Pre Number Sense - Development (age 0 - six)
Concepts, sample activities, and assessment ideas
See also [ Concepts for Number sense (age 6 - age 8)] [Counting summary]
Classification
Classification or categorizing is integral to knowing and using number sense. Different ways to categorize and classify must be known to understand different ways a group or groups can be made before understanding how to represent them numerically, as a quantity (cardinality) or other representation. Classification is seen in children's play as they sort, organize, group and regroup objects. Classify by one property (or attribute) using general properties and specific properties. Selecting objects using classification to duplicate a pattern. Get confused when objects have multiple properties with which they are classifying and will eventually learn to classify groups (sets) of objects by multiple attributes (properties) and represent them with Venn diagrams. See classification concepts and misconceptions.
Counting numerals and synchrony
After children classify, they will question: How many objects are in the different groups they create. Learning to count takes time and is initially aided or hampered by our visual spatial skills. Counting is motivated by questions students ask such as: how many? are there more? less? or are they the same?
Counting is first learned as connected words by rote. The sequence of number words are memorized. "onetwothreefourfivesixseven" At first words may not be differentiated and few values are understood.
Later the words are differentiated in the sequence "one, two, three, four, five, six, seven..." A few values may be understood. Responding with a counting sequence depends entirely on memory, not on the value of numbers (cardianlity) in the sequence.
Synchrony is the attempt to use one word for every object (1 - 1 tagging). It is developed before one-to-one correspondence. 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.
Young children will count objects without thinking:
- An object ought to be selected to start the counting (starting line).
- An object ought to be selected to stop the counting (finish line).
- A path ought to be selected from the beginning object to the finishing object so each object can be included.
- The path for counting should be such that each object is counted once and only once (one-to-one coorespondence).
- Objects can be rearranged to faciliatate counting (conservation).
- About the total value (cardinality).
Suggestions:
When students learn to count, they want to use it to solve all problems. This is detrimental for learning and using mathematics effectively, efficiently, and with understanding. Once students memorize number words sequentially, they should stop counting and we should help them move toward a greater understanding of number sense and number value and how values related to different operations.
One-to-one correspondence
One-to-one correspondence is when one numeral is related to one object being counted. Students pair words with objects. Students usually start by pointing to each object and moving each object as they count. Later point without touching.
One, two, three, four, five, six, seven
x........x......x......x......x....x......x
One-to-one correspondence is constructed before conservation of numbers.
Suggestions:
Once students know how to count they should be encouraged to develop more efficient strategies(subitizing and construction) to determine quantities (cardinality).
Activities
- One-one
- Number trails
Counting systematically
With experience students will count objects in an organized manner to help them remember with what object to start counting, a path of objects to count each and every object, and with what object to stop the counting. Additionally they will understand the value will not change if objects are moved, there by allowing them to put objects into pairs, groups of five, or tens to assist counting and accuracy.
Subitizing
Subitizing is being able to identify the value of a group of objects by looking at it without actually counting the objects. Children can do this at a very young age and seem to be able to understand one, two, and three as a perceptual magnitude not cardinality. Being able to perceive two or three as a whole without doing mathematical thinking can be done by birds and some other animals. Therefore, young children can label small groups, as two, three, or four, accurately by subitizing, but no cardinality maybe recognized for the group. However, as children mature and gain experience it can also be a confident judgement of cardinality for a small number of objects.
Subitize, comes from the Latin subitus meaning sudden, which represents the sudden dawning of that's three or four. It was suggested in 1949 by E. L. Kaufman, M. W. Reese, T. W. Volkmann, and J. Volkmann.
It is possible for children to identify two fingers as two fingers. Or say they have five fingers. Or that you are holding up three fingers, without knowing the value of the the numbers. Particularly for five and above. They can recognize the value of a group through perceptual recognition of visual patterns of the objects' relative positions (subitizing) or through procedural counting without understanding the group of object's cardinality.
Benefits, people can learn to subitize and develop skill with practice. The benefit of drilling students with subitization exercises, rather than counting or basic fact operations, will increase the student's ability in number sense. Relationships of number quantity, cardinality, conservation of numbers, and proportion.
Also being able to recognize and apply it in different instances is a first step toward being able to use mathematics in a flexible and creative way. With practice students will be able to quickly subitize values of subgroups within a larger group and mentally join the subgroups to find the total value of the larger group.
Suggestions:
Additional ideas on subitizing
Conservation of numbers
Conservation of numbers must be constructed before cardinality. Children conserve numbers when they know that the value (cardinality) of a group of five is five no matter where the five objects are placed. No matter if the five horses are gathered by the fence or are in the barn or if they are spread out across a whole 10 acre field. Or the value of a group of five objects is the same as the value of a different group of five objects. Even if the size of objects is different. Say five elephants or five mice. See also notes on development...
Activities
Number value or Cardinality
Cardinality can be determind by counting, pattern recognition (recognition of objects' relative positions relative to a cardinality related to the positioning) or by construction and deconstruction of subsets (recognize five dots on a die as 3 + 2). Amounts such as one, two, and three can be recognized and comnbined in a variety of ways for quick recognition of larger groups without counting. Similary five can be recognized or constructed as an anchor and combined with other groupings.
Young children who conserve and understand cardinality usually count each object in the selected group and repeat the cardinal value after counting the objects in a group. 1, 2, 3, 4, 5..... Five.
To understand cardinality, students must know how to classify or create a group, number sequence (counting sequence), one-to-one correspondence, conservation of number, systematic organization of objects, and numbers have values when connected to real objects.
Resources: cardinality rubric
- Listing of activities
- Number - numeral associations. Numbers can be represented with symbols. Cardinality can be represented with symbols (numerals) recogniton and formation of numerals checklist .
- number keepers,
- Unifix cubes,
- Groovy boards,
- Counting bears, Fingers,
- Matching cards and objects... Numeral cards, Number word cards .
- Global summary checksheet for number value
Zero and numbers beyond ten
Value of zero as absence of something or a starting point.
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 are said as ten anc one, ten and two, ten and three, and so on ...