Whole Number Sense - Development for age 6 - 8+ Concepts, Activities, Assessment, and Evaluation

Overview

While there may not always be an exact demarcation between PreNumber sense and number sense the understanding of cardinality, which includes: knowing how to classify or create a group, number sequence, counting sequence, counting systematically, synchrony, one-to-one correspondence, conservation of number, systematic organization of objects, and numbers have values when connected to real objects, is being used here. For information See PreNumber sense (0 - age 6)

Number sense includes these ideas and extends them by including the idea that numbers can be represented in multiple ways and have relationships to other numbers.

Cardinality

Counting objects results in a cardinal result (numbered value). Students will repeat or emphasize the last word to differentiate between the counting number and the total value of the group of objects to identify the value (cardinality) with great assurance.

Will say: "one two three four five six seven, seven."
Will represent seven --->>> x x x x x x x
or recognize a group of five and a group of two and say seven.

Numbers have values when connected to real objects and measures.

A student may count two quantities with one quantity being one more or less than the other and know the cardinalities (values), but not know that the second quantity is one more or one less than the first.

Ordinality

Objects can be ordered by number values (numerically) according to their relative position.

Number notation

• Symbols can be used to represent numbers (number words (one...) and numerals (1, 2, ...)
• Symbols can be used to represent relationships (< > =) equal, not equal, greater than, less than or number or sets.
• Symbols can be used to represent operations (+ - * /) with numbers.
• Simple number sentences include numbers, relationships and operations.
• Equations are number sentences, which claim to state equal or unequal relationships (equality or inequality).

Number relationships

• Numbers are related to other numbers through a variety of relationships. These relationships of numbers can be compared as less (<), greater (>), equal (=), and unequal (≠).
• Equivalency is a comparison of two groups with the same value (cardinality).
• Greater than and less than is understood when:
1. the value of numbers is known (cardinality),
2. conservation of numbers is possible,
3. ordinality is understood,
4. and can be represented symbolically if the symbols and their meaning are known and understood.

Simple one to one relationships

• Numbers can be compared in value as greater than, less than, or equal to another.
• A relationship of one more exists for all numbers.
• One less exists for all whole numbers except zero.
• Two more exists for all numbers, three more, ...
• Two less exists for all whole numbers greater than two, three less, ...

Relationships of one to many, many to one, and many to many

• Numbers can be represented in different forms. (four, 4, 2+2, 3+1, 4+0, 3+1=4+0,...)
• Smaller sequences can be included in larger sequences (inclusion) (1,2,3; is also in 1,2,3,4 and so on). (number sequence and counting).
• Sequences of counting can be joined to make longer sequences. Counting from one to four can be joined with counting from one to three and it is the same as counting from one to seven. Or counting from 1 to 3 and 1 to 4 is the same as from 1 to 7.
• Addends have independent cardinal values (5 and 3) and are part of the sum (8) of the two numbers (5 + 3 = 8).
• Simple patterns can be recognized in more complicated patterns (inclusion) (squares in squares, 1=0+1,1+0; 2=0+2,1+1,2+0; 3=0+3,1+2,2+1,3+0).
• Parts can be compared to other parts and wholes (7 = 3 + 4 and 3 + 4 = 5 + 2).
• Equivalency is a comparison of one number or more numbers combined with an operation to another number or two or more numbers combined by an operation (5 = 2 + 3; 4 + 1 = 5; 3 + 2 = 0 + 5.

Hierarchical inclusion

Hierarchical inclusion of a counting sequence

The idea that all numbers preeceding a number can be or are systematically included in the value of another selected number. The idea that yellow cube isn't just the third object in a group of four cubes, but also represents all three cubes in the set of four cubes. Or the numbers 1, 2, 3 are also included in 1, 2, 3, 4. Or every number below any selected number is included in a counting sequence when counting to that number.

Know that every number contains the sequence of all the numbers that are smaller than the largest number in any sequence. A totally embedded sequence for cardinality of all numbers.

• 1
• 1 2
• 1 2 3
• 1 2 3 4

For hierarchical inclusion students must know what numbers come before and after a number in sequence (one more and one less). The student must be able to conserve and know cardinality. Know that ordinality is contain within cardinality and there is a total embedded reversibility of sequences for cardinality and ordinality of numbers.

At a begining level of hierarchical inclusion a student could be asked to count some objects (seven) and put them in a container. Then a person takes some out (three), covers the container with the remaining, and asks how many are remaining. If they can answer correctly (four), then they have constructed hierarchical inclusion.

Counting on

• The sequence words can be cardinal values and are recognized as embedded in the total.
• To count on students must know cardinality and hierarchical inclusion of the counting sequence.

Instead of counting to four for the following group of o and x's.

...........0 0 0 0........................... x ...x..... x
They say four and count............five six seven

Count back

Imagine a set of 8 objects with the eighth, seventh, and sixth object being removed (8 - 3 = 5), leaving 5. Or a group of eight objects with 8, 7, 6 being removed to leave 5.

Hierarchical inclusion of a counting sequence related to number value and addition

• A part can be compared to a whole. Whole being 5 can be composed of 3 + 2 = 5; therefore, in this case, a part (3) is less than the whole (5), 3 < 5; or 3 is two less than 5.
• Parts can be compared to wholes, part part whole (2 + 3 compared to 5 )
• Hierarchical inclusion for additon is knowing that each number has all combinations of addends for numbers before it:
5 = 0+5, 1+4, 2+3, 3+2, 4+1, 5+0.

one two three four = four ........one two three = three;
x......x......x.....x................ ......o.....o.....o; is the same as seven
one two three four five six seven = seven;
x......x.....x......x.....o....o.....o

Number value and operations

It is helpful to compare numbers to groups of five and ten (five as an anchor and significance of ten in place value) .

A missing part can be found if the total and left over amounts are known

The sum of three addends in a sequence are equal to the sum of two addends in an equal sequence: 6 + 6 + 1 = 7 + 6. Used for solving the problem of 7 + 6 by thinking of it as 12 + 1 = 13; because the 7 can be decomposed into 6 + 1, and 6 + 6 can be thought of as 12, then add the one. Simplified as 6 + 6 + 1 = 13, therefore 6 + 7 = 13

Resources

Zero and numbers beyond ten

• Value of zero as absence of something or a starting point.
• Zero is equivalent to n - n

Activities

• Use an empty plate to represent zero objects on the plate.
• Ask questions such as how many elephants in the room, zero.
• Use bags of objects to sequence numbers starting with an empty bag for zero. Later combine objects in one bag to zero objects in another bag. Suggest several problems with zero as an addend and include both kinds of problems 3 + 0 = 3 and 0 + 3 = 3. Have others make and share problems.

Use of zero as a place saver is problematic for students not only as a place saver, but because it has two functions: place saver, and value of nothing. Similarly it becomes problematic for students to realize that as other numerals are moved into different positions they can simultaneously represent values other than the counting value they originally learned for them.

It is easier for children to understand the values of the numerals 11, 12, 13,... than the value of the numeral 10. It is easier to understand the values of 14, 15, 16,... than 11, and 12. In some languages these values said as ten an one, ten and two, ten and three ...

Nominal numbers

Nominal numbers are used for names. She is number four on the team.

Infinite

• Infinite has no bounds or limits
• There can be infinite number systems (Roman, Mayan, Egyptian, and Greek

Next see development of Place value concepts