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## Mathematics

### Barney and Friends

Practice math skills with Barney and his friends. Match the Shapes

### Mister Rogers

Kids develop spatial sense with this up/down game. Up and Down Fun

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").

### Operations on Numbers

• Throughout this year, some children will still be learning how to nonverbally and mentally determine that one item added to another makes "two," and that one item taken away or subtracted from "two" makes "one." Also during this year, some children 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 be able 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). The average child can do this during the second half of this year.

• During the first part of this year, the average child intuitively recognizes 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 three-year-olds will trade several small items for a larger one (e.g., trades four small candies for a candy bar).

### Geometry and Spatial Sense

• During the first half of this year, some three-year-olds may still be learning how to complete simple "insert" puzzles (e.g., completes a three-piece simple puzzle where pieces are whole objects). Also, some children may be learning how to remove a part from a toy (e.g., a wheel) and replace it.

• 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 one type of item (e.g., a cube) and/or multiple types of items (e.g., a rectangular prism, cube and arches).

• During the first half of this year, the average child creates pictures using one shape, but doesn't yet use shapes in combination.

• Throughout this year, a small number of three-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, some children can use a model of a room or simple picture maps to locate where an object is hidden in a real room.

• Throughout this year, some three-year-olds can informally create two-dimensional shapes and three-dimensional buildings that have symmetry.

### Measurement

• During the first half of this year, the average child will discover attributes of objects by filling a container with solids or liquids (e.g., ice cubes or water). The average child will also understand that different sized containers will hold more or less.

• Throughout this year, some children 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).

• During the first half of this year, some children understand the concepts of "same" and "different," and describe how items are the same or different. The average child can do this during the second half of the year. Also during the first half of this year, a few children begin to make comparisons between several objects based on a single attribute (e.g., says, "She has a bigger piece of cake than I do."). During the second 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").

• Throughout the year, children continue to develop their sense of time through their participation in daily activities (e.g., knows the basic sequence of the day). During the second half of this year, the average child understands daily time concepts like "morning," "afternoon," "night," "earlier," "later," and "soon." The average three-year-old is also able to identify basic concepts associated with night/day and seasons, but sometimes confuses "yesterday," "today," and "tomorrow." Also during the second 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).

### Patterns, Reasoning, and Algebra

• Throughout this year, some children will still be developing a firm understanding of daily time sequence (e.g., time to eat, nap time, etc.). At the same time, some children will be just discovering basic patterns in the environment (e.g., day follows night, patterns in carpeting or clothing, etc.). They also use the terms, "tomorrow," and "yesterday." During the second 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.").

• Throughout this year, some children are just showing interest in patterns or sequence (e.g., attempts to follow patterns with stringing beads, magnetic shapes, peg boards). Children can also 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 very small number of three-year-olds may recognize (from working concretely with small collections) the following general rules and may begin to summarize them with natural language: "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.").

• Throughout the year, some three-year-olds 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).

• Throughout this year, some children will still be learning how to classify, label, and sort familiar objects by a known group (e.g., hard v. soft, large v. small, heavy v. light). A few children will also 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.

• During the second 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).

### Statistics and Probability

• Throughout 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 (e.g., What eye color is most common in the family?) or making a decision of personal importance (e.g., Which ice cream shop has the most flavors?).

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