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

See also PreNumber sense ( age 0 - 6)

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.


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.


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

Number notation

Number relationships

Simple one to one relationships

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

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.

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

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

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;

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


Zero and numbers beyond ten


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.


Next see development of Place value concepts

Addition and subtraction checklist


Dr. Robert Sweetland's notes
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