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# Child Development Tracker: Mathematics From Age 3 to 4

Three-year-olds improve their logical reasoning skills as they play. They can put together simple puzzles, and understand that a whole object can be separated into parts. They identify and describe objects that are the "same" or "different". Three-year-olds can count up to "five," and begin to recognize written numerals "0" through "9." When counting items in a collection, they can now label each object with just one number word to determine the total ("one to one correspondence").

### Numbers

• At 36 months, the average child understands the words "one," "two," and "three" (e.g., distinguishes "one," "two," and "three" from "many"; can identify collections of three items as "three"; asks for "three" of something; knows age; can put out "one," "two," or "three" items upon request). Some children at the beginning of this year may also understand number words up to "four." Some three-year-olds will still be developing an understanding of the words "one" and "two," and may not grasp these concepts until age four.
• While some children are still learning how to verbally count by ones in the correct order up to "three," the average child can count up to "five." Some three-year-olds will also be able to verbally count by ones up to "ten," and possibly beyond, but not necessarily in the correct order. A very few children will be able to use the "teen" counting pattern to count up to "twenty."
• At the beginning of this year, some children will accurately determine the number of items in a collection of up to five items by using one-to-one counting, or "enumeration" (i.e., the child labels each item in a collection with one and only one number word from the counting sequence to determine the total number of items in the collection). The average child will be able to do this in the second half of this year. A few children may also recognize that the last number word used to count (enumerate) a collection has special significance because it represents the total number of items in the collection.
• Some three-year-olds will be able to use the counting sequence to accurately count out a collection up to five items, in response to a verbal request (e.g., if asked, "Give me three blocks," counts out three blocks from a pile, labeling each in turn, "One, two, three.").
• In the second half of this year, some children will be able to name a number that comes after a specific count term from "one" to "nine" when given a running start (e.g., "What comes after 1, 2, 3, 4, 5?"). A very few three-year-olds will be able to name a specified count term between "one" and "nine" without being given the preceding sequence.
• At 36 months, some children may still be learning how to correctly use the size terms "many" and "same" when making comparisons. The average child is able to use the size word "more" to identify the larger of two obviously different-sized collections. Some children may also be able to use the size word, "fewer," to appropriately identify the smaller of two obviously different-sized collections. The average child understands the meaning of "fewer" during the second half of this year.
• In the second half of this year, a very small number of children may begin to use the "larger-number principle" (i.e., the later a number appears in the counting sequence, the larger the quantity represented) and number-after knowledge to determine which of two "neighboring" numbers up to "five" (e.g., "three" and "four") is "more" (e.g., "Which number is more, 'two' or 'three'?").
• Some three-year-olds are capable of effectively using the ordinal terms "first" and "last."
• During this year, a very few children may start to draw objects, make a tally, or use some other informal symbol to represent a spoken number.
• During the second half of this year, some children will be able to recognize or read one-digit numerals "0" to "9" (e.g., is able to point out a "three" given a choice of five numerals, or identifies the numeral "3" as "three"). Also in the second half of this year, a few children may be able to connect at least some numerals to both number words and the quantities they represent (e.g., uses one-digit written numerals to represent the value of a collection, identifies the larger of two written numerals, recognizes that "0" can mean "none").

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