Chapter 02

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The Foundation of All Math Learning:

Representations of Early Number Concepts

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Mathematical Routine: Tell me Fast

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Mathematical Routine: Tell me Fast

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Mathematical Routine: Tell me Fast

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Mathematical Routine: Tell me Fast

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Mathematical Routine: Tell me Fast

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Conversation in Mathematics

• Discuss what the child’s misconception was around the use of tools.

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Concrete & Abstract Representations

• Pedagogy

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What are Mathematical Tools?

• Often called manipulatives• “Manipulatives” can be interpreted as something

to play with in your hands whereas “tools” indicate that they are to assist in the learning of a math concept.

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Issues Regarding Tools

• Availability• Knowing how to use them• Management• Free choice or teacher initiated

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Constructing Meaning for Tools

• A tool does not have meaning inherent in it. It must be constructed by the learner.

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Moving From Concrete to Abstract Representations

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Making the Connections

Build Draw Record27 + 4 (3+1)

10

20+10+1=31

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Moving Toward More Efficient Strategies

• Students share their work• Students make connections between different

strategies• Students decide on most efficient strategies • Students try on other students’ strategies

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Early Number Concepts

• Content

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Emergent:Oral counting and identification of numbers

Matching:Counting- by 1’s, Number relationships- more/less; relating one number to another

Quantifying:Counting - on/backSkip countingNumber Relationships - part/part/whole

Partitioning:Tens and ones -making tens -expanded notation -ten more/less

Factoring:Multiplication & Division

Operating:Multiplication & Division

Operations with fractions

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Emergent Phase

• Numbers Signify quantity• Rote Count to 10 with words in order• Identify by sight 1-5 objects• Beginning to write numerals• Know more/less/same and that a change will

result in either bigger or smaller (but not by how much)

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Counting in the Matching Phase

What we look for when assessing/observing1-1 correspondencehow they keep track?do they remember a quantity thatthey counted?can they pull out a quantity that you ask?cardinality principledirect modeling

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Student Interview

• We will assess the counting skills of two kindergarten students.

• Identify each of the student’s strengths• What is the number range that each student

should be working in?

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Counting in the Quantifying phase

Counting on, counting back

Counting in groups

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Number Relationships in the Matching Phase

• More / less-

one more / one less

• Relating one number to another

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Number Relationships in the Quantifying Phase

• Part-Part-Whole (“There are numbers within numbers”)-

grab bag subtraction

- building 2 color trains

- snap it

- the tub game

• Number relationships (doubles/doubles plus one)

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The Continuum of Number Concepts Developing from Kindergarten to Algebra