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Balanced Math K-8 Framework October 2010 Saville-Brock 2010 1 Norms • Be prompt Return from breaks on time • Be respectful Put cell phones on silent. • Be a polite and positive participant Speak in a normal tone of voice, and listen attentively. • Be a problem solver Saville-Brock 2010 2 Who are these guys? This is who I am by the numbers….. 1985, 25, 155551 Saville-Brock 2010 3 By the numbers……. • A significant year in my life was 1985 • I’ve spent 25 wonderful years with MNPS. • My favorite palindrome is 155551. Saville-Brock 2010 4 Meet your neighbor by the numbers… • Select 5 numbers that are meaningful to you that will help someone understand who you are. • Then write a sentence or question for each number, leaving a blank line where the number should go. • Share you numbers and sentences with your neighbor. See if he or she can match the correct number to the line. Saville-Brock 2010 5 Activity: Group and Label • Write each of your numbers on a post it. One number per post it. • Place all of your numbers from your table in the middle and eliminate any duplicates. • Then group your numbers and label them according to some common characteristics. Then turn you labels over. • Visit another table and try to figure out there groupings. • Discuss how you can use this activity in your own classroom. Saville-Brock 2010 6 Why a Balanced Approach? Saville-Brock 2010 7 Why a Balanced Approach? The National Mathematics Advisory Panel States… “The mutually reinforcing benefits of conceptual understanding, procedural fluency, and automatic, i.e. quick recall of facts.” Saville-Brock 2010 8 Conceptual Understanding Computational Problem Fluency Solving Saville-Brock 2010 9 Why a Balanced Approach? Traditional Approach to Teaching Math • Large group instruction • All students work on the same level • Primarily instruction and practice from text book • Emphasis on paper and pencil work • One correct answer • Individual work Saville-Brock 2010 10 Why a Balanced Approach? Effectiveness of the Traditional Approach • Successful for some students • Even less successful for struggling students • Encourages emphasis on computation skills • Little opportunity for communication • More emphasis on evaluation, rather than assessment for learning Saville-Brock 2010 11 Why a Balanced Approach? The Statistics • Fifth graders spend more than 90 percent of their time in their seats listening to the teacher or working alone and only about 7 percent of their time working in groups. • The average fifth grader received five times as much instruction in basic skills as instruction focused on problem solving or reasoning. The ratio was 10:1 in first and third grades. From a study published by Robert Pianta et al. The Global Achievement Gap by Tony Wagner. Saville-Brock 2010 12 Why a Balanced Approach? To teach math more effectively, teachers must… • reach students at all levels of achievement • provide diverse methods of learning • allow more opportunities for observation and communication by students • encourage active engagement by students Saville-Brock 2010 13 Teaching Procedures What the Research and Experts Say According to the National Mathematics Advisory Panel, students should understand key concepts achieve automaticity as appropriate (e.g., with addition facts) develop flexible, accurate, and automatic execution of standard algorithms Computational Proficiency is dependent on automatic recall of math facts a solid understanding of core concepts National Mathematics Advisory Panel. Foundations for Success: The Final Report of the National Mathematics Advisory Panel, U.S. Department of Education: Washington, DC, 2008. Saville-Brock 2010 14 Conceptual vs. Procedural Striking the Balance Vocabulary Four development operations Manipulatives Standard- algorithms Multiple representations Step-by-step methods Pictures Math facts Real-life contexts Release of responsibility Practice Saville-Brock 2010 15 Day 15 min 40 minutes 5 minutes 1 Math Mental Concept Lesson Closure/ Review Math Problem solving, manipulatives, small groups, centers math journals 2 Math Mental Concept Lesson Closure Review Math Problem solving, manipulatives, small groups, centers 3 Math Mental Concept Lesson Closure Review Math Problem solving, small groups , centers 4 Math Mental Concept Lesson Closure Review Math Problem solving, manipulatives, small groups, centers 5 Math Facts Practice/ Problem-based activities, centers, games, small groups Math Review Quiz 6 Math Mental Concept Lesson Closure Review Math Problem solving, manipulatives, small groups, centers 7 Math Mental Concept Lesson Closure Review Math Problem solving, manipulatives, small groups, centers 8 Math Mental Concept Lesson Closure Review Math Problem solving, manipulatives, small groups, centers 9 Math Mental Concept Lesson Closure Review Math Problem solving, manipulatives, small groups, centers 10 Assessment/Math Assessment Saville-Brock 2010 16 Review Quiz Saville-Brock 2010 17 Saville-Brock 2010 18 •One more/one less, before/after, a given number •Counting by twos, fives, tens •Doubles •Fact families •Measurement (time, money, calendar, feet, etc.) •Math Vocabulary/Math Word Wall •Addition &/ or Subtraction Facts •Estimation •Math Around the Room Saville-Brock 2010 19 Math Review and Mental Math- Rally Coach Partners Take turns, one solving a problem while the other coaches. Each pair needs one set of problems and one pencil. Person A Person B • Partner A solves the first • Partner B watches and problem. Talking out listens, checks, coaches their thinking. if necessary and praises. • Partner A watches and • Partner B solves the listens, checks, coaches next problem. Talking if necessary, and praises. out their thinking. Saville-Brock 2010 20 Saville-Brock 2010 21 Balanced Math What is Conceptual Understanding? It is the underlying knowledge behind the concept Teaching techniques include: concrete models, vocabulary connections, problem solving, real-life applications, etc. Conceptual understanding is important for two main reasons: In order to apply knowledge to new situations Subsequent math concepts rely on students’ ability to understand the current concept Saville-Brock 2010 22 Conceptual Understanding Saville-Brock 2010 23 Vocabulary Development Content vocabulary words are used within the subject matter you are teaching (e.g., fractions, decimals). Academic vocabulary is the higher-level language needed to understand the content (e.g., analyze, identify). Saville-Brock 2010 24 Vocabulary Development Word Wall Bulletin board display of key vocabulary or concept words Students can be involved in their creation Include illustrations, photos, examples Refer to the words often during instruction http://www.kealakehe.k12.hi.us/amilwordsmedia/FWWordWall/FWWordW all-Images/21.jpg Saville-Brock 2010 25 Vocabulary Development Developing Math Vocabulary to Make Concepts Accessible Identify Decide on Engage Your Vocabulary Teaching Students! Words Strategies Saville-Brock 2010 26 Group and Label Activity Purpose: Group and Label asks students to conceptualize their way to deep understanding by organizing mathematical data into meaningful categories. Students analyze a collection of mathematical information, group the items into categories, and label each category in a way that explains why the items go together. Finally, students use their labeled groups to generate a set of hypotheses or generalizations, which they revisit periodically and refine in light of new Learning. Saville-Brock 2010 27 Saville-Brock 2010 28 Problem Solving A Starting Point for Problem Solving Many children think of mathematics as a subject that relies on their memorizing facts and practicing skills. But, the true test of children’s success in mathematics is when they can’t remember a fact or have forgotten a skill, they are able to think, reason, and solve problems and make sense of mathematical ideas. Marilyn Burns Saville-Brock 2010 29 Thought is STRATEGIC… Therefore, we need a classroom that models instructional practices and strategies that enhance students’ ability to think. We can't solve problems by using the same kind of thinking we used when we created them. -- Albert Einstein Saville-Brock 2010 30 Teaching through Problem Solving Saville-Brock 2010 31 How Much Does Matt Weigh? What We Know……. • Matt’s head weighs 15 lbs • His torso weighs as much as his head and legs together • His legs weigh as much as his head and half his torso Saville-Brock 2010 32 Using Manipulatives Saville-Brock 2010 33 Using Manipulatives What Are Manipulatives? Manipulatives are colorful, intriguing materials constructed to illustrate and model mathematical ideas and relationships and are designed to be used by students in all grades (Burns and Sibley 2008). Saville-Brock 2010 34 Using Manipulatives Why Use Manipulatives? Using Manipulatives, “helps students understand the mathematical concepts and processes, increases thinking flexibility, provides tools for problem-solving, and can reduce math anxiety for some students. (The Education Alliance 2006). Saville-Brock 2010 35 Using Manipulatives Classroom Management Tips for Manipulatives • Organize manipulatives. • Set expectations. • Give students time to • Model examples. practice. Saville-Brock 2010 36 Laura went shopping Saville-Brock 2010 37 Using Manipulatives Phases of Instruction and Learning C-R-A -Moving with Math Saville-Brock 2010 38 Using Manipulatives Concrete Hands-on teaching method using manipulatives such as: • Algebra Tiles • Toothpicks • Counting blocks • Unifix Cubes • Cuisenaire Rods • Food • Algeblocks • Balance scales • Hands-On Equations Saville-Brock 2010 39 Using Manipulatives Representational Uses: • Pictures • Tally marks • Diagrams • Drawings • Maps • Graphs • Charts Relates directly to the manipulatives Saville-Brock 2010 40 Using Manipulatives Abstract A teaching method using written words and symbols. • Graphs (meaning) • Matrices • Estimation • Predictions • Tables (ex. slope) • Oral explanations • Systems of equations Saville-Brock 2010 41 Differentiating to Meet Students Needs Mathematical Thinking Tasks Think about the enduring understandings. – What do you want students to learn during the activity? – All students should work toward the same objective. Objective: Students will demonstrate understanding of the relationship between factor, product, and multiple. Saville-Brock 2010 42 Differentiating to Meet Students Needs • Create the activity you want your on-grade-level students to complete. 1) On graph paper, draw an array to represent the multiplication problem 3 x 4. What is the product? How do you know? 2) Make an array with a different length and width, but with the same product as 3 x 4. 3) Write a multiplication number sentence to represent your array. 4) For both arrays, complete the sentences below by writing the number in the blank. This array has ____ rows by _____ columns. This array has _____groups of _____. 5) List the factors shown in both arrays. 6) The product is also called a multiple. How is the multiple related to its factors? 7) List some other multiples and their factors. Saville-Brock 2010 43 Differentiating to Meet Students Needs • Revise the activity for your English language learners. 4 1) An array is putting objects in equal rows and 2 columns to make a rectangle. On graph paper, draw an array to show the multiplication problem 3 x 4. array 2) The product is the answer to a multiplication problem. What is the product of 3 x 4? How do you know? 3x4= 3) Draw an array with a different length and width. Use product the same number of squares, so the array has the same product as 3 x 4. length width 4) An example of a multiplication number sentence is 2 x 4 = 8. Write a multiplication number sentence for your array? Saville-Brock 2010 44 Differentiating to Meet Students Needs Revise the activity for your English language learners. column 5) For both arrays, complete the sentences by writing the number in the blank. Row This array has ___ rows by ___ columns. This array has groups of ____. 6) The factors are the number of objects in a row and a column of the array. Write the factors shown in the row of columns of both arrays. 7) The product is also called a multiple. How is the multiple related to its factors. You can show your answer with pictures. 8) Draw arrays showing other multiples and their factors. Saville-Brock 2010 45 Differentiating to Meet Students Needs • Revise the activity for your below-grade-level students. 1) Write a number sentence with repeated addition to represent this array. 2) Write a multiplication number sentence to represent the array. 3) What is the product of the number sentence? How do you know? 4) For the array, complete the sentences by writing the number in the blank. This array has ____ rows by _____ columns. This array has _____groups of _____. 5) What are the factors of the product shown in this array? 6) The product is also called a multiple. How is the multiple related to its factors 7) Using manipulatives, make an array with a different factors, but with the same product as 3 x 4. Draw this array. Name the factors and the multiple. Saville-Brock 2010 46 Differentiating to Meet Students Needs • Revise the activity for your above-grade-level students. 1) On graph paper, draw an array to represent the multiplication problem 6 x 4. What is the product? How do you know? 2) Make at least two arrays with different factors, but with the same product as 6 x 4. 3) Write multiplication number sentences to represent your arrays. 4) List the factors shown in all of your arrays. Does the product or multiple have any other factors, not shown in your arrays? 5) How is the multiple related to its factors? 6) Write a real-life scenario to represent one of your arrays. What are the factors, product, and multiple in your scenario? Saville-Brock 2010 47 Game Time! Concept-Based Games Saville-Brock 2010 48 • Math Journals • Reflect individually and with whole-group • Record representation of key concepts • Opportunity to use math vocabulary/word wall in context • Pose questions • Making their thinking visible! Saville-Brock 2010 49 Three-Way-Tie 1. Along each side of the triangle, the student writes a sentence that clearly relates the two terms. 2. Have students use their three sentences to develop a brief summary of the concept. 3. Allow students time to share and explain what they wrote on their organizers. PRODUCT, FACTORS, MULIPLES Saville-Brock 2010 50 Saville-Brock 2010 51 FORMATIVE—checking on learning as students progress SUMMATIVE—checking on learning at the end of the learning experience Saville-Brock 2010 52 Assessment “When the cook tastes the soup, that’s formative; when the guests taste the soup, that’s summative.” (Stake, 2005) Saville-Brock 2010 53 Bringing It Altogether Saville-Brock 2010 54 Bringing It Altogether Mix-Freeze-Share • What resources do you use to teach mathematics? • How do you determine what to teach? • What is the process you use to write your lesson plans? Saville-Brock 2010 55 Bringing It Altogether Things to Consider • Amount of instructional time for each phase • Meeting students’ needs • Grouping of students • Strategies for engaging students • Differentiation strategies to be used • How are students’ assessed on the TCAP? Saville-Brock 2010 56 Balanced Math Lesson • In a grade level group, develop a Balanced Math Lesson that you could use in your class this year. • We will share with the group in a gallery walk at the end of the day. • On chart paper, include your math review/mental math, strategies and methods for teaching the concept, and the closure. Saville-Brock 2010 57