Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Making Sense of The CT Mathematics Standards (Common Core State Standards) GRADES K – 2 ATOMIC CONFERENCE NOVEMBER 29, 2011 Kathy St. Onge Ann Spinelli M a r y S a n t i l l i M a rc i a Fe rre i ra Intent of the Common Core Same goals for all students Coherence Focus Clarity, rigor and specificity Opportunities for broadening the discussion about the teaching and learning of mathematics 45 states have adopted (as of December 2011) CCSS Assessment Projects SBAC SMARTER Balanced Assessment Consortium - 30 states http://www.k12.wa.us/smarter/ PARCC (Partnership for the Assessment of Readiness for College and Careers) 25 states - http://www.achieve.org/PARCC “These Standards are not intended to be new names for old ways of doing business.” CCSSM, p. 5 Organization of the CCSS Standards for Mathematical Practice Math Content Standards Domains Clusters Standards Connecting the Practices to the Content Math content standards describe what students should understand and be able to do. Math practices describe ways in which students should interact with mathematics. Curricula, assessment and professional development should be focused on connecting the mathematical practices and the content standards. (CCSS p. 8) Standards for Mathematical Practices 8 Mathematical Practices Related to the NCTM Process Standards (2000) and the Strands of Mathematical Proficiency (Adding It Up, 2001) Standards for Mathematical Practice The standards for mathematical practices are located in the front of the mathematics standards and within the “nature of mathematics” section at each grade level. The standards for mathematical practice illustrate the connection between 21st century skills and mathematical content and instruction. The standards for mathematical practices should be considered when creating curricula, assessments, and professional development for teachers, and administrators. Standards for Mathematical Practice “…describe the varieties of expertise that mathematics educators at all levels should seek to develop in their students.” Mathematically Proficient Students Will… Adapted from Inside Mathematics Standards for Mathematical Practice 1. Make sense of problems and persevere in solving them 6. Attend to precision Overarching habits of mind of a productive mathematical thinker. 2. Reason abstractly and quantitatively Reasoning and explaining 3. Construct viable arguments and critique the reasoning of others 4. Model with mathematics 5. Use appropriate tools strategically Modeling and using tools 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning Seeing structure and generalizing 1. Make Sense of Problems and Persevere in Solving Them Engage in problem solving on a regular basis Foster a “productive disposition” - build success early on Involve students in sharing solutions, methods, and reasoning Frame the class environment to encourage student interaction and conversation – math discourse Allow students to “struggle” with the mathematical tasks – avoid rescuing too soon to diminish the cognitive load Emphasize equivalent representations of a given situation or mathematical relationship 2. Reason Abstractly and Quantitatively Mathematical Problem Decontextualize Represent as symbols, abstraction Refer back to the situation Contextualize 4 x x x x 5 2 =? Teach concepts in context – symbols have meaning Base instruction on making sense and select practice that involves the application of concepts being learned Emphasize reasoning as opposed to only learning procedures Allow students to develop a representation of mathematical problems on a regular basis 3. Construct Viable Arguments and Critique the Reasoning of Others Encourage interaction and conversation on a regular basis Use problem-based activities – rich tasks Practice the language of “argument,” conjecture, and discourse while students are engaged in mathematical tasks Facilitate student discourse – “talk moves” * Encourage taking risks, defending solutions Have students present solutions and ideas on a regular basis *Classroom Discussions: Using Math Talk to Help Students Learn, 2nd edition, Grades K-6 by Suzanne Chapin, Catherine O’Connor and Nancy Anderson, Math Solutions, 2009. 4. Model with Mathematics Problems in everyday life… Use physical objects, drawings and physical gestures to represent math situations Encourage student verbal descriptions Encourage representing the same situation in different ways Guide students to see similarities in different ways to represent the same situations 5. Use Appropriate Tools Strategically Provide mathematical tools in the classroom Ensure that students know how to use the appropriate tools effectively Discuss criteria to help make a decision as to when to use a mathematical tool Encourage students use their rationale for using a tool in their explanation of their solution 6. Attend to Precision Make mathematical tools available in the classroom Display and provide instruction on mathematical vocabulary – interactive word wall Hold students accountable for using vocabulary in discussion and written explanations Embed instruction about math symbols (7, +, =, >,) Discuss answers in terms of the context of the problems to give students experience with the idea of a “reasonable” answer Review processes for computational skills; include error analysis and feedback to develop accuracy and proficiency 7. Look for and Make Use of Structure Encourage students to always look for patterns to help develop conceptual understanding Provide opportunities for students to generalize Use mental math to practice patterns in our number system Provide opportunities to work on tasks that generate data that can be used to develop a generalization Foster a class environment that values and encourages student reasoning as opposed to teacher “telling” 8. Look for and Express Regularity in Repeated Reasoning Encourage students to always look for patterns or an opportunity to generalize about computational skills Use mental math to practice patterns in our number system that can be used to develop more efficient computation methods Incorporate lessons and activities that use pattern or structure to help develop conceptual understanding Foster a class environment that values and encourages student reasoning as opposed to teacher “telling” what to notice or how to do a skill The Leadership and Learning Center Seminar- “Digging Deeper into the Common Core State Standards” Incorporating the Practice Standards… Examine the math problems. Think about the Mathematical Practices that students would engage in when solving the problems. Share with someone next to you your reasoning. Summary All Standards for Mathematical Practice will not be demonstrated with every math exercise given, but multiple standards should be evident in every mathematics lesson. Common Core State Standards K-12 Mathematics Learning Progressions Kindergarten 1 2 3 4 5 6 7 8 Counting and Cardinality HS Number and Quantity Number and Operations in Base Ten Number and Operations: Fractions Operations and Algebraic Thinking The Number System Ratios and Proportional Relationships (6 and 7) Expressions and Equations Functions Geometry Measurement and Data Algebra Functions Geometry Geometry Statistics and Probability Statistics and Probability http://education.ohio.gov/GD/Templates/Pages/ODE/ODEDetail.aspx?page=3&TopicRelationID=1704&ContentID=83475&Content=102764 Priorities in Mathematics Grade K–2 3–5 6 7 8 Priorities in Support of Rich Instruction and Expectations of Fluency and Conceptual Understanding Addition and subtraction, measurement using whole number quantities Multiplication and division of whole numbers and fractions Ratios and proportional reasoning; early expressions and equations Ratios and proportional reasoning; arithmetic of rational numbers Linear algebra http://commoncoretools.wordpress.com/ Key Fluencies Grade Required Fluency K Add/subtract within 5 1 Add/subtract within 10 Add/subtract within 20 2 3 Add/subtract within 100 (pencil and paper) Multiply/divide within 100 Add/subtract within 1000 4 Add/subtract within 1,000,000 5 Multi-digit multiplication 6 Multi-digit division Multi-digit decimal operations 7 Solve px + q = r, p(x + q) = r 8 Solve simple 22 systems by inspection http://commoncoretools.wordpress.com/ Content Standards K-2 Grade Level Overview Critical Areas of Focus A description of the key areas where instruction & learning time should be focused. Mathematics | Kindergarten In Kindergarten, instructional time should focus on two critical areas: (1) representing, relating, and operating on whole numbers, initially with sets of objects; (2) describing shapes and space. More learning time in Kindergarten should be devoted to number than to other topics. (1) Students use numbers, including written numerals, to represent quantities and to solve quantitative problems, such as counting objects in a set; counting out a given number of objects; comparing sets or numerals; and modeling simple joining and separating situations with sets of objects, or eventually with equations such as 5 + 2 = 7 and 7 – 2 = 5. (Kindergarten students should see addition and subtraction equations, and student writing of equations in kindergarten is encouraged, but it is not required.) Students choose, combine, and apply effective strategies for answering quantitative questions, including quickly recognizing the cardinalities of small sets of objects, counting and producing sets of given sizes, counting the number of objects in combined sets, or counting the number of objects that remain in a set after some are taken away. (2) Students describe their physical world using geometric ideas (e.g., shape, orientation, spatial relations) and vocabulary. They identify, name, and describe basic two-dimensional shapes, such as squares, triangles, circles, rectangles, and hexagons, presented in a variety of ways (e.g., with different sizes and orientations), as well as three-dimensional shapes such as cubes, cones, cylinders, and spheres. They use basic shapes and spatial reasoning to model objects in their environment and to construct more complex shapes. Format of Pre-K-8 Standards Domain Standard 2.NBT.1 (code) Cluster Process Used to Develop Framework for District Curriculum Work Grade Level Band Teams • • • • K–2 3–5 6–8 High School A Frame for District Curriculum Work Prioritized Standards Vertical Alignment Named Units & Assigned Standards Developed a Suggested Sequence of Instruction Created Sample Assessment Items Developed Unit Pacing Plan ALL Standards are Important KINDERGARTEN SECOND GRADE Counting and Cardinality Number and Operations in Base Ten Know number names and the count sequence. 1. Count to 100 by ones and by tens. 2. Count forward beginning from a given number within the known sequence (instead of having to begin at 1). 3. Write numbers from 0 to 20. Represent a number of objects with a written numeral 0-20 (with 0 representing a count of no objects). Understand place value. 1. Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. 2. Count within 1000; skip-count by 5s, 10s, and 100s. 3. Read and write numbers to 1000 using base-ten numerals, number names, and expanded form. Priority Standards Supporting Standards Units of Study Critical Areas of Focus: Key areas where instruction & learning time should be focused. Kindergarten Units Counting and Matching Numerals 0-5 with Comparing Counting and Matching Numerals 6- 10 with Comparing Counting and Matching Numerals 11-20 Teen Numbers (11-19) & Counting to 100 Fluency with Addition & Subtraction within 5 Exploring Addition & Subtraction within 10 Identify & Describe 2D & 3D Shapes Compare, Analyze and Compose 2D & 3D Shapes Measurement by Direct Comparison Suggested Pacing Timeframe • • • • Mapped out year using school calendars Developmentally appropriate Critical Areas of Focus for grade level Time to Process & Practice FIRST GRADE Suggested Unit Sequence 1 Fluency with Addition & Subtraction within 10 Pacing 5 weeks 2 Exploring Addition & Subtraction within 20 4 weeks 3 Counting & Place Value 5 weeks 4 Exploring Addition and Subtraction within 100 5 weeks 5 Defining Attributes of 2D & 3D Shapes 2 weeks 6 Partitioning Circles & Rectangles 2 weeks 7 Measuring Length with Non-Standard Units 2 weeks 8 Time to the Hour and Half-Hour 2 weeks Unit Planning Organizer Development of Unit Planning Organizer (in process) Mathematical Practices Domain & Standards Overview Priority & Supporting CCSS Explanations & Examples Concepts Students Need to Know Skills Students Need to Be Able to Do Bloom’s Taxonomy Levels Unit Assessment Items Transition Guide Transition Guide: Displaced Grade Level Concepts Assessment Grade Level Band Teams Assessment Items Based on CCSS Unit Planning Organizer Items for use during instruction when appropriate Kindergarten Assessment Items Unit 1 - Counting and Matching Numerals 0 – 5 with Comparing Test Mode: Administer one on one Rote Count Teacher: Count out loud starting at 1 and count as high as you can. Record highest number student accurately counts to. Ex: Child counts from 1-15 accurately, then skips 16. Stop student and record last correct number stated. Kindergarten Assessment Items Unit 2 - Counting and Matching Numerals 6 – 10 with Comparing Match Numerals Preparation: In advance, teacher puts out groups of objects (ex: counters, unifix cubes or bears) and numeral cards 0-10. Objects should be arranged in groups of 3, 5, 8 and 10. Teacher: Give students the shuffled set of numeral cards. Count each group. Put the matching numeral card next to each set. Observe and record ( or - ) if student correctly matches all four sets. Kindergarten Assessment Items Unit 4 - Fluency with Addition and Subtraction within 5 There are 5 apples in a bowl. Some apples are red. Some apples are green. •How many of each color apple could be in the bowl? ___ red apples ___ green apples •Find a different answer. ___ red apples ___ green apples Grade 1 - Assessment Items Unit 4 - Exploring Addition and Subtraction within 100 Constructed Response Write a number sentence and solve the problem. Use manipulatives (base-ten blocks, hundreds chart, number lines) or a drawing to show how to solve this problem. Mrs. Jones needs 42 cupcakes for the class picnic. She has 32 cupcakes. How many more cupcakes does she need to buy? This is how Joe found the answer to 29 + 30 + 1 29 + 30 + 1 = 30 + 30 = 60 What did Joe do to solve the problem? Grade 2 - Assessment Items Unit 2 - Place Value to 1,000 Multiple Choice Circle all the statements that are equal to this number. 823 a) 8 hundreds and 23 tens c) 7 hundreds, 12 tens and 3 ones e) 8 hundreds and 23 ones What is another way to show 729? 700 + 2 + 90 700 + 20 + 9 70 + 200 + 9 7 + 20 + 900 b) 823 ones d) 82 tens and 3 ones f) 7 hundreds and 23 tens Grade 2 - Assessment Items Unit 3 - Fluency with Addition and Subtraction within 100 Constructed Response Solve the problem. 54 - 29 Show or explain how to find the answer two different ways. Write an equation for this problem. Solve the equation to find the answer. The teacher is 70 inches tall. The student is 47 inches tall. How much taller is the teacher than the student? QUESTIONS