Four-year-olds have an increased capacity for learning math concepts. They use logical reasoning to solve everyday problems, and can effectively use language to compare and describe objects and shapes. They can count to "ten," recognize written numerals "0" to "9," and add and subtract using numbers up to "four." Four-year-olds know some variations of a circle, square, triangle and rectangle. They know days of the week, months, and the seasons, but still cannot tell time.

**Numbers**

- At age four, some children may still be gaining an understanding of the number words up to "four" (e.g., distinguishes one-four items from "many"; can identify collections of up to four items with a corresponding number; asks for up to "four" of something; knows age; can put out "one," "two," "three," or "four" items upon request).

- While some children at the beginning of this year are still learning how to verbally count by ones in the correct order up to "five," the average child can count up to "ten," and possibly beyond, but not necessarily in the correct order. A few four-year-olds will be able to use the "teen" counting pattern to accurately count up to "twenty." In the second half of this year, there may also be a few children who use repeating patterns to correctly count up to "forty-two."

- At the beginning of this year, some children may still be learning how to accurately determine the number of items in a collection of up to five items 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). A few may be able to do so with a collection of up to ten items. Also, the average child recognizes 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.

- Around the fourth birthday, the average child will accurately count out up to five items in response to a verbal request. Some children may be able to accurately count out up to ten items, and a few may be able to count out up to 20 items.

- The average child can verbally start counting from a number other than "one" during the second part of this year. Some children may be able to do this earlier in the year.

- At the beginning of this year, the average child can 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?"). Some children will also be able to name a specified count term between "one" and "nine" without being given the preceding sequence. Finally, a few children will be able to name the number after a specified count term between "ten" and "40."

- During the second half of this year, some children will be able to name a number that comes before count terms from "two" to "ten" (e.g., What number comes before "three"?).

- In the second half of this year, a few children will be able to verbally count backwards from "five" or "ten," and/or count by tens up to "100."

- A very small number of children will understand terms related to estimation (e.g., "about," "near," "closer to," "between," "a little less than") around age four-and-a-half.

- In the second half of this year, some children will make a reasonable estimate of the number of items in a collection involving up to five items, and a few children may be able to do this with a collection of up to ten items.

- At the beginning of this year, some children may still be learning how to use the size words "more" and "fewer" to identify the larger of two obviously different-sized collections. A very small number of children will also be able to use formal relational terms correctly, such as "greater than," "less than," and "equal to."

- At the beginning of this year, some children may be able to determine which of two numbers less than "ten" and widely separated in the counting sequence (e.g., "nine" and "three") is "more" by understanding the "larger-number principle" (i.e., the later a number appears in the counting sequence, the larger the quantity represented). The average child will successfully make such determinations during the second half of this year. Some older fours may even be able to determine which of two numbers widely separated in the counting sequence is "less."

- During the fourth year, some children may begin to use the larger-number principle and number-after knowledge to determine which of two "neighboring" numbers (e.g. "three" and "four") in the counting sequence is "more," working with numbers up to "five," and in the second half of the year, working with numbers up to "ten" (e.g. "Which number is more, 'seven' or 'eight'?").

- In the second half of this year, some children will be able to use a mental number line to determine the relative proximity of one-digit numbers (e.g., recognizes that "five" is closer to "three" than to "nine").

- At the beginning of this year, the average child will be able to understand and effectively apply the ordinal terms "first" and "last."

- During this year, a few children may start to draw objects, make a tally, or use some other informal symbol to represent a spoken number.

- A very few children may begin to use informal and symbolic representations (e.g., drawings of objects, a tally, etc.) to represent the number of items in a collection up to "nine."

- At the beginning of this year, some children will be able to recognize or read 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"). The average child will be able to recognize these numbers in the second half of this year. Some children may be able to copy or write numerals "0" to "9." Throughout this year, some 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").

Reprinted with the permission of PBS. © PBS 2003 - 2008, all rights reserved.