Tier II Instruction for Whole-Number Concepts in Grades K-2

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Tier II Instruction for Whole-Number Concepts
in Grades K-2

• Matt Hoskins, NCDPI
• Tania Rollins, Ashe County Schools
• Denise Schulz, NCDPI

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WelcomeWhos in the Room?
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Advanced Organizer
Counting and Cardinality Strengthening
connections between quantity, language, and
symbols
TIER II Features of supplemental instruction
Unitizing Strengthening understanding of the
base-ten number system
Computation Progression of strategy use and
structures
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Layering of Support
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Tier II Mathematics Instruction

• Should focus intensely on in-depth treatment of
  whole numbers in kindergarten through grade 5
• Should be explicit and systematic. This includes
  providing models of proficient problem solving,
  verbalization of thought processes, guided
  practice, corrective feedback, and frequent
  cumulative review
• Should include opportunities for students to work
  with visual representations of mathematical ideas
  and interventionists should be proficient in the
  use of visual representations of mathematical
  ideas
• Should devote about 10 minutes in each session to
  building fluent retrieval of basic arithmetic
  facts
• Gersten, R., Beckmann, S., Clarke, B., Foegen,
  A., Marsh, L., Star, J. R., Witzel, B., 2009

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Kindergarten Kindergarten
Major Clusters Supporting/Additional Clusters
Counting and Cardinality Know number names and the count sequence. Count to tell the number of objects. Compare numbers.   Operations and Algebraic Thinking Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.   Number and Operations in Base Ten Work with numbers 1119 to gain foundations for place value.   Measurement and Data Describe and compare measurable attributes. Classify objects and count the number of objects in categories.   Geometry Identify and describe shapes. Analyze, compare, create, and compose shapes.
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First Grade First Grade
Major Clusters Supporting/Additional Clusters
Operations and Algebraic Thinking Represent and solve problems involving addition and subtraction. Understand and apply properties of operations and the relationship between addition and subtraction. Add and subtract within 20. Work with addition and subtraction equations.   Number and Operations in Base Ten Extend the counting sequence. Understand place value. Use place value understanding and properties of operations to add and subtract.   Measurement and Data Measure lengths indirectly and by iterating length units.   Measurement and Data Tell and write time. Represent and interpret data.   Geometry Reason with shapes and their attributes.  
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Second Grade Second Grade
Major Clusters Supporting/Additional Clusters
Operations and Algebraic Thinking Represent and solve problems involving addition and subtraction. Add and subtract within 20. Work with equal groups of objects to gain foundations for multiplication.   Number and Operations in Base Ten Understand place value. Use place value understanding and properties of operations to add and subtract.   Measurement and Data Measure and estimate lengths in standard units. Relate addition and subtraction to length.   Measurement and Data Work with time and money. Represent and interpret data.   Geometry Reason with shapes and their attributes.    
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What is the most powerful predictor of GROWTH in
math achievement?

• Measured Intelligence (IQ test)
• External Motivation
• Internal Motivation
• Deep Learning Strategies
• Murayama, Pekrun, Lichtenfeld, and vom Hofe, 2013

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Kindergarten Teachers In case you didnt know

• The gap in knowledge of number and other aspects
  of mathematics begins well before kindergarten!

I NEED TO LEARN MATH ALREADY!?
Romani Seigler, 2008
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Early Identification / Rapid Response

• Students who enter and leave kindergarten below
  the 10th percentile
• 70 remain below the 10th percentile at the end
  of 5th grade
• Students who enter kindergarten below the 10th
  percentile and leave kindergarten above the 10th
  percentile
• 36 are below the 10th percentile in 5th grade
• Morgan et al., 2009

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Within the first weeks of kindergartenWe need
to knowwho most likely has it and who most
likely does not!
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Early Identification

• Screening Tools
• Rote Counting
• Number Identification
• Quantity Discrimination
• Missing Number

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• Counting and Cardinality

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A Progression
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Number Sense

• Quantity Number Names Symbols

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Instructional Strategies to Develop this
Conceptual Bridge

• Subitizing

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Subitizing

• Conceptual
• Apprehension of numerosity through part-whole
  relationships (one three and one three form a six
  on a domino)
• Supports addition and subtraction

• Perceptual
• Apprehension of numerosity without using other
  mathematical processes (e.g., counting)
• Supports cardinality
• Clements, 1999

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Lets try

• Provide a choral response of the number name.

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Lets try

• Write the number on your white board.

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Lets try

• Hold the number with you fingers behind your head
  like bunny ears.

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Lets try

• On your white board, write the number that is two
  more.

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Lets try

• On your white board, Create a different dot
  pattern using two colors that represents an
  equivalent number of dots.

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Lets Try anchors.

• Write the number that represents how far away
  this quantity is from 5.

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Number Sense

• Quantity Number Names Symbols

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Geary and Hoard, Learning Disabilities in Basic
Mathematics from Mathematical Cognition, Royer,
Ed.
Gellman and Gallistels (1978) Counting Principles

• 1-1 Correspondence
• Stable Order
• Cardinality
• Abstraction
• Order-Irrelevance

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Briars and Siegler (1984)Unessential features of
counting

• Standard Direction
• Adjacency
• Pointing
• Start at an End

Geary and Hoard, Learning Disabilities in Basic
Mathematics from Mathematical Cognition, Royer,
Ed.
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Students who are Identified with a Math
Disability

• Error Double Counts

Working memory is a key factor!
Geary and Hoard, Learning Disabilities in Basic
Mathematics from Mathematical Cognition, Royer,
Ed.
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Cardinality Principle

• Children learn how to count (matching number
  words with objects) before they understand the
  last word in the counting sequence indicates the
  amount of the set
• Fosnot Dolk, 2011

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The Cardinality Principle
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Using the Progression

• From counting to counting objects
• Orally say the counting words to a given number
• Attain fluency with the sequence of the counting
  words so they can focus attention on making a
  one-to-one correspondence
• To count a small set, students pair each word
  said with an object, usually by pointing or
  moving objects
• They learn to count small sets of objects in
• A line
• A rectangle
• A circle
• A scattered array
• Count out a given number of object in a scattered
  array
• The Common Core Standards Writing Team, 2011

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Model and Feedback
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Model and Feedback
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Model and Feedback
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From Cardinality to Counting On

• Materials deck of cards (1-7), a die, a paper
  cup, and counters
• Directions The first player turns over the top
  number card and places the indicated number of
  counters in the cup. The card is placed next to
  the cup as a reminder of how many are inside.
  The second player rolls the die and places that
  many counters next to the cup. Together, they
  decide how many counters in all.

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From Cardinality to Counting On
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Composing and DecomposingNumber

• To conceptualize a number as being made up of two
  or more parts is the most important relationship
  that can be developed about numbers
• Van de Walle, Karp, Bay-Williams, 2013

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Quick Activities

• Finger Games Ask students to make a number with
  their fingers (hands should be placed in lap
  between tasks),
• Show eight with your fingers. Tell your partner
  how you did it. Now do it a different way. Show
  your partner.
• Now make eight with the same number in each hand.
  
• Now make five without using your thumbs.
• Show seven with bunny ears behind your head.
• Make three with one hand. How many fingers are
  up? How many fingers are down?
• Clements Sarama, 2014

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Quick Activities

• Make a Number
• Students decide on a number to make (e.g.,
  seven). They then get three decks of cards and
  take out all cards numbered seven or more. The
  students take turns drawing a card and try to
  make a seven by combining it with another face up
  card if they can, they keep both cards. If
  they cant, they must place it face up beside the
  deck. When the deck is gone, the player with the
  most cards wins.
• Clements Sarama, 2014

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Jumping Frogs
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Jumping Frogs
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Lets try anchors

• Write the number that represents how far away
  this quantity is from 10.

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Ten Frame Flash

• How many?
• How many more to make ten?
• Say one more/one less/two more/two less?
• Say the 10 fact. For example, six and four make
  ten.

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• Unitizing

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Unitizing

• Unitizing is complex, it requires students to
  simultaneously hold two ideas they must think
  of a group as one unit and as a collection.
• -Richarsdon, 2012

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Circuit Number Lines
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Building Two-Digit NumbersTens Frames
First Grade Exploring Two-Digit Numbers Unit
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Building Two Digit-NumbersBase-Ten Blocks and
Arrow Cards
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I HaveWho Has?
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I HaveWho Has?

• Possible modifications
• Who has 7 ones and 2 tens?
• Who has 13 ones and 2 tens?
• Visual representations for I Have.

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• Computation

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Fluency with Computation

• NMP / IES Practice Guide Recommendation
• Computational fluency is an instructional target
  that leads to success in algebra
• For students who are not fluent, 10 minutes of
  instruction daily should be devoted to improving
  computational fluency

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So, this means
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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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Those Pesky Facts
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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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• 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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Typical Development of Strategy Use
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Using Progressions
http//www2.ups.edu/faculty/woodward/publications.
htm
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0 1 2 3 4 5 6 7 8 9
0 0 1 2 3 4 5 6 7 8 9
1 1 2 3 4 5 6 7 8 9 10
2 2 3 4 5 6 7 8 9 10 11
3 3 4 5 6 7 8 9 10 11 12
4 4 5 6 7 8 9 10 11 12 13
5 5 6 7 8 9 10 11 12 13 14
6 6 7 8 9 10 11 12 13 14 15
7 7 8 9 10 11 12 13 14 15 16
8 8 9 10 11 12 13 14 15 16 17
9 9 10 11 12 13 14 15 16 17 18
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Plus 0
0 1 2 3 4 5 6 7 8 9
0 0 1 2 3 4 5 6 7 8 9
1 1 2 3 4 5 6 7 8 9 10
2 2 3 4 5 6 7 8 9 10 11
3 3 4 5 6 7 8 9 10 11 12
4 4 5 6 7 8 9 10 11 12 13
5 5 6 7 8 9 10 11 12 13 14
6 6 7 8 9 10 11 12 13 14 15
7 7 8 9 10 11 12 13 14 15 16
8 8 9 10 11 12 13 14 15 16 17
9 9 10 11 12 13 14 15 16 17 18
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Plus 1
0 1 2 3 4 5 6 7 8 9
0 0 1 2 3 4 5 6 7 8 9
1 1 2 3 4 5 6 7 8 9 10
2 2 3 4 5 6 7 8 9 10 11
3 3 4 5 6 7 8 9 10 11 12
4 4 5 6 7 8 9 10 11 12 13
5 5 6 7 8 9 10 11 12 13 14
6 6 7 8 9 10 11 12 13 14 15
7 7 8 9 10 11 12 13 14 15 16
8 8 9 10 11 12 13 14 15 16 17
9 9 10 11 12 13 14 15 16 17 18
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Plus 2
0 1 2 3 4 5 6 7 8 9
0 0 1 2 3 4 5 6 7 8 9
1 1 2 3 4 5 6 7 8 9 10
2 2 3 4 5 6 7 8 9 10 11
3 3 4 5 6 7 8 9 10 11 12
4 4 5 6 7 8 9 10 11 12 13
5 5 6 7 8 9 10 11 12 13 14
6 6 7 8 9 10 11 12 13 14 15
7 7 8 9 10 11 12 13 14 15 16
8 8 9 10 11 12 13 14 15 16 17
9 9 10 11 12 13 14 15 16 17 18
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Plus 0, Plus 1, Plus 2
0 1 2 3 4 5 6 7 8 9
0 0 1 2 3 4 5 6 7 8 9
1 1 2 3 4 5 6 7 8 9 10
2 2 3 4 5 6 7 8 9 10 11
3 3 4 5 6 7 8 9 10 11 12
4 4 5 6 7 8 9 10 11 12 13
5 5 6 7 8 9 10 11 12 13 14
6 6 7 8 9 10 11 12 13 14 15
7 7 8 9 10 11 12 13 14 15 16
8 8 9 10 11 12 13 14 15 16 17
9 9 10 11 12 13 14 15 16 17 18
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Doubles
0 1 2 3 4 5 6 7 8 9
0 0 1 2 3 4 5 6 7 8 9
1 1 2 3 4 5 6 7 8 9 10
2 2 3 4 5 6 7 8 9 10 11
3 3 4 5 6 7 8 9 10 11 12
4 4 5 6 7 8 9 10 11 12 13
5 5 6 7 8 9 10 11 12 13 14
6 6 7 8 9 10 11 12 13 14 15
7 7 8 9 10 11 12 13 14 15 16
8 8 9 10 11 12 13 14 15 16 17
9 9 10 11 12 13 14 15 16 17 18
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0 1 2 3 4 5 6 7 8 9
0 0 1 2 3 4 5 6 7 8 9
1 1 2 3 4 5 6 7 8 9 10
2 2 3 4 5 6 7 8 9 10 11
3 3 4 5 6 7 8 9 10 11 12
4 4 5 6 7 8 9 10 11 12 13
5 5 6 7 8 9 10 11 12 13 14
6 6 7 8 9 10 11 12 13 14 15
7 7 8 9 10 11 12 13 14 15 16
8 8 9 10 11 12 13 14 15 16 17
9 9 10 11 12 13 14 15 16 17 18
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Using Progressions

• Concrete and Visual Representations

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Using Progressions
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Double /- 1 (Near Doubles)
0 1 2 3 4 5 6 7 8 9
0 0 1 2 3 4 5 6 7 8 9
1 1 2 3 4 5 6 7 8 9 10
2 2 3 4 5 6 7 8 9 10 11
3 3 4 5 6 7 8 9 10 11 12
4 4 5 6 7 8 9 10 11 12 13
5 5 6 7 8 9 10 11 12 13 14
6 6 7 8 9 10 11 12 13 14 15
7 7 8 9 10 11 12 13 14 15 16
8 8 9 10 11 12 13 14 15 16 17
9 9 10 11 12 13 14 15 16 17 18
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0 1 2 3 4 5 6 7 8 9
0 0 1 2 3 4 5 6 7 8 9
1 1 2 3 4 5 6 7 8 9 10
2 2 3 4 5 6 7 8 9 10 11
3 3 4 5 6 7 8 9 10 11 12
4 4 5 6 7 8 9 10 11 12 13
5 5 6 7 8 9 10 11 12 13 14
6 6 7 8 9 10 11 12 13 14 15
7 7 8 9 10 11 12 13 14 15 16
8 8 9 10 11 12 13 14 15 16 17
9 9 10 11 12 13 14 15 16 17 18
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Up Over 10
0 1 2 3 4 5 6 7 8 9
0 0 1 2 3 4 5 6 7 8 9
1 1 2 3 4 5 6 7 8 9 10
2 2 3 4 5 6 7 8 9 10 11
3 3 4 5 6 7 8 9 10 11 12
4 4 5 6 7 8 9 10 11 12 13
5 5 6 7 8 9 10 11 12 13 14
6 6 7 8 9 10 11 12 13 14 15
7 7 8 9 10 11 12 13 14 15 16
8 8 9 10 11 12 13 14 15 16 17
9 9 10 11 12 13 14 15 16 17 18
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0 1 2 3 4 5 6 7 8 9
0 0 1 2 3 4 5 6 7 8 9
1 1 2 3 4 5 6 7 8 9 10
2 2 3 4 5 6 7 8 9 10 11
3 3 4 5 6 7 8 9 10 11 12
4 4 5 6 7 8 9 10 11 12 13
5 5 6 7 8 9 10 11 12 13 14
6 6 7 8 9 10 11 12 13 14 15
7 7 8 9 10 11 12 13 14 15 16
8 8 9 10 11 12 13 14 15 16 17
9 9 10 11 12 13 14 15 16 17 18
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Using Progressions
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Making 10 Facts within 20
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5 13
Associative Property of Addition
__8__ ( 2 3 ) (8 2) 3
(10) 3 13
Makes 10
Left over
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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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Whats My Number?
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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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