Early Number Sense
The “Phonics” of Mathematics
Presenters:
Lisa Zapalac, Head of Lower School
Kevin Moore, 4 th Grade Math
Brooke Carmichael, Kindergarten
November 19, 2010
10:15 – 11:30 a.m.
www.austintrinity.org

3,996 + 4,246

Is this how you simplified it?
1 1 1 1
3,996
+4,246
8,242

2 nd Grader Simplifying 3,996 + 4,246
Example of 2nd Grader Using Compensation

How did the 2 nd grader simplify the expression?
He used an addition strategy called compensation, but there are many underlying concepts that are embedded in compensation
1) He noticed that 3,996 is 4 less than 4,000
2) He recognized 4,246 as being equivalent to 4,242+ 4
3) He then associated (3,996 + 4) + 2,242

Recognizes unreasonable conclusions
Possesses a repertoire of mental computation strategies
Number
Sense
Recognizes values in their various forms
Demonstrates proficiency with estimation and evaluation of quantities

50 x 48
4th Grade Video

4
48
4 x 50
00
+2400
48 x50
2400
240 0

4 th Graders Simplifying
48 (50)
Example 1
Example 2
Example 3

Solve 76 x 89
4th Grader

Number Sense…. How do we build it?

Using “Strings” to Develop Number Sense
Strings are a set of arithmetic problems in which the children are developing very specific strategies. Strings are generally done mentally. Each string begins with a known expression and moves towards the unknown, scaffolding the development of key strategies.
The following slides contain examples of strings at various grade levels.

st
5 + 5
5 + 6
6 + 6
6 + 7
7 + 7
7 + 8
8 + 8
9 + 7
6 + 8

Building Number Sense through Facts
Possesses a repertoire of mental computation strategies
• Doubles plus or minus 1
– Ex. 6 + 7 = 6 + 6 + 1 (or 7 + 7 – 1) = 13
• Doubles plus or minus 2
– Ex. 5 + 7 = 5 + 5 + 2 (or 7 + 7 - 2)
• Working with the structure of five
– Ex. 6 + 7 = 5 + 1 + 5 + 2 = 10 + 3 = 13
• Making tens
– Ex. 8 + 4 = 8 + 2 + 2
• Using tens to solve nines
– Ex. 9 + 7 = 10 + 7 - 1
• Using compensation
– Ex. 6 + 8 = 7 + 7 (adding one to one addend, while subtracting one from the other addend)
5 + 5
5 + 6
6 + 6
6 + 7
7 + 7
7 + 8
8 + 8
9 + 7
6 + 8

Using Tools and Models to
Develop Number Sense
Recognizes values in their various forms
The rekenrek, or arithmetic rack, is a tool consisting of two rows of ten beads with two sets of five in each. The rekenrek was developed by Adri Treffers, a researcher at the Freudenthal Institute in the Netherlands, and it provides a powerful model for exploring the composing and decomposing of number (Treffers 1991)

Recognizes values in their various forms
Kindergarten String
Example
Kindergarten String
5 on the top, 5 on the bottom
7 on the top, 3 on the bottom
4 on the top, 6 on the bottom
6 on the top, 4 on the bottom
8 on the top, 2 on the bottom
Possesses a repertoire of mental computation strategies

2
Moving Beyond Facts
Modeling 38 + 42
40
38 40
80
The open number line is a tool used to model students’ thinking. In this problem, 38 + 42, a student might solve it by moving to a landmark number first. Or, they might first make jumps of ten.
40 2
38
78 80

Example of 2 nd Grade String
Big Idea: Keeping One Number Whole and Taking Leaps of 10
2nd Grade String
75 + 20
75 + 25
75 + 24
55 + 30
55 + 39
69 + 21
69 + 29

Building Number Sense with Multiplication
Constructing facts through relationships and models
4(4) = 16 2[(4)2] or 2(8)

Multiplication
(3 rd & 4 th Grade Strategies)
• Doubling
▪ 6 x 6 = 2 x 3 x 6
• Halving and doubling
▪ 4 x 3 = 2 x 6
• Using the distributive property
▪ 7 x 8 = (5 x 8) + (2 x 8), or
▪ 7 x 8 = (8 x 8) – (1 x 8)
• Using the commutative property
▪ 5 x 8 = 8 x 5
Possesses a repertoire of mental computation strategies

Example of 4 th Grade String
4th Grade Multiplication String
4 x 8
14 x 8
6 x 9
26 x 9
12 x 13
15 x 24

4 th Grade String Revisited –
Connecting to Algebra a (8)
(a + b) 8
(3a)2
(2a + c) (5)
(a + 3) (a + 2)

Number sense is the bridge between arithmetic and algebra
Number Sense
Arithmetic Algebra

Resources
Books
Ma, L. (1999). Knowing and Teaching Elementary Mathematics. Mahwah, NJ: Lawrence Erlbaum Associates, Inc.
Devlin, K. (2000). The Math Gene. Great Britain: Weidenfeld & Nicolson
Stigler & Hiebert (1999). The Teaching Gap. New York, NY: The Free Press
Fosnot, C., & Dolk, M. (2001). Young Mathematicians at Work: Constructing Number Sense, Addition, and Subtraction. Portsmouth, NH:
Heinemann
Fosnot, C., & Dolk, M. (2001). Young Mathematicians at Work: Constructing Multiplication and Division. Portsmouth, NH: Heinemann
Carpenter, T., Franke, M., & Levi, L. (2003). Thinking Mathematically: Integrating Arithmetic and Algebra in Elementary School. Portsmouth, NH:
Heinemann
Fosnot, C. & Uittenbogaard, W. (2007). Minilessons for Early Addition and Subtraction. Portsmouth, NH: Heinemann
Fosnot, C. & Uittenbogaard, W. (2007). Minilessons for Extending Addition and Subtraction. Portsmouth, NH: Heinemann
Fosnot, C. & Uittenbogaard, W. (2007). Minilessons for Early Multiplication and Division. Portsmouth, NH: Heinemann
Fosnot, C. & Uittenbogaard, W. (2007). Minilessons for Extending Multiplication and Division. Portsmouth, NH: Heinemann
Articles
Faulkner, V. (2009). The Components of Number Sense – An Instructional Model for Teachers. – Teaching Exceptional Children, Vol. 41, No. 5,
24-30
Gersten, R. & Chard, D. (2010). Validating a Number Sense Screening Tool for Use in Kindergarten and First Grade: Prediction of Mathematics
Proficiency in Third Grade – School Psychology Review, Vol. 39, No. 2, 181-195
Harel, G. & Rabin, J. (2010). Teaching Practices Associated With the Authoritative Proof Scheme – Journal for Research in Mathematics
Education, Vol. 41, No. 1, 14-19
Web Sites
DreamBox Learning www.dreambox.com
To order a rekenrek: www.eNasco.com