Introduction to Common Core State Standards for Mathematics

Principal's Workshop: How Does the Common Core Change What We Look For in the Math Classroom? Panama City, FloridaJanuary 22 & 23, 2013Presenter: Elaine Watson, Ed.D.1Hunt Institute Video: The Importance of Mathematical Practicehttp://vimeo.com/album/1702025/video/295680081. Make Sense of Problems and Persevere in SolvingIts not that Im so smart, its just that I stay with problems longer. Albert Einstein1. Make Sense of Problems and Persevere in Solving5th Grade Perseverance

1. Make Sense of Problems and Persevere in SolvingMathematically proficient students: Explain to self the meaning of a problem and look for entry points to a solutionAnalyze givens, constraints, relationships and goalsMake conjectures about the form and meaning of the solution

1. Make Sense of Problems and Persevere in SolvingMathematically proficient students:

Plan a solution pathway rather than simply jump into a solution attemptConsider analogous problemsTry special cases and simpler forms of original problem1. Make Sense of Problems and Persevere in SolvingMathematically proficient students:Monitor and evaluate their progress and change course if necessaryDoes this approach make sense?

1. Make Sense of Problems and Persevere in SolvingMathematically proficient students:Persevere in Solving by:Transforming algebraic expressionsChanging the viewing window on a graphing calculatorMoving between the multiple representations of:Equations, verbal descriptions, tables, graphs, diagrams

1. Make Sense of Problems and Persevere in SolvingMathematically proficient students:Check their answersDoes this answer make sense?Does it include correct labels?Are the magnitudes of the numbers in the solution in the general ballpark to make sense in the real world?

1. Make Sense of Problems and Persevere in SolvingMathematically proficient students:Check their answersVerify solution using a different methodCompare approach with others:How does their approach compare with mine?SimilaritiesDifferences

2. Reason Abstractly and QuantitativelyMathematically proficient students:Make sense of quantities and their relationships in a problem situationBring two complementary abilities to bear on problems involving quantitative relationships: The ability to decontextualizeto abstract a given situation, represent it symbolically, manipulate the symbols as if they have a life of their owncontextualizeto pause as needed during the symbolic manipulation in order to look back at the referent values in the problem

2. Reason Abstractly and QuantitativelyMathematically proficient students:Reason Quantitatively, which entails habits of:Creating a coherent representation of the problem at handconsidering the units involvedAttending to the meaning of quantities, not just how to compute themKnowing and flexibly using different properties of operations and objects

Watch the video and note where you see evidence of Middle School Classifying Equations1. Make Sense of Problems and Persevere in Solving2. Reason Abstractly and Quantitatively

3.Construct viable arguments and critique the reasoning of othersMathematically proficient students:

Understand and use stated assumptions, definitions, and previously established results when constructing arguments

3.Construct viable arguments and critique the reasoning of othersIn order for students to be practicing this standard, they need to be talking to each other, so teachers need to plan lessons that include a lot of large group and small group discussions.

A classroom culture must be cultivated in which it is as safe to disagree as it is to agree.

3.Construct viable arguments and critique the reasoning of othersHere are some sentence structures from the first video:I agree that ______________ because __________I disagree because __________________________How can we be sure?What do you think?Are you convinced?Do we all agree?

3.Construct viable arguments and critique the reasoning of othersWhat SMPs do you observe the students practicing?Here's the Problem from the video:Write several different types of equations for 2.4. Draw some different types of pictures to represent 2.4. Is 2.4 the same thing as the quotient 2 remainder 4? Why or why not?http://youtu.be/EA3YkawKEWc

4. Model with MathematicsModeling is both a K - 12 Practice Standard and a 9 12 Content Standard.4. Model with MathematicsMathematically proficient students:Use powerful tools for modeling:Diagrams or graphs SpreadsheetsAlgebraic Equations

4. Model with MathematicsMathematically proficient students:Models we devise depend upon a number of factors:How precise do we need to be?What aspects do we most need to undertand, control, or optimize?What resources of time and tools do we have?

4. Model with MathematicsMathematically proficient students:Models we devise are also constrained by:Limitations of our mathematical, statistical, and technical skillsLimitations of our ability to recognize significant variables and relationships among them

Modeling CycleThe word modeling in this context is used as a verb that describes the process of transforming a real situation into an abstract mathematical model.Modeling CycleProblemFormulateComputeInterpretValidateReportModeling CycleProblemIdentify variables in the situationSelect those that represent essential featuresModeling CycleFormulateSelect or create a geometrical, tabular, algebraic, or statistical representation that describes the relationships between the variables

Modeling CycleComputeAnalyze and perform operations on these relationships to draw conclusions

Modeling CycleInterpretInterpret the result of the mathematics in terms of the original situation

Modeling CycleValidateValidate the conclusions by comparing them with the situationModeling Cycle ValidateRe - FormulateReport on conclusions and reasoning behind themModeling CycleProblemFormulateComputeInterpretValidateReport6. Attend to precisionMathematically proficient students:Try to communicate precisely to others:Use clear definitionsState the meaning of symbols they useUse the equal sign consistently and appropriatelySpecify units of measureLabel axes

6. Attend to precisionMathematically proficient students:Try to communicate precisely to othersCalculate accurately and efficientlyExpress numerical answers with a degree of precision appropriate for the problem contextGive carefully formulated explanations to each otherCan examine claims and make explicit use of definitions

6. Attend to precisionStudents Practicing and Discussing PrecisionUse the Standards for Mathematical Practice Lesson Alignment Template.What SMPs do you see?http://ummedia04.rs.itd.umich.edu/~dams/umgeneral/seannumbers-ofala-xy_subtitled_59110_QuickTimeLarge.mov

7. Look for and make use of structureMathematically proficient students:Look 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

8. Look for and express regularity in repeated reasoning.Mathematically proficient students:

Notice if calculations are repeatedLook for both general methods and for shortcutsMaintain oversight of the process while attending to the details.

What Practice Standards do you see?Sean: Is 6 Even and Odd?Do All 8 Practice Standards Need to be Used in Every Lesson?There are some rich problems that elicit all 8 of the Practice Standards. However, these types of problems cant be done on a daily basis. Instructional time still needs to be balanced between building the students technical skills and

Nobut the teacher should plan so that over the span of a few days, the students are given learning opportunities to of the practicing standardsA Balanced Approach math facts how to approach and a novel situation procedures mathematically

Math Facts and Procedures

Memorizing Math Facts and Naked Number Exercises are Important!Practice Standards that apply:#2 Reason Quantitatively#6 Attend to Precision#7 Look for and Use Structure#8 Use Repeated Reasoning

Does Every Worthwhile Problem Have to Model a Real World Situation?

What SMPs Do You Observe Maya Practicing? What errors do you notice? What would you do to have Maya notice the errors?Maya Representing 52

Lets Practice Some ModelingStudents can: start with a model and interpret what it means in real world terms ORstart with a real world problem and create a mathematical model in order to solve it.Possible or Not? Here is an example of a task where students look at mathematical models (graphs of functions) and determine whether they make sense in a real world situation.

Possible or Not?Questions:Mr. Hedman is going to show you several graphs. For each graph, please answer the following:

A. Is this graph possible or not possible?

B.If it is impossible, is there a way to modify it to make it possible?

C. All graphs can tell a story, create a story for each graph.

One

A. Possible or not?

B. How would you modify it?

C. Create a story.Two

A. Possible or not?

B. How would you modify it?

C. Create a story.Three

A. Possible or not?

B. How would you modify it?

C. Create a story.Four

A. Possible or not?

B. How would you modify it?

C. Create a story.Five

A. Possible or not?

B. How would you modify it?

C. Create a story.Six

A. Possible or not?

B. How would you modify it?

C. Create a story.Seven

A. Possible or not?

B. How would you modify it?

C. Create a story.Eight

A. Possible or not?

B. How would you modify it?

C. Create a story.Nine

A. Possible or not?

B. How would you modify it?

C. Create a story.Ten

A. Possible or not?

B. How would you modify it?

C. Create a story.All 10 Graphs

What do all of the possible graphs have in common?And now...For some brief notes on functions!!!!

Lesson borrowed and modified from Shodor.Musical Notes borrowed from Abstract Art Pictures Collection.Pyramid of PenniesHere is an example of a task where students look at a real world problem, create a question, and create a mathematical model that will solve the problem.Dan Meyers 3-Act ProcessAct IShow an image or short video of a real world situation in which a question can be generated that can be solved by creating a mathematical model.Pyramid of Pennies 3-Acthttp://mrmeyer.com/threeacts/pyramidofpennies/Dan Meyers 3-Act ProcessAct I (continued)1. How many pennies are there?2. Guess as close as you can. 3. Give an answer you know is too high.4. Give an answer you know is too low.

Dan Meyers 3-Act ProcessAct 2Students determine the information they need to solve the problem.The teacher gives only the information students ask for.Dan Meyers 3-Act ProcessWhat information do you need to solve this problem?Dan Meyers 3-Act ProcessAct 2 continuedStudents collaborate with each other to create a mathematical model and solve the problem. Students may need find text or online resources such as formulas.Dan Meyers 3-Act ProcessGo to it!Dan Meyers 3-Act ProcessAct 3

The answer is revealed.

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 Procedure and UnderstandingThey shift the focus to ensuremathematical understanding over computation skills

Standards for Mathematical PracticeStudents 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.

Think back to the Pyramid of Pennies. At what point during the problem did you do the following?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.

See Inside Mathematics Videos for good examples of the Practice Standards in action.http://www.insidemathematics.org/index.php/mathematical-practice-standards

Resources for Rich Mathematical Taskshttp://illustrativemathematics.org/The Go-To site for looking at the Content Standards and finding rich tasks, called Illustrations that can be used to build student understanding of a particular Content Standard.Resources for Rich Mathematical Taskshttp://insidemathematics.org/index.php/homeis a website with a plethora of resources to help teachers transition to teaching in a way that reflects the Standards for Mathematical Practice.Its worth taking the 6:19 minutes to watch the Video Overview of the Video Tours to familiarize yourself to all of the resources. There are more video tours that can be accessed by clicking on a link below the overview video.

Resources for Rich Mathematical Taskshttp://map.mathshell.org/materials/stds.phpThere are several names that are associated with the website: MARS, MAPS, The Shell Centerhowever the tasks are usually referred to as The MARS Tasks. The link above will show tasks aligned with the Practice StandardsThey have been developed through a partnership with UC Berkeley and the University of NottinghamResources for Rich Mathematical Taskshttp://commoncoretools.me/author/wgmccallum/Tools for the Common Core is the website of Bill McCallum, one of the three principle writers of the CCSSM. Highlights of this site are the links (under Tools) to the Illustrative Mathematics Project, the Progressions Documents, and the Clickable Map of the CCSSM.77Resources for Rich Mathematical TasksMy blog WatsonMath has a lot of resources listed in the right hand columnThis powerpoint (with the links removed, but the URLs for the links included) can be accessed on watsonmath.com78