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## Numbers and Operations in Base Ten

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**Success Implementing CCSS for**K-2 Math Numbers and Operations in Base Ten**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**Why CCSS?**Greta’s Video Clip**Common Core State Standards**Source: www.corestandards.org • 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**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**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**Focuson the Major Work of the Grade**Two levels of focus ~ • What’s in/What’s out • The shape of the content**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.**Shift Two: CoherenceThink across grades, and link to major**topics within grades • 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.**CoherenceAcross and Within Grades**It’s about math making sense. The power and elegance of math comes out through carefully laid progressions and connections within grades.**CoherenceThink across grades, and link to major topics**within grades • 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.**Coherence Across the Grades?**Varied problem structures that build on the student’s work with whole numbers 5 = 1 + 1 + 1 + 1 +1 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**Coherence Within A Grade**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**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.**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.**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)**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**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**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**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.**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.**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**Mathematical Progression:K-5 Number and Operations in Base**Ten • Read through the progression document at your grade level. • Discuss with your grade level team and record the following on your poster: • Big ideas • Progression within the grade level • What is this preparing students for?**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**Tens, Ones and Fingers**• Where does this activity fall in the progression and what clusters does this address? • How can this activity be adapted?**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**The Standards for Mathematical Practice**Return to your home group and share out your practices.**http://www.learner.org/resources/series32.html?pop=yes&pid=873**http://www.learner.org/resources/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**What makes a rich task?**Is the task interesting to students? Does the task involve meaningful mathematics? Does the task provide an opportunity for students to apply and extend mathematics? Is the task challenging to all students? Does the task support the use of multiple strategies and entry points? Will students’ conversation and collaboration about the task reveal information about students’ mathematics understanding? Adapted from: Common Core Mathematics in a PLC at Work 3-5 Larson,, et al**Environment for Rich Tasks**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**Bring it all together**• 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?**http://vimeo.com/45953002**Bringing It All TogetherCounting Collections Video**Adapting a Task**In your group, think of ways to adapt this problem**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?**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