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Balanced Math Framework August 15, 2013 Getting to Know You... Math Style • • Grab a bingo card from the middle of your table Circulate the room searching for teachers who can "sign" a box on your bingo card (one signature per box please) Math Workshop Readers and Writers = Workshop Framework Framework Math Workshop Math Review is............. • • • • • • Time to reinforce a previously taught concept Formative and based on daily student understanding Work that is de-briefed and discussed Used to guide instruction 3 to 6 review problems (based on grade level) An opportunity to circulate and observe Math Review is ............... • • • • • • Not time to teach a new concept or trick the students Not pre-printed or planned by yearlong or unit objectives Not work completed without discussion Not used as a grade or graded by others Not more than six problems Not busy work Math Review and Mental Math Problem Solving happens daily in the classroom. Conceptual Understanding This is where you teach your curriculum. You will use Math Expressions, Glencoe and DMI experiences as a resource. Standards for Mathematical Practice Mathematically Proficient Students... 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeating reasoning. The Standards for Mathematical Practice Take a moment to examine the first three words of each of the 8 mathematical practices... what do you notice? Mathematically Proficient Students... The Standards for [Student] Mathematical Practice What are the verbs that illustrate the student actions of each mathematical practice? Mathematical Practice #3: Construct viable arguments and critique the reasoning of others Mathematically proficient students: • understand and use stated assumptions, definitions, and previously established results in constructing arguments. • make conjectures and build a logical progression of statements to explore the truth of their conjectures. • analyze situations by breaking them into cases, and can recognize and use counterexamples. • justify their conclusions, communicate them to others, and respond to the arguments of others. • reason inductively about data, making plausible arguments that take into account the context from which the data arose. • compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and-if there is a flaw in an argument-explain what it is. • construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until In the SJSD curriculum... Standards for Mathematical Practice Buttons Task Standards for [Student] Mathematical Practices • "Not all tasks are created equal, and different tasks will provoke different levels and kinds of student thinking." ~ Stein, Smith, Henningsen, & Silver, 2000 • "The level and kind of thinking in which students engage determines what they will learn." ~ Hiebert, Carpenter, Fennema, Fuson, Wearne, Murray, Oliver, & Human 1997 Comparing Two Mathematical Tasks Martha was re-carpeting her bedroom which was 15 feet long and 10 feet wide. How many square feet of carpeting will she need to purchase? ~ Stein, Smith, Henningsen, & Silver, 2000, p. 1 Comparing Two Mathematical Tasks Ms. Brown's class will raise rabbits for their spring science fair. They have 24 feet of fencing with which to build a rectangular rabbit pen in which to keep the rabbits. 1. If Ms. Brown's students want their rabbits to have as much room as possible, how long would each of the sides of the pen be? 2. How long would each of the sides of the pen be if they had only 16 feet of fencing? 3. How would you go about determining the pen with the most room for any amount of fencing? Organize your work so that someone else who read it will understand it. ~ Stein, Smith, Henningsen, & Silver, 2000, p.2 Comparing Two Mathematical Tasks Discuss: How are Martha's Carpeting Task and the Fencing Task the same and how are they different? Comparing Two Mathematical Tasks Lower-Level Tasks Higher-Level Tasks Reflection My definition of a good teacher has changed from "one who explains things so well that students understand" to "one who gets students to explain things so well that they can be understood." (Steven C. Reinhart, "Never say anything a kid can say!" Mathematics Teaching in the Middle School 5, 8 [2000]: 478) Richard Schaar What I learned in school may be growing increasingly obsolete today, but how I learned to learn is what helps me keep up with the world around me. I have the study of mathematics to thank for that. Rigor and Relevance Rigor & Relevance Framework Relevance makes RIGOR possible, but only when trusting and respectful relationships among students, teachers, and staff are embedded in instruction. Relationships nurture both rigor and relevance. Rigor is... Article: Tips for Using Rigor, Relevance and Relationships. Rigor is... Work that requires students to work at high levels of Bloom's Taxonomy combined with application to the real world. 3 Misconceptions of Rigor •MORE – does not mean more rigorous. •DIFFICULT – increased difficulty does not mean increased rigor. •RIGID – “all assignments are due by… no exception.” RIGOR Relevance Why do I need to know this? Misconceptions of Relevance •COOL – relevance doesn’t exclusively mean what the students do for “fun” •EXCLUSIVE – relevance without rigor does not ensure success. Relevance Application Model 1. Knowledge in one discipline 2. Application within discipline 3. Application across disciplines 4. Application to real-world predictable situations 5. Application to real-world unpredictable situations Putting it all together Activity Rigor and Relevance Card Sort Six Questions All Students Must Be Able to Answer When seeking rigor, relevance, and relationships, all students should be able to answer the following questions: 1. What is the purpose of this lesson? 2. Why is this important to learn? 3. In what ways are you challenged to think in this lesson? 4. How will you apply, assess, or communicate what you've learned? 5. Do you know how good your work is and how you can improve it? 6. Do you feel respected by other students in this class? Mastery of Math Facts After students have reached conceptual understanding, the following fluencies are required by the CAS: • • • • • • • K K.OA.5 Add/subtract within 5 1 1.OA.6 Add/subtract within 10 2 2.OA.2 Add/subtract within 20 (know single digit products from memory) 2.NBT.5 Add/subtract within 100 3 3.0A.7 Multiply/divide within 100 (know single-digit products from memory). 3.NBT.2 Add/subtract within 1000 4 4.NBT.4 Add/subtract within 1,000,000 5 5.NBT-5 Multi-digit multiplication 6 6.NS.2,3 Multi-digit division Multi-digit decimal operations Common Formative Assessment • • • • • • • Kindergarten and First Grade Mathematics Interviews Math Fact Fluency - Reflex Conference Notes - anecdotal records and Math Reasoning Inventory Performance Tasks Mathematics Predictive Exams Math Review and Mental Math For Session 1: Please read Casebook pages 13-28 Cases 3, 4, 5 Lunch Developing Mathematical Ideas August 15 & 16, 2013 DMI is about developing YOUR mathematical understanding “If our goal is to create mathematically powerful children then we must also create mathematically powerful teachers.” --Lance Menster What is DMI? Developing Mathematical Ideas (DMI) is a professional development curriculum presented through a series of seminars. The premise of the DMI materials is that the art of teaching involves helping students move from where they are into the content to be learned. DMI Premises DMI seminars bring together teachers from kindergarten through middle grades to: • learn mathematics content • learn to recognize key mathematical ideas with which their students are grappling • learn how core mathematical ideas develop across the grades • learn how to continue learning about children and mathematics DMI is a Process • This year we are working through the first module: Building a System of Tens • Today and tomorrow we are working in the first 3 sessions Session 1: Analyzing Addition Strategies Session 2: Place Value and Multiplication Session 3: The Mathematics of Algorithms Session One: Building a System of Tens Student's Addition and Subtraction Strategies • • • Mathematical Goals for Session One I can use multiple strategies relying on the base ten structure and properties of operation to add and subtract multi-digit computations. I can use a logical visual or physical representation, such as a number line, base ten blocks, arrays, etc. to explain why my strategy works. I can express the same amount in different ways using powers of 10. For example, I can decompose numbers using the powers of 10 (100 is 100 ones, or ten tens, or one ten and 90 ones, etc.). Mental Math 57 + 24 Mental Math 83-56 • • • Mathematical Goals for Session One I can use multiple strategies relying on the base ten structure and properties of operation to add and subtract multi-digit computations. I can use a logical visual or physical representation, such as a number line, base ten blocks, arrays, etc. to explain why my strategy works. I can express the same amount in different ways using powers of 10. For example, I can decompose numbers using the powers of 10 (100 is 100 ones, or ten tens, or one ten and 90 ones, etc.). Second Grade Strategies 40 - 26 Seventh Grade Strategies 123 - 76 • • • Mathematical Goals for Session One I can use multiple strategies relying on the base ten structure and properties of operation to add and subtract multi-digit computations. I can use a logical visual or physical representation, such as a number line, base ten blocks, arrays, etc. to explain why my strategy works. I can express the same amount in different ways using powers of 10. For example, I can decompose numbers using the powers of 10 (100 is 100 ones, or ten tens, or one ten and 90 ones, etc.). Break Chapter 1 Case Discussion In your group, examine Focus Questions 3, 4, and 5. Use any manipulatives or chart paper you need to work through these questions. Small Group Discussion Whole Group Discussion • • • Mathematical Goals for Session One I can use multiple strategies relying on the base ten structure and properties of operation to add and subtract multi-digit computations. I can use a logical visual or physical representation, such as a number line, base ten blocks, arrays, etc. to explain why my strategy works. I can express the same amount in different ways using powers of 10. For example, I can decompose numbers using the powers of 10 (100 is 100 ones, or ten tens, or one ten and 90 ones, etc.). Math Activity: Close to 100 Game The object of the game is to create two 2-digit numbers whose sum is as close to 100 as possible. Each game has five rounds. At the end of five rounds the player with the lowest total score wins. • • • Mathematical Goals for Session One I can use multiple strategies relying on the base ten structure and properties of operation to add and subtract multi-digit computations. I can use a logical visual or physical representation, such as a number line, base ten blocks, arrays, etc. to explain why my strategy works. I can express the same amount in different ways using powers of 10. For example, I can decompose numbers using the powers of 10 (100 is 100 ones, or ten tens, or one ten and 90 ones, etc.). For Session 2, please be sure to read: Case studies 6, 7, & 10 Exit cards 1. What mathematical ideas did this session highlight for you? 2. What was this session like for you as a learner? 3. What burning questions do you have about this session? Session Two: Building a System of Tens The Base Ten Structure of Numbers August 16, 2013 Mathematical Goal for Session Two • The value of a number is determined by multiplying the value of each digit by the value of the place that it occupies and then summing. For whole numbers, the value of the place farthest to the right is 1; the value of every other place is 10 times the value of the place to its right. Math Activity Small Group: Representing Multiplication Whole-group Discussion: Sharing Representations Mathematical Goal for Session Two • The value of a number is determined by multiplying the value of each digit by the value of the place that it occupies and then summing. For whole numbers, the value of the place farthest to the right is 1; the value of every other place is 10 times the value of the place to its right. DVD: Interview with Three Students Mathematical Goal for Session Two • The value of a number is determined by multiplying the value of each digit by the value of the place that it occupies and then summing. For whole numbers, the value of the place farthest to the right is 1; the value of every other place is 10 times the value of the place to its right. Break Case Discussion Think about: 1. What is right about the student's thinking? 2. Where has the student's thinking gone awry? Small-Group: Ideas about the Number System Whole-Group: Number Lines Mathematical Goal for Session Two • The value of a number is determined by multiplying the value of each digit by the value of the place that it occupies and then summing. For whole numbers, the value of the place farthest to the right is 1; the value of every other place is 10 times the value of the place to its right. During your working lunch please be sure to read Case 14 (pages 65 - 70 Casebook). Lunch Session Three: Building a System of Tens Making Sense of Addition and Subtraction Algorithms Mathematical Goals for Sessions Three • Extend students’ knowledge of place value (ones, tens, hundreds) to solving addition and subtraction problems efficiently. • Understand how place value underlies the traditional algorithms for addition and subtraction. Whole Group: Addition and Subtraction Strategies Investigating addition strategies: • Creating Verbal Descriptions • Visual Representations • Story Context Small-Group: Addition and Subtraction Strategies Creating Subtraction Posters Gallery Walk: addition and subtraction strategies Mathematical Goals for Sessions Three • Extend students’ knowledge of place value (ones, tens, hundreds) to solving addition and subtraction problems efficiently. • Understand how place value underlies the traditional algorithms for addition and subtraction. Break! DVD: Addition and Subtraction Mathematical Goals for Sessions Three • Extend students’ knowledge of place value (ones, tens, hundreds) to solving addition and subtraction problems efficiently. • Understand how place value underlies the traditional algorithms for addition and subtraction. Small Group Discussion: Addition and Subtraction Algorithms Case 14 Focus Questions 3 and 4 Whole Group Discussion: Addition and Subtraction Algorithms • What is the same about the two strategies in case 14? • What is different about the two strategies? • What are the mathematical principles underlying each of the strategies the students use? Mathematical Goals for Sessions Three • Extend students’ knowledge of place value (ones, tens, hundreds) to solving addition and subtraction problems efficiently. • Understand how place value underlies the traditional algorithms for addition and subtraction. Task #1 - Read and discuss the article "Orchestrating Discussions" Task #2 - Do the Writing Assignment: A math interview Exit Cards.............. • What was important or significant to you in the mathematics discussed at this session? • What mathematics are you still wondering about from this session? • What do you want to tell us about how the seminar is working for you?