Mathematics

By 9 months, most children:

• Explore the size and shape of objects through various means (e.g., banging, mouthing, dropping, etc.)
• Explore the way objects move by tracking objects with their eyes and head
• Explore distance by reaching for something
• Explore their spatial sense through movement, both involuntary and voluntary (e.g., being picked up, scooting, and pulling up)

By 18 months, most children:

• Explore how differently shaped objects fit or do not fit together by manipulating such things as nesting cups or stacking cones
• Explore barriers to movement when not able to walk or push past something
• Compare the attributes of objects or pictures and identify those that are similar
• Explore their spatial sense (e.g., by bumping into things; squeezing into a tight space; or looking at an adult or a toy from a different angle, when bending over, or with head turned)

By 24 months, most children:

• Match familiar shapes (e.g., circle, square, triangle) that have the same size and the same orientation
• Attempt to stack blocks as high or higher than themselves
• Match simple shapes (e.g., placing a shape on a shape board)
• Explore gravity (e.g., push toy cars down an incline, such as a slanted board)
• Respond to spatial directions, such as “come here,” “go over there,” and “get down on the floor,” especially if the words are accompanied by gestures, such as pointing

By 36 months, most children:

• Link or sort (put together) familiar shapes (e.g., circle, square, triangle) that have different sizes and orientation
• Build pictures or designs with two-dimensional shapes and create buildings or structures with three-dimensional shapes, such as building blocks
• Recognize and name some familiar shapes
• Understand and use basic language related to locations (e.g., “above,” “below,” “under,” “over”)

By 48 months, most children:

• Build familiar two-dimensional shapes from components or parts (e.g., using a set of circle, rectangle, and line shapes to create an image of a snowman)
• Combine and separate shapes to make designs or pictures (e.g.,  completing shape puzzles)
• Build simple examples of buildings, structures, or areas (e.g., their classroom or playground) with three-dimensional shapes, such as building blocks
• Name familiar two- dimensional shapes (circle, triangle, square, rectangle), regardless of their size or orientation
• Use basic language to describe their location (e.g., “I am under the bed.)
• Correctly follow directions involving their own positions in space (e.g., “move forward,” “sit behind,” etc.)

By 60 months, most children:

• Describe and compare shapes using their attributes (e.g., “a triangle has three sides, but a square has four.”)
• Combine and separate shapes to make other shapes (e.g., using two triangles to make a square)
• Build more complex models of buildings, structures, or areas (e.g., their classroom or playground) with three-dimensional shapes, such as building blocks
• Correctly name familiar shapes (e.g., circle, triangle, and square) and  less familiar shapes (e.g., hexagon, trapezoid, and rhombus)
• Correctly name some three- dimensional shapes (e.g., cube, cone, cylinder)
• Understand and use language related to directionality, order, and the position of objects, such as “up, ”“down,” “in front,” and “behind”