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Teaching Numbers
Focus Questions:1. How does childrens number sense
develop?
2. How can technology help children
develop number sense?
3. What characteristics are associated with
the different counting stages?
4. How can teachers use questions topromote childrens thinking about
numbers?
Discuss what you already know and what you want to learn more about.
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Number Sense
A good intuition about numbers and their
relationships. It develops gradually as a
result of exploring numbers, visualizingthem in a variety of contexts, and relating
them in ways that are not limited by
traditional algorithms (Howden, 1989).
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Number Sense
Number Sense includes:
An understanding of number concepts and
operations on these numbers. The development of useful strategies for
handling numbers and operations.
The facility to compute accurately andefficiently, to detect errors, and to recognizeresults as reasonable.
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Number Sense
The ability and inclination to use this
understanding in flexible ways to make
mathematical judgments.
An expectation that numbers are useful and
that work with numbers is meaningful and
makes sense.
Brainstorm examples for each bullet item
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The meanings for the number five
suggested by young children
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Excerpts of Expectations from
theNumber and OperationsStandardGrades Pre-K-2
Principles and Standards for School
Mathematics
National Council of Teachers of
Mathematics
2000
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Understand numbers, ways of representing
numbers, relationships among numbers, and
number systems
Count with understanding and recognize how
many in sets of objects
Develop understanding of the relative position andmagnitude of whole numbers and of ordinal and
cardinal numbers and their connections.
Connect number words and numerals to the
quantities they represent, using various physical
models and representations
Review the Snapshot of a Lesson. Discuss the counting and early numberconcepts displayed.
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Prenumber Concepts are those that do not
rely on numbers but provide a foundation for
later number concepts and skills.
Prenumber Concepts
Classification
Patterns
Comparisons
Early Number Development Conservation
Group Recognition
[Brainstorm activities and materials that may be used to develop the five concepts]
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Prenumber Concepts
Classification
Patterns
Comparisons
Early Number Development
Conservation
Group Recognition
[Complete In the Classroom 7-1 and 7-2.
Refer to Figure 7-5. Discuss the different levels of involvement andunderstanding experienced in a whole class demonstration versus a small group,
hands on setting.
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Prenumber Concepts
Classification
- size, colour, shape,
Patterns
- copying a pattern
- finding the next one
- extending a pattern- create their own patterns
Comparisons
[Complete In the Classroom 7-1 and 7-2.
Refer to Figure 7-5. Discuss the different levels of involvement andunderstanding experienced in a whole class demonstration versus a small group,
hands on setting.
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Early Number Development
Conservation
Group Recognition - subitizing
[Complete In the Classroom 7-1 and 7-2.
Refer to Figure 7-5. Discuss the different levels of involvement andunderstanding experienced in a whole class demonstration versus a small group,
hands on setting.
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Conservation of Number
Groups of 5 counters are arranged in the
following patterns
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Student A
T: Are there more red, blue, or yellow counters? S: More blue.
T: How do you know?
S: I can tell by looking.
T: How many of each?
S: One, two, three, four, five... five red. One,
two, three, four, five...five blue. One, two,three, four, five...five yellow.
T: Five of each?
S: Yes.
T: Do you still think there are more blue?
S: Yes, I can just see there's more blue.
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Student B
T: Are there more red, blue, or yellow counters?
S: They're the same.
T: How do you know?
S: I counted them.
T: How many of each? S: One, two, three, four, five...Five red. Five
blue. Five yellow.
T: Five of each?
S: Yes.
Determine the children's conservation of number
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Group RecognitonSubitizing
Rectangular
Linear
Circular Scrambled
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Counting
Counting Principles
One-to-One Rule
Stable Order Rule
Order Irrelevance Rule
Cardinality Rule
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Counting
Counting Stages
Rote Counting
- involves only the ability to recite the number
names in sequence
Rational Counting
- requires the child to make a one-to one-correspondence between each number
name and one-and-only-one object.
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Counting Stages Rote Counting
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Counting Stages
Rational Counting
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Counting
Counting Strategies
Counting On
Counting Back
Skip Counting
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In the following slides, analyze the
children's use of the counting principles.
Focus on what the children know ratherthan what they do not know.
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Counting Principles: Student A
T: Here are some blocks in a row. Startwith this one on the end and count them.
S: One, two, three, four, five, SIX. Thereare six blocks.
T: What if you start at the other end of therow and count them?
S: One, two, three, four, five, SIX. Thereare six.
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Counting Principles: Student B
T: Here are some red blocks in a row. Start
with this one on the end and count them.
S: (Touches each of the 5 blocks) One, two,three, five, six. Six red blocks
T: Now count these blue blocks.
S: (Touches each of the 4 blocks) One, two,
three, five. Five blue blocks.
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Counting Principles: Student C
T: Here are some blocks in a row. Start
with the one on this end and count them.
S: One, two, three, four, five, six. There aresix.
T: What if you start at the other end of the
row and count them?
S: I already counted them! There are six!
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Counting Principles: Student D
T: Here are some red blocks (4) in a row.
Start with this one on the end and count
them. S: (Points to each but says two numbers
with each point) One, two, three, four, five,
six, seven, eight. Eight red blocks.
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Construct a Pattern
The color spinner will be used to select the colorsto be used in constructing the pattern.
Using the colors chosen, make a pattern that
repeats atleast twice in the time allotted.
When the time is up, compare patterns with others.Did you come up with a unique pattern idea?
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Construct a Pattern
Try another pattern with new colors ordifferent types of manipulatives.
Why should students have an opportunity toparticipate in patterning activities?
Do you have other suggestions forpatterning activities?
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Try out In the Classroom 7-3, Numbers on aCalendar. Then Make up additional problems andshare them with the class.
Use calculators to complete In the Classroom 7-4and 7-5. Discuss other calculator counting activitiesthat could be done with children.
Complete In the Classroom 7-6 and 7-7. Then usethe calculator and crayons to shade some of the100's charts. Look for patterns that emerge.
Use the five- and ten-frames
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Relationships Among Numbers
Spatial Relationships
One and Two More, One and Two Less
Number Benchmarks
Part-part-whole Relationships
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Relationships Among Numbers
Spatial Relationships
Children can learn to recognize sets of objects in patterned
arrangements and tell how many without counting. Prior to
counting, children are aware of small numbers of things: one
nose, two hands, three wheels on a tricycle.
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Relationships Among Numbers
One and Two More, One and Two Less
The two-more-than and two-less than relationships
involve more than just the ability to count on two or
count back two. Children should know that 7, for
example, is 1 more than 6 and also 2 less than 9.
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Relationships Among Numbers
Number Benchmarks
1. Benchmarks or anchors give students a referencepoint. Since 10 plays such a large role in ournumeration system and because two fives make up10, it is very useful to develop relationships for the
numbers 1 to 10 to the important anchors of 5 and10. (e.g. 8 is 5 and 3 more or two away from 10)
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Relationships Among Numbers
Part-part-whole Relationships
To conceptualize a number as being
made up of two or more parts.
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Writing numerals: samples
Samples taken from kindergarten
children during the third month of
school. The children were asked tobegin with one and write their numbers
to ten. The samples are intended to
show the wide range of writing skillsencountered within one class of
students.
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Chad: Third Month of Kindergarten
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Cody: Third Month of Kindergarten
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Steven: Third Month of
Kindergarten
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Elizabeth: Third Month of
Kindergarten
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Andrea: Third Month of
Kindergarten
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Jason: Third
Month of
Kindergarten
Joshua: Fifth Month of Second
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Joshua: Fifth Month of Second
Grade
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Writing Numerals
Start with very clear, very strong models.
Focus on one number at a time.
Provide maximum guidance at first.
Be accepting of initial efforts.
Gently reduce the amount of guidance.
Reward correct performance.
Review previously-learned material at regular
intervals.
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Frame for Development of Number
Emergent: pre-number reasoning
Early: counting all
Developing: counting on
Expanding: early grouping
Established: later grouping
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