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7/28/2015
1
Pennsylvania Training and Technical Assistance Network
Early Numeracy &
Beginning Math Concepts
Jared Campbell
Willow Hozella
Educational Consultants, PaTTAN Harrisburg
PaTTANβs Mission
The mission of the Pennsylvania
Training and Technical Assistance
Network (PaTTAN) is to support the
efforts and initiatives of the Bureau of
Special Education, and to build the
capacity of local educational agencies
to serve students who receive special
education services.
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PDEβs Commitment to Least Restrictive Environment (LRE)
Our goal for each child is to ensure
Individualized Education Program (IEP)
teams begin with the general
education setting with the use of
Supplementary Aids and Services
before considering a
more restrictive environment.
3
Session Outline
Early numeracy concepts and skills are essential
for continued achievement in mathematics. Structuring
studentsβ earliest experiences with mathematics in a
CRA sequence can help them conceptualize the
concept of number and provide for more fluent
and flexible counting and computation.
Objectives
Participants will be able to model whole numbers using place value
concepts.
Participants will understand the importance of the ability to subitize
and apply to skill to teach addition and subtraction.
Participants will be able to utilize various tools (ten-frame, rek-n-rek,
etc.) to model mathematical concepts.
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Tech Connection
5
wiggio.com
group name: pattan math
password: ptnmath
Session Outline
1. Quasi-History of Math
π. Concept of Number
π. Number Bonds
π. Ten-Frames
π. Rekenrek
π. Fractions
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Pennsylvania Training and Technical Assistance Network
Early Numeracyβ¦
PA Core: Early Numbers/Operations Standards
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Pennsylvania Training and Technical Assistance Network
A Quasi-History
What is this?
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20,000 years agoβ¦
β Tally Systems
β Grouping structure
A Quasi-History of Number
(Czechoslovakia, 1937)
A Quasi-History of Number
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Tally Systems
Grouping structure
A Quasi-History of Number
(Czechoslovakia)
Uruk c. 4000BC
Place tokens in ball
Bake to prevent tampering
Mark outside with symbols to preserve records
Some time passes⦠local systems converge
Babylonian Number Systems c.1950 BC
A Quasi-History of Number
Chinese Number System
BASE-10
3 β 100 1 β 101 5 β 102
3 10 + 500 +
513
Positional Base System (oracle bone script c. 1400 BC)
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A Quasi-History of Number
Germanic / Irish / Britain / Roman (Base 12)
12 π‘πππ¦ ππ§. = 1 π‘πππ¦ ππ. 12 πππππ = 1 π βππππππ
π·ππ§ππ = 12
πΊπππ π = 12 Γ 12 = 144
πΊππππ‘ πΊπππ π = 12 Γ 12 Γ 12 = 1728
TIME
12 Γ 2 βππ’ππ = 1 πππ¦ 12 ππππ‘βπ = 1 π¦πππ
12 π§πππππ π ππππ πΆβππππ π πΆπππππππ
π©ππππππ β¦ ππ Γ· π = ππ!
βShang-styleβ Counting
50 + 8
5 π‘πππ , 8 60 + 9
6 π‘πππ , 9
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Language & Number
The numeric systems invented vary across time and place, and there is no
doubt that the properties of such a system can facilitate or impede the
development of childrenβs mathematical understanding.
Chinese (and Asian languages based on ancient Chinese) are organized such
that the numerical names are compatible with the traditional 10-base
numeration system. So spoken numbers correspond exactly to their written
equivalent: 15 is spoken as "ten five" and 57 as "five ten seven."
Most European systems of number words are irregular up to 100. For
example in French, 92 is said as "four twenty twelve," corresponding to
4 Γ 20 + 12.
The more complicated the number word system is, the harder it is for
children to learn the counting sequence.
http://web.media.mit.edu/~stefanm/society/som_final.html
Interpreting Numbers
1. What is this number?
2. What is the meaning of this number?
18
3264
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Decimal (base 10)
19
3264 100 101 102 103
3 Γ 103 + 2 Γ 102 + 6 Γ 101 + (4 Γ 100)
3000 + 200 + 60 + 4
3 Γ 1000 + 2 Γ 100 + 6 Γ 10 + (4 Γ 1)
Language of Number
26 π π¦πππππ ππππ π ππ’ππ π€πππ
10 π π¦πππππ ππππ ππ’πππ‘ππ‘π¦ ππ’ππππ
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Symbols & Meaning
β’ Two ways to understand lettersβ¦
β βBβ is the letter βbeeβ and makes the sound /b/
β’ What about numbers?
β Names are taught
β Meaning is based on place value (base 10)
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Number Names & Meanings
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βa childβs fluidity and flexibility with numbers, the sense of what numbers mean, and an ability to perform mental mathematics and to look at the world and make comparisonsβ
(Gersten & Chard, 1999)
What is Number Sense?
CRA Sequence of Instruction
βflexibility with mental mathβ
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CRA
β’ Concrete (sense making by moving)
β’ Representational (sense making by drawing)
β’ Abstract (sense making with symbols)
CONSISTENT LANGUAGE
Rationale β Doing What Works
Research-based studies show that students who use
concrete materials develop more precise and more
comprehensive mental representations, often
show more motivation and on-task behavior,
understand mathematical ideas, and better apply these
ideas to life situations.
(Harrison, & Harrison,1986)
(Suydam & Higgins,1977)
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Why would CRA be effective?
β’ Multimodal forms of math acquisition to aid
memory and retrieval
β’ Meaningful manipulations of materials allows
students to rationalize abstract mathematics
β’ Procedural accuracy; provides alternate to
algorithm memorization
(Witzel, Riccomini, & Scheider, 2008)
Other Research.
β’ Direct Instruction
β’ Errorless Teaching
β’ Formative Assessment
β’ Correct Feedback
β’ Improved Teacher Content Knowledge
β Task Analyze
β Instruct on Specific Skills or Process
β Monitor progress
β Correct errors
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Something hereβ¦
β’ Multimodal forms of math acquisition to aid
memory and retrieval
β’ Meaningful manipulations of materials allows
students to rationalize abstract mathematics
β’ Procedural accuracy; provides alternate to
algorithm memorization
Students having difficulties with math...
β’ Counting seen as rote, mechanical, left to right,
1: 1 correspondence only; INEFFECTIVE
β’ Automaticity problems take up working memory,
inhibit discourse & algebraic thinking
(Gersten, Jordan, & Flojo, 2005)
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Pennsylvania Training and Technical Assistance Network
Concept of Number
βWhat does three really mean? What is three-nessβ
-M
βa childβs fluidity and flexibility with numbers, the sense of what numbers mean, and an ability to perform mental mathematics and to look at the world and make comparisonsβ
(Gersten & Chard, 1999)
What is Number Sense?
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What does β3β really mean?
three 3 βthreeβ
"1 β¦ 2 β¦ 3! "
3 π’πππ‘π
βone more than 2β βone less than 4β
βis betweenβ¦ β βis more thanβ¦ β
βis less thanβ¦ β βis the same asβ¦ β
Teaching each symbol or Teaching the collection
Each Symbol
β’ Name β Meaning β Quantity
β’ Ability to Subitize
Collection
β’ Count Sequence
β’ Magnitude
β’ Missing Number
β’ Applications
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Subitize
The ability to see a quantity and
know how many, without βcounting.β
Perceptual and Conceptual
Subitizing & Conceptual Counting
3 + 2 = π
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Elementary Classroom β Conceptual Addition
Teaching each symbol or Teaching the collection
Each Symbol
β’ Name β Meaning β Quantity
β’ Ability to Subitize
Collection
β’ Count Sequence
β’ Magnitude
β’ Missing Number
β’ Applications
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Basic Principles of Counting
One-to-one β Counting one βthingβ at a time; transfer from
uncounted group to counted group (π: π πͺπππππππππ ππππ)
Stable-order β Establishes consistent sequence
Cardinal β The last count represent the quantity in the counted
group (πͺπππ πππππππ)
Abstraction β applying counting to like objects, actions, sounds,
etcβ¦
Order-irrelevance β Can count in any order
Stages of Early Arithmetical Learning
Wright, R., Martland. J, Stafford, A., & Stanger, G. (2006). Teaching Number: Advancing Childrenβs Skills and Strategies. London: Sage.
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βa childβs fluidity and flexibility with numbers, the sense of what numbers mean, and an ability to perform mental mathematics and to look at the world and make comparisonsβ
(Gersten & Chard, 1999)
What is Number Sense?
Pennsylvania Training and Technical Assistance Network
From Counting to Computation
β¦ or more efficient counting
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What is the sum?
999 + 32 = ?
1000 + 32 = 1032 β 1
999 + 1 + 31 = 1000 + 31
1031
Decomposition & Compensation
Decomposition β decomposing numbers to
compute faster
make a π
make a ππ
doubles (Β±π)
Compensation β Adjust the problem to compute,
then readjust the answer
may utilize known facts.
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The Mathematics Framework, Appendix F
Pennsylvania Training and Technical Assistance Network
Number Bonds
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Composing & Decomposing Numbers
Composing & Decomposing Numbers
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Number Bonds β Fact Families
4 = 2 + 2
4 = 1 + 3
4 = 0 + 4
Number Bonds β Fact Families
π
π
π
π + π = π
π + π = ?
π + ? = π
? +π = π π β π = ?
π β ? = π
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Partner Practice (C or R)
β’ Count on
β’ Making 5
β’ Making 10
β’ Doubles (Β±1)
2 + 3
3 + 6
7 + 2
1 + 7
4 + 3
8 + 3
2 + 9
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Partner Practice (C or R)
β’ Take Away
β’ Count on (Think +)
β Missing addend
β’ Compensation β πΉπππ 5
β πΉπππ 10
β’ Doubles (Β±1)
3 β 1
4 β 2
6 β 4
8 β 7
4 β 2
8 β 4
9 β 3
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Pennsylvania Training and Technical Assistance Network
Ten-Frames
Decomposition
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Decomposition
π + π = π +π
"π‘π€π πππ πππ ππππ β¦ "
"π‘π€π πππ’π πππ πππππ β¦ "
"π‘π€π πππ’π πππ πππ’πππ β¦ "
Decomposition
π ππ π‘βπ ππππ‘π & π ππ π‘βπ π€βπππ
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Purpose of 10-frame
β’ See sets of 5
β’ See sets of 10
β’ Organize in rectangular array
β’ Subitize
β’ Reduces need to βcountβ
β’ Visually decompose numbers in sets of 5
π β πππππ
ππ β πππππ
ππ°π¨ ππ β ππππππ
Subitizing the 10-frame support
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Ten-Frame Variations
Help students Subitize on the ππ β πππππ.
π + π
π
What do you see?
ππ β π
π β π π + π + π
π + π
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π + π = π
Modeling on a ten frame
π + π =
π β π = π
π
π + π = π
Modeling on a ten frame
π + π = π
π β π = π
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π + π =
ππ + π =
ππ
Modeling on a ten frame
π + π
π + π =
ππ + π =
ππ
Modeling on a ten frame
π + π
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Partner Practice (C)
β’ Count All or Take Away
β’ Counting on
β Subtraction: Missing addend
β’ Making 5
β’ Making 10
β’ Doubles (Β±1)
8 + 9
4 + 6
7 β 3
3 + 4
7 + 8
12 β 4
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Teaching facts w/ 10-frame support
Ten-Frame Progression
2 7 + 3 6
6 + 5 10 + 1
11
6 3
60 + 3
1
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Ten Frame Ideas
Pennsylvania Training and Technical Assistance Network
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Vocabulary
73
The Rekenrek (also called an arithmetic rack) has emerged as perhaps the
most powerful of all models for young learners.
Developed by mathematics education researchers at the highly regarded
Freudenthal Institute in the Netherlands.
Designed to reflect the natural intuitions and informal strategies that young
children bring to the study of numbers, addition, and subtraction.
Provides a visual model that encourages young learners to build numbers by
β’ groups of five
β’ groups of ten
β’ doubling and halving strategies
β’ counting-on from known addition/subtraction
Rekenrek
74
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Some activitiesβ¦
β’ See & Slide β Given #, make in 1 move.
β’ Build a Number β move first row, how
many more on second row
β’ Show Me β Give number, make combination
β’ Flash Attack β Show beads, get number
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Rekenrek
One more ideaβ¦
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Pennsylvania Training and Technical Assistance Network
Fractions
Early on in Fractionsβ¦
2 3 4 5
6
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Early on in Fractionsβ¦
π
π = π Γ
1
π πππ’ππ‘πππ "
1
πβ²π "
3
"π‘βππππ "
Vocabulary
ππ’πππππ‘ππ
πππππππππ‘ππ
Fraction β from Latin: fractus, βbrokenβ
enumerate
denomination
count
size / value what is being
counted
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Interpreting Fractions β βcountingβ
Definitions Models
Part of whole
Ratio
Measurement
Operator/Quotient
Area
circles, pattern blocks, graph/dot
paper, paper folding
Length
Fraction strips, Cuisenaire rods,
line segments, number line
Sets
Objects, groups or arrays
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What & When?
Early on in Fractionsβ¦
π
π = π Γ
1
π πππ’ππ‘πππ "
1
πβ²π "
1
2 =
2
4 =
4
8
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Early on in Fractionsβ¦
π
π = π Γ
1
π πππ’ππ‘πππ "
1
πβ²π "
1
2 =
2
4 =
4
8
π π
Early on in Fractionsβ¦
π
π = π Γ
1
π πππ’ππ‘πππ "
1
πβ²π "
π π
π
π
π
π
π
π
π
π
β¦
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CRA Sequence of Instruction
βflexibility with mental mathβ
CRA
β’ Concrete (sense making by moving)
β’ Representational (sense making by drawing)
β’ Abstract (sense making with symbols)
CONSISTENT LANGUAGE
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Pennsylvania Training and Technical Assistance Network
Resources
Tech Connection
92
wiggio.com
group name: pattan math
password: ptnmath
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93
Pittsburgh Harrisburg King of Prussia
Early Numeracy ππ/ππ/ππ ππ/π/ππ ππ/ππ/ππ
Addition &
Subtraction ππ/π/ππ ππ/ππ/ππ ππ/ππ/ππ
Multiplication &
Division π/ππ/ππ π/ππ/ππ π/ππ/ππ
Fractions π/ππ/ππ π/ππ/ππ π/ππ/ππ
Integers &
Equations π/π/ππ π/ππ/ππ π/π/ππ
Contact Information www.pattan.net
Jared Campbell jcampbell@pattan.net
Willow Hozella whozella@pattan.net
Educational Consultants
Commonwealth of Pennsylvania
Tom Wolf, Governor
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