Alabama Explorations Guide: Grade 3 Mathematics ©2010 Wisconsin Cooperative Educational Service Agencies (CESAs) School Improvement Services Permission is granted to the Alabama Department of Education for dissemination and use in any whole or part in any form within the Alabama Department of Education region.

Today’s Agenda Foundations of the Standards Exploring Grade Level Intent Exploring the Content Standards’ Structure Exploring Standards for Mathematical Practice Exploring Mathematical Understanding Exploring the Expectations of Understanding Exploring Two Standards Exploring Vertical Connections Determining Implications and Next Steps

Outcomes 1. To understand the foundation of the Alabama College-and Career-Ready Standards 2. To explore the critical focus areas by grade level 3. To explore the grade level standards 4. To explore mathematical understanding 5. To reflect on implications to your practice

The Message Cannot/should not be rushed–a marathon, not a race. Your LEAs teacher leaders are needed. Our focus–to learn HOW to explore these standards. We aren’t exploring all standards today. You will be given a process that can be duplicated in your school. We won’t be aligning today–because alignment cannot be done effectively without careful exploration.

To explore, you will need … Copy of the 2010 Alabama Course of Study: Mathematics The Explorations Guide Highlighters Pen or pencil Tables for group work Timer/timekeeper

Ground Rules for Today Information-Giving Group Work & Recording Attentive listening Open mindset to receive new ideas and information Note-taking Open mindset Professional conversations Careful note-taking (for taking back) Deep thinking Record questions–to be addressed later

Now … for some background information

Development of Common Core State Standards Joint initiative of: Supported by:

The College-and Career-Ready Standards are not intended to be new names for old ways of doing business. They are a call to take the next step. It is time to recognize that standards are not just promises to our children, but promises we intend to keep.

What Is College and Career Readiness? College readiness means students are prepared for credit-bearing academic courses at 2-year or 4-year postsecondary schools. Career readiness means students are prepared to succeed in workforce training programs in careers that: 1) Offer competitive, livable salaries above the poverty line 2) Offer opportunities for career advancement 3) Are in a growing or sustainable industry

Why are college- and career-ready standards good for students? College & Career Focus Consistent Mobility Student Ownership

How can you ensure that the work in your classroom is preparing students to be ready for essential workforce and college responsibilities?

Which Standards Are In The Course Of Study? + = Alabama Added Content

Alabama Added Content + + Grades K- 12 Grades K-8 Grades 9-12

FOCUS Identifies key ideas, understandings and skills for each grade or course Stresses deep learning (addresses mile-wide, inch-deep issue) Connects topics and standards within grade or course Requires applying concepts and skills within same grade or course

FOCUS: Increased Clarity and Specificity “It is important to recognize that “fewer standards” are no substitute for focused standards. Achieving “fewer standards” would be easy to do by resorting to broad, general statements. Instead, these Standards aim for clarity and specificity.” CCSS page 3.

COHERENCE Provides the opportunity to make connections between mathematical ideas Occurs both within a grade and across grades Is necessary because mathematics instruction is not just a checklist of topics to cover, but a set of interrelated, powerful ideas

LEARNING PROGRESSIONS Learning Trajectories – sometimes called learning progressions – are sequences of learning experiences hypothesized and designed to build a deep and increasingly sophisticated understanding of core concepts and practices within various disciplines. The trajectories are based on empirical evidence of how students’ understanding actually develops in response to instruction and where it might break down. Daro, Mosher, & Corcoran, 2011 Insert hyperlink from Hunt Institute on Learning Progressions

Learning Progression Framework Ending Point Ending Point Starting Point Starting Point K 1 2 3 4 5 6 7 8 HS Counting and Cardinality Number and Operations in Base Ten Ratios and Proportional Relationships Number and Quantity Number and Operations – Fractions The Number System Operations and Algebraic Thinking Expressions and Equations Algebra Functions Geometry Measurement and Data Statistics and Probability

Value of Learning Progressions/Trajectories to Teachers Know what to expect about students’ preparation. Manage more readily the range of preparation of students in your class. Know what teachers in the next grade expect of your students. Identify clusters of related concepts at grade level. Provide clarity about the student thinking and discourse to focus on conceptual development. Engage in rich uses of classroom assessment.

A Vision for Implementation Administrator and teacher awareness of new standards, grade-specific training in the critical areas of focus, structure of the standards, verbs and vocabulary, student practice standards, etc… District curricula, assessments, and instruction (the exterior and interior) Alabama College- and Career-Ready Standards (2010 Alabama Course of Study: Mathematics)

Exploring the Standards: Grade 3 Mathematics

Activity #1 Focus Area Narratives Important descriptions at the beginning of each grade level. Provide the intent of the mathematics at each grade. Provide 3-4 critical focus areas for the grade level . Provide a sense of … The sophistication for mathematical understanding at the grade level. The learning progressions for the grade. Extensions from prior standards. What’s important at the grade level. Focus area narratives are important descriptions at the beginning of each grade level. They express the intent of the standards for the grade and specify 3 to 4 critical mathematical areas of focus.  

Turn to page 27 in your ACOS Activity #1 Grade-Level Intent Grade 3 Narrative Turn to page 27 in your ACOS for Grade 3.

Activity #1: Exploring Grade Three Intent

Components of the Mathematics Course of Study Activity #2 Components of the Mathematics Course of Study Number & Quantity Algebra Functions Modeling Geometry Statistics & Probability Mathematical Practice Standards Mathematical Content Standards

Structure of the Standards Activity #2 Structure of the Standards Standards for Mathematical Practice Carry across all grade levels Describe habits of mind of a mathematically expert student ACOS – pages 6-8 Make sense of problems and persevere in solving them Reason abstractly and quantitatively Construct viable arguments & 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

The Practices are also listed on the Grade 3 Overview. Activity #2 Grade 3 Overview Standards for Mathematical Practice are provided in detail on pages 6-8 of the ACOS. The Practices are also listed on the Grade 3 Overview.

K-12 Standards for Mathematical Content Activity #2 K-12 Standards for Mathematical Content Refer to the ACOS K-8 standards presented by grade level Organized into domains that progress over several grades Grades K-8 introductions give 2 to 4 focal points at each grade level High school standards presented by conceptual theme (Number & Quantity, Algebra, Functions, Modeling, Geometry, Statistics & Probability)

Structure of the Standards Activity #2 Structure of the Standards Content standards define what students should understand and be able to do Clusters are groups of related standards Domains are larger groups that progress across grades Content Standard Identifiers Domain Cluster Statement The actual standards are written in a format that will be new to Alabama teachers. As we mentioned a few moments ago, you will see domains in each grade level. These are akin to the content strands that most math teachers have been planning their instruction with for years. All of the standards that relate to that domain are organized by clusters. The individual standards are akin to learning objectives. Within each grade level you find, numbered content “standards,” preceded by the phrase “students will” that define what students should know and be able to do. Related standards are grouped into “clusters,” which are housed within “domains.” Domains are larger groups which progress across grade levels. Direct participants to look at page 10 in ACOS 2010.   Point out the content standard identifier at the end of each standard the correlates with CCSS numbers. Cluster Standards

Grade-Level Standards Activity #2 Grade-Level Standards “…grade placements for specific topics have been made on the basis of state and international comparisons and the collective experience and collective professional judgment of educators, researchers and mathematicians.” (2010 Alabama Course of Study, p.1)

Activity #2: Exploring Structure of Content Standards Grade 3

Mathematical Practice Activity #3 Standards for Mathematical Practice “The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students.” (2010 Alabama Course of Study, p. 6)

Underlying Frameworks Activity #3 Underlying Frameworks National Council of Teachers of Mathematics 5 PROCESS Standards Problem Solving Reasoning and Proof Communication Connections Representations NCTM (2000M). Principles and Standards for School Mathematics. Reston, VA: Author.

Underlying Frameworks Activity #3 Underlying Frameworks National Research Council Strands of Mathematical Proficiency Conceptual Understanding Procedural Fluency Strategic Competence Adaptive Reasoning Productive Disposition NRC (2001). Adding It Up. Washington, D.C.: National Academies Press.

Refer to pages 6-8 in the ACOS Activity #3 Standards for Mathematical Practice Make sense of problems and persevere in solving them Reason abstractly and quantitatively Construct viable arguments & 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 Refer to pages 6-8 in the ACOS

Make sense of problems and persevere in solving them #1 When presented with a problem, I can make a plan, carry out my plan, and evaluate its success. AFTER… CHECK Is my answer correct? How do my representations connect to my algorithms? EVALUATE What worked? What didn’t work? What other strategies were used? How was my solution similar to or different from my classmates? BEFORE… EXPLAIN the problem to myself. Have I solved a problem like this before? ORGANIZE information. What is the question I need to answer? What is given? What is not given? What tools will I use? What prior knowledge do I have to help me? DURING… PERSEVERE MONITOR my work CHANGE my plan if it isn’t working out ASK myself, “Does this make sense? “

Reason abstractly and quantitatively I can use reasoning habits to help me contextualize and decontextualize problems. #2 CONTEXTUALIZE I can take numbers and put them in a real-world context. For example, if given 3 x 2.5 = 7.5, I can create a context: I walked 2.5 miles per day for 3 days. I walked a total of 7.5 miles. DECONTEXTUALIZE I can take numbers out of context and work mathematically with them. For example, if given I walked 2.5 miles per day for 3 days, How far did I walk? I can write and solve 3 x 2.5 = 7.5.

Construct viable arguments and critique the reasoning of others #3 I can make conjectures and critique the mathematical thinking of others. I can critique the reasoning of others by… listening comparing arguments identifying flawed logic asking questions to clarify or improve arguments I can construct, justify, and communicate arguments by… considering context using examples and non- examples using objects, drawings, diagrams and actions

Model with mathematics #4 I can recognize math in everyday life and use math I know to solve everyday problems. concrete models I can… make assumptions and estimate to make complex problems easier. identify important quantities and use tools to show their relation- ships. evaluate my answer and make changes if needed. pictures symbols Represent Math oral language real-world situations

Use appropriate tools strategically #5 I know when to use certain tools to help me explore and deepen my math understanding. I have a math toolbox. I know HOW to use math tools. I know WHEN to use math I can reason: “Did the tool I used give me an answer that makes sense?”

I can use precision when solving problems and communicating my ideas. #6 Attend to precision I can use precision when solving problems and communicating my ideas. Problem Solving I can calculate accurately. I can calculate efficiently. My answer matches what the problem asked me to do–estimate or find an exact answer. Communicating I can SPEAK, READ, WRITE, and LISTEN mathematically. I can correctly use… math symbols. math vocabulary. units of measure.

Look for and make use of structure. I can see and understand how numbers and spaces are organized and put together as parts and wholes. #7 SHAPES For example: Dimension Location Attributes Transformation NUMBERS For example: Base 10 structure Operations and properties Terms, coefficients, exponents

Look for and express regularity in repeated reasoning I can notice when calculations are repeated. Then, I can find more efficient methods and short cuts. #8 Patterns: 1/9 = 0.1111…. 2/9 = 0.2222… 3/9 = 0.3333… 4/9 = 0.4444…. 5/9 = 0.5555…. I notice the pattern which leads to an efficient shortcut!!!

The Mathematical Practices Activity #3 The Mathematical Practices These practices rest on important “processes and proficiencies” with longstanding importance in mathematics education. The first of these are the NCTM process standards of problem solving, reasoning and proof, communication, representation, and connections. Lets look at mathematical practices in action in a classroom. Video Clip What Mathematical Practices did you see the students use?

Activity #3: Exploring Standards for Mathematical Practice Grade 3

Activity #4 Exploring Domains Domains are common learning progressions that can progress across grade levels. Domains do not dictate curriculum or teaching methods. Topics within domains are not meant to be taught in the order presented. Teachers must present the standards in a manner that is consistent with decisions that are made in collaboration with their K-12 mathematics team.

Mathematical Language Activity #4 Mathematical Language Mathematical language may be different than everyday language and other disciplinary area language. Questions may arise about the meaning of the mathematical language used. This is a good opportunity for discussions and sense making in the ACOS. Questions about mathematical language can be answered by investigating the progression of the concepts in the standards throughout other grades.

Activity #4: Exploring Domains Grade 3

Outline of Grade 3 Math Standards Domain Clusters Standards Operations & Algebraic Thinking 4 9 Number and Operations in Base Ten 1 3 Number and Operations--Fractions Measurement and Data 8 Geometry 2 TOTAL 25 Total

Mathematics Understanding Activity #5 Mathematics Understanding The Alabama College- and Career-Ready Standards for mathematics provide a major focus on UNDERSTANDING. Questions to think about … What is meant by understanding? How do we see it in students? How do we teach it?

Activity #5: Exploring Understanding

Mathematical Understanding Reflected in the Standards Activity #6 Mathematical Understanding Reflected in the Standards Interpretation Explanation Application Mathematics Procedural Skills From Kindergarten through to Grade 12, there is a strong emphasis and specificity on ways that students will be expected to show their understanding.

explain … interpret …apply Activity #6 Students who understand a concept can: explain … interpret …apply For example, they can … use it to make sense of and explain quantitative situations (see "Model with Mathematics" in Practices) incorporate it into their own arguments and use it to evaluate the arguments of others (see " Construct viable arguments and critique the reasoning of others" in Practices) bring it to bear on the solutions to problems (see "Make sense of problems and persevere in solving them") make connections between it and related concepts

Activity #6: Exploring the Expectations of Understanding Grade 3

Three Important Lenses to Explore the Standards Activity #7 Three Important Lenses to Explore the Standards Lens #1: Student-Friendly Language Lens #2: Key Vocabulary Lens #3: Mathematical Practices 56 56

Lens #1: Student-Friendly Language Activity #7 Lens #1: Student-Friendly Language Explaining the intended learning in student-friendly terms at the outset of a lesson is the critical first step in helping students know where they are going...Students cannot assess their own learning or set goals to work toward without a clear vision of the intended learning.  When they do try to assess their own achievement without understanding the learning targets they have been working toward, their conclusions are vague and unhelpful. -Stiggins, Arter, Chappuis & Chappuis, 2004, pp. 58-59

Activity #7 Lens #2: Key Vocabulary Why identify key vocabulary in the standards for instruction? To clarify the teacher’s understanding To activate prior vocabulary in context To make connections to the prior learning and experiences of students To observe how vocabulary is developed in the learning progressions of the standards What implications does the vocabulary of the standards hold for teacher professional development?

Lens #3: Mathematical Practices Activity #7 Lens #3: Mathematical Practices “…those content standards which set an expectation of understanding are potential ‘points of intersection’ between the Standards for Mathematical Content and the Standards for Mathematical Practice.” “…attend to the need to connect the mathematical practices to mathematical content in mathematics instruction.” (2010 Alabama Course of Study: Mathematics, page 9)

Activity #7: Exploring the Content Standards Grade 3

Activity #8 Vertical Connections All Standards in mathematics have a connection to early and subsequent concepts and skills. The flow of those connections is documented by how a student develops the concepts. Prior Standards Prior Standards Current Standard Future Standards Future Standards

Big Ideas that Carry Across the Document (K-12) Activity # 8 Big Ideas that Carry Across the Document (K-12) (from Phil Daro, one of three lead writers on the Common Core Standards for Mathematics) Properties of operations: their role in arithmetic and algebra Mental math and algebra vs. algorithms (Inspection) Units and unitizing Operations and the problems they solve Quantities Variables Functions Modeling (As a sequence across grades) Number Operations Expressions Equations (As a sequence across grades) Modeling Practices

Fractions Progression Activity #8 K-2 3-5 6-8 Understanding that arithmetic of fractions draws upon four prior progressions that informed the CCSS Number Line in Quantity and Measurement Equal Partitioning Fractions Rational numbers Properties of Operations Rational Expressions Unitizing in Base 10 and in Measurement Rates, proportional and Linear Relationships

Vertical Connections (example) Fractions, Grades 3–6 Activity #8 Vertical Connections (example) Fractions, Grades 3–6 Gr. 3. Develop an understanding of fractions as numbers. Gr. 4. Extend understanding of fraction equivalence and ordering. Gr. 4. Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers. Gr. 4. Understand decimal notation for fractions, and compare decimal fractions. Gr. 5. Use equivalent fractions as a strategy to add and subtract fractions. Gr. 5. Apply and extend previous understandings of multiplication and division to multiply and divide fractions. Gr. 6. Apply and extend previous understandings of multiplication and division to divide by fractions.

Functions and Equation Progression Activity #8 Functions and Equation Progression K-2 3-6 7-12 Quantity and Measurement Operations and Algebraic Thinking Ratio and Proportional Relationships Expressions and Equations Functions Modeling Practices Modeling (with Functions)

Activity #8: Exploring Vertical Connections Grade 3

Determining Implications and Next Steps Activity #9 Determining Implications and Next Steps We’ve been exploring the standards–now, what do we do?

Activity #9: Determining Implications

Activity #10: Determining Next Steps

Exit Ticket Please complete the exit ticket as instructed. 3. Write three significant things that you learned today. 2. Write about two areas that are still confusing to you. 1. Write one immediate step you will take to implement the 2010 Alabama Course of Study: Mathematics when you return to your LEA.