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## Mathematics

### Early Math

The next time you make a pancake breakfast, use cookie cutters to have fun with geometry! Cookie Cutter Pancakes

### Sesame Street

Color with your toddler and explore new shapes! Coloring Pages

Two-year-olds learn important math skills from their play and routines. They show symbolic thinking with pretend play, and recognize patterns with daily activities. They also understand what "tomorrow" and "yesterday" mean. Two-year-olds are just beginning to use logical reasoning to solve everyday problems. They can sort shapes, complete puzzles with eight pieces or less, and stack a set of rings on a peg by size. They also understand addition and subtraction with the numbers "one" and "two."

### Numbers

• Around 24 months, the average child uses a few number words without understanding quantity (e.g., imitates a simple counting rhyme). Some may not do so until later this year.

• On average, an older two-year-old understands the words "one" and "two" (e.g., distinguishes "one" or "two" from many; can identify pairs of items as "two"; identifies three or more items as "many" rather than as "one" or "two"; asks for "one" or "two" of something; knows age; responds appropriately to the request, "Take just one," or "Give me two."). A few two-year-olds may also begin to understand the word "three," and a very few may grasp the concept of "four."

• At the beginning of this year, some children will be able to verbally count by ones up to "three," and sometimes beyond, but not necessarily in the correct order. The average child will be able to do this later this year. A few two-year-olds may even be able to count in the correct order up to "five."

• In the second half of this year, a very few children will be able to 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).

• At 24 months, some children can correctly use the size terms "many" and "same" when making comparisons. Some children will also begin to appropriately use the size word, "more," to identify the larger of two obviously different-sized collections.

### Operations on Numbers

• At 24 months, some children will begin 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." The average child, however, will be able to do this during the second half of this year.

• In the second half of this year, a few children may also 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).

• Throughout this year, some two-year-olds intuitively recognize that if you change the size of a part of a collection, then you also change the size of the whole collection.

### Geometry and Spatial Sense

• Throughout this year, some two-year-olds will be able 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.).

• During the first half of this year, some children will still be learning how to recover an object that has been hidden in one place, and then visibly moved to a second position and re-hidden. At the same time, some children may still be learning how to recover an object that has been covered with one item, and then covered with something else while remaining in the same position.

• Between 24 and 30 months, the average child can informally identify and play with solid objects (e.g., picks out a familiar object by touch when that object is placed in a bag with two other objects). Some children will learn to informally identify objects in this way during the second half of this year. During the first half of this year, some two-year-olds will be able to sort, order and build with solids (e.g., when playing with a ring-stacking toy, ignores any forms that have no hole and stacks only rings or other objects with holes; can stack rings on a peg in order of size). The average child can do this during the second half of this year.

• Between 24-36 months, children are typically able to work simple "insert" puzzles (e.g., completes a three-piece simple puzzle where pieces are whole objects). Also, children can 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) 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).

• At age two, some children create pictures using one shape, but don't yet use shapes in combination.

• Throughout this year, a very small number of two-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").

• Throughout this year, a few two-year-olds will be able to informally create two-dimensional shapes and three-dimensional buildings that have symmetry.

### Measurement

• Between 24-30 months, some children still believe that a cracker broken into many pieces is more food

• During the first half of this year, the average two-year-old explores objects by filling and emptying containers (e.g., with sand or water). Such explorations continue for some children throughout the year. Some children will also fill a shape with solids or liquids (e.g., ice cubes or water), and know that different sized containers will hold more or less.

• Throughout this year, a few two-year-olds 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).

• In the second half of this year, some children understand the concepts of "same" and "different," and describe how items are the same or different.

• Throughout the year, children continue to develop their sense of time through their participation in daily activities (e.g., knows about when it is time to eat, time to go home, nap time, etc.). Children's sense of time continues to build gradually over the next year.

### Patterns, Reasoning, and Algebra

• Between 24-30 months, some children will still be learning that there is an order to the day (e.g., Mommy comes to get me after storytime). In the first half of the year, some children show a greater understanding of daily time sequence (e.g., time to eat, nap time, etc.). The average child develops this understanding during the second half of this year.

• During the first half of the year, some children notice patterns in the environment (e.g., day follows night, patterns in carpeting or clothing, etc.). They also use the terms, "tomorrow," and "yesterday." The average child understands these patterns and terms during the second half of this year.

• In the first half of this year, some children show interest in patterns or sequence (e.g., attempts to follow patterns with stringing beads, magnetic shapes, peg boards). The average child develops this interest during the second half of the year.

• Between 30-36 months, a small number of two-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).

• Between 30-36 months, some children can classify, label and sort familiar objects by a known group (e.g., hard v. soft, large v. small, heavy v. light). The average child logically sorts and classifies during the second half of this year.

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