NUMBER CONCEPT | AT 3 YEARS CHILDREN MAY: | AT 4 YEARS CHILDREN MAY: | AT 5 YEARS CHILDREN MAY: |

Verbal Counting. Learning the standard sequences of number words. | Count 1 to 10 | Count one to 30, with emphasis on counting patterns; for instance, knowing that "twenty- one, twenty- two..." is parallel to "one, two ..." | Count one to 100, with emphasis on patterns (e.g., "60, 70, "parallel "six, seven"; and "14" to "19" parallel "four" to "nine") |

Object Counting. Creating a one-to-one correspondence between a number word and an item. | Count one to four items, maintaining one-to-one correspondence | Count one to 10 items, knowing that the last counting word tells "how many" | Count one to 20 items |

"Seeing" Numbers. Instantly "seeing how many" supports counting, comparing, and adding. | See groups of one to three | See groups of one to five | See groups of one to six; regular patterns up to 10 |

Comparing Numbers. Comparing and ordering build on nonverbal knowledge and experience with real collections. | Identify whether collections are the "same" number or which is "more" visually | Use counting or matching to compare two collections one to five, despite distracting appearances | Use counting to compare two collections one to 10, using words "equal," "more," "less," and "fewer" |

Adding and Subtracting. Solving problems using informal strategies is critical in learning arithmetic. | Use nonverbal adding and subtracting with very small numbers of objects | Solve and make word problems using concrete modeling with sums to five | Pose and solve word problems using counting-based strategies such as counting on, sums to 10 |

GEOMETRY AND MEASUREMENT | |||

Shapes. Geometric shapes can be used to represent and understand objects in the world around us. | Match shapes, first with same size and orientation, then with different sizes and orientation | Recognize and name some variations of the circle, square, triangle, rectangle | Recognize and name circle, square, triangle, rectangle, in any size or orientation (varying shapes for triangles and rectangles) |

Putting Together Shapes. Shapes can be decomposed and composed into other shapes and structures. | Use shapes in isolation to make a picture | Cover an outline with shapes without leaving gaps by trial-and-error | Cover an outline with shapes without leaving gaps with foresight Makes a picture by combining shapes |

Locations, Directions, and Coordinates. Mathematics can precisely specify directions, routes, and locations in the world. | Understand and use ideas such as over , under , above , on , beside , next to , between | Learn a simple route from a map placed in direct relation to the space | Place toy objects in correct relative position to make a map of the classroom |

Symmetry. Symmetry can be used to analyze, understand, and create shapes in geometry and art. | Show awareness of symmetry in block buildings | Informally create 2-D shapes and 3-D buildings that have symmetry | Identify and create shapes that have line or rotational symmetry |

Measurement. Measuring can be used to specify and compare "how much." | Develop language such as bigger , longer , and taller | Discuss and compare attributes informally, including comparing gross differences | Compare length using another object. Measure with multiple copies of a unit (such as block) |

Patterns. Patterns weave through all other topics in mathematics. | Notice simple repeating patterns, such as a wall of blocks with long, short, long, short, long ... | Copy simple repeating patterns | Notice and discusses patterns in arithmetic (such as adding one to any number results in the next "counting number") |