Developing Number Concepts in K-2 Learners

Developing and Assessing Number Sense By Michelle Flaming [email_address]

How Does One Build Number Sense There is NO silver bullet. It takes time. Several components or Building blocks involved.

Design of the class http://8.6.89.92/classroom/portal/ essdack Define each building block. Discuss Examples. View Classroom Vignettes Classroom Activities Diagnostic Assessment Tool

Rote Counting Knowing how to recite numbers in correct order. It is the simplest of counting concepts to learn. Examples: 1,2,3,4,… 22, 32, 42, 52, … 2,4,6,8,…

Rote Counting Tend to memorize through songs, finger plays and rhymes. Different groupings are critical. Tie in other senses, especially motor.

One-to-One Correspondence Definition: When a student says or thinks one number word for each object. One-to-One correspondence is matching one word with one object. Children who are insecure with this concept will say the number words faster or slower than they point to an object.

Subsidizing Definition: Often referred to as “magnitude of a group”. It is one’s ability to look at a group of objects (usually 2-5 objects) and know how many is in the group, and which group has more without even counting. It is the visual recognition of number size.

Subsidizing Example: A student can look at the group of objects and say “four” without actually counting the objects.

Subsidizing U.S. textbooks often do not address this skill. Minilessons Tens frames Arrays Dominoes Sticky Dots

Keeping Track Definition: Keeping track of which numbers or objects they have already counted. It requires another level of sophistication in children’s conceptual understanding of counting.

Keeping Track Example: A student counts the following sets of objects “one”, “two”, “three”, “four”, and “five” and recognized when all objects have been counted, only once, and does not duplicate a count.

Keeping Track Strategies Place in a pile (in groups of five, or ten, etc.) Sliding objects to other side of page Random order: Line, circle, random

Conservation of Number Definition: A “number” means an “amount” and that amount does not change no matter how you arrange the objects.

Conservation of Number The amount is “seven” and doesn’t change.

Conservation of Number NOT dependent upon spatial arrangement. Sometimes referred to as “invariance of number”. Further enhance this skill - Hidden Numbers

Hierarchal Inclusion Definition: An understanding that 19 is inside of twenty, the numbers are nested inside each other and that the numbers grow one each time. Example: 20 is the same as 19+1. If you remove one the number goes back to 19.

Hierarchal Inclusion “1” “2” “3”

Hierarchal Inclusion “Beyond labeling individual objects in a collection with a name, counting eventually involves a further mental act of relating the individual objects into wholes of increasing size. - Labinowics 1980.

Hierarchal Inclusion “Constructing the number as a unit.” Child is able to “see” the number as a unit, while at the same time “seeing it made up of it’s parts”. Richards, Steffe, and von Glaserfeld

Compensation Directly linked to Hierarchal Inclusion. Definition: When working with numbers you can take an amount from one set and add it to another set, the total amount does not change.

Compensation Example: 6+1 = 7. I can take one away from 6 and make it 5, as long as I add the 1 back with the other 1 and make it 2, 5 + 2 = 7. The total amount does not change.

Compensation IMPORTANT conceptual skill. Referred to as “compose and decompose” numbers. Flexibility with numbers

Compensation Suppose the problem is 44 - 28. Many problems with give us the same answer. 43 - 27; 36 - 20 42 - 26; 41 - 25; 40 - 24; 39 - 23; 38 - 22; 37 - 21;

Compensation Strategy Shift both numbers to amounts that don’t require regrouping. Students MUST understand a strategy to be competent with it.

Part/Whole Relationships Definition: The ability to reason with numbers and to work with numbers flexibly, to chose the most appropriate representation of a number for a given circumstance.

Part/Whole Relationships Example: The number “seven” can be represented as: 5 + 2 3 + 4 7 + 0 9 - 2 1 + 6 Etc….

Unitizing/Place Value Definition: Unitizing is the place value understanding that ten can be represented and thought of as one group of ten or ten individual units. HUGE shift in thinking for children. 47 4 tens and 7 ones; 3 tens and 17 ones; 2 tens and 27 ones; 1 ten and 37 ones; 47 ones.

Unitizing/Place Value The number 34 can be represented as: 3 tens, 4 ones 2 tens, 14 ones 1 ten, 24 ones 0 tens, 34 ones Etc…

Unitizing/Place Value “Big Idea” in mathematics. Shift in reasoning, perspective, logic, and in mathematical relationships. Connected to part/whole relationships. Important skill for all operations.

Relationships Definition: Repeated subtraction is the equivalent to division and repeated addition is equivalent to multiplication. The relationship between the operations is necessary before facts can be automatic.

Relationships Research-based Strategy: Cognitively Guided Instruction (Thomas Carpenter)

A Numerically Powerful Child: Decompose of break apart numbers in different ways. Knows how numbers are related to other numbers. Understands how the operations are connected to each other. Connects numerals with situations from life experiences. Creates appropriate representation for numbers/operations.

What mathematical concept does this child have, what concepts are lacking?

Diagnostic Tool - Spreadsheet Contact Information Michelle Flaming - michellef@essdack.org

Developing Number Concepts in K-2 Learners

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Developing Number Concepts in K-2 Learners