Transcript Tier II Instruction for Whole-Number Concepts in

Tier II Instruction for
Whole-Number Concepts
in Grades K-2
Matt Hoskins, NCDPI
Tania 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 baseten number system
Computation:
Progression of strategy use
and structures
TIER II
Features of
supplemental
instruction
Resources
LAYERING OF SUPPORT
Student Needs
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 Clusters
Counting and Cardinality
Measurement and Data
 Know number names and the
 Describe and compare measurable
count sequence.
attributes.
 Count to tell the number of
 Classify objects and count the
objects.
number of objects in categories.
 Compare numbers.
Geometry
Operations and Algebraic Thinking
 Identify and describe shapes.
 Understand addition as putting
 Analyze, compare, create, and
together and adding to, and
compose shapes.
understand subtraction as taking
apart and taking from.
Number and Operations in Base Ten
 Work with numbers 11–19 to gain
foundations for place value.
First Grade
Major 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.
Supporting/Additional Clusters
Measurement and Data
 Tell and write time.
 Represent and interpret data.
Geometry
 Reason with shapes and their attributes.
Second Grade
Major 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.
Supporting/Additional Clusters
Measurement and Data
 Work with time and money.
 Represent and interpret data.
Geometry
 Reason 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 know
who 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
• Conceptual
– Apprehension of
numerosity without
using other
mathematical
processes (e.g.,
counting)
– Apprehension of
numerosity through
part-whole
relationships (one
three and one three
form a six on a domino)
– Supports cardinality
– Supports addition and
subtraction
Clements, 1999
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
Gellman and Gallistel’s (1978)
Counting Principles
• 1-1 Correspondence
• Stable Order
• Cardinality
• Abstraction
• Order-Irrelevance
Geary and Hoard, Learning Disabilities in Basic Mathematics
from Mathematical Cognition, Royer, Ed.
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.
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.
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 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
Model and Feedback
Model and Feedback
Model and Feedback
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.
From Cardinality to Counting On
Composing and Decomposing
Number
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
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Plus 0, Plus 1, Plus 2
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Doubles
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Using Progressions
• Concrete and Visual Representations
Using Progressions
Double +/- 1 (Near Doubles)
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Using Progressions
Making 10: Facts within 20
8
+
5 =
13
Associative Property of
Addition
__8__ +
(2 + 3)
=
Makes 10
(8 + 2) + 3=
(10) + 3=
13
Left over
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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