Developmental of understanding Number Sequence or Counting

See also PreNumber sense (0 - age 6+) and Number sense (age 6-8+)

The following focuses on counting, however counting is a useless procedure without associating it with a particular purpose (number value, place value, ...). Further a focus on counting is detrimental to both understanding and skill in the use of mathematics.

  • Connected words
  • Synchrony
  • One-to-one correspondence
  • Cardinality
  • Smaller sequence inclusion
  • Hierarchical inclusion
  • Addends as cardinal values
  • Counting on
  • Total hierarchical inclusion
  • Sum three addends
  • Count back

Connected words

Sequence of number words are memorized. "onetwothreefourfivesixseven" Words are not differentiated and very few values are understood.

Sequence of words are differentiated. "one two three four five six seven" A few values are understood.


Students connect one word to one object but not one-to-one correspondence yet. May not be able to order objects so some objects are counted more than once, or missed, may not know the sequence of all the numbers needed to count a set of objects (10-20), ...

One-to-one correspondence

One - to - one correspondence is understood. Students pair words with objects. Students may point to each object and move each object as they count.

One two three four five six seven



Counting objects can result in a cardinal result (numbered value). Students will repeat or emphasize last word to mean the total value of a set of objects.

one two three four five six seven = seven

x x x x x x x

Smaller sequences are included in larger sequences.

Students will recognize that four plus three are both included in seven.

Hierarchical inclusion

They will count out four (cardinality for four). Count out three more (cardinality for three). Put them together and count the total (seven). They will recognize that both sequences make up the sequence of seven.

Addends can be independent cardinal values and part of the sum of the two numbers.

one two three four = four two three = three

x.. . ... x. .... x....... x................ ....... o.... o..... o

one two three four five six seven = seven

x ......x..... x....... x.... o... o..... o

Counting on

The sequence words can be cardinal values and are recognized as embedded in the total.

0 0 0 0......... x ...x..... x

four ............five six seven

Totally embedded cardinality of numbers Hierarchical inclusion

Know each number can be creted as combinations of addends for all numbers below.

5 = 0+5, 1+4, 2+3, 3+2, 4+1, 5+0

Total embedded reversibility sequences for cardinality and ordinality of numbers.

Know that every number contains the sequences of all the numbers that are smaller than the largest number in the sequence.


1 2

1 2 3

1 2 3 4

The sum of three addends in a sequence are equal to the sum of two addends in an equal sequence

7 + 6 = 12 + 1 = 13

because 7 = 6 + 1, and 6 + 6 = 12, and 6 + 6 + 1 = 13, therefore 6 + 7 = 13

Count back

8 - 3 = 5

Imagine a set of 8 objects with the eight, seventh, and sixth object being removed, leaving 5.

8, 7, 6 removed and 5 left

Dr. Robert Sweetland's notes
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