Numbers and Operations in Base Ten

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Numbers and Operations in Base Ten
Success Implementing CCSS for K-2 Math Numbers and Operations in Base Ten

Introductions

K – 2 Objectives Reflect on teaching practices that support the shifts (Focus, Coherence, & Rigor) in the Common Core State Standards for Mathematics. Deepen understanding of the progression of learning and coherence around the CCSS-M for Number and Operations in Base 10 Analyze tasks and classroom applications of the CCSS for Number and Operations in Base 10

Success Implementing CCSS for K-2 Math
Number and Operations in Base 10 Success Implementing CCSS for K-2 Math

Why CCSS? Greta’s Video Clip

Common Core State Standards
Define the knowledge and skills students need for college and career Developed voluntarily and cooperatively by states; more than 46 states have adopted Provide clear, consistent standards in English language arts/Literacy and mathematics Explain the value of a coordinated effort in the sharing of resources nationwide….ask participants to share what they notice. Source:

What We are Doing Doesn’t Work
Almost half of eighth-graders in Taiwan, Singapore and South Korea showed they could reach the “advanced” level in math, meaning they could relate fractions, decimals and percents to each other; understand algebra; and solve simple probability problems. In the U.S., 7 percent met that standard. Results from the 2011 TIMMS

Theory of Practice for CCSS Implementation in WA
2-Prongs: The What: Content Shifts (for students and educators) Belief that past standards implementation efforts have provided a strong foundation on which to build for CCSS; HOWEVER there are shifts that need to be attended to in the content. The How: System “Remodeling” Belief that successful CCSS implementation will not take place top down or bottom up – it must be “both, and…” Belief that districts across the state have the conditions and commitment present to engage wholly in this work. Professional learning systems are critical

WA CCSS Implementation Timeline
Phase 1: CCSS Exploration Phase 2: Build Awareness & Begin Building Statewide Capacity Phase 3: Build State & District Capacity and Classroom Transitions Phase 4: Statewide Application and Assessment Ongoing: Statewide Coordination and Collaboration to Support Implementation Remove this slide

Transition Plan for Washington State
K-2 3-5 6-8 High School Year 1- 2 School districts that can, should consider adopting the CCSS for K-2 in total. K – Counting and Cardinality (CC); Operations and Algebraic Thinking (OA); Measurement and Data (MD) 1 – Operations and Algebraic Thinking (OA); Number and Operations in Base Ten (NBT); 2 – Operations and Algebraic Thinking (OA); Number and Operations in Base Ten (NBT); and remaining 2008 WA Standards 3 – Number and Operations – Fractions (NF); Operations and Algebraic Thinking (OA) 4 – Number and Operations – Fractions (NF); Operations and Algebraic Thinking (OA) 5 – Number and Operations – Fractions (NF); Operations and Algebraic Thinking (OA) and remaining 2008 WA Standard 6 – Ratio and Proportion Relationships (RP); The Number System (NS); Expressions and Equations (EE) 7 – Ratio and Proportion Relationships (RP); The Number System (NS); Expressions and Equations (EE) 8 – Expressions and Equations (EE); The Number System (NS); Functions (F) Algebra 1- Unit 2: Linear and Exponential Relationships; Unit 1: Relationship Between Quantities and Reasoning with Equations and Unit 4: Expressions and Equations Geometry- Unit 1: Congruence, Proof and Constructions and Unit 4: Connecting Algebra and Geometry through Coordinates; Unit 2: Similarity, Proof, and Trigonometry and Unit 3:Extending to Three Dimensions

Focus, Coherence & Rigor

The Three Shifts in Mathematics
Focus: Strongly where the standards focus Coherence: Think across grades and link to major topics within grades Rigor: Require conceptual understanding, fluency, and application This is a reminder of the three shifts that are required by the Common Core State Standards for Mathematics. Read slide

Focus on the Major Work of the Grade
Two levels of focus ~ What’s in/What’s out The shape of the content There are two levels of focus. The first level is the focus of what is in versus what is out; what is being taught at each grade level compared to what is not. It is because of this level of focus that teachers will have the time to go deeper with the math that is most important. Compared to the typical state standards of the past (which in some cases were literally volumes of standards that would have taken years to “cover,” even one grade’s worth of math), the Common Core State Standards for Mathematics have fewer standards which are manageable and it is clear what is expected of the teachers and students at each grade level. That is the 1st level of focus. The other level of focus is the shape of the content that is in each grade or course. What that means is that if you look at the “focused” list, say for Kindergarten, you can see the list in terms of shades. There are things that are really sharp and focused in the middle, that are the major content for that grade. The other topics are there in a supporting way and help to support that major work. So, even within the list that exists, there is focus. That is the 2nd level of focus.

Shift #1: Focus Key Areas of Focus in Mathematics
Grade Focus Areas in Support of Rich Instruction and Expectations of Fluency and Conceptual Understanding K-2 Addition and subtraction - concepts, skills, and problem solving and place value 3-5 Multiplication and division of whole numbers and fractions – concepts, skills, and problem solving 6 Ratios and proportional reasoning; early expressions and equations 7 Ratios and proportional reasoning; arithmetic of rational numbers 8 Linear algebra and linear functions

Focus on Major Work In any single grade, students and teachers spend the majority of their time, approximately 75% on the major work of the grade. The major work should also predominate the first half of the year. This is from the Achieve the Core document Publisher’s Criteria p. 7 Find areas where minor clusters support the major cluster in fractions from PARCC

Carefully connect the learning within and across grades so that students can build new understanding onto 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.

Carefully connect the learning within and across grades so that students can build new understanding onto 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. You have just purchased an expensive Grecian urn and asked the dealer to ship it to your house. He picks up a hammer, shatters it into pieces, and explains that he will send one piece a day in an envelope for the next year. You object; he says “don’t worry, I’ll make sure that you get every single piece, and the markings are clear, so you’ll be able to glue them all back together. I’ve got it covered.” Absurd, no? But this is the way many school systems require teachers to deliver mathematics to their students; one piece (i.e. one standard) at a time. They promise their customers (the taxpayers) that by the end of the year they will have “covered” the standards. ~Excerpt from The Structure is the Standards Phil Daro, Bill McCallum, Jason Zimba

Varied problem structures that build on the student’s work with whole numbers 5 = builds to 5/3 = 1/3 + 1/3 + 1/3 + 1/3 + 1/3 and 5/3 = 5 x 1/3 Conceptual development before procedural Use of rich tasks-applying mathematics to real world problems Effective use of group work Precision in the use of mathematical vocabulary You will see evidence of these throughout the session Learning Progressions

Use addition and subtraction within 100 to solve word problems involving lengths that are given in the same units, e.g., by using drawings (such as drawings of rulers) and equations with a symbol for the unknown number to represent the problem. 2.MD.5 In the 2nd grade standard 2.NBT.5, students are required to add and subtract within MD.5 strengthens this for students by asking them to solve word problems involving length as a context for adding and subtracting within Here we see standards that support each other so math makes sense to students and that the math they are doing is related to other math that they are doing, rather than a endless list of discrete topics to learn.

Rigor: Illustrations of Conceptual Understanding, Fluency, and Application
Here rigor does not mean “hard problems.” It’s a balance of three fundamental components that result in deep mathematical understanding. There must be variety in what students are asked to produce. Rigor, as defined here, does not mean hard problems. It doesn’t mean more difficult. Rigor, here, means something very specific. We are talking about the balance of these components of conceptual understanding, fluency, and application. We are going to look at a set of problems; some assess fluency, some require conceptual understanding, and some are examples of application. By working through these problems, we can start seeing what this looks like.

Some Old Ways of Doing Business
Lack of rigor Reliance on rote learning at expense of concepts Severe restriction to stereotyped problems lending themselves to mnemonics or tricks Aversion to (or overuse) of repetitious practice Lack of quality applied problems and real-world contexts Lack of variety in what students produce E.g., overwhelmingly only answers are produced, not arguments, diagrams, models, etc. Either teachers don’t allow for repetitious practice or spend too much on repetitious practice Greta will add a slide about new ways of doing business

Redefining what it means to be “good at math”
Expect math to make sense wonder about relationships between numbers, shapes, functions check their answers for reasonableness make connections want to know why try to extend and generalize their results Are persistent and resilient are willing to try things out, experiment, take risks contribute to group intelligence by asking good questions Value mistakes as a learning tool (not something to be ashamed of) We need to redefine what it means to be “good at math” or a good math student

What research says about effective classrooms
The activity centers on mathematical understanding, invention, and sense-making by all students. The culture is one in which inquiry, wrong answers, personal challenge, collaboration, and disequilibrium provide opportunities for mathematics learning by all students. The tasks in which students engage are mathematically worthwhile for all students. A teacher’s deep knowledge of the mathematics content she/he teaches and the trajectory of that content enables the teacher to support important, long-lasting student understanding Begin session with research-based best practices…we would like to encourage you to engage in personal challenge throughout the sessions, collaborate with your colleagues, welcome disequilibrium and wrong answers, and be open to questions…take them as a challenge to your ideas not to you personally. if this is what needs to occur in the classroom, we are going to try to model this in the workshop. offering you suggestions and techniques to establish a classroom culture that is conducive to learning mathematics. You will notice that these best practices encompass the Standards for Mathematical practice. Could link with the shifts in classroom practice (7) if you are using that in your session.

What research says about effective classrooms
The activity centers on mathematical understanding, invention, and sense-making by all students. The culture is one in which inquiry, wrong answers, personal challenge, collaboration, and disequilibrium provide opportunities for mathematics learning by all students. The tasks in which students engage are mathematically worthwhile for all students. A teacher’s deep knowledge of the mathematics content she/he teaches and the trajectory of that content enables the teacher to support important, long-lasting student understanding Begin session with research-based best practices…we would like to encourage you to engage in personal challenge throughout the sessions, collaborate with your colleagues, welcome disequilibrium and wrong answers, and be open to questions…take them as a challenge to your ideas not to you personally. if this is what needs to occur in the classroom, we are going to try to model this in the workshop. offering you suggestions and techniques to establish a classroom culture that is conducive to learning mathematics. You will notice that these best practices encompass the Standards for Mathematical practice. Could link with the shifts in classroom practice (7) if you are using that in your session.

What research says about effective classrooms
The activity centers on mathematical understanding, invention, and sense-making by all students. The culture is one in which inquiry, wrong answers, personal challenge, collaboration, and disequilibrium provide opportunities for mathematics learning by all students. The tasks in which students engage are mathematically worthwhile for all students. A teacher’s deep knowledge of the mathematics content she/he teaches and the trajectory of that content enables the teacher to support important, long-lasting student understanding Begin session with research-based best practices…we would like to encourage you to engage in personal challenge throughout the sessions, collaborate with your colleagues, welcome disequilibrium and wrong answers, and be open to questions…take them as a challenge to your ideas not to you personally. if this is what needs to occur in the classroom, we are going to try to model this in the workshop. offering you suggestions and techniques to establish a classroom culture that is conducive to learning mathematics. You will notice that these best practices encompass the Standards for Mathematical practice. Could link with the shifts in classroom practice (7) if you are using that in your session.

What research says about effective classrooms
The activity centers on mathematical understanding, invention, and sense-making by all students. The culture is one in which inquiry, wrong answers, personal challenge, collaboration, and disequilibrium provide opportunities for mathematics learning by all students. The tasks in which students engage are mathematically worthwhile for all students. A teacher’s deep knowledge of the mathematics content she/he teaches and the trajectory of that content enables the teacher to support important, long-lasting student understanding. Begin session with research-based best practices…we would like to encourage you to engage in personal challenge throughout the sessions, collaborate with your colleagues, welcome disequilibrium and wrong answers, and be open to questions…take them as a challenge to your ideas not to you personally. if this is what needs to occur in the classroom, we are going to try to model this in the workshop. offering you suggestions and techniques to establish a classroom culture that is conducive to learning mathematics. You will notice that these best practices encompass the Standards for Mathematical practice. Could link with the shifts in classroom practice (7) if you are using that in your session.

Effective implies: Students are engaged with important mathematics.
Lessons are very likely to enhance student understanding and to develop students’ capacity to do math successfully. Students are engaged in ways of knowing and ways of working consistent with the nature of mathematicians ways of knowing and working. Teachers Development Group-Best Practices Workshop Donovan and Bransford, eds., 2005; Weiss et al, 2003; Kilpatrick et al, 2001; Glenn et al, 2000;

Mathematical Progression: K-5 Number and Operations in Base Ten
Everyone skim the OVERVIEW (p. 2-4) Divide your group so that everyone has at least one section: position, base-ten units, computations, strategies and algorithms, and mathematical practices Read your section carefully, share out 2 big ideas and give at least one example from your section Facilitator Note: There is a progression recording sheet that you can use if your participants aren’t keeping the progression document. If they are bringing their own or you are handing out progressions that they get to keep, then you may not want to use the recording sheet handout.

Mathematical Progression: K-5 Number and Operations in Base Ten

Big Ideas Rather than learn traditional algorithms, children’s struggle with the invention of their own methods of computation will enhance their understanding of place value and provide a firm foundation for flexible methods of computation. Computation and place value development need not be entirely separated as they have been traditionally. Van de Walle 2006 This quote is to be used as a summary of the progression discussion

Tens, Ones and Fingers Where does this activity fall in the progression and what clusters does this address? How can this activity be adapted? This is a quick activity from our activity handout. Used as an energizer for the group. Instead of calling up participants, have them respond as groups. The animations are structured so just the title shows up first for the activity part, then click for the questions to appear.

Standards for Mathematical Practices
Graphic Jigsaw. Create 4 expert groups and assign each group a set of SMP based on the groupings above.

The Standards for Mathematical Practice
Skim The Standards for Mathematical Practice Read The Standards for Mathematical Practice assigned to you Reflect: What would this look like in my classroom? Review the SMP Matrix for your assigned practices Add to your recording sheet if necessary Facilitator: Handout SMP in the recording sheet have the expert groups fill-out the sheet for the two standards assigned to them. Handout SMP Matrix They aren’t going to fill-out the last column about student friendly language yet.

The Standards for Mathematical Practice

Mathematical Practices in Action
esources/series32.html? pop=yes&pid=873 Using the matrix, what Mathematical Practices were included in these centers? What major and supporting clusters are addressed? Mathematical Practices in Action Teachers watch the video – online click included. Have them locate classroom examples of proficient mathematical practices found on the matrix. Discuss. If you have trouble connecting to the video, it is under learner.org and is video #4 – (scroll down) Place Value Centers for 1st Grade You can have groups write the SMP in friendly language on chart paper. If time is short, you can give them pre-created ones (Sue B sent them out)

Learners not passive recipients of mathematical knowledge Learners are active participants in creating understanding and challenge and reflect on their own and others understandings Instructors provide support and assistance through questioning and supports as needed

Depth of Knowledge (DOK)
Review the DOK matrix and hand it out

Bring it all together Is this a meaningful mathematical lesson?
Divide into triads Watch and reflect based on: “What Makes a Rich Task?” DOK Standards (which clusters and SMP were addressed?) As a group, using all three pieces of information, decide Is this a meaningful mathematical lesson? Hand out task analysis sheet. Have them split up into triads to watch the video.

Bringing It All Together Counting Collections Video
Counting Collections Video After viewing the video, allow time for groups to discuss and share out whole group is this a worthwhile math task?

Estimating Groups of Tens and Ones
Where does this activity fall in the progression and what clusters does this address? What mathematical practices are used? What makes this a good problem? What is the DOK? How can this activity be adapted? Task is from Van de Walle handouts

Let’s Analyze a Task This is typical place value practice given in text books. Use the Task Analysis Template to elicit a brief conversation with participants about the rigor and DOK level. Have participants brainstorm ways that DOK and rigor could be increased and ways in which the implementation could highlight a SMP. Share out as a group.

Do the same with this one.

Base-Ten Riddles Where does this activity fall in the progression and what clusters does this address? What mathematical practices are used? What makes this a good problem? What is the DOK? How can this activity be adapted? Van de Walle problem

Patterns on a 100 chart Have them analyze this task – handout then adapt it together – we will have notes for this…

One example: On a blank 100 chart find 27 on the 100 chart. Have students share out how they found 27 (did they count by ones, did they go to 7, then go down 2 rows, etc…) have them fill in the neighbors of 27 and “ask them what do you notice”, “Is this true with all numbers?”… Have them investigate with a different number… See “More and Less on the Hundreds Chart” from Van de Walle – next activity

More and Less on the Hundred Chart
Where does this activity fall in the progression and what clusters does this address? What mathematical practices are used? What makes this a good problem? What is the DOK? How can this activity be adapted? Van de Walle task – if you are tight on time, you can skip this activity.

Big Ideas A good base-ten model for ones, tens and hundreds is proportional. That is, a ten model is physically ten times larger than the model for one, and a hundred model is ten times larger than the ten model. No model, including a group able model, will guarantee that children are reflecting on the ten-to-one relationship in the materials. With pre-grouped models, we need to make an extra effort to see that children understand that a ten piece really is the same as ten ones. Van de Walle 2006 This is to summarize ideas – if time allows, you can have discussions around the two bullets.

Reflection What is your current reality around classroom culture?
What can you do to enhance your current reality? Handout

Next Meeting… Questions? Contact? Next Meeting?
Here you would have to include your contact info, next meeting dates, other logistics, etc.

Operations and Algebraic Thinking
Success Implementing CCSS for K-2 Math Operations and Algebraic Thinking

Today’s Objectives Develop a deeper understanding of how students progress in their understanding of the Common Core clusters related to operations and algebraic thinking in grades PreK-2. Learn engaging instructional strategies through hands on activities that connect content to the mathematical practices.

Quiz What is your definition of Operations and Algebraic Thinking?

Collaboration Protocol-Looking at Student Work
1. Individual review of student work samples (10 min) All participants observe or read student work samples in silence, making brief notes on the form “Looking at Student Work” 2. Sharing observations (15 min) The facilitator asks the group What do students appear to understand based on evidence? Which mathematical practices are evident in their work? Each person takes a turn sharing their observations about student work without making interpretations, evaluations of the quality of the work, or statements of personal reference. 3. Discuss inferences -student understanding (15 min) Participants, drawing on their observation of the student work, make suggestions about the problems or issues of student’s content misunderstandings or use of the mathematical practices. Adapted from: Steps in the Collaborative Assessment Conference developed by Steve Seidel and Project Zero Colleagues Select one group member to be today’s facilitator to help move the group through the steps of the protocol. Teachers bring student work samples with student names removed. 4. Discussing implications-teaching & learning (10 min) The facilitator invites all participants to share any thoughts they have about their own teaching, students learning, or ways to support the students in the future. How might this task be adapted to further elicit student’s use of Standards for Mathematical Practice or mathematical content. 5. Debrief collaborative process (5 min) The group reflects together on their experiences using this protocol. Scheduled for 55 minutes. Please remind teachers that they will first focus on evidence for students use of the Mathematical Practices and understanding of content.

Mathematical Progression: Operations and Algebraic Thinking

Major and Supporting Clusters: A Quick Review
One of the shifts from our current Math Standards to the Common Cores State Standards is the idea of focus. Students spend more time learning deeply with fewer topics. Grade level concepts have been divided into major, supporting and additional clusters. An example of the Grade 1 Cluster document is on the next slide!

Bring it all together… AGAIN!
Divide into triads Watch and reflect based on: “What Makes a Rich Task?” DOK Standards (which clusters and SMP were addressed?) As a group, using all three pieces of information, decide Is this a meaningful mathematical lesson? Hand out task analysis sheet. Have them split up into triads to watch the video.

Bringing It All Together Wheels Video
Wheels Video Go to the site (Annenberg Media) and it is the 14th video down After viewing the video, allow time for groups to discuss and share out whole group is this a worthwhile math task?

Lunch

Eyes, Fingers, and Legs Where does this activity fall in the progression and what clusters does this address? What mathematical practices are used? What is the DOK? How might students solve these problems? What student misconceptions might arise? How can this activity be adapted? Hand-out the Eyes, Fingers, and Legs activity. Let participants complete the activity and then answer the questions on the slide. If crunched for time, assign each group a question to answer.

Welcome Back Session 4

Objectives for the day Develop a deeper understanding of how students progress in their understanding of the Common Core clusters related to operations and algebraic thinking in grades PreK-2. Learn engaging instructional strategies through hands on activities that connect content to the mathematical practices.

With a partner, go through the Deep Task Analysis Sheet on the task you adapted Consider: Do you achieve the outcome you expected? How well did your students engage in the task? Where their any surprises? What questions would ask students based on their work? How would you change this task for next time? This slide is hidden if you could day 2 and day 3 in one sitting. Otherwise, you would have asked teachers to adapt a task that is in his/her current curriculum.

Reviewing the SMP Reflect on the work we have done and your students have done and write the SMP assigned to your table in student friendly language. Record your thoughts on a poster. Assign 2 SMP to each group and have them write student friendly language. The purpose of this task is to reflect on their learning during the first two sessions and what they have seen their students do in their classrooms and add to their charts. At this point, they should work together to come up with student friendly language for the SMP.

WA Kids This slide is objective 20 which is the objective that aligns with kindergarten Counting and Cardinality and Operations and Algebraic Thinking used in the WaKIDS assessment. There are two other objectives that are not used in this training. Hand out the WaKIDS/TS Gold Objectives handout. Have participants look over the objectives paying special attention to the beginning of kindergarten. Note, the bold print is what kids should be able to know/do, the bullets are suggestions for assessment purposes.

Smarter-Balanced Assessment Consortium (SBAC)
In your groups, work through the assessment task Consider – what standards are necessary for students to master in my grade level so they can be successful with this task?

Consider WA Kids and SBAC…
What are some implications of WA Kids and SBAC in mathematics? Look at your major and supporting clusters and the progression documents– what are the milestones students need to attain in order to prepare for the 3rd grade assessment? Where should we focus instruction if students aren’t coming to us prepared? What can we do to ensure our students are prepared? The 3 bullet points are possible discussion questions and/or prompts for the group

Classroom Connections….
Students using Quick Images: As you watch this video Reflect on the following: Where does this fit on our progression? How can this be adapted to meet the needs of your students? What mathematical practices did you observe? What can be assessed from this activity? https://www.teachingchannel.org/videos/visualizing-number- combinations?fd=1 Classroom Connections…. You will find this video on the Teaching Channel, Quick Images: Visualizing Number Combinations.

There were 7 children at the park. Then 4 more showed up. How many children were at the park all together? There were 7 children at the park. Some more showed up. Then there were 11 children in all. How many more children came? There were some children at the park. Four more children showed up. Then there were 11 children at the park. How many children were at the park to start with? Now consider different ways you can use these scenarios and what makes one more difficult than another. One might ask students….. Addition and Subtraction Situations

Which equation matches?
+ 4 = = 6 + =10 6 boys and 4 girls were at the party. How many children were at the party in all? 10 children were at the party. 6 were boys and the rest were girls. How many girls were at the party? 10 children were at the party. There were some boys and 4 girls. How many boys were at the party?

Teacher Questioning Strategies
How can the questions you ask move students’ thinking forward? How should it differ for struggling or high achieving students? Using the DOK, identify the level of some of the questions on the matrix provided. What can students do to engage in the mathematics more deeply? When you look at the website resources, what might work in your classroom? df Teacher Questioning Strategies Have them look at page 2 of the deep task analysis sheet and hand-out the matrix Instructional Implementation Matrix… Also have them look at their SMP practice sheet and see what they thought teachers/students should be doing… Another good resource is to look at the web site: It has great sentence starters and things to think about. One good idea (and also crosses over to speaking and listening skills, is to have students question each other. :

Connecting Literature with Math Concepts….
There are many different ways you can work with this idea. One suggestion is to ask individuals to share books that are good connections for math. You could run off the list of suggested literature and discuss them. You could find some books, set them at tables and have partners write math stories that could arise from the story. Folks share out. Another idea is to read a book aloud (one great one is called Moira’s Birthday by Robert Munsch) and have the group suggest rich tasks that could be written. Determine what works best with the time you have You can also pass out the OA Activities Handout if time allows that give directions for the examples above. Other classroom connection ideas for OA? The “hand game”, Go Fish, Memory, Race to 10….

Evaluation and Next Steps….