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").

**Operations on Numbers**

- During the first half of this year, the average child will be able to nonverbally and mentally determine sums up to "four" and their subtraction counterparts (e.g., "3 + 1," "4 - 1," "2 + 1," "3 - 2").

- Some children will still be learning how to use informal knowledge gained from everyday experiences to nonverbally estimate sums up to "five" (e.g., for "3 + 2," puts out four to six items to estimate the answer) and their subtraction complements (e.g., for "5 - 2," puts out around three items to estimate the answer). In the second half of this year, some children will be able to use informal knowledge to estimate the sums of addition word problems or their subtraction complements up to "ten."

- Throughout this year, some children will be able to use concrete counting strategies to solve addition word problems (e.g., for a problem involving three and two more, the child counts out three items, puts out two more items, and then counts all the items to determine the answer) and concrete take away strategies to solve subtraction word problems (e.g., for a problem involving five take away two, counts out five items, removes two, and counts the remaining three items to determine the answer) with sums up to "ten" and corresponding differences.

- In the second half of this year, some children can use various addition strategies to mentally determine sums up to "nine."

- In the second half of this year, some children can use existing knowledge and reasoning strategies to logically determine unknown sums up to "18" and their subtraction counterparts, including the "additive and subtractive identity" rule (e.g., "n + 0 = n" and "n - 0 = n"), the" number-after" rule (e.g., "7 + 1" equals the number eight when we count), and the notion that addition doubles have an even sum or form part of the skip count by two's sequence (e.g., "3 + 3 = 6," "4 + 4 = 8," "5 + 5 = 10"...).

- During the first part of this year, some children may still be learning to understand that if you change the size of a part of a collection, then you also change the size of the whole collection. Throughout this year, some children may intuitively recognize that adding to a collection creates a sum greater than the starting amount.

- During the second half of this year, a few children will see that a part is less than the whole as they solve addition word problems (e.g., Bret had three cookies. His mother gave him some more, and now he has five cookies. How many cookies did Bret's mother give him?). A few children will also see that the whole is larger than its composite parts as they solve subtraction word problems (e.g., Chico had five cookies. He ate some, and now he has three left. How many cookies did Chico eat?).

- During the second half of this year, a few four-year-olds can use up to ten objects to construct number partners up to "5" (e.g., 5 = "1 + 4," "2 + 3," "3 + 2," "4 + 1"), and doubles partners up to "10" (e.g., "3 + 3 = 6").

- During the second half of the year, a few children will understand the "part-whole" relationship of addition and will be able to informally solve "part-part-whole" word problems that have a missing whole and sums up to "10" (e.g., Deborah had five chocolate chip cookies and three ginger snap cookies. How many cookies did she have altogether?).

- During the first half of this year, the average child will trade several small items for a larger one (e.g., trades four small candies for a candy bar). Throughout this year, some children will group objects into 5's or 10's, and recognize that the position of a digit in a number affects its value (e.g., recognizes that "23" and "32" are different). During the second half of this year, some children can break down a larger unit (especially "10" and "100") into smaller units, and can combine smaller units into a larger unit.

- During the second half of this year, a few four-year-olds will be able to accurately read multidigit numerals up to "19." At the same time, a very few children may even be able to accurately read multidigit numerals up to "99."

- During the second half of this year, a few four-year-olds may be able to write multidigit numerals up to "99" (e.g., writes "twenty-four" as "24" and not "204").

- During the second half of this year, there may be a small number of four-year-olds who can meaningfully represent multidigit numerals up to "100" in different forms, such as with numerals and grouping/place-value models (e.g., recognizes that "2" in "27" represents two "tens" and "7" indicates seven "ones").

- During the second half of this year, a few four-year-olds will be able to use informal strategies to solve "divvy-up/fair-sharing" problems where up to "10" items are distributed evenly to two or three people (e.g., if Este and Freeha share fairly the "12" cookies they baked, how many cookies would each get?).

**Geometry and Spatial Sense**

- During the first half of this year, some children may still be learning to match shapes, first with same size and orientation, then with different sizes and orientation (e.g., matches simple shapes in form boards and puzzles, sorts simple shapes in a sorter box, etc.). At the same time, some children may be able to recognize and name some variations of a circle, square, triangle and rectangle. The average child can recognize and name these shapes during the second half of this year.

- During the first half of the year, some four-year-olds can build, copy and informally describe two-dimensional shapes. The average child can do this during the second half of the year. Throughout the year, some children will also be able to copy a shape from memory after seeing a model for several seconds.

- In the first half of this year, some four-year-olds will intuitively recognize congruence by matching shapes with other objects that have the same shape and size. The average child will be able to do this during the second half of this year. During the second half of this year, a few four-year-olds can explicitly define the term "congruent" as two shapes with the same size and shape.

- Throughout the year, children can complete increasingly complex puzzles (e.g., four-piece interlocking to eight- or ten-piece puzzles, to puzzles with smaller and up to 15 pieces) and progress in their abilities to put together and take apart shapes (e.g., understands that a whole object such as a pizza can be separated into parts). Children also build three-dimensional structures using multiple types of items (e.g., a rectangular prism, cube and arches). In the second half of the year, some children may create drawings that involve more than two geometric forms.

- In the first half of the year, some children may still create pictures using one shape, but won't yet use shapes in combination. Others may make a picture by combining shapes. By the second half of this year, the average four-year-old will be able to combine shapes to form a picture. Throughout this year, some children may also be able to cover an outline of a shape with other shapes without leaving gaps, first with trial-and-error, and then with foresight.

- Throughout the year, some four-year-olds can break apart simple two-dimensional shapes that have obvious clues for breaking them apart.

- Throughout the year, some children can find some shapes "hidden" in arrangements in which the shapes overlap each other, but are not embedded inside one another.

- Throughout this year, some four-year-olds will understand and use words representing physical relations or positions (e.g., "over," "under," "above," "on," "beside," "next to," "in front," "behind," "in," "inside," "outside," "between," "up," "down," top," "bottom," "front," "back," "near," "far," "left," "right").

- During this year, the average child can use a model of a room or simple picture maps to locate where an object is hidden in a real room.

- During the first half of this year, some four-year-olds will be able to orient objects vertically or horizontally. The average child will be able to do this during the second half of the year.

- Throughout this year, some four-year-olds will informally recognize when a two-dimensional shape has been turned, flipped or otherwise moved, and will also move such shapes in this way.

- During the first half of this year, the average child can informally create two-dimensional shapes and three-dimensional buildings that have symmetry.

**Measurement**

- During the first half of this year, some four-year-olds will still be discovering attributes of objects by filling a container with solids or liquids (e.g., ice cubes or water). These children will also figure out that different sized containers will hold more or less.

- During the first part of this year, the average child will recognize, informally discuss, and develop language to describe attributes such as "big" or "small" (height/area/volume), "long" and "tall" or "short" (length/height), "heavy" or "light" (weight), and "fast" or "slow" (speed).

- Throughout this year, some children will still be learning the concepts of "same" and "different," as well as how to describe the ways in which items are the same or different. Also during the first half of this year, some children compare a single attribute of several objects (e.g., The child says, "She has a bigger piece of cake than I do."). The average child makes such comparisons during the second half of this year. During the first half of this year, some children can order objects from smallest to largest (e.g., lines up from shortest to tallest, nests cups, etc.) and describe relationships among objects (e.g., "big," "bigger," "biggest"). The average child develops these skills during the second half of this year.

- Throughout the year, some children may still be developing their sense of time through their participation in daily activities (e.g., knows the basic sequence of the day). By the end of this year, children should understand daily time concepts like "morning," "afternoon," "night," "earlier," "later," and "soon." Children should also be able to identify basic concepts associated with night/day and seasons, but may still confuse "yesterday," "today," and "tomorrow." During the first half of this year, some children can recite the days of the week and seasons, but cannot tell time. Some children this age also recognize that a specific time is associated with certain events (e.g., favorite TV show comes on at 4:00). The average child understands these things during the second half of this year. Finally, during the second half of this year, some children will have developed a strong sense of time and will know when events close to them take place. They will know the days of the week, the months, and the seasons, but will still be learning how to tell time.

- During the first half of this year, some children may solve a problem by comparing lengths directly (e.g. placing two sticks side by side to see which is longer). The average child does this during the second half of this year.

- During the first half of this year, some children may compare the areas of two objects by placing one object on another. The average child does this during the second half of this year.

- Throughout this year, some four-year-olds will make informal comparisons and estimates (e.g., says, "I'm as tall as the yellow bookshelf.").

**Patterns, Reasoning, and Algebra**

- During the first half of the year, some children will understand a sequence of events when it is clearly explained (e.g., parent says, "First we plug the drain, then we run the water, and finally we take the bath."). The average child shows such understanding during the second half of this year. In addition, children during the second half of this year recognize regularities in a variety of contexts (e.g., events, designs, shapes, sets of numbers).

- Throughout this year, some children can identify the "core" of simple repeating patterns (i.e., the basic sequence or building block that is repeated) and extend the pattern by replicating the core (e.g., for the pattern "red/blue/red/blue/red/blue," the child will add "red/blue"). Children show varying levels of progress with this skill through age six. This development is also true for when children imitate pattern sounds and physical movements (e.g., clap, stomp, clap, stomp...).

- During the second half of this year, a few four-year-olds will discover the concepts of "even" numbers (i.e., a number of items that can be shared fairly between two people), and "odd" numbers (i.e., sharing between two people results in a leftover item).

- During the second half of this year, a very small number of four-year-olds may be able to use letters to represent the "core" of a repeating pattern (i.e., the basic sequence or building block that is repeated) of up to "three" elements (e.g., "ABC" for "123123123...").

- Throughout this year, a few four-year-olds may begin to summarize with natural language the ideas of "additive identity" (e.g., says, "You did not add anything, so it is still the same"), "subtractive identity" (e.g., says, "You did not take anything away, so it is still the same"), and "subtractive negation" (e.g., says, "You took it all; there is nothing left."). During the second half of this year, a small number of children may also summarize "additive commutativity" (e.g., says, "You can add numbers in any order.").

- During the first half of the year, the average four-year-old will be able to use deductive reasoning (using what we know to logically reason out a conclusion about what we do not know) to solve everyday problems (e.g., figures out which child is missing by looking at children who are present).

- During the first half of this year, some children will move beyond using arbitrary rules (e.g., creating a category for "because I like it") to complete an adult-imposed classification task. Instead, these children can stick with one feature (e.g., color, shape, size) in sorting objects into a class. The average child can stick with one feature when classifying during the second half of this year. Throughout this year, a small number of children will be able to sort and classify on the basis of one or more characteristics (e.g., color, size, etc.), and can articulate why items are grouped together.

- During the first half of this year, some children will be able to reason "transitively" (e.g., if Abby is older than Betsy, and Betsy is older than Charlene, then Abby is also older than Charlene). The average child can reason this way during the second half of this year. A small number of children throughout the year will also be able to sequence events chronologically.

**Statistics and Probability**

- During this year, some children will recognize that some questions, issues, or areas of disagreement are "empirical questions" that cannot be answered without first collecting data. Also, children will be able to collect relevant data for addressing a question or making a decision of personal importance.

- Throughout this year, children will learn to organize and describe data (e.g., by constructing real or picture graphs) to address a question (e.g., What eye color is most common in the family?) or make a decision of personal importance (e.g., Which ice cream shop has the most flavors?).

- Children will develop skills to read and interpret real graphs or picture graphs that summarize information needed to address a question, make a prediction, communicate to others, or make a decision of personal importance.

- Children will begin to understand that some events are more likely to occur than others (e.g., snow is more likely in winter than in summer). They will also begin to understand and use the language of probability (e.g., "certain" or "sure," "uncertain" or "unsure," "likely" or "probable," "unlikely" or "improbable," "maybe" or "possible," and "impossible").

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

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