making sense of the new standards

Making SenseOf The New StandardsK-12 Alliance Staff Developer TrainingJanuary 25, 2013

K-12 Alliance/WesEd 13

New Opportunities for All LearnersCalifornia Common Core State Standards (ELA and Math)Next Generation Science Standards21st Century Skills


Prior KnowlegeWith your group discuss what you know about 21st Century Skills, CCSS, and NGSSWhat connections do you see?Chart your ideas. You will return to your thinking on Sunday.


In The ClassroomMathScience21st Century SkillsELA


Activity Choose a secret number (dont say it aloud) add 5 to that numberdouble that sumsubtract 4 from that productdivide the results by 2 subtract your secret number from the quotient you have 3.


How Did That Happen?What is behind this trick?Is it foolproof?Discuss your thoughts at your table.


Explain this trickUse the tools at your table to explain what you think is happening.Remember:You may all have had different secret numbers.No matter what number you selected, you all ended up with 3.


Transfer to a WhiteboardThink about how to use the tools to demonstrate how the trick was done.Use the white board to depict your demonstration.


Connect action to a direct model with a specific Secret # choice

InstructionsWhat I didMy modelChoose a secret numberI chose 2 beans (2 beans)Add 5I added 5 more Double that sumI multiplied by two Subtract 4Subtracted 4 beans Divide by 2I pulled half away Subtract secret numberI pulled off 2 beans You have threeI was astonished!


Connect action to a model for any choice (A set of all secret numbers)

InstructionsWhat I didMy modelChoose a secret numberI chose a cup to represent anyCup Add 5I added 5 beansCup Double that sumI multiplied by twoCup Cup Subtract 4Subtracted 4 beansCup Cup Divide by 2I pulled half away Cup Subtract secret numberI pulled off the cup You have threeVoila!


WordsManipulativesAlgebra Expressions/ResultChoose a secret numberCupnAdd 5Double that sumSubtract 4Divide by 2Subtract secret numberYou have three


WordsManipulativesAlgebra Expressions/ResultChoose a secret numberCupnAdd 5Cup N+5Double that sumSubtract 4Divide by 2Subtract secret numberYou have three


WordsManipulativesAlgebra Expressions/ResultChoose a secret numberCupnAdd 5Cup N+5Double that sumCup Cup 2(n+5) = 2n + 10Subtract 4Divide by 2Subtract secret numberYou have three


WordsManipulativesAlgebra Expressions/ResultChoose a secret numberCupnAdd 5Cup N+5Double that sumCup Cup 2(n+5) = 2n + 10Subtract 4CupCup2n + 10 4= 2n+6 Divide by 2Subtract secret numberYou have three


WordsManipulativesAlgebra Expressions/ResultChoose a secret numberCupnAdd 5Cup N+5Double that sumCup Cup 2(n+5) = 2n + 10Subtract 4CupCup2n + 10 4= 2n+6 Divide by 2Cup(2n+6)/2 = n + 3 Subtract secret numberYou have three


WordsManipulativesAlgebra Expressions/ResultChoose a secret numberCupnAdd 5Cup N+5Double that sumCup Cup 2(n+5) = 2n + 10Subtract 4CupCup2n + 10 4= 2n+6 Divide by 2Cup(2n+6)/2 = n + 3 Subtract secret numberN + 3 n = 3 You have three


WordsManipulativesAlgebra Expressions/ResultChoose a secret numberCupnAdd 5Cup N+5Double that sumCup Cup 2(n+5) = 2n + 10Subtract 4CupCup2n + 10 4= 2n+6 Divide by 2Cup(2n+6)/2 = n + 3 Subtract secret numberN + 3 n = 3 You have threeYes, I do.3


8th Grade Expectation2(n + 5) 4 2(n + 5) 4 -n

WordsResultsCumulative ExpressionChoose a secret numbernnAdd 5N+5N+5Double that sum2(n+5) = 2n+102(n+5)Subtract 42n + 10 4 = 2n+62(n + 5) 4 Divide by 2(2n+6)/2 = n + 3 2

Subtract secret numberN + 3 n = 3 2

You have three 32(n + 5) 4 -n = 3 2


Language Of Doing MathWhich of these words best describes what I asked you to do:Represent Solve EvaluateSimplify Determine Compare Find Explain


Instructional Words =implication for process

Represent To portray in another abstract way.Solve Finding an unknown quantity or expression.Evaluate To find the value of by substituting values in an expression or function.Simplify To make a math expression simpler.Determine Find or chooseCompare Relate values of 2 or more expressionsExplain Describe a process.Justify Support the logic of a process or solution


Standards for Mathematical Content K-8How the grade level standards are organized Standards Clusters Domains


7th Grade Common Core Standard Connection

7.EE.4. Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.


Common Core Connections7 ------> 67th Grade (p. 31) 7.EE.4. Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.

6TH: (P. 26) 6.EE.6. Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.


Follow the Standard Back54321Kinder!Keep track of the standards you determine to be in the line from Kindergarten to 5th grade and on to Middle school.

Remember the key idea is representing (modeling) a mathematical idea in a manner chosen by the student.


Common Core Standards for MathematicsTwo Types of StandardsMathematical Practice (recurring throughout the grades)Mathematical Content (different at each grade level)


Standards for Mathematical PracticeMake 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 reasoningWhich of These Describe your work on the Trick?


Common Core Math Standards


1997 CA. Math Standards


*CCSS Requires Three Shifts in MathematicsFocus: 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


*Shift #1: Focus Strongly where the Standards FocusSignificantly narrow the scope of content and deepen how time and energy is spent in the math classroom.Focus deeply on what is emphasized in the standards, so that students gain strong foundations.


*Key Areas of Focus in Mathematics

GradeFocus Areas in Support of Rich Instruction and Expectations of Fluency and Conceptual UnderstandingK2Addition and subtraction - concepts, skills, and problem solving and place value35Multiplication and division of whole numbers and fractions concepts, skills, and problem solving6Ratios and proportional reasoning; early expressions and equations7Ratios and proportional reasoning; arithmetic of rational numbers8Linear algebra


Grade Shifts: Examples

Concept1997 StandardsCCSSCompose simple shapes to form larger shapes (e.g., 2 triangles to form a rectangle)Grade2KIntroduction to ProbabilityGrade3Grade7Introduction of fractions as numbersGrade2Grade3Add and subtract simple fractionsGrade3Grade4


*Shift #2: Coherence: Think Across Grades, and Link to Major Math Topics Within GradesCarefully connect the learning within and across grades so that students can build new understanding on foundations built in previous years. Begin to count on solid conceptual understanding of core content and build on it. Each standard is not a new event, but an extension of previous learning.


Shift #3: Rigor: Expect fluency, deep understanding, and application*The CCSSM require a balance of:Solid conceptual understandingProcedural skill and fluencyApplication of skills in problem solving situationsPursuit of all threes requires equal intensity in time, activities, and resources.


Solid Conceptual UnderstandingTeach more than how to get the answer and instead support students ability to access concepts from a number of perspectivesStudents are able to see math as more than a set of mnemonics or discrete proceduresConceptual understanding supports the other aspects of rigor (fluency and application)*


FluencyThe 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

*


*Required Fluencies in K-6

GradeStandardRequired FluencyKK.OA.5Add/subtract within 511.OA.6Add/subtract within 1022.OA.22.NBT.5Add/subtract within 20 (know single-digit sums from memory)Add/subtract within 10033.OA.73.NBT.2Multiply/divide within 100 (know single-digit products from memory)Add/subtract within 100044.NBT.4Add/subtract within 1,000,00055.NBT.5Multi-digit multiplication66.NS.2,3Multi-digit divisionMulti-digit decimal operations


ApplicationStudents 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.*


**Our goal is to focus on the connections between and among these content areas. Today we will dip into the math circle; tomorrow ELA and science and on Sunday we will discuss how all of the circles are related.*Mention that we may have skipped the part where students show there thinking and model with tools. This is a typical chart used. The standards addressed are many. *Mention that we may have skipped the part where students show there thinking and model with tools. This is a typical chart used. The standards addressed are many. *Mention that we may have skipped the part where students show their thinking and model with tools. This is a typical chart used. The standards addressed are many. *Because I say so.*Have teachers jot down choices and discuss why in small groups*K-5 Mathematics Presentation CTA - CLAB: Developed by SCFIRD with support from ELCSD, SCALD, and AAD***The standards for grades K8 share a similar organization. Standards define what students should understand and be able to do. Clusters are groups of related standards. Domains are larger groups of related standards.The kindergarten through grade eight standards are grouped by grade level and are organized into domains that vary slightly by grade with four or five domains encountered at each grade level, which provide focus.

Red emphasis is focus on meaning. Italics follow and since that is a second or in this case secondary part of the standard. One can represent problems and then also solve those. *It is NOT following directions and mimicking how to represent. It is being able to represent abstractly. One may have been taught this but the representation is theirs in design.*Overview Presentation CTA - CLAB: Developed by SCFIRD with support from ELCSD, SCALD, and AAD*They include two types standards:Standards for Mathematical Practice which remain constant throughout the grades andStandards for Mathematical Content which are different for each grade level.

Overview Presentation CTA - CLAB: Developed by SCFIRD with support from ELCSD, SCALD, and AAD*A summary of the eight Common Core Standards for Mathematical Practice are listed here.The Mathematical Practice standards define how students develop mathematical understanding as they make sense of a problem, reason abstractly, construct arguments, model with mathematics, use tools strategically, attend to precision, and look for structure and repeated reasoning.

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 Im (were) telling you, these are things Im (were) 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 anyones message against these touchstones. They are meant to be succinct, and easy to remember; well discuss them each in turn. What does it mean to focus? {Read slide}So, focus in the standards quite directly means that the scope of content is to be narrowed. This is that notion of The power of the eraser. After a decade of NCLB, we have come to see narrowing as a bad word and when it means cutting arts programs and language programs, it is. But meanwhile, math has swelled in this country. It has become a mile wide and an inch deep. The CCSSM is telling us that math actually needs to lose a few pounds.Just as important is that we are deepening expectations as well. So rather than skating through a lot of topics covering the curriculum, we are going to have fewer topics on our list, but the expectations in those topics are much deeper. Without focus, deep understanding of core math concepts for all students is just a fantasy. A study of the standards demonstrates that there are areas of emphasis already engineered into the standards at each grade level. Focus in the Common Core Standards means two things. What is in versus what is out, but also what the main focus of the standards are for each grade. This chart shows what the major focus areas are for K-8 math. These are the concepts which demand the most time, attention and energy throughout the school year. It is through focus in these key areas in K-8, that students will be best be prepared for further studies of math in HS and consequently, college and career ready.It is important to note that these are not topics to be checked off a list during an isolated unit of instruction, but rather these priority areas will be present throughout the school year through rich instructional experiences. Overview Presentation CTA - CLAB: Developed by SCFIRD with support from ELCSD, SCALD, and AAD*Although the CCSS maintain the current focus on operations with whole numbers, fractions and decimals at the early grades, with full implementation of the CCSS, some topics will be taught at different grades.Here are some examples of topics moving both up and down one or more grade levels.Notice that the introduction to probability moves from grade 3 in the 1997 standards to grade 7 in the CCSS.The introduction of fractions as numbers moves from grade two to grade three. Although introduced later, the CCSS addresses the development of fractions in a very focused and coherent manner.

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

Typically, current math curriculum spends as much as 25% of the instructional school year on review and re-teaching of previous grade level expectations not as an extension but rather as a re-teaching because many students have very little command of critical concepts.

Just as there are two ways to look at focus, there are two elements of coherence: The coherence across grades and the coherence that links topics to the major work of the grade.

{read slide}

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. 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 cant 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. 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 youre fluent, you flow. Fluent isnt 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. 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 cant just focus solely on applicationwe 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 whats at stake hereif we attempt this with a lack of conceptual knowledge and procedural fluency, the problem just becomes three times harder. At the same time, we dont 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.

making sense of the new standards

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