Introduction to Common Core State Standards for Mathematics

North Country InserviceHS MathematicsCommon Core State StandardsMathematicsPractice and Content StandardsDay 1Friday, October 19, 2012Presenter: Elaine Watson, Ed.D.

1IntroductionsShare What feeds your soul personally?What is your professional role?What feeds your soul professionally?

Volunteers for BreaksI need volunteers to remind me when we need breaks! Every 20 minutes, we need a 2-minute movement break to help our blood circulate to our brains.Every hour we need a 5-minute bathroom break.

Formative AssessmentHow familiar are you with the CCSSM?

Setting the Stage Dan Meyers TED TalkMath Class Needs a MakeoverGo to link: watsonmath.comNorth Country High School Math Inservice October 19, 2012

CCSSM Equally Focuses on

Standards for Mathematical PracticeStandards for Mathematical ContentSame for All Grade LevelsSpecific to Grade Level8 Practice StandardsLook at the handout SMP Lesson Alignment TemplateFor an electronic copy to use later, go to watsonmath.com North Country High School Math Inservice October 19, 2012

Standards for Mathematical PracticeDescribe ways in which student practitioners of the discipline of mathematics increasingly ought to engage with the subject matter as they grow in mathematical maturity

Standards for Mathematical PracticeProvide a balanced combination of Procedureand UnderstandingShift the focus to ensuremathematical understanding over computation skills

Standards for Mathematical Practice

Students will be able to: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.

Video of NYC High School Piloting the CCSSWatch first 5 minutes on MathSee link in watsonmath.com

Standards for Mathematical PracticeSome of the following slides on the Practice Standards have been adapted from slides presented in several online EdWeb Webinars

in February through May 2012 discussing that focused on the Practice Standards bySara Delano Moore, Ph.D.

The 8 Standards for Mathematical Practice can be divided into 4 CategoriesOverarching Habits of Mind of a Mathematical Thinker (# 1 and # 6)Reasoning and Explaining (# 2 and # 3)Modeling and Using Tools (# 4 and # 5)Seeing Structure and Generalizing (# 7 and # 8)

The 8 Standards for Mathematical Practiceare fluidly connected to each other.

One action that a student performs, either internally or externally, when solving a problem can take on characteristics from several of the 8 Practice Standards.

Overarching Habits of Mind of a Mathematical ThinkerMake sense of problems & persevere in solving them.Reason abstractly and quantitatively.Construct viable arguments & critique the reasoning of others.Model with mathematics.Use appropriate tools strategically.6. Attend to precision.Look for & make use of structure.Look for & express regularity in repeated reasoning.Start with Good ProblemsCharacteristicsExample from Illustrative Mathematics (F-BF.A.1.a, F-IF.B.4, F-IF.B.5 ) Context relevant to studentsIncorporates rich mathematicsEntry points/solution pathways not readily apparent

Mike likes to canoe. He can paddle 150 feet per minute. He is planning a river trip that will take him to a destination about 30,000 feet upstream (that is, against the current). The speed of the current will work against the speed that he can paddle.Let s be the speed of the current in feet per minute. Write an expression for r(s), the speed at which Mike is moving relative to the river bank, in terms of s. Mike wants to know how long it will take him to travel the 30,000 feet upstream. Write an expression for T(s), the time in minutes it will take, in terms of s.What is the vertical intercept of T? What does this point represent in terms of Mikes canoe trip?At what value of s does the graph have a vertical asymptote? Explain why this makes sense in the situation.For what values of s does T(s) make sense in the context of the problem?

Make Sense of Problems (part I)Mathematically proficient studentsExplain the meaning of the problem to themselvesLook for entry points to the solutionAnalyze givens, constraints, relationships, goals

Mikes Canoe TripExplain the meaning of the problemEntry pointsGivens, constraints, relationships, goalsMike likes to canoe. He can paddle 150 feet per minute. He is planning a river trip that will take him to a destination about 30,000 feet upstream (that is, against the current). The speed of the current will work against the speed that he can paddle.Let s be the speed of the current in feet per minute. Write an expression for r(s), the speed at which Mike is moving relative to the river bank, in terms of s. Mike wants to know how long it will take him to travel the 30,000 feet upstream. Write an expression for T(s), the time in minutes it will take, in terms of s.What is the vertical intercept of T? What does this point represent in terms of Mikes canoe trip?At what value of s does the graph have a vertical asymptote? Explain why this makes sense in the situation.For what values of s does T(s) make sense in the context of the problem?

Persevere in Solving ThemMathematically proficient students.Plan a solution pathwayConsider analogous cases and alternate formsMonitor progress and change course if necessary

Persevere in Solving ThemIt's not that I'm so smart, it's just that I stay with problems longer.

- Albert Einstein

Mikes Canoe TripPossible solution pathways/strategiesConsider analogous cases & alternate formsMonitor progress and change course if neededMike likes to canoe. He can paddle 150 feet per minute. He is planning a river trip that will take him to a destination about 30,000 feet upstream (that is, against the current). The speed of the current will work against the speed that he can paddle.Let s be the speed of the current in feet per minute. Write an expression for r(s), the speed at which Mike is moving relative to the river bank, in terms of s. Mike wants to know how long it will take him to travel the 30,000 feet upstream. Write an expression for T(s), the time in minutes it will take, in terms of s.What is the vertical intercept of T? What does this point represent in terms of Mikes canoe trip?At what value of s does the graph have a vertical asymptote? Explain why this makes sense in the situation.For what values of s does T(s) make sense in the context of the problem?

Make Sense of Problems (part II)Mathematically proficient studentsExplain correspondence and search for trendsCheck their answers using alternate methodsContinually ask themselves, Does this make sense? Understand the approaches of othersWhat can teachers do?Select rich mathematical tasksConnected to rigorous mathematics content

Resources for rigorous mathematical tasks can be found onwww.watsonmath,.com North Country High School Math Inservice October 19, 2012

Illustrative MathematicsMARS TasksInside Mathematics3 Act Math TasksDan MeyerAndrew StadelOthersWhat can teachers do?Ask good questionsIs that true every time? Explain how you know. Have you found all the possibilities? How can you be sure?Does anyone have the same answer but a different way to explain it?Can you explain what youve done so far? What else is there to do?

What can teachers do?Communicate to students the final solution to a problem is less important than the skills they develop during the process of finding the solution.The skills developed in working through the process are long-lasting skills that will serve them in other areas of life.Attend to PrecisionIn VocabularyIn Mathematical SymbolsIn ComputationIn MeasurementIn Communication

How is the teacher ensuring that students are making sense of problem and attending to precision?See Video: Discovering Properties of QuadrilateralsonWatsonmath.comChallenges to PrecisionVocabularySimilar, adjacentMathematical Symbols=Computation and MeasurementAccurate computation Estimating when appropriateAppropriate units of measureCommunicationFormulate explanations carefullyMake explicit use of definitions

Mikes Canoe TripVocabularyMathematical SymbolsComputation & MeasurementCommunicationMike likes to canoe. He can paddle 150 feet per minute. He is planning a river trip that will take him to a destination about 30,000 feet upstream (that is, against the current). The speed of the current will work against the speed that he can paddle.Let s be the speed of the current in feet per minute. Write an expression for r(s), the speed at which Mike is moving relative to the river bank, in terms of s. Mike wants to know how long it will take him to travel the 30,000 feet upstream. Write an expression for T(s), the time in minutes it will take, in terms of s.What is the vertical intercept of T? What does this point represent in terms of Mikes canoe trip?At what value of s does the graph have a vertical asymptote? Explain why this makes sense in the situation.For what values of s does T(s) make sense in the context of the problem?

Reasoning and ExplainingMake sense of problems & persevere in solving them.2. Reason abstractly and quantitatively.3. Construct viable arguments & critique the reasoning of others.Model with mathematics.Use appropriate tools strategically.Attend to precision.Look for & make use of structure.Look for & express regularity in repeated reasoning.2. Reason abstractly & quantitativelyMathematics in and out of contextWorking with symbols as abstractionsQuantitative reasoning requires number senseUsing properties of operations and objectsConsidering the units involvedAttending to the meaning of quantities, not just computation

In and out of context inspire pageNumber sense fractions vs whole numbers, positive vs negative, place value and scaleProperties of operations multiplication makes bigger vs multiplication as areaConsider the units money, measurement school bus problem - meanings31Construct viable argumentsUnderstand and use assumptions, definitions, and prior resultsThink about precision (MP6)Make conjectures and build logical progressions to support those conjecturesNot just two column proofs in high schoolAnalyze situations by casesPositive values of X and negative values of XTwo-digit numbers vs three-digit numbersRecognize & use counter-examplesMaximum area problemCounter-examples max area problems lead to squares not when one side is the barn rather than fencing32How do we help children learn how to reason and explain?Provide rich problems where multiple pathways and solutions are possibleCelebrate multiple pathways to the same answer Monitor students as they work to choose approaches to share with the whole classProvide plenty of opportunities for students to talk to each other Recognize the difference between a viable argument and opinionProvide scaffolds for thembut not too many!How do we help children learn how to reason and explain?Provide plenty of opportunities for students to talk to each other.Create a classroom culture in which all students feel safe to express their thinkingMake sure students recognize the difference between a viable argument and an opinionCreate a classroom culture where its safe to critique each other in a respectful wayProvide scripts (sentence frames)for them to use such as those from Accountable Talk (see resources on watsonmath.com)Teacher Moves in Group Discussion By scaffolding students' responses and contributions,

teachers can quickly make a difference in the level of rigor and productivity in classroom talk.

Teachers can bring everyone's attention to a key point

By "marking" a student's contribution "that's an important point

By asking the student to repeat the remarkor restating it in their own wordsand indicating why the point is important.

From ACCOUNTABLE TALK SOURCEBOOK: FOR CLASSROOM CONVERSATION THAT WORKS Version 3.1 By Sarah Michaels (Clark University), Mary Catherine OConnor (Boston University), Megan Williams Hall (University of Pittsburgh), with Lauren B. Resnick (University of Pittsburgh)

Teacher Moves in Group Discussion If someone asks a thought-provoking question,

the teacher might turn the question back to the group Good question, what do you think? as a way to encourage students to push their own thinking.

By citing facts and posing counterexamples, teachers can challenge students to elaborate or clarify their arguments

"but what about...?

From ACCOUNTABLE TALK SOURCEBOOK: FOR CLASSROOM CONVERSATION THAT WORKS Version 3.1 By Sarah Michaels (Clark University), Mary Catherine OConnor (Boston University), Megan Williams Hall (University of Pittsburgh), with Lauren B. Resnick (University of Pittsburgh)

Teacher Moves in Group Discussion Teacher can model what desirable behaviors and habits of mind look like,

Heres what good problem solvers do when they're monitoring their own process."

Teachers can focus the group's thinking by recapping or summarizing key points that have been brought up in a discussion.

From ACCOUNTABLE TALK SOURCEBOOK: FOR CLASSROOM CONVERSATION THAT WORKS Version 3.1 By Sarah Michaels (Clark University), Mary Catherine OConnor (Boston University), Megan Williams Hall (University of Pittsburgh), with Lauren B. Resnick (University of Pittsburgh)

Teacher Moves That Support Accountability to the Learning Community

Accountability to the learning community requires that students listen to one another, attending carefully so that they can use and build on one another's ideas. Students and teachers agree and disagree respectfully, challenging a claim, not the person who made it. To support this kind of accountability, teachers must establish a classroom environment where everyone can hear each other, and where everyone knows how important it is to hear and be heard.

From ACCOUNTABLE TALK SOURCEBOOK: FOR CLASSROOM CONVERSATION THAT WORKS Version 3.1 By Sarah Michaels (Clark University), Mary Catherine OConnor (Boston University), Megan Williams Hall (University of Pittsburgh), with Lauren B. Resnick (University of Pittsburgh)

Teacher Moves That Support Accountability to the Learning Community

Keeping the channels open: "Did everyone hear that?" Keeping everyone together: "Who can repeat...?" Linking contributions: "Who wants to add on...?" Verifying and clarifying: "So, are you saying...? Teacher Moves That Support Accountability to Accurate Knowledge Pressing for accuracy: "Where can we find that?"

Building on prior knowledge: "How does this connect?" Teacher Moves That Support Accountability to Rigorous Thinking Pressing for reasoning: "Why do you think that?

Expanding reasoning: "Take your time; say more." Teacher Moves That Support Accountability to Rigorous Thinking Pressing for reasoning: "Why do you think that?

Expanding reasoning: "Take your time; say more." Other Ideas For Getting Students TalkingPose a question and then say,

Turn and talk to your neighbor Bring the discussion back to the whole group.

Pose a question to the whole class and then draw names out of a hatThe Importance of Wait TimeIncreased wait time of at least 2.7, and preferably at least 3, seconds can have these effects on students: 1) The length of student responses increases between 300% and 700%. 2) More inferences are supported by evidence and logical argument. 3) The incidence of speculative thinking increases. 4) The number of questions asked by students increases.

The Importance of Wait TimeIncreased wait time of at least 2.7, and preferably at least 3, seconds can have these effects on students:5) Student-student exchanges increase; teacher-centered show and tell behavior decreases. 6) Failures to respond decrease.

7) Disciplinary moves decrease.

The Importance of Wait TimeIncreased wait time of at least 2.7, and preferably at least 3, seconds can have these effects on students: 8) The variety of students participating voluntarily increases. Also the number of unsolicited, but appropriate contributions by students increases. 9) Student confidence, as reflected in fewer inflected responses, increases. 10) Achievement improves on written measures where the items are cognitively complex.

Teacher Moves in Group Discussion How can you show that your computation is correct?Use a different tool or strategyCompare your work with someone elseHow can you explain why your answer is best?What possibilities did you consider?What criteria did you use?Why did you reject some options?What made you choose this option?Embedding logic into your thinkingDoes one part depend on another part?Does changing one aspect of the problem change the result?What are you sure about? What comes next?and critique the reasoning of othersCompare two plausible argumentsDistinguish correct from flawed reasoningExplain/correct the flawAsk useful questions to clarify and improve argumentsModeling and Using ToolsMake sense of problems & persevere in solving them.Reason abstractly and quantitatively.Construct viable arguments & critique the reasoning of others.4. Model with mathematics.5. Use appropriate tools strategically.Attend to precision.Look for & make use of structure.Look for & express regularity in repeated reasoning.4. Model with MathematicsModeling is both a K - 12 Practice Standard and a 9 12 Content Standard.

Modeling DefinedModeling is the process of choosing and using appropriate mathematics and statistics to:analyze empirical situationsunderstand them better improve decisionsModeling DefinedThe word modeling in this context is used as a verb that describes the creative process of transforming a real situation into an abstract mathematical model.Modeling DefinedA model can be very simple, such as:writing total cost as a product of unit price and number bought, using a geometric shape to describe a physical object like a coin. Even such simple models involve making choices. It is up to us whether to model a coin as a three-dimensional cylinder, or whether a two-dimensional disk works well enough for our purposes. Modeling DefinedOther situations may need more elaborate models:modeling a delivery route, modeling a production schedule, modeling a comparison of loan amortizations

Knowledge from other disciplines may be necessary

Technology may or may not be neededModeling DefinedReal-world situations are not organized and labeled for analysis.

Formulating tractable models, representing such models, and analyzing them is a creative process.

Like every such process, this depends on acquired expertise as well as creativity.

Modeling DefinedThere is no one correct model.The models students create increase in sophistication, elegance, and efficiency as they mature mathematicallyModeling is best done as a collaborative activityModeling DefinedThe models devised depend on a number of factors: How precise an answer do we want or need?What aspects of the situation do we most need to understand, control, or optimize?What resources of time and tools do we have?

Model with MathematicsApply mathematics to solve problems in the real worldMake assumptions and approximations to simplifyIdentify important quantities and map relationships among themInterpret mathematical results in context and make adjustments to the model60Use appropriate tools strategicallyConsider available mathematical toolsKnow enough about tools to make informed decisionsRecognize strengths and limitations of toolsIdentify and use relevant external tools, including technological resources61What are mathematical tools?Tools for doing mathematicsPaper and pencilCalculators & SpreadsheetsMeasuring ToolsTools for learning mathematicsTools for doing mathematicsPhysical ManipulativesTechnological toolsUsing Manipulatives for Learning MathematicsConcreteRepresentationalAbstractSeeing Structure and Generalizing Make sense of problems & persevere in solving them.Reason abstractly and quantitatively.Construct viable arguments & critique the reasoning of others.Model with mathematics.Use appropriate tools strategically.Attend to precision.7. Look for and make use of structure.8. Look for and express regularity in repeated reasoning.Look for and make use of structurePatterns, patterns, patternsProperties of Operations3 + 7 = 7 + 37 x 8 = 7 x 5 + 7 x 3Geometric StructureSorting geometric shapesReasoning about the attributes of shapes

65Look for and make use of structureLook closely to discern a pattern or structureIn x2 + 9x + 14, can see the 14 as 2 x 7 and the 9 as 2 + 7Can see complicated algebraic expressions as being composed of several objects: 5 3 (x y)2 is seen as 5 minus a positive number times a square, so its value cant be more than 5 for any real numbers x and y

66Look for and make use of structureWatch the Algebra video. (see watsonmath)

Use your SMP Lesson Plan Template

Record the SMPs the teachers are using.

How is structure used in these lessons?67Using Properties to See StructureProperties of OperationsCommutative Property of multiplication and additionDistributive Property of multiplication over additionIdentity Property of multiplication and additionProperties of EqualityTransitive Property (if a=b and b=c, then a=c)Properties of InequalityExactly one of the following is true:a > b, a = b, a < bVan Hiele Levels of Geometric Thinking shows how students increasing see more structure in shapes as they mature mathematically Level 0 (Pre-recognition)Students do not yet see shapes clearly enough to compare with prototypesLevel 1 (Visualization)Students understand shapes by comparing to prototypesStudents do not see propertiesStudents make decisions based on perception, not reasoningLevel 2 (Analysis)Students see shapes as collections of propertiesStudents do not identify necessary and sufficient propertiesVan Hiele Levels (cont)Level 3 (Abstraction)Students see relationships among figures and propertiesStudents can create meaningful definitions and reason informallyLevel 4 (Deduction)Students can construct proofsStudents understand necessary & sufficient conditionsLevel 5 (Rigor)Students can understand non-Euclidean systemsStudents can use indirect proof and formal deduction

Look for and express regularity in repeated reasoningFocus on computation here1 3 =Examining points on a line and slope(1,2), m=3 (y-2)/(y-1) = 3Attending to intermediate results71Look for and express regularity in repeated reasoningThese practices are about seeing the underlying mathematical principles and generalizations.

These practices have more subtlety, and are often hard to distinguish between each other.

Pyramid of PenniesWork through Dan Meyers Pyramid of Pennies Problem

See link on watsonmath.com

Use your SMP Lesson Planning Template and fill out what Math Practices you used when solving the problem.

Content Standard ActivityWork with a partner and on your own laptop, go toIllustrative Mathematics on watsonmath.com Go to the HS StandardsCheck out the illustrations. Which SMPs would they reinforce?Which illustrations would you use in your classroom?Check out the modeling standards (those with a star)Why do you think these standards were chosen as modeling standards?

Last But Not LeastFormative AssessmentWatch the two videos on watsonmath.com:My Favorite NoDaily Assessment with Tiered Exit Cards

ResourcesFor resources used and/or discussed in this presentation, go to:www.watsonmath.comNorth Country High School Math Inservice October 19, 2012Check out other resources available on watsonmath.com: archives of past posts resource links in the right hand columnContact Elaine at elaine.watson0729@gmail.com