Transcript What is Number Sense?

Supporting Rigorous Mathematics
Teaching and Learning
Developing Students’ Sense of Quantity: A
Means to Mathematical Understanding
Tennessee Department of Education
Elementary School Mathematics, Kindergarten
December 7, 2012
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Rationale
Number sense refers to a person’s general understanding of numbers
and operations and the ability to handle daily life situations that include
numbers. This includes the ability to develop useful, flexible, and efficient
strategies (i.e., mental computation or estimation) for handling numerical
problems.
Howden, 1989; McIntosh, Reys & Reys, 1992; Reys, 1991; Reys & Yang, 1998;
Sowder, 1992a, 1992b; Treffers, 1991; Yang, 2002a, 2002b
Students who participate in well-designed activities are more likely to
develop number sense than students who receive instruction focusing on
the development of standard written algorithms and computation
proficiency.
Sowder, 1941; Reys, 2001
In this session, we will learn to recognize indicators of students who are
developing number sense and instruction that is fostering number sense.
We will also consider the role of the principal in supporting teachers so
they can develop students’ number sense.
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Session Goals
Participants will learn:
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The Common Core Standards for Mathematical Content.
Methods of developing students’ sense of quantity.
Indicators of number sense.
Activities for developing number sense.
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Overview of Activities
• Read and discuss the CCSS Standards, specifically
the Counting and Cardinality Standards and the
Operation and Algebraic Thinking Standards.
• Read classroom cases and discuss teaching and
learning that supports and illustrates the
development of Number Sense.
• Read and discuss ways in which hands-on activities
support students in developing Number Sense.
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The CCSS for Mathematical Content: Kindergarten
Counting and Cardinality
K.CC
Know number names and the count sequence.
Count to 100 by ones and by tens.
K.CC.1
K.CC.2
Count forward beginning from a given number within the known
sequence (instead of having to begin at 1).
K.CC.3
Write numbers from 0 to 20. Represent a number of objects with a
written numeral 0-20 (with 0 representing a count of no objects).
Count to tell the number of objects.
Understand the relationship between numbers and quantities; connect
K.CC.4
counting to cardinality.
K.CC.4a
When counting objects, say the number names in the standard order,
pairing each object with one and only one number name and each number
name with one and only one object.
K.CC.4b
Understand that the last number name said tells the number of objects
counted. The number of objects is the same regardless of their
arrangement or the order in which they were counted.
K.CC.4c
Understand that each successive number name refers to a quantity that is
one larger.
Common Core State Standards, 2010, p. 11, NGA Center/CCSSO
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The CCSS for Mathematical Content: Kindergarten
Counting and Cardinality
K.CC
Count to tell the number of objects.
K.CC.5
Count to answer “how many?” questions about as many as 20 things
arranged in a line, a rectangular array, or a circle, or as many as 10
things in a scattered configuration; given a number from 1–20, count
out that many objects.
Compare numbers.
K.CC.6
Identify whether the number of objects in one group is greater than,
less than, or equal to the number of objects in another group, e.g., by
using matching and counting strategies.
K.CC.7
Compare two numbers between 1 and 10 presented as written numerals.
Common Core State Standards, 2010, p. 11, NGA Center/CCSSO
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The CCSS for Mathematical Practice
1.
Make sense of problems and persevere in solving
them.
2.
Reason abstractly and quantitatively.
3.
Construct viable arguments and critique the reasoning
of others.
4.
Model with mathematics.
5.
Use appropriate tools strategically.
6.
Attend to precision.
7.
Look for and make use of structure.
8.
Look for and express regularity in repeated reasoning.
Common Core State Standards, 2010, p. 6-8, NGA Center/CCSSO
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What is Number Sense?
• Read the excerpts on Number Sense.
• What are the authors describing?
• In what ways do you give students opportunities to
develop Number Sense?
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Recognizing Opportunities that
Develop a Sense of Quantity
(Private Work Time)
Read each mini-case.
• In what ways is the teacher giving students
opportunities to develop Number Sense?
• Do the students have a sense of quantity? If so, why
do you say they have a sense of quantity? What in the
description indicates a sense of quantity?
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Sharing Our Observations
(Small Group Discussion)
• Share your observations with each other.
• Make a poster with a list of strategies that you and
other teachers can use to develop students’ Number
Sense.
• Identify the indicators of Number Sense.
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Sharing Strategies for Developing
Number Sense
(Whole Group Discussion)
• Listen to one team’s observations.
• Be prepared to add teacher moves or student
indicators that group one did not identify.
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Strategies for Developing Number
Sense
Read the list of strategies for developing Number Sense.
Check the strategies that you have identified already.
Circle those that we did not discuss.
Are there other strategies that you would add to the list
now that you have read the excerpts from research?
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Early Numeracy Strategies
• Developing spatial relationships involving hands-on
experiences (i.e., provide the sensory input that helps students
develop mental imagery).
• Focusing on the meaning of sets in the context of problems.
• Developing visual cues such as dot cards and patterns on the
die help students see relationships.
• Building mental imagery expands children’s ability to think in
flexible ways.
• Recording students’ ideas as they share them can reinforce
concepts and help students make the connection between the
concrete items and the abstract numbers.
• Asking students to compare quantities (i.e., Which is more?
How do you know? Which is less? How do you know?).
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Early Numeracy Strategies
• Solving problems involving joining, separating,
grouping, and sharing helps students see how sets
come together and are taken apart.
• Counting and showing objects to 120 helps students to
hear the number pattern and to see quantities.
• Counting forward and backward.
• Ordering/sequencing sets, pictures, and numbers from
least to greatest.
• Matching numerals to objects.
• Being exposed to part/whole relationships.
• Showing students sets and asking them to make
estimations of the quantities. Talking about the
estimates that are closest and furthest from the
amount.
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Hands-On Activities
In what ways might work with hands-on activities help
students to develop a sense of quantity?
Use each activity and discuss what students might learn
from the activity.
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Numeracy Activities
• Oral Counting and Writing Numbers and Noticing
Patterns
• Dice Recognition
• Ten-Frame Recognition and Addition
• Beads
• Dominos
• Spill the Pennies
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Setting Goals
What are some ways you will work to develop students’
sense of quantity?
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