them were able to correctly identify the number of white buttons in stages 5 and 6. The two exceptions to stages 5 and 6 were learners who scored the special case points for counting the black button along with the white buttons.13 out of 28 learners incorrectly identified 33 as the number of total buttons in stage 11. These learners also went on to incorrectly list 72 as the number of total buttons in stage 24. This error, I believe, solely to be a critical reading miscue. These learners were leaving out the black button and missing the language “total number of buttons.” I base this inference on the abundance of learner work which successfully describes the growth of the white buttons.The learners were mostly successful at generalizing the pattern in their own words or through the use of number sentences. Some learners were describing the pattern as adding on:“I added 4 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 [ to get stage 11.]”Other learners were moving towards thinking multiplicatively about the +3-ness of the growth:“ I counted by 3’s for each pattern...”Less than half of the class chose to represent the growth with a number sentence such as:“I [did] (11 x 3) +1 = 34 buttons [stage 11.] I added one for the black button.”Overall, my 6th grade class performed very well on this task, which is why we can use this reengagement lesson to begin looking at multiple representations of functions.All of the teacher tools you see at work in my classroom have accumulated over the years of observing other math teachers in video and in person. It is easy to become isolated in our profession, but there is a lot to be gained from observing others at work in their craft.
  • Grade Band Math Tasks Handout from Step 2 (Illustrative Mathematics Problems)How do you see this progressing through the grade levels?
  • Rich problems provide an opportunity for students to solve again and again with increasing expectations by changing details to highlight new learning. Task progressions will be explored in future sessions.
  • Words within the transition documents:“Continue to…” – these are things that are within current WA standards“Move to…” – this is a new content for the grade level“No longer responsible for…” - with full implementation this content will be removed or covered somewhere else“Partially…” – the portions that will not be taught are highlighted
  • Posted on Orting website
  • Common Core State Standards Math Workgroup Training

    1. 1. Math Common Core State Standards Dr. Marci Shepard Orting School District CCSS Math Workgroup April 2012Includes information from OSPI, ESDs, NCTM, Ohio Department of Education and other sources Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    2. 2. Did you miss previous sessions? http://www.orting.wednet.edu/education/components/layout/default.php?sectionid=374& Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    3. 3. Common Core State Standards The Big Ideas in MATH Office of Superintendent of Public Instruction Randy I. Dorn, State Superintendent
    4. 4. Focusing on the Foundation… Washington’s Implementation Timeline & Activities 2010-11 2011-12 2012-13 2013-14 2014-15Phase 1: CCSS ExplorationPhase 2: Build Awareness & BeginBuilding Statewide CapacityPhase 3: Build Statewide Capacityand Classroom TransitionsPhase 4: Statewide Application andAssessmentOngoing: Statewide Coordinationand Collaboration to SupportImplementation January 2012 CCSS Webinar Series Part 2: Mathematics 4
    5. 5. http://www.youtube.com/watch?v=dnjbwJdcPjE&list=UUF0pa3nE3aZAfBMT8pqM5PA&index=5&feature=plcp Orting School District * Teaching, Learning and Assessment * 2012
    6. 6. Content Progressions and Major ShiftsMajor ShiftsFocus Coherence Application• Fewer big ideas --- • Articulated progressions of topics • Being able to apply learn more and performances that are concepts and skills to• Learning of concepts developmental and connected to new situations is emphasized other progressions January 2012 CCSS Webinar Series Part 2: Mathematics 6
    7. 7. Structural Comparison: WA Standards vs. CCSS Mathematics WA Mathematics Standards Common Core State Standards Grades K-8, high school standards presented through six mathematical Presentation of Grade K-8, high school standards presented domains including specially noted Standards in traditional and integrated pathways. STEM standards - denoted by (+) symbols. Grade-level standards are broken into Grade-level standards are broken into core clusters of learning under several Organization content areas, additional key content, and domains and all have Standards for mathematical processes. Mathematical Practice. Standards are accompanied by explanatory Standards have occasional examples Examples comments and examples. in italics.Kindergarten | Grade 1 | Grade 2 | Grade 3 | Grade 4 | Grade 5 | Grade 6 | Grade 7 |Grade 8Transition for Algebra I | Transition for Geometry | Integrated Math I | Integrated Math II
    8. 8. Reading Literacy Standards Grades 6-8 Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    9. 9. What does literacy look like in the mathematics classroom?• Learning to read mathematical text• Communicating using correct mathematical terminology• Reading, discussing and applying the mathematics found in literature• Researching mathematics topics or related problems• Reading appropriate text providing explanations for mathematical concepts, reasoning or procedures• Applying readings as citing for mathematical reasoning• Listening and critiquing peer explanations• Justifying orally and in writing mathematical reasoning• Representing and interpreting data Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    10. 10. Organization of the Standards Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    11. 11. CCSS Design and Organization Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    12. 12. Format of K-8 Standards Grade Level Domain Standard Cluster Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    13. 13. Cross-cuttingGrade Level Introduction themes Critical Area of Focus Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    14. 14. Grade Level OverviewGrade 4 Overview Mathematical PracticesOperations and Algebraic Thinking 1. Make sense of problems and Use the four operations with whole numbers to solve problems. persevere in solving them Gain familiarity with factors and multiples. 2. Reason abstractly and Generate and analyze patterns. quantitativelyNumber and Operations in Base Ten 3. Construct viable arguments and Generalize place value understanding for multi-digit whole numbers. critique the reasoning of others Use place value understanding and properties of operations to perform multi-digit arithmetic. 4. Model with mathematicsNumber and Operations—Fractions 5. Use appropriate tools Extend understanding of fraction equivalence and ordering. strategically Build fractions from unit fractions by applying and extending previous 6. Attend to precision understandings of operations on whole numbers. Understand decimal notation for fractions, and compare decimal 7. Look for and make use of fractions. structureMeasurement and Data 8. Look for and express regularity Solve problems involving measurement and conversion of in repeated reasoning measurements from a larger unit to a smaller unit. Represent and interpret data. Geometric measurement: understand concepts of angle and measure angles.Geometry Draw and identify lines and angles, and classify shapes by properties of their lines and angles. Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    15. 15. CCSS for High School Mathematics• Organized in “Conceptual Categories” – Number and Quantity – Algebra – Functions – Modeling – Geometry – Statistics and Probability• Conceptual categories are not courses• Additional mathematics for advanced courses indicated by (+)• Standards with connections to modeling indicated by (★) Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    16. 16. Format of High School StandardsDomainClusterStandardAdvanced Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    17. 17. Format of StandardsDr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    18. 18. Conceptual Category Introduction Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    19. 19. Conceptual Category OverviewDomainCluster Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    20. 20. High School Mathematical Pathways Typical• Two main pathways: in U.S. – Traditional: Two algebra courses and a geometry course, with statistics and probability in each – Integrated: Three courses, each of which includes algebra, geometry, statistics, and probability Typical outside U.S.• Both pathways: – Complete the Common Core in the third year – Include the same “critical areas” – Require rethinking high school mathematics – Prepare students for a menu of fourth-year courses Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    21. 21. Two Main Pathways Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    22. 22. Pathway OverviewDr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    23. 23. Course Overview: Critical Areas (units) Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    24. 24. Course Detail by Unit (critical area) Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    25. 25. Understanding the Math Common Core State Standards QUESTIONS 1-4 Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    26. 26. ContentDr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    27. 27. Critical Areas in Mathematics Priorities in Support of Rich Instruction andGrade Expectations of Fluency and Conceptual Understanding Addition and subtraction, measurement usingK–2 whole number quantities Multiplication and division of whole numbers3–5 and fractions Ratios and proportional reasoning; early 6 expressions and equations Ratios and proportional reasoning; arithmetic 7 of rational numbers 8 Linear algebra Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    28. 28. Activity 2: K-8 Critical Areas of Focus HS Critical Areas• Read a K-8 grade level’s Critical Areas of Focus or HS Critical Area – What are the concepts? – What are the skills and procedures? – What relationships are students to make? Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    29. 29. Concepts, Skills and ProceduresConcepts• Big ideas• Understandings or meanings• Strategies• RelationshipsUnderstanding concepts underlies the development and usage of skills and procedures and leads to connections and transfer.Skills and Procedures• Rules• Routines• AlgorithmsSkills and procedures evolve from the understanding and usage of concepts. Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    30. 30. Concepts, Skills and ProceduresGrade 4 Number and Operations in Base TenGeneralize place value understanding for multi-digit whole numbers.• Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right. For example, recognize that 700  70 = 10 by applying concepts of place value and division.• Read and write multi-digit whole numbers using base- ten numerals, number names, and expanded form. Compare two multi-digit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.• Use place value understanding to round multi-digit whole numbers to any place. Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    31. 31. Activity 2 Critical Areas• Read the grade level Critical Areas of Focus or HS Critical Areas What are the concepts? What are the procedures and skills? What relationships are students to make?• Look at the domains, clusters and standards for the same grade(s) or High School Course How do the Critical Areas inform their instruction? Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    32. 32. Critical Areas of Focus Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    33. 33. Digging into the Standards…. Focusing on the Domain• Read using a highlighter to identify language someone might have difficulty with• Develop parent friendly language and/or examples for 2nd column of template Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    34. 34. Understanding the Math Common Core State Standards QUESTION 5 Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    35. 35. Progressions Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    36. 36. • Progressions – Describe a sequence of increasing sophistication in understanding and skill within an area of study• Three types of progressions – Learning progressions – Standards progressions – Task progressions Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    37. 37. Learning Progression for Single-Digit AdditionFrom Adding It Up: Helping Children Learn Mathematics, NRC, 2001. Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    38. 38. Learning Progressions Document for CCSSMhttp://ime.math.arizona.edu/progressions/• Narratives• Typical learning progression of a topic• Childrens cognitive development• The logical structure of mathematics• Math Common Core Writing Team with Bill McCallum as Creator/Lead Author Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    39. 39. Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    40. 40. CCSS Domain Progression K 1 2 3 4 5 6 7 8 HSCounting &Cardinality Ratios and Proportional Number and Operations in Base Ten Relationships Number & Number and Operations – Quantity The Number System Fractions Expressions and Equations Algebra Operations and Algebraic Thinking Functions Functions Geometry Geometry Statistics & Measurement and Data Statistics and Probability Probability Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    41. 41. Standards Progression:Number and Operations in Base Ten Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    42. 42. Use Place Value UnderstandingGrade 1 Grade 2 Grade 3Use place value understanding and Use place value understanding and Use place value understanding andproperties of operations to add and properties of operations to add and properties of operations to performsubtract. subtract. multi-digit arithmetic.4. Add within 100, including adding a 5. Fluently add and subtract within 100 1. Use place value understanding totwo-digit number and a one-digit using strategies based on place value, round whole numbers to the nearest 10number, and adding a two-digit number properties of operations, and/or the or 100.and a multiple of 10, using concrete relationship between addition and 2. Fluently add and subtract within 1000models or drawings and strategies based subtraction. using strategies and algorithms based onon place value, properties of operations, 6. Add up to four two-digit numbers place value, properties of operations,and/or the relationship between using strategies based on place value and/or the relationship betweenaddition and subtraction; relate the and properties of operations. addition and subtraction.strategy to a written method and explain 7. Add and subtract within 1000, using 3. Multiply one-digit whole numbers bythe reasoning used. concrete models or drawings and multiples of 10 in the range 10–90 (e.g.,Understand that in adding two-digit strategies based on place value, 9 × 80, 5 × 60) using strategies based onnumbers, one adds tens and tens, ones properties of operations, and/or the place value and properties ofand ones; and sometimes it is necessary relationship between addition and operations.to compose a ten. subtraction; relate the strategy to a5. Given a two-digit number, mentally written method. Understand that infind 10 more or 10 less than the number, adding or subtracting three digitwithout having to count; explain the numbers, one adds or subtractsreasoning used. hundreds and hundreds, tens and tens,6. Subtract multiples of 10 in the range ones and ones; and sometimes it is10-90 from multiples of 10 in the range necessary to compose or decompose10-90 (positive or zero differences), tens or hundreds.using concrete models or drawings and 8. Mentally add 10 or 100 to a givenstrategies based on place value, number 100–900, and mentally subtractproperties of operations, and/or the 10 or 100 from a given number 100–relationship between addition and 900.
    43. 43. High School Pathways• The CCSSM Model Pathways – Two models that organize the CCSSM into coherent, rigorous courses – NOT required. The two sequences are examples, not mandates• Pathway A: Consists of two algebra courses and a geometry course, with some data, probability and statistics infused throughout each (traditional)• Pathway B: Typically seen internationally that consists of a sequence of 3 courses each of which treats aspects of algebra, geometry and data, probability, and statistics. Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    44. 44. Flows Leading to Algebra Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    45. 45. (later in presentation)TASK PROGRESSION Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    46. 46. Understanding the Math Common Core State Standards QUESTIONS 6-14 Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    47. 47. PracticesDr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    48. 48. 8 CCSSM Mathematical Practices Standards for Mathematical Practice – 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 Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    49. 49. Standards for Mathematical PracticesGraphic Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    50. 50. Standards for Mathematical Practices Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    51. 51. Standards for Mathematical PracticesTake a moment to examine the first three wordsof each of the 8 mathematical practices… whatdo you notice? Mathematically Proficient Students… Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    52. 52. Standards for Mathematical Practices• Consider the verbs that illustrate the student actions each practice.• For example, examine Practice #3: Construct viable arguments and critique the reasoning of others. Highlight the verbs. Discuss with a partner: What jumps out at you? Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    53. 53. Mathematical Practice #3: Construct viablearguments and critique the reasoning of othersMathematically proficient students understand and use stated assumptions,definitions, and previously established results in constructing arguments. They makeconjectures and build a logical progression of statements to explore the truth of theirconjectures. They are able to analyze situations by breaking them into cases, and canrecognize and use counterexamples. They justify their conclusions, communicate themto others, and respond to the arguments of others. They reason inductively aboutdata, making plausible arguments that take into account the context from which thedata arose. Mathematically proficient students are also able to compare theeffectiveness of two plausible arguments, distinguish correct logic or reasoning fromthat which is flawed, and if there is a flaw in an argument-explain what it is.Elementary students can construct arguments using concrete referents such asobjects, drawings, diagrams, and actions. Such arguments can make sense and becorrect, even though they are not generalized or made formal until later grades. Laterstudents learn to determine domains to which an argument applies. Students at allgrades can listen or read the arguments of others, decide whether they make sense,and ask useful questions to clarify or improve the arguments. Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    54. 54. Mathematical Practice #3: Construct viablearguments and critique the reasoning of othersMathematically proficient students:• understand and use stated assumptions, definitions, and previously established results inconstructing arguments.• make conjectures and build a logical progression of statements to explore the truth of theirconjectures.• analyze situations by breaking them into cases, and can recognize and use counterexamples.• justify their conclusions, communicate them to others, and respond to the arguments of others.• reason inductively about data, making plausible arguments that take into account the contextfrom which the data arose.• compare the effectiveness of two plausible arguments, distinguish correct logic or reasoningfrom that which is flawed, and-if there is a flaw in an argument-explain what it is.• construct arguments using concrete referents such as objects, drawings, diagrams, and actions.Such arguments can make sense and be correct, even though they are not generalized or madeformal until later grades.• determine domains to which an argument applies.• listen or read the arguments of others, decide whether they make sense, and ask usefulquestions to clarify or improve the arguments. Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    55. 55. Mathematical Practice #3: Construct viablearguments and critique the reasoning of othersMathematically proficient students:• understand and use stated assumptions, definitions, and previously established results inconstructing arguments.• make conjectures and build a logical progression of statements to explore the truth of theirconjectures.• analyze situations by breaking them into cases, and can recognize and use counterexamples.• justify their conclusions, communicate them to others, and respond to the arguments of others.• reason inductively about data, making plausible arguments that take into account the contextfrom which the data arose.• compare the effectiveness of two plausible arguments, distinguish correct logic or reasoningfrom that which is flawed, and-if there is a flaw in an argument-explain what it is.• construct arguments using concrete referents such as objects, drawings, diagrams, and actions.Such arguments can make sense and be correct, even though they are not generalized or madeformal until later grades.• determine domains to which an argument applies.• listen or read the arguments of others, decide whether they make sense, and ask usefulquestions to clarify or improve the arguments. Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    56. 56. Observations• What do you notice? • What will students be doing differently? • What will teachers be doing differently? Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    57. 57. The Standards for [Student] Mathematical Practice On a scale of 1 (low) to 6 (high), to what extent is your school/our districtpromoting students’ proficiency in Practice 3? Evidence for your rating? Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    58. 58. The Standards for [Student] Mathematical Practice• SMP1: Explain and make conjectures…• SMP2: Make sense of…• SMP3: Understand and use…• SMP4: Apply and interpret…• SMP5: Consider and detect…• SMP6: Communicate precisely to others…• SMP7: Discern and recognize…• SMP8: Notice and pay attention to… Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    59. 59. Standards for Mathematical Practice…describe the thinking processes, habits of mind anddispositions that students need to develop adeep, flexible, and enduring understanding ofmathematics; in this sense they are also a means to anendSP1. Make sense of problems“….they [students] analyzegivens, constraints, relationships and goals. ….theymonitor and evaluatetheir progress and change course if necessary. …. andthey continually ask themselves “Does this make sense?” Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    60. 60. Standards for Mathematical PracticeAND…. describe mathematical content students need to learnSP1. Make sense of problems“……. students can explain correspondences betweenequations, verbal descriptions, tables, and graphs ordraw diagrams of important features andrelationships, graph data, and search for regularity ortrends.” Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    61. 61. Buttons TaskGita plays with her grandmother’s collection of black & white buttons.She arranges them in patterns. Her first 3 patterns are shown below.Pattern #1 Pattern #2 Pattern #3 Pattern #41. Draw pattern 4 next to pattern 3.2. How many white buttons does Gita need for Pattern 5 and Pattern 6?Explain how you figured this out.3. How many buttons in all does Gita need to make Pattern 11? Explainhow you figured this out.4. Gita thinks she needs 69 buttons in all to make Pattern 24. How doyou know that she is not correct? How many buttons does she need tomake Pattern 24? Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    62. 62. Buttons Task1. Individually complete parts 1 - 3.2. Then work with a partner to compare yourwork and complete part 4. (Look for as manyways to solve parts 3 and 4 as possible.) Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    63. 63. Buttons TaskGita plays with her grandmother’s collection of black & white buttons.She arranges them in patterns. Her first 3 patterns are shown below.Pattern #1 Pattern #2 Pattern #3 Pattern #41. Draw pattern 4 next to pattern 3. 15 buttons and 18 buttons2. How many white buttons does Gita need for Pattern 5 and Pattern 6?Explain how you figured this out. 34 buttons3. How many buttons in all does Gita need to make Pattern 11? Explainhow you figured this out. 73 buttons4. Gita thinks she needs 69 buttons in all to make Pattern 24. How doyou know that she is not correct?How many buttons does she need to make Pattern 24? Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    64. 64. Buttons Task Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    65. 65. Buttons TaskWhich mathematical practices are needed completethe task?Indicate the primary practice.1. Make sense of problems and persevere in solving them.2. Reason abstractly and quantitatively.3. Construct viable arguments and critique the reasoning of others.4. Model with mathematics.5. Use appropriate tools strategically.6. Attend to precision.7. Look for and make use of structure.8. Look for and express regularity in repeated reasoning. Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    66. 66. Standards for [Student] Mathematical Practice“Not all tasks are created equal, and differenttasks will provoke different levels and kindsof student thinking.”Stein, Smith, Henningsen, & Silver, 2000“The level and kind of thinking in whichstudents engage determines what theywill learn.”Hiebert, Carpenter, Fennema, Fuson, Wearne, Murray, Oliver, & Human, 1997 Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    67. 67. The Nature of Tasks Used in the Classroom ……Will Impact Student Learning! Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    68. 68. But, WHAT TEACHERS DO with the tasks matters too!The Mathematical Tasks FrameworkStein, Grover & Henningsen (1996)Smith & Stein (1998)Stein, Smith, Henningsen & Silver (2000) Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    69. 69. http://www.insidemathematics.org/index.php/classroom-videovisits/public-lessons-numerical-patterning/218-numerical-patterninglesson-planning?phpMyAdmin=NqJS1x3gaJqDM-1-LXtX3WJ4e8 Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    70. 70. Learner A Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    71. 71. Learner B Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    72. 72. Buttons Task RevisitedWhat might a teacher get out of using the samemath task two days in a row, rather thanswitching to a different task(s)? – Address common misconceptions – Support students in moving from less to more sophisticated solutions Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    73. 73. Buttons Task RevisitedWhich of the Standards of Mathematical Practice did thestudents engage in when they revisited the task?Indicate the primary practice.1. Make sense of problems and persevere in solving them.2. Reason abstractly and quantitatively.3. Construct viable arguments and critique the reasoning of others.4. Model with mathematics.5. Use appropriate tools strategically.6. Attend to precision.7. Look for and make use of structure.8. Look for and express regularity in repeated reasoning. Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    74. 74. But, WHAT TEACHERS DO with the tasks matters too!The Mathematical Tasks FrameworkStein, Grover & Henningsen (1996)Smith & Stein (1998)Stein, Smith, Henningsen & Silver (2000) Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    75. 75. Standards for [Student] Mathematical PracticeThe 8 Standards for Mathematical Practice –place an emphasis on student demonstrationsof learning… Equity begins with an understanding of how the selection of tasks, the assessment of tasks, the student learning environment creates great inequity in our schools… Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    76. 76. Standards for [Student] Mathematical Practice To what extent do all students in your class, school or our district have the opportunity to engage in tasks thatpromote attainment of the mathematical practices on a regular basis? Please rate on a scale of 1 (low) to 6 (high). Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    77. 77. Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    78. 78. Content and Practices Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    79. 79. Cognitive Complexity Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    80. 80. Depth of Knowledge Levels Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    81. 81. Sorting Activity• Categorize tasks into level 1, 2, 3, or 4 using Cognitive Complexity Levels. Record your responses on the provided worksheet.• Share results and come to consensus at your table. One person will record results on the “master” copy.• Share results and review criteria groups used for low and high levels. Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    82. 82. Sorting questions to ponder…• How did you determine between levels 2 & 3?• Does a task presented as a word problem always have a high level of cognitive complexity?• If a task requires an explanation, does it have a high level of cognitive complexity? Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    83. 83. Changing the Cognitive Complexity Level• Pick out a task that was placed in level 1 or 2. Determine how you would modify your task to be a level 3 task.• Share task out with whole group. Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    84. 84. Cognitive Complexity and Mathematical PracticesWhich levels of cognitive complexity allowstudents to develop the mathematicalpractices? Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    85. 85. Task Progression• A rich mathematical task can be reframed or resized to serve different mathematical goals Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    86. 86. Are there various levels of CognitiveComplexity in your instructional materials?• Review several types of problems/tasks found in your instructional materials.• What level of cognitive complexity are these tasks? – Level 1 (recall) – Level 2 (skill/concept) – Level 3 (strategic thinking) – Level 4 (extended thinking) Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    87. 87. Are there various levels of CognitiveComplexity in your instructional materials?Share the types of problems/tasks you found.• What are the prevalent levels of complexity in your instructional materials?• How will this impact meeting the standards for mathematical practice? Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    88. 88. Gas Mileage Problem• With scaffolding• Without scaffolding Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    89. 89. Who’s Doing the Work? TEDtalk: Dan Meyer Video http://www.youtube.com/watch?v=BlvKWEvKSi8 Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    90. 90. Video Debrief• How much is too much support; how much is too little?• How does scaffolding interfere/promote standards for mathematical practice?• Compare/contrast Gas Mileage activities Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    91. 91. Appendix AUnderstanding the Math Common Core State Standards Questions 15-22 Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    92. 92. TablesUnderstanding the Math Common Core State Standards Question 23 Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    93. 93. Transition Plans Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    94. 94. Three-Year Transition Plan for Common Core State Standards for Mathematics by Grade Level Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    95. 95. Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    96. 96. State Resources for TransitionGrade-level transition documents describe: – What standards to continue – What standards to remove – What standards to move to Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    97. 97. OSD RESOURCESMATH COMMON CORE STATE STANDARDS Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    98. 98. OSD Teaching, Learning and Assessment Website: Common Core State Standards: http://www.orting.wednet.edu/education/components/scrapbook/default.php?sectiondetailid=3910&Common Core State Standards for Math: http://www.k12.wa.us/CoreStandards/Mathematics/pubdocs/CCSSI_MathStandards.pdfDesigning High School Mathematics Courses (Appendix A): http://www.corestandards.org/assets/CCSSI_Mathematics_Appendix_A.pdf Illustrative Math: http://illustrativemathematics.org/ Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    99. 99. Mathematical Practices by Grade Level: http://www.azed.gov/standards-practices/files/2011/10/2010mathglossary.pdf3-Year Transition Plan: http://www.k12.wa.us/CoreStandards/pubdocs/Three-YearDomainImplementation.pdf Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    100. 100. Transition Plans by Grade Level: Kindergarten | Grade 1 | Grade 2 | Grade 3 | Grade 4 | Grade 5 | Grade 6 | Grade 7 | Grade 8 Transition for Algebra I | Transition for Geometry | Integrated Math I | Integrated Math IIProgressions Documents: http://ime.math.arizona.edu/progressions/Videos on CCSS-M: http://www.youtube.com/playlist?list=PLD7F4C7DE7CB3D2E6&feature=plcp Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
    101. 101. 1 CCSS, 2010, p. 5 Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  20122 PARCC – Draft Content Framework - 2011

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