Curriculum Standards

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Basic Facts Big Idea or Mindless Task?

Linda Gojak President, NCSM
Copies of this presentation are available at
www.jcu.edu/cmsett
www.ncsmonline.org
lgojak_at_jcu.edu
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Three Phases in Learning Basic Facts

• Using Counting Strategies
• Use objects to count to determine the answer
• fingers, blocks, tally marks, counters
• Using Reasoning Strategies
• Use known information to determine the answer of
  an unknown combination
• 7 8 15 because (7 7) 1 14 1 15
• Mastery
• Quick recall and accuracy

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Two Points of View

• Conventional Approach
• Mastery grows out of
• memorizing individual
• facts by rote through
• repeated practice and
• reinforcement.

• Number Sense
• Mastery that underlies
• computational fluency
• grows out of discovering
• the patterns and
• relationships that
• interconnect the basic facts.

Baroody, TCM 8/2006
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Memorize this sequence of numbers
25811141720

258 111 417 20
2 5 8 11 14 17 20
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Two Points of View

• Conventional Approach
• Difficulties with
• mastering facts are due to
• deficits inherent with the
• learner.

• Number Sense
• Difficulties are due to
• deficits inherent in
• conventional instruction.

Baroody, TCM 8/2006
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Peas and Carrots
Tonights dinner includes peas and carrots. You
HATE peas and carrots! In order to delay eating
the peas and carrots, you count them! You find
out you have 12 vegetables on your plate. How
many peas could you have? How many carrots could
you have?
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Reasonable Expectation

• By the end of the K-2 Program Demonstrate
  fluency in addition and subtraction facts with
  addend through 9
• By the end of the 3-4 program Demonstrate
  fluency in multiplication and division facts with
  factors through 10

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Biggest Idea!!!!!!

• Drill of inefficient methods does not produce
  mastery!

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What is mastery?

• According to VandeWalle, mastery of a basic fact
  means that a child can give a quick response (in
  about 3 seconds) without resorting to inefficient
  means such as counting.
• All children are able to master the basic facts
  if they can construct efficient mental models
  that will help them.

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Big Ideas

• Number relationships provide the foundation for
  strategies that help students remember the facts.

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Big Idea Number 1 Composition Numbers can be
put together in different ways

• How many ways can you stack 5 cubes using only 2
  colors?
• What is 9 8? How do you know this is true?
• 9 8 9 1 7 10 7 17

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Big Idea 2 DecompositionNumbers can be
taken apart in different ways
Empty the Bowl Game
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Empty the Bowl

• 1 9
• 2 10
• 3 11
• 4 12
• 5
• 6
• 7
• 8

• You need
• Small bowls
• 12 tiles per pair of students
• One die per pair of students
• List as shown to the right

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One person rolls the die and removes that number
of tiles from the bowl. The other player records
that number on a piece of paper. Continue until
all of the tiles have been removed from the
bowl. Example 3 4 2 5 14 Players switch
jobs. Record the number on each roll until all
of the tiles have been removed. Play the game 5
times. Be sure to record your rolls for each
game. When you have played 5 times record the
number of rolls it took to empty the bowl on the
class chart.
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Questions What is the most rolls it took to
empty the bowl? What is the least number of rolls
it took to empty the bowl? What number(s) of
rolls came up most often? Could you empty the
bowl in one roll? Why or why not? What is the
greatest number of rolls it could take to empty
the bowl?
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Big Idea 3 Commutivity
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Big Idea 4 Relationships
More Than Less Than Same As (Equality)
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Big Idea 4 Relationships
Addition
Multiplication
Division
Subtraction
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Levels of Understanding/Skill

• From bottom to top
• The big base -- conceptual understanding of what
  the operation means and number sense.
• The middle layer -- strategies for learning the
  facts
• The top layer -- the ultimate goal of instant
  recall. The child can answer, "What is 7 8?"
  as quickly as if asked, "What is your name?"

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Three Steps for Mastery

• Help children develop a strong understanding of
  the operations and of number relationships.
• Develop efficient strategies for fact retrieval
  through practice.
• Then provide drill in the use and selection of
  those strategies once they have been developed.

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Materials to develop number sense and
relationships

• Dominoes
• Dot Cards/Plates
• Number Cards
• Dice
• Ten Frames
• Calculators

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Development of Strategies

• An efficient strategy is one that can be done
  mentally and quickly. Counting is not efficient.
• The use of strategies is not new. Research has
  been going on since 1930s.
• You may think you just know the facts but it is
  because you have developed such efficient
  strategies for retrieval and they are now
  automatic.
• For your students to develop efficient
  strategies,you must have command of as many good
  strategies as possible even if you have never
  used them. This will help you to recognize your
  students strategies.

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Hints for Helping Students to Master Basic Facts

• Avoid premature drill
• Practice strategy selection/retrieval
• Make strategies explicit
• Drill established strategies
• games
• manipulatives
• worded problems
• Individualize

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Addition Basic Facts
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Experiences to support ONE MORE THAN TWO MORE
THAN
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Strategies for Addition Facts
One more thantwo more than
3 1 3 2 8 1 8 2
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Silly Stories Adding Zero
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Strategies for Addition Facts
Facts with Zero
2 0 0 2 5 0 0 5
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• Experiences for Doubles
• Seeing Double
• Finding Doubles
• Calculator

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Strategies for Addition Facts
Doubles
2 2 6 6 7 7 9 9
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• Experiences for Doubles Plus One and Doubles Plus
  2
• Dice
• Roll the die, double and add one to the
  number, that is your score
• Card Games
• Dot Cards
• Number Cards
• Match

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Strategies for Addition Facts
Near Doubles
2 3 6 5 6 7 7 8
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• Experiencing Combinations for Ten
• Tens Frames
• Adding 9
• Adding 8

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Strategies for Addition Facts
Combinations for ten
3 7 6 4 1 9 8 2
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Strategies for Addition Facts
The final few.
5 3 6 3 7 3 4 7 5 7

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Strategies For Addition Facts

• 1.One-More-Than and Two-More-Than Facts
• 2. Facts with zero
• 3. Doubles
• 4. Near Doubles
• 5. Make-Ten-Facts
• 6. A Generic Task
• 7. Doubles-plus-two or Two-Apart Facts
• 8. Make Ten Extended
• 9. Counting On
• 10. Ten Frames Facts

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Your Turn

• 6 7
• 4 9
• 6 6

7 8 5 4 2 7
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Subtraction Facts
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Strategies for Subtraction Facts
Think Addition Sums to 10
8 - 5

Think 5 and how many more make 8?
The total is 8. how many are on The other side?
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Strategies for Subtraction Facts
Think Addition
13 - 6

How many are under the rectangle?
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Strategies for Subtraction Facts
Build to 10

14 - 8

• Think Start with
• How much to get
• to 10? How much
• more to get to 14?

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Strategies for Subtraction Facts
Back to 10

16 - 7
Think Start with 16 -- take off 6 to get to
10. Take off 1 more.
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Strategies for Subtraction Facts Extend Think
Addition
7
?
7
8
4
4
12
?
15
14
9
5
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Strategies for Subtraction Facts

• Subtraction as Think-Addition
• Subtraction Facts with Sums to 10
• Build up to 10
• Back Down to 10
• Extend Think Addition

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Number Relationships

• Number Relationships play a significant role in
  fact mastery. To solve 6x7 it is efficient to
  think of 5x7 and 7 more. The efficiency can be
  lost if they have to count on from 35. Children
  with number sense can think of 35 and 5 more is
  40 and 2 more is 42.
• The commutative (turnaround) properties for
  addition and multiplication reduce the number of
  addition and multiplication facts from 100 to 55.
• Without a strong command of number relationships
  and concepts, students will not master basic
  facts.

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Multiplication Basic Facts
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Candy Boxes
You have just gone to work for the Charming
Chocolate Company. Your job is to figure out how
to put the candy into rectangular boxes. You
cannot stack the pieces on top of each other (or
they will melt.) You have 12 pieces of candy.
Show how you can arrange the candy in the boxes.
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Strategies for Multiplication Facts
Doubles
2 x 3 33 2 x 9 99 6 x 2 2 x 6 66
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Strategies for Multiplication Facts
The Fives.

• Nickels
• Clocks
• Hundreds Chart
• Patterns
• Jump Rope

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Strategies for Multiplication Facts
Zeros and Ones
Silly Stories Emphasize the meaning of
multiplication
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Strategies for Multiplication Facts
Patterns with Nines
0 x 9 1 x 9 2 x 9 3 x 9 4 x 9
5 x 9 6 x 9 7 x 9 8 x 9 9 x 9
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Strategies for Multiplication Facts
Square Numbers
3 x 3
6 x 6
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Strategies for Multiplication Facts
Halving and Doubling
6 x 8
6 x 8
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Your Turn

• 5 x 3
• 8 x 7
• 6 x 6
• 9 x 6

• 0 x 8
• 3 x 8
• 7 x 9
• 1 x 7

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Division Facts
24 6
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Use Factor Families

14 7 x 2 2 x 7
24
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3 x 8 4 x 6 6 x 4 8 x 3
7
?
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Games
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Make Strategies Explicit in the Classroom

• When students suggest strategies be sure everyone
  understands them. Explore other facts that work
  with the new strategy.
• Provide lots of opportunities to make a strategy
  their own. Write new strategies on a poster.
• No student should be forced to adopt someone
  elses strategy but every student should
  understand all strategies discussed.

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Review

• Make strategies explicit in the classroom.
• Drill established strategies
• Individualize
• Practice strategy selection

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Effective Drill

• Drill without strategy development and number
  sense has repeatedly been deemed as ineffective.
• It is unreasonable to expect all children to be
  comfortable with the same strategies. Listen for
  strategies used by individuals.
• A child that has not mastered addition facts is
  not ready for subtraction practice.

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Fact Remediation

• Recognize that more drill does not work!
• Inventory the known and unknown facts for each
  student in need.
• Diagnose strengths and weaknesses.
• Build in Success!!!!

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Timed Tests???

• Teachers that use timed tests believe that the
  tests help the kids 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 practices under pressure. Also
  they can become fearful about or negative toward
  their mathematical learning!
• - Marilyn Burns

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Facts are not a Barrier to Good Mathematics

• Mathematics is about reasoning and patterns and
  making sense of things. Mathematics is problem
  solving. There is no reason to exclude children
  who have not mastered basic facts from real
  mathematics experiences.
• The experts agree that the calculator will not
  impede basic fact mastery. On the contrary, the
  more students use the calculator, the more
  proficient they will be. (VandeWalle)

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