tier ii instruction for whole-number concepts in grades k-2 matt hoskins, ncdpi tania rollins, ashe...
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Tier II Instruction for Whole-Number Concepts
in Grades K-2
Matt Hoskins, NCDPITania Rollins, Ashe County Schools
Denise Schulz, NCDPI
Welcome“Who’s in the Room?”
Advanced Organizer
Counting and Cardinality:Strengthening connections
between quantity, language, and symbols
Unitizing:Strengthening
understanding of the base-ten number system
TIER II
Features of supplemental
instruction
Computation:Progression of strategy use
and structures
Student Needs
Reso
urc
es
LAYERING OF SUPPORT
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
Kindergarten
Major Clusters Supporting/Additional ClustersCounting and CardinalityKnow number names and the count sequence.Count to tell the number of objects.Compare numbers. Operations and Algebraic ThinkingUnderstand addition as putting together and adding to, and understand subtraction as taking apart and taking from. Number and Operations in Base TenWork with numbers 11–19 to gain foundations for place value.
Measurement and DataDescribe and compare measurable attributes.Classify objects and count the number of objects in categories. GeometryIdentify and describe shapes.Analyze, compare, create, and compose shapes.
First GradeMajor Clusters Supporting/Additional Clusters
Operations and Algebraic ThinkingRepresent 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 TenExtend the counting sequence.Understand place value.Use place value understanding and properties of operations to add and subtract. Measurement and DataMeasure lengths indirectly and by iterating length units.
Measurement and DataTell and write time.Represent and interpret data. GeometryReason with shapes and their attributes.
Second GradeMajor Clusters Supporting/Additional Clusters
Operations and Algebraic ThinkingRepresent 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 TenUnderstand place value.Use place value understanding and properties of operations to add and subtract. Measurement and DataMeasure and estimate lengths in standard units.Relate addition and subtraction to length.
Measurement and DataWork with time and money.Represent and interpret data. GeometryReason with shapes and their attributes.
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
Kindergarten Teachers: In case you didn’t 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
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
Within the first weeks of kindergarten:
We need to knowwho most likely has it and who most likely does not!
Early Identification
• Screening Tools
– Rote Counting
– Number Identification
– Quantity Discrimination
– Missing Number
Counting and Cardinality
A Progression
Number Sense
Quantity Number Names Symbols
Instructional Strategies to Develop this Conceptual Bridge
Subitizing
Subitizing
• Perceptual
– Apprehension of numerosity without using other mathematical processes (e.g., counting)
– Supports cardinality
Clements, 1999
• Conceptual
– Apprehension of numerosity through part-whole relationships (one three and one three form a six on a domino)
– Supports addition and subtraction
Let’s try…
Provide a choral response of the number name.
Let’s try…
Write the number on your white board.
Let’s try…
Hold the number with you fingers behind your head like bunny ears.
Let’s try…
On your white board, write the number that is two more.
Let’s try…
On your white board, Create a different dot pattern using two colors that represents an equivalent number of dots.
Let’s Try anchors….
• Write the number that represents how far away this quantity is from 5.
Number Sense…
Quantity Number Names Symbols
Geary and Hoard, Learning Disabilities in Basic Mathematics from Mathematical Cognition, Royer, Ed.
•1-1 Correspondence
•Stable Order
•Cardinality
•Abstraction
•Order-Irrelevance
Gellman and Gallistel’s (1978)Counting Principles
• Standard Direction
• Adjacency
• Pointing
• Start at an End
Briars and Siegler (1984)Unessential features of counting
Geary and Hoard, Learning Disabilities in Basic Mathematics from Mathematical Cognition, Royer, Ed.
Error: Double Counts
Students who are Identified with a Math Disability
Geary and Hoard, Learning Disabilities in Basic Mathematics from Mathematical Cognition, Royer, Ed.
Working memory is a key factor!
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
The Cardinality Principle
Using the ProgressionFrom 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
Model and Feedback
Model and Feedback
Model and Feedback
From Cardinality to Counting OnMaterials: 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.
From Cardinality to Counting On
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
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
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 can’t, they must place it face up beside the deck. When the deck is gone, the player with the most cards wins.
Clements & Sarama, 2014
Jumping Frogs
Jumping Frogs
Lets try anchors…
Write the number that represents how far away this quantity is from 10.
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.
Unitizing
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
Circuit Number Lines
Building Two-Digit Numbers:Tens Frames
First Grade: Exploring Two-Digit Numbers Unit
Building Two Digit-Numbers:Base-Ten Blocks and Arrow Cards
I Have…Who Has?
I Have…Who Has?
• Possible modifications:
– Who has 7 ones and 2 tens?
– Who has 13 ones and 2 tens?
– Visual representations for I Have.
Computation
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
So, this means…
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
Those Pesky “Facts”
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
• 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
Typical Development of Strategy Use
Using Progressions
http://www2.ups.edu/faculty/woodward/publications.htm
+ 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
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
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
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
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
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
+ 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
Using Progressions
• Concrete and Visual Representations
Using Progressions
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
+ 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
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
+ 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
Using Progressions
Makes 10 Left over
8 + 5 = 13
__8__ + ( 2 + 3 ) =
(8 + 2) + 3= (10) + 3= 13
Associative Property of Addition
Making 10: Facts within 20
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
What’s My Number?
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 Isn’t Smarter: Messages about Math, Teaching, and Learning in the 21st Century, pg. 95
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