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Welcome This webinar will begin at 3:30

Welcome This webinar will begin at 3:30. While you are waiting, please: mute your sound. During the webinar, please: type all questions in the question/chat box in the go-to task pane on the right of your screen. This webinar will be available on the NCDPI Mathematics Wiki:

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Welcome This webinar will begin at 3:30

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  1. Welcome This webinar will begin at 3:30 • While you are waiting, please: mute your sound. • During the webinar, please: type all questions in the question/chat box in the go-to task pane on the right of your screen. This webinar will be available on the NCDPI Mathematics Wiki: http://maccss.ncdpi.wikispaces.net/Webinars

  2. Elementary Mathematics Webinar November 2013 Computational Fluency, Algorithms, and Mathematical Proficiency

  3. Due to the number of participants in attendance we ask for all questions about this webinar to be typed into the question box to the right of your screen. If you have other math questions not pertaining to this webinar please feel free to email: kitty.rutherford@dpi.nc.gov or denise.schulz@dpi.nc.gov

  4. Computational Fluency • Strategies vs. Algorithms • Drill and Practice • Memorization or Automaticity • Timed Tests

  5. Computational Fluency

  6. What does it mean to have computational fluency?

  7. Computational fluency refers to having efficient and accurate methods for computing. Students exhibit computational fluency when they demonstrate flexibility in the computational methods they choose, understand and can explain these methods, and produce accurate answers efficiently. NCTM, Principles and Standards for School Mathematics, pg. 152

  8. The computational methods that a student uses should be based on mathematical ideas that the student understands well, including the structure of the base-ten number system, properties of multiplication and division, and number relationships. NCTM, Principles and Standards for School Mathematics, pg. 152

  9. “Computational fluency entails bringing problem solving skills and understanding to computational problems.” Bass, Computational Fluency, Algorithms, and Mathematical Proficiency: One Mathematician’s Perspective, Teaching Children Mathematics, pgs. 322-327

  10. How do we develop computational fluency in students?

  11. Developing fluency requires a balance and connection between conceptual understanding and computation proficiency. • Computational methods that are over-practiced without understanding are forgotten or remembered incorrectly. • Understanding without fluency can inhibit the problem solving process. NCTM, Principles and Standards for School Mathematics, pg. 35

  12. Conceptual Understanding: • Important component of proficiency, along with factual knowledge and procedural facility • Essential component of the knowledge needed to deal with novel problems and settings NCTM, Principles and Standards for School Mathematics, pg. 20

  13. What Are the Expectations for Students?

  14. Strategies vs. Algorithms

  15. Why should we spend time teaching strategies instead of teaching only the standard algorithm?

  16. The CCSSM distinguish strategies from algorithms: • Computation strategy: purposeful manipulations that may be chosen for specific problems, may not have a fixed order, and may be aimed at converting one problem into another • Computation algorithm: a set of predefined steps applicable to a class of problems that gives the correct result in every case when the steps are carried out correctly Progressions for the Common Core State Standards in Mathematics, K-5 Number and Operations in Base Ten, pg. 3

  17. Building Strategies “Strategy” emphasizes that computation is being approached thoughtfully with an emphasis on student sense making. Fuson & Beckman, Standard Algorithms in the Common Core State Standards, NCSM Fall/Winter Journal, pgs. 14-30

  18. Instruction Should Focus On: • Strategies for computing with whole numbers so students develop flexibility and computational fluency • Development and discussion of strategies, so various “standard” algorithms arise naturally or can be introduced by the teacher as appropriate NCTM, Principles and Standards for School Mathematics, pg. 35

  19. Addition/Subtraction Strategies • One-More-Than/Two-More-Than • Facts with zero • Doubles • Near Doubles • Make 10 • Think-Addition • Build up through 10 • Back down through 10

  20. Multiplication/Division Strategies • Commutative Property • Doubles • Fives Facts • Helping Facts • Double and Double Again • Double and one more set • Near facts • Looking for patterns

  21. Students who used invented strategies before they learned standard algorithms demonstrated a better knowledge of base-ten concepts and could better extend their knowledge to new situations. • When students compute with strategies they invent or choose because they are meaningful, their learning tends to be robust—they are able to remember and apply their knowledge. NCTM, Principles and Standards for School Mathematics, pg. 86

  22. Common school practice has been to present a single algorithm for each operation. However, more than one efficient and accurate computational algorithm exists for each arithmetic operation. If given the opportunity, students naturally invent methods to compute that make sense to them. NCTM, Principles and Standards for School Mathematics, pg. 153

  23. “In mathematics, an algorithm is defined by its steps and not by the way those steps are recorded in writing. With this in mind, minor variations in methods of recording standard algorithms are acceptable.” Progressions for the Common Core State Standards in Mathematics, K-5 Number and Operations in Base Ten, pg. 13

  24. Kamii, Young Children Reinvent Arithmetic, pg 8

  25. 456 + 167 How would you solve this problem?

  26. Fuson & Beckman, Standard Algorithms in the Common Core State Standards, NCSM Fall/Winter Journal, pgs. 14-30

  27. The standard algorithms are especially powerful because they make essential use of the uniformity of the base-ten structure. Fuson & Beckman, Standard Algorithms in the Common Core State Standards, NCSM Journal, pgs. 14-30

  28. Students use strategies for addition and subtraction in grades K-3. Students are expected to fluently add and subtract whole numbers using the standard algorithm by the end of grade 4. Progressions for the Common Core State Standards in Mathematics, K-5 Number and Operations in Base Ten, pg. 3

  29. For students to become fluent in arithmetic computation, they must have efficient and accurate methods that are supported by an understanding of numbers and operations. “Standard” algorithms for arithmetic computation are one means of achieving this fluency. NCTM, Principles and Standards for School Mathematics, pg. 35

  30. Drill and Practice

  31. How does drill and practice impact a student’s ability to become proficient in math?

  32. We know quite a bit about helping students develop fact mastery, and it has little to do with quantity of drill or drill techniques. If appropriate development is undertaken in the primary grades, there is no reason that all children cannot master their facts by the end of grade 3. Van de Walle & Lovin, Teaching Student-Centered Mathematics Grades K-3, pg. 94

  33. The kinds of experiences teachers provide clearly play a major role in determining the extent and quality of students’ learning. • Students’ understanding can be built by actively engaging in tasks and experiences designed to deepen and connect their knowledge. • Procedural fluency and conceptual understanding can be developed through problem solving, reasoning, and argumentation. NCTM, Principles and Standards for School Mathematics, pg. 21

  34. Meaningful practice is necessary to develop fluency with basic number combinations and strategies with multi-digit numbers. • Practice should be purposeful and should focus on developing thinking strategies and a knowledge of number relationships rather than drill isolated facts. NCTM, Principles and Standards for School Mathematics, pg. 87

  35. Do not subject any student to fact drills unless the student has developed an efficient strategy for the facts included in the drill. Van de Walle & Lovin, Teaching Student-Centered Mathematics Grades K-3, pg. 117

  36. Memorizing facts with flashcards or through drill and practice on worksheets will not develop important relationships: • Double plus or minus • Working with the structure of five • Making tens • Using compensations • Using known facts Fosnot& Dolk, Constructing Number Sense, Addition, and Subtraction, pg. 98

  37. Drill can strengthen strategies with which students feel comfortable—ones they “own”—and will help to make these strategies increasingly automatic. Therefore, drill of strategies will allow students to use them with increased efficiency, even to the point of recalling the fact without being conscious of using a strategy. Drill without an efficient strategy present offers no assistance. Van de Walle & Lovin, Teaching Student-Centered Mathematics Grades K-3, pg. 117

  38. Memorization or Automaticity?

  39. How should students develop automaticity? Through drill, practice, and memorization, or through a focus on relationships?

  40. February 25, 2013 *H146-v-1* A BILL TO BE ENTITLED AN ACT TO REQUIRE THE STATE BOARD OF EDUCATION TO ENSURE INSTRUCTION IN CURSIVE WRITING AND MEMORIZATION OF MULTIPLICATION TABLES AS A PART OF THE BASIC EDUCATION PROGRAM. The General Assembly of North Carolina enacts: SECTION 1. G.S. 115C-81 is amended by adding new subsections to read: (l) Multiplication Tables. – The standard course of study shall include the requirement that students enrolled in public schools memorize multiplication tables to demonstrate competency in efficiently multiplying numbers." SECTION 2. This act is effective when it becomes law and applies beginning with the 2013-2014 school year.

  41. Teaching for Memorization: refers to committing the results of unrelated operations to memory so that thinking is unnecessary • Teaching for Automaticity: refers to answering facts automatically, in only a few seconds without counting, but thinking about the relationships within facts is critical Fosnot & Dolk, Constructing Number Sense, Addition, and Subtraction, pg. 98

  42. There are no “tricks” in math. • Understanding math makes it easier • Setting up opportunities for students to discover rules or generalizations allows them to exercise reasoning skills as they are making sense of math concepts. • O’Connell & SanGiovanni, Putting the Practices Into Action: Implementing the Common Core Standards for Mathematical Practice K-8, pg 124.

  43. It’s not wise to focus on learning basic facts at the same time children are initially studying an operation. A premature focus gives weight to rote memorization, instead of keeping the emphasis on developing understanding of a new idea. • When learning facts, children should build on what they already know and focus on strategies for computing. Burns , About Teaching Mathematics: A K-8 Resource, pg.191

  44. Memorization of basic facts usually refers to committing the results of unrelated operations to memory so that thinking is unnecessary. • Isolated additions and subtractions are practiced one after another as if there were no relationships among them. • The emphasis is on recalling the answers. • Children who struggle to commit basic facts to memory often believe that there are hundreds to be memorized because they have little or no understanding of the relationships among them. Fosnot& Dolk, Constructing Number Sense, Addition, and Subtraction, pg. 98

  45. When Relationships are the Focus: • Fewer facts to remember • Big ideas: compensation, hierarchical inclusion, part/whole relationships • Strategies for quickly finding answers when memory fails Fosnot & Dolk, Constructing Number Sense, Addition, and Subtraction, pg. 99

  46. Is Memorization Faster? • A comparison of two first grade classrooms • Classroom A focused on relationships and working toward automaticity • Classroom B memorized facts with drill sheets and flashcards • Students in Classroom A significantly outperformed the traditionally taught students in being able to produce correct answers to basic addition facts within three seconds (76% vs 55%) Fosnot & Dolk, Constructing Number Sense, Addition, and Subtraction, pg. 99

  47. Students who memorize facts or procedures without understanding often are not sure when or how to use what they know, and such learning is often quite fragile. NCTM, Principles and Standards for School Mathematics, pg. 20

  48. Timed Tests

  49. Is it necessary to assign a time limit for students to demonstrate knowledge of math facts?

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