Building conceptual knowledge in mathematics: grades 3 - 5

1. Building conceptual knowledge in mathematics: grades 3 - 5 Fluency Strategies December 7, 2012 Presenters: Andrea Tamarazio & Steven Graser Erie 1 BOCES

2. MAD MINUTE! • READY, SET, GO • When asked to begin, take one minute to complete as many questions as you can

3. Anticipatory guide • Read each statement • Circle True / False • Write an sentence / comment about each statement • Complete Frayer Model for “fluency”

4. SPRINT!! • READY, SET, GO! • When asked to begin, take one minute to complete as many questions as you can Sprint Video

5. Sprint vs. Mad Minute • Take a minute to complete the graphic organizer • Likes and Dislikes of each activity • Share Out

6. MATHEMATICAL SHIFTS

7. MATHEMATICAL PRACTICES • Make sense of problems and persevere in solving them • Reason abstractly and quantitatively • Construct viable arguments and 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

8. Fluency Practice (common Core Inc.) Fluency in designed to promote automaticity by engaging students in practice in ways that get their adrenaline flowing. Automaticity is critical so students avoid using up to many of their attention resources with lower level skills when they are addressing lower level problems. Computational foundations enables deep understanding.

9. Sample Task • What fluencies do students need to solve this problem? See Handout for problem

10. Common Core Standards • Professional Development Needs • Curriculum • Assessments • Instruction (Teacher Capacity) • What Common Core aligned PD has your district provided in regards to the items listed above?

11. Math Fluency Experts agree that the ability to recall basic math facts fluently is necessary for students to attain higher-order math skills. *Drop off in student performance in the middle and high school years can be attributed to math instruction at the elementary level. Many students memorize facts and can pass elementary assessments, however the lack of computational fluency becomes apparent in the secondary grades.

12. Warm up: Mental Math Activity • 7 Questions • Solve mentally • Explain the process used to solve each problem

13. Jigsaw Activity • In small groups, read assigned articles • Complete Graphic Organizer for your article

14. Mental Math • Mental Math is one component of Computational Fluency • Understanding Basic Number Combinations • Number Bonds • Dot Plates • Understanding Base Ten Number System • 10 times more, 10 times less • In the number 220, the 2 in the hundreds place is 10 times greater than the 2 in the tens place • Recognize and Use related problems • Fact Families • Inverse Functions; solve for missing addend, subtrahend, and minuend. • Understand and Use Properties • Commutative and Associative • Making Connections between operations • Inverse Functions

15. Math models • Place Value & Standard Algorithm • Bundles • Place Value Chart • Base Ten Blocks • Money • Number Disks with Place Value Chart • Fractions • Number Line • Area Model • Multiplication • Array • Area Model • Rekenrek (Number Rack) • Word Problems • Bar Diagram

16. Fluency vs. Memorization • Memorization • Single procedure and / or recall • Fluency • Understanding of operations and their relationship to each other (inverse function; fact families) • Number Relationships (compatible numbers – fives and tens) • Base Ten System • For Example: recognizing the relationship between 5 + 9 and 50 + 90

17. Computational Fluency Computational Fluency includes three ideas: efficiency, accuracy, and flexibility • Efficiency implies that the student does not get bogged down in too many steps or lose track of the logic of the strategy. An efficient strategy is one that the student can carry out easily, keeping track of subproblems and making use of intermediate results to solve the problem. • Accuracy depends on several aspects of the problem-solving process, among them careful recording, knowledge of number facts and other important number relationships, and double-checking results. • Flexibility requires the knowledge of more than one approach to solving a particular kind of problem, such as two-digit multiplication. Students need to be flexible in order to choose an appropriate strategy for the problem at hand, and also to use one method to solve a problem and another method to double-check the results.

18. Assessment of Computational Fluency • The student draws on basic facts and number relationships • Properties • Student uses the structure of the base ten system • 2 + 5 = 7; 20 + 50 = 70; 200 + 500 = 700 • Student recognizes related problems • Students relationships among related operations • Student knows what each number and numeral means in a problem • Student can explain the steps he or she uses to solve even recall problems • Same student explains twice • Student can clearly record and track his or her procedure • Student may have multiple approaches to solve a problem

19. “Drill & Kill” ≠ Computational Fluency • Memorization of math facts is not the same as understanding number concepts • Regardless of time and energy spent with “drill & kill”, some students will not master facts • Memory • Memory problem interferes with ability to retrieve basic facts quickly • Short Term Retention • Cognitive • Understanding relationship between numbers • Solving word problems • Understanding number system & effective counting strategies • Visual / Spatial • Misalignment of numerals in a column • Problems with Place Value

20. Common Core Fluency

21. FACT FLUENCY INSTRUCTIONAL FLOW • Number Relationships & Concepts • 40 – 50% of Instructional Time • Contextualizing math fact problems • Creating arrays and drawing problems • Decomposing or using known facts • Using compensation strategies • Fact Strategy Development • 30 - 40% of Instructional Time • Doubles, Near Doubles, Combinations to 10 • Drill • 15 – 20% of Instructional Time • Fact Fluency related games and flash cards • Fluency Assessment • Less than 3% of Instructional Time • Timed Test

22. CCLS Fluency Recommendation Distribution of Instructional Minutes

23. Number Relationships & Concepts • Contextualize Math Facts • Reading, Writing, and Symbolizing Math Problems • High Number Toss Game • Creating Arrays and Drawing Pictures (word to picture; number to picture; picture to word or number) • Place Value Literacy • Decomposing Numbers & Listing their Values • Number Line Up • Build a Number

24. Fluency Activities • Adding Mentally • Whole Numbers, Fractions & Decimals • 1, 2, 3, sit on 20 • This can be modified for multiple grade levels – skip counting, multiplication facts, etc • Take the 1 Out • Fluency Counting • This can be modified for multiple grade levels • Skip Counting by Fractions • White Board Equivalent Fractions • Written or Oral

25. Fact Strategy Development (Choral White Board) • One-More-Than & Two-More-Than Facts • Facts with Zero • Doubles / Near Doubles (Doubles Plus 1, Doubles Plus 2) • Compensation (Plus 1, Minus 1; Plus 2 Minus 2) • Compatible Numbers (Combinations to 5 or 10)

26. Drills • Around the World • Math Fact Race • Math Fact War • BINGO • I Have You Have

27. Activity Share out • Take time to create activities to use in your classroom • What do you have in your classroom that you would like to share? • Activities • Strategies

28. Questions / Comments • If you have any questions or comments, please feel free to contact us. Andrea Tamarazio atamarazio@e1b.org Steven Graser sgraser@e1b.org