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Computational Fluency, Algorithms, and
Mathematical Proficiency

• Elementary Mathematics Webinar
• November 2013

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• 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_at_dpi.nc.gov or
• denise.schulz_at_dpi.nc.gov

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

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• Computational Fluency

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• What does it mean to have computational fluency?

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• 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

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• 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

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• Computational fluency entails bringing problem
  solving skills and understanding to computational
  problems.
• Bass, Computational Fluency, Algorithms, and
  Mathematical Proficiency One Mathematicians
  Perspective, Teaching Children Mathematics, pgs.
  322-327

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• How do we develop computational fluency in
  students?

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• 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

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• 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

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What Are the Expectations
for Students?
Grade Level Common Core Standard Required Fluency
K K.OA. 5 Add/subtract within 5
1 1.OA.6 Add/subtract within 10
2 2.OA.2 2.NBT.5 Add/subtract within 20 Add/subtract within 100
3 3.OA.7 3.NBT.2 Multiply/divide within 100 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 6.NS.3 Multi-digit division Multi-digit decimal operations
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Kindergarten

• Understand addition, and understand subtraction.
• K.OA.A.5 Fluently add and subtract within 5

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First Grade

• Add and subtract within 20.
• 1.OA.C.5 Relate counting to addition and
  subtraction (e.g., by counting on 2 to add 2).
• 1.OA.C.6 Add and subtract within 20,
  demonstrating fluency for addition and
  subtraction within 10. Use strategies such as
  counting on making ten (e.g., 8 6 8 2 4
  10 4 14) decomposing a number leading to a
  ten (e.g., 13 4 13 3 1 10 1 9)
  using the relationship between addition and
  subtraction (e.g., knowing that 8 4 12, one
  knows 12 8 4) and creating equivalent but
  easier or known sums (e.g., adding 6 7 by
  creating the known equivalent 6 6 1 12 1
  13).

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Second Grade

• Add and subtract within 20.
• 2.OA.B.2 Fluently add and subtract within 20
  using mental strategies.2 By end of Grade 2, know
  from memory all sums of two one-digit numbers.
• Use place value understanding and properties of
  operations to add and subtract.
• 2.NBT.B.5 Fluently add and subtract within 100
  using strategies based on place value, properties
  of operations, and/or the relationship between
  addition and subtraction.

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Third Grade

• Multiply and divide within 100.
• 3.OA.C.7 Fluently multiply and divide within 100,
  using strategies such as the relationship between
  multiplication and division (e.g., knowing that 8
  5 40, one knows 40 5 8) or properties of
  operations. By the end of Grade 3, know from
  memory all products of two one-digit numbers.
• Use place value understanding and properties of
  operations to perform multi-digit arithmetic.¹
• 3.NBT.A.2 Fluently add and subtract within 1000
  using strategies and algorithms based on place
  value, properties of operations, and/or the
  relationship between addition and subtraction.

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Fourth Grade

• Use place value understanding and properties of
  operations to perform multi-digit arithmetic.
• 4.NBT.B.4 Fluently add and subtract multi-digit
  whole numbers using the standard algorithm.
• 4.NBT.B.5 Multiply a whole number of up to four
  digits by a one-digit whole number, and multiply
  two two-digit numbers, using strategies based on
  place value and the properties of operations.
  Illustrate and explain the calculation by using
  equations, rectangular arrays, and/or area
  models.
• 4.NBT.B.6 Find whole-number quotients and
  remainders with up to four-digit dividends and
  one-digit divisors, using strategies based on
  place value, the properties of operations, and/or
  the relationship between multiplication and
  division. Illustrate and explain the calculation
  by using equations, rectangular arrays, and/or
  area models.

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Fifth Grade

• Perform operations with multi-digit whole numbers
  and with decimals to hundredths.
• 5.NBT.B.5 Fluently multiply multi-digit whole
  numbers using the standard algorithm.
• 5.NBT.B.6 Find whole-number quotients of whole
  numbers with up to four-digit dividends and
  two-digit divisors, using strategies based on
  place value, the properties of operations, and/or
  the relationship between multiplication and
  division. Illustrate and explain the calculation
  by using equations, rectangular arrays, and/or
  area models.
• 5.NBT.B.7 Add, subtract, multiply, and divide
  decimals to hundredths, using concrete models or
  drawings and strategies based on place value,
  properties of operations, and/or the relationship
  between addition and subtraction relate the
  strategy to a written method and explain the
  reasoning used.

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• Strategies vs. Algorithms

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

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• 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

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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

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• 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

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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

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Multiplication/Division Strategies

• Commutative Property
• Doubles
• Fives Facts
• Helping Facts
• Double and Double Again
• Double and one more set
• Near facts
• Looking for patterns

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• 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 robustthey are able to
  remember and apply their knowledge.

NCTM, Principles and Standards for School
Mathematics, pg. 86
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• 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

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• 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

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Kamii, Young Children Reinvent Arithmetic, pg 8
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• Standard algorithms for base-ten computations
  with the four operations rely on decomposing
  numbers written in base-ten notation into
  base-ten units. The properties of operations then
  allow any multi-digit computation to be reduced
  to a collection of single-digit computations.
  These single-digit computations sometimes require
  the composition or decomposition of a base-ten
  unit.
• Fuson Beckman, Standard Algorithms in the
  Common Core State Standards, NCSM Journal, pgs.
  14-30

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• 456 167
• How would you solve this problem?

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

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• 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

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• 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

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• Drill and Practice

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• How does drill and practice impact a students
  ability to become proficient in math?

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x a b c d e f g
a h i j k l m n
b i o p q r s t
c j p u v w x y
d k q v z aa ab ac
e l r w aa ad ae af
f m s x ab ae ag ah
g n t y ac af ah ai
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• 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

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• 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

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• 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

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• 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

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• 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

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• Drill can strengthen strategies with which
  students feel comfortableones they ownand
  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

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• Memorization or Automaticity?

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

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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.
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• 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

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• 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.
• OConnell SanGiovanni, Putting the Practices
  Into Action Implementing the Common Core
  Standards for Mathematical Practice K-8, pg 124.

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• Its 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

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• 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

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• 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

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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

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• 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

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• Timed Tests

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

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• If teachers highly value speed in mathematics,
  what are the potential gains for student
  learning? The potential barriers?
• Seeley, Faster Isnt Smarter Messages about
  Math, Teaching, and Learning in the 21st Century,
  pg. 95

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Faster Isnt Smarter

• As part of a complete and balanced mathematics
  program it is useful to be able to add, subtract,
  multiply, and divide quickly.
• It is important to know basic addition and
  multiplication facts without having to figure
  them out or count on your fingers.
• Asking students to demonstrate this knowledge
  within an arbitrary time limit may actually
  interfere with their learning.
• Seeley, Faster Isnt Smarter Messages about
  Math, Teaching, and Learning in the 21st Century,
  pg. 93

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• Computational recall is important, but it is only
  part of a comprehensive mathematical background
  that includes more complex computation, and
  understanding of mathematical concepts, and the
  ability to think and reason to solve problems.
• Measuring one aspect of mathematicsfact
  recallusing timed tests is both flawed as an
  assessment approach and damaging to many
  students confidence and willingness to tackle
  new problems.
• Seeley, Faster Isnt Smarter Messages about
  Math, Teaching, and Learning in the 21st Century,
  pg. 93

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• Those who use timed tests believe that the tests
  help children learn basic facts. This makes NO
  instructional sense.
• Children who perform well under time pressure
  display their skills
• Children who have difficulty with skills, or who
  work more slowly, run the risk of reinforcing
  wrong learning under pressure
• Children can become fearful and negative toward
  their math learning.
• Burns , About Teaching Mathematics A K-8
  Resource, pg.191

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• Overemphasizing fast fact recall at the expense
  of problem solving and conceptual experiences
  gives students a distorted idea of the nature of
  mathematics and of their ability to do
  mathematics.
• Seeley, Faster Isnt Smarter Messages about
  Math, Teaching, and Learning in the 21st Century,
  pg. 95

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• Timed tests do not measure childrens
  understanding.
• An instructional emphasis on memorizing does not
  guarantee the needed attention to understanding
• Timed tests do not ensure that students will be
  able to use the facts in problem-solving
  situations.
• Timed tests convey to children that memorizing is
  the way to mathematical power, rather than
  learning to think and reason to figure out
  answers
• Burns , About Teaching Mathematics A K-8
  Resource, pg.192 

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• Timed Tests
• Cannot promote reasoned approaches to fact
  mastery
• Will produce few long-lasting results
• Reward few
• Punish many
• Should generally be avoided
• Van de Walle Lovin, Teaching Student-Centered
  Mathematics Grades K-3, pg. 119

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In Conclusion

• After looking at research from experts in the
  field, reflect on your practices.
• Computational Fluency
• Strategies vs. Algorithms
• Drill and Practice
• Memorization or Automaticity
• Timed Tests
• Does the research affirm your current teaching
  practices?
• Are there practices you would like to revisit?

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• Fluency refers to having efficient, accurate,
  and generalizable methods (algorithms) for
  computing that are based on well-understood
  properties and number relationships.
• NCTM, Principles and Standards for School
  Mathematics, pg. 144

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• What questions do
• you have?

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maccss.ncdpi.wikispaces.net
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Contact Information

• Kitty Rutherford kitty.rutherford_at_dpi.nc.gov

Denise Schulz denise.schulz_at_dpi.nc.gov
Website maccss.ncdpi.wikispaces.net
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• For all you do for our students!

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Algorithms

• Standard algorithm
• for each operation, there is a particular
  mathematical approach that is based on place
  value and properties of operations
• an implementation of the particular mathematical
  approach is called the standard algorithm for
  that operation

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• Students who understand the structure of numbers
  and the relationship among numbers can work with
  them flexibly
• NCTM, Principles and Standards for School
  Mathematics, pg. 149

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• Once children understand the process of
  multiplication and can represent multiplication
  situations with symbols, they are ready to focus
  on the number patterns and relationships that
  will help them internalize the basic
  multiplication facts. They should spend much of
  their time exploring and recording multiplication
  patterns. The search of patterns and
  relationships will help children learn
  multiplication facts in a much more powerful way
  than they would by simply memorizing the times
  table.(Richardson)

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