Notes on Development of Number Value From pre K Ð Third and Beyond ???? (Linear, but ALL Over Town)
Associating names to objects, values, sets, actionsÉ Memorizing or generating steps for procedure sequences. The pink dot on the wall is number one, the yellow is number two, the green is number threeÉ 
Associating a concrete physical operation to a transformation through all phases of transformation from start to finish and back again (reversibility). There are three dots. The pink dot is number one, the yellow is number two, the green is number threeÉ Or you could count the green dot as number one, the yellow as two, and the pink is three. It doesnÕt matter. 
Not only can any dot be counted but there can be pieces of dots represented between each dot to get fractional dots for numbers. 
Formal understanding includes transformation, reversible understanding beyond concrete representations to theoretical mental constructs. The dots are not needed. Counting represents a system of associating a number value from an infinite continuous sequence of numbers that can represent any number value or measurement no matter how small or large. 

Classification 
Measuring 
Counting 
Number value 
Operations 
Objects have properties. Properties include size, shape, color, textureÉ Objects are identified by names. Objects have more than one property. Objects with similar properties are the same. Objects with different properties are different. Objects can have properties that are the same and different, but still be the same (triangles  same shape, different size, color). Objects can be sorted by one similar property/ attribute. A set of objects can be sorted in different ways, with one property. Objects can be grouped into sets/ groups by a single property. Sorted sets can be described by a common property. A set can be made with the label of other to sort objects without the distinguishing property of objects in other the sets. 
Without conservation students use perception of size (length, volume, surface area) to order objects and make comparisons (>, <, =). Children seem to interpret number value similar to length. 
Counting as connected words learned by rote memory. The sequence of number words is memorized such as: "onetwothreefourfivesixseven" Words are not differentiated and their values, if understood, are not associated with the process of counting. Students count objects without a plan as to where to start or stop and do not arrange objects with a clear beginning and end. Nor do they see a need to do so when observing others who do. The strategy of putting objects into groups to count is also seen as not necessary. 
Children seem to be able to understand 1, 2, 3, and sometimes 4 equality by perception not cardinality. Perceive one or two or three objects as a whole and can associate a value with the objects. Most likely without mathematical thinking (birds can also). Numbers are names used for symbols (not value). 

Visual and spatial patterns can be similar and different. Visual and spatial patterns can be represented in sketches or drawings. 
Value as measurement. Length is a property of an object that extend the farthest in two directions. Length can be measured by direct comparison. Same as, is seen as a comparison of two or more objects of equal lengths. More, is seen as the longest length in the comparison of two objects. Less, is seen as the shortest length in the comparison of two objects. Least, is seen as the smallest length in a comparison of three or more objects. Most, is a comparison of three or more objects with longest value. 
Counting words are separate words. "one two three four five six seven" A few values may be understood, (14) and maybe larger, but can be easily confused if based on memory and not on understanding of number value. Synchrony is the attempt to use one word for every object (onetoone correspondence). Developed before onetoone correspondence. Children can connect one word to one object, before actually realizing onetoone correspondence. May not order objects so that 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 (1020) 
Two is one more than one. One is one less than two. Three is one more than two. Two is one less than three. Four is one more than three. Three is one less than four. Three is two more than one. Four is two more than two. Two is two less than four. 
Two is one more than one. One is one less than two. Three is one more than two. Two is one less than three. Four is one more than three. Three is one less than four. Three is two more than one. Four is two more than two. Two is two less than four. 
Sets have cardinality. Cardinality of sets can be the same or different. Objects with similar properties that change sequentially can be ordered by that property. Classification by common properties creates similar groups. Sorted sets can be described by their common properties. 
The position of one object to another is important when making comparisons. The relative position of objects is important when making comparisons among them. 
Onetoone correspondence is one numeral related to one object for each object being counted. Students pair words with objects. To count objects a clear beginning and ending point needs to be determined. Students usually start by pointing to each object and moving each object as they count. Later point without touching. Total embedded reversibility and sequences for cardinality and ordinality of numbers. (Example One through four is embedded in one through seven. One two three four five six seven x ..... x .... x É. x ... x .. x .... x ) Onetoone correspondence is constructed before conservation of numbers. Once students know how to count they should be encouraged to develop more efficient strategies to determine cardinality. Objects can be grouped (mentally and physically) for more efficient counting. 
Conservation of number must be constructed before cardinality. Counting tells how many objects in a set (cardinality). Cardinality can be determined through pattern recognition and counting. Numbers (number values, cardinalities) can be represented in different forms (numerals/ symbols, numeral words, pictures, É). Numbers have values when connected to real objects and measures Sets can be created of any size. Quantities can be represented with numbers, words, numerals, sets of objects, picturesÉ 
Counting up can be represented with physical objects and drawings of a growing set up to 5. Objects can be arranged physically or in a drawing in different ways to represent the same (equivalent) number value (cardinality). (15) Counting up can be represented with physical objects and drawings of a growing set up to 10. Objects can be arranged physically or in a drawing in different ways to represent the same (equivalent) number value (cardinality). (110) 
Objects in a group share some characteristics while differing in others. A group of objects may be sub classified as members of an ascending hierarchy. A group or set can include ideas as well as things. Thinking about things as groups or sets means looking at how every element relates to other members of the group or set. Objects may have properties of two different groups or sets. Order is created by properties that change sequentially 
Less is the number with the smallest cardinality (value) when comparing two numbers. More is the number with the largest cardinality (value) when comparing two numbers. Least is the number with the lowest cardinality (value) when comparing three or more numbers. Most is the number with the largest cardinality (value) when comparing three or more numbers. Less, least, more, and most can be determined by value and/ or measurement 
Recognize numerals and number words. A procedure for counting matches numerals and numbers to what is being counted in a onetoone relationship. Different strategies for counting and methods for keeping track of quantities can be used to maximize accuracy to (10). Students will repeat or emphasize last word to mean the total value of a set of objects. Zero is the absence of objects. 
Less and more can be determined by value and/ or measurement As many as, equivalency, or equal is a comparison with the same cardinality. Greater than and less than is understood when: 1. the value of numbers is known, 2. conservation of numbers is possible, and 3. the symbols, their meaning, and use is understood. Value of zero 
Changes in number value (increases, decreases, growthÉ) can be kept track of with pictures, words, symbolsÉ (1 to 10). 
Ordinals are first, second, third, É Ordinal words can be used to describe ordered sequences A procedure for ordering, matches ordinal words and ordinality to what is being ordered in a onetoone relationship and order. Different strategies for ordering and methods for keeping track of order can be used to maximize accuracy (110). 
Objects can be ordered with numbers (ordinality). Quantities can be ordered from least to most and from most to least. 

Greater than, less than, longer, shorter, longest, shortest, or other size terms related to measurement is understood when: 1. the value of numbers is known, 2. conservation of numbers is possible, 3. conservation of length, area, mass, volume, É 4. units of measurement 5. visualize units of measurement 6. iterate a unit of measurement 7. Know the symbols, their meaning, and use. 
A procedure for counting matches numerals and numbers to what is being counted in a onetoone relationship. Different strategies for counting and methods for keeping track of quantities can be used to maximize accuracy to (20). 
One and two more and one or two less are helpful in comparison of numbers. Numbers can be represented in different forms. Simple patterns can be recognized in more complicated patterns (inclusion). Numbers are related to other numbers through a variety of relationships. 
Objects can be arranged physically or in a drawing in different ways to represent the same (equivalent) number value (cardinality). (020) Counting up can be represented with physical objects and drawings of a growing set from 020. Changes in number value (increases, decreases, growthÉ) can be kept track of with pictures, words, symbolsÉ (0 to 20). 

Cardinality can be determined through pattern recognition of the cardinality of all subsets and adding them or transforming them into patterns with equivalent cardinality. 
Numbers can be combined (added) (110). 

Function matches one set to another set. Represent twotoone correspondence ((function) shoes Ð peopleÉ) 

1 2 3 4(=four) + 1 2 3 (=three) x x x x + o o o 1 2 3 4 5 6 7 (=seven) x x x x o o o Addends can be independent cardinal values and part of the sum of the two numbers. For students to know hierarchical inclusion they must know what numbers come before and after a number in sequence, conserve numbers, and know cardinality. Can assess by counting seven objects and putting them in a container. Take some out and cover the container. Then ask how many are remaining in the container. If a student consistently answers correctly, they have constructed hierarchical inclusion. 
1 2 3 4(=four) + 1 2 3 (= 3) x x x x + o o o 1 2 3 4 5 6 7 (=seven) x x x x o o o Addends can be independent cardinal values and part of the sum of the two numbers. For students to know hierarchical inclusion they must know what numbers come before and after a number in sequence, conserve numbers, and know cardinality. Can assess by counting seven objects and putting them in a container. Take some out and cover the container. Then ask how many are remaining in the container. If a student consistently answers correctly, they have constructed hierarchical inclusion. 
Smaller sequences can be included in larger sequences. Addends can be independent cardinal values and part of the sum of the two numbers (5 + 3 = 8). Simple patterns can be recognized in more complicated patterns. Parts can be compared to other parts and wholes (7 = 3 + 4 and 5 + 2). The sum of three addends in a sequence are equal the sum of two addends in an equal sequence. Totally embedded sequence for cardinality of numbers. Total ordinality within cardinality. Sequences can make up other sequence(s). Until students recognize that both sequences make up the sequence of seven. Students will count out four (cardinality for four). Count out three more (cardinality for three). Put them together and count the total (seven). There is total embedded reversibility sequences for cardinality and ordinality of numbers 

Counting on. Sequence words (counting numbers) are cardinal values that are embedded in a total. To count on students must know cardinality and hierarchical inclusion. 0 0 0 0 x x x four five six seven Each number is a combination of all the counting numbers that come before it (combinations of addends) 5 = 1 + 4, 2 + 3, 3 + 2, 4 + 1 Every number contains the sequences of all the numbers that are smaller than the largest number in the sequence. 1 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 6 + 6 + 1 = 13 6 + 7 = 13 or 7 + 6 = 12 + 1 = 13 because 7 = 6 + 1, and 6 + 6 = 12, 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 It is helpful to compare numbers to multiples of five and ten Zero is equivalent to n Ð n 
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. A missing part can be found if the total and left over amounts are known. Parts can be compared to other parts and wholes (7 = 3 + 4 and 5 + 2). A missing part can be found if the total and left over amounts are known. One more, two more and one less and two less are helpful in comparison of numbers. It is helpful to compare numbers to multiples of five and ten. Ordinality objects can be ordered with numbers. Objects can be ordered with numbers. Add and subtract quantities to 20 Problems can have more than one solution. Solutions can be written in number sentences 

Symbols can be used to represent numbers (number words and numerals) and relationships (< >) or equality (equal, not equal, greater than, less than, construct equal sets, sets, less than, or greater than) Operational symbols (+  * /) can be used to represent operations. 

Place value concepts Sets of ten objects can be thought of as groups of ten. (significance of ten) It is easier for children to understand 11, 12, 13,... than 10. Count by tens, know 10 more and ten less Unitize (students count each group as one unit (1 of 10, 2 of 10, 3 of 10 ...) and realize that each unit has and must have an equivalent number of objects in it. Don't need to count by tens to know that five groups is 50. Understands that groups can be regrouped as other equivalent groups (one group of ten is also two groups of five). Until about third grade students act as if a group of ten the ten individual objects in the group are two different entities. A group of ten straws and ten straws are not different they are the same thing. Students that persist in representing the two in 23 with two blocks instead of twenty blocks are demonstrating this phenomenon. Young students that inventory books in the classroom by putting rubber bands around ten books and counting the books as groups of ten and extras probably are not using place value to determine a total number of books, but are thinking of each group of ten as a convenient collection of ten blocks, not one group of ten units as well as ten units. If this idea is expanded to one group of hundreds can be unitized as one group (100Õs), ten groups (10Õs), or one hundred groups (1Õs). Numbers to 1 000 Multiple 100 charts can be used to represent number values to 1 000. What about an number roll to 1 000? Infinite has no bounds or limits There can be infinite number systems (Roman, Mayan, Egyptian, and Greek 

Fractional values
Fractions are equal parts of a whole. 
Dr. Robert Sweetland's Notes ©