0–9 months – See this age across all domains Infants

What educators might observe

• Holds one object in each hand and then drops one when someone holds out a third object
• Explores one toy at a time by holding and mouthing it

Ways educators might support the child's development

• Sing number songs to the child
  • “Ten fingers, ten toes, two little eyes, one little nose…”
  • “Five little ducks went out to play...”
  • “Uno, dos, tres amigos...”
• Reference numbers and quantity as the educator interacts with the baby (“I see one little baby looking at me.”)
• As the child picks up, holds on to, and releases objects from their hands, name the quantity in each hand

8–16 months – See this age across all domains Infants

What educators might observe

• Tries to hold onto two large balls while reaching for third ball
• Fills a bucket with objects and dumps it out
• Holds a block in each hand and bangs them together
• Shakes head “no” when asked if they want more food
• Puts multiple blocks in each cup of a muffin tin

Ways educators might support the child's development

• Use comparative vocabulary as you interact with children (“Carla has more cars than Shante.”)
• Slowly count with children throughout the day. How many
  • Steps on the staircase?
  • Keys on a ring?
  • Wheels on a car?

15–24 months – See this age across all domains Toddlers

What educators might observe

• Puts three animals in a row when playing
• Understands and communicates the concepts of “more” and “all done” through words or gestures
• Identifies which group has “more” with collections in which one has much more than the other
• Names groups of one or two (e.g., when shown a pair of shoes, says “two shoes.”)

Ways educators might support the child's development

• Encourage counting out loud in multiple languages
• Read books about numbers that reinforce concepts of one-to-one correspondence, numerical order, and mathematical concepts of more, less, too many, and all gone
• Use number and quantity vocabulary during mealtimes (“two crackers, one apple, and one cup of milk), and count as you pour each cup of water (“one, two, three, and stop”)
• Name groups of two objects, and identify groups that do not include two items (“That’s not two, that’s three!”)

24–36 months – See this age across all domains Toddlers

What educators might observe

• Starts counting with “one”
• Says all the number words they know when counting objects, even when the quantity of objects is small
• Reuses the number words they know if the quantity of objects is greater than the number words they know
• Uses “a little” and “a lot” to describe quantity
• When told, “Give me one,” hands educator just one object
• When educator says, “Just take two,” takes only two slices of apple from the bowl
• Demonstrates one-to-one correspondence with objects (e.g., puts one block in each muffin tin)

Ways educators might support the child's development

• Encourage children to set the table for snack or lunch with one cup, one plate, and one napkin at each chair
• Use language to describe quantity throughout the day (“We have a lot of apples for snack! We have many more apples than we can eat.”)
• Incorporate counting into representational play with children (“There are three bears in the house. One, two, three.”)
• Provide egg cartons, ice cube trays, and muffin tins with small blocks so children can practice one-to-one correspondence

3-year-olds – See this age across all domains Preschool

What educators might observe

• Recognizes and names some numerals when pointing to them
• Recognizes that numerals can be written and pretends or attempts to write some
• Verbally counts to 10 with some correspondence, but skips some objects and counts others twice
• Produces sets of three (e.g., “Bring me three paint brushes from the art center”)
• When asked for five of something, counts out three and says, “One, two, five.”
• Changes word order or gives up on one-to-one correspondence to make the answer the one they had predicted or their favorite number (e.g., “one, two, three, four, 100!”)
• Matches small sets (one to four) of different items and shows they are the same quantity

Ways educators might support the child's development

• Use numbers and counting throughout the day, such as counting how many children are at the sand table
• Present sets of one to four objects and ask “How many?” questions; encourage children to name the sets before counting (“How many blocks do I have? That’s right I have two blocks.”)
• Point out quantities of items that the child possesses, such as two shoes, one hat, one jacket, or three stuffed animals; count each item out loud to correspond the number with the quantity of the items
• Model placing the objects that the child wants to count in a line
• Give child simple tasks that involve counting (“Give every child one cracker”)

4-year-olds – See this age across all domains Preschool

What educators might observe

• Attempts to count to 20 by ones
• Recognizes the number of objects in groups of four without counting
• Accurately counts five to 10 objects in a line and answers the “how many” question with the last number counted
• Can match a number of objects with a written numeral 0 to 5 (0 representing a count of no objects)
• When asked for five of something, counts out five objects
• Compares groups of one to six objects by matching (e.g., gives each child a bagel and says that there are the same number of bagels as children)
• Counts the number of objects in a group so they can compare and say which has more, up to five objects; however, if the objects in one group are larger, may say that group has more (e.g., says three large trucks are more than five small cars

Ways educators might support the child's development

• Observe children’s use of counting, oneto-one correspondence, estimating, and simple arithmetic in their play; connect to and extend this learning
• Offer games that ask children to match a numeral to a number of things
  • String six beads on a pipe cleaner
  • P ick a card and fill the basket with that number of objects
  • Roll a dice and do a movement that many times
  • Jump, clap, or stomp one to five times and ask the children to hold up the number of fingers that matches the number of movements
• Include books related to number and quantity in the learning environment
  • The Doorbell Rang , by Pat Hutchins
  • Anno’s Counting Book , by Mitsumasa Ann
  • Ten Black Dots , by Donald Crews
  • City by Numbers , by Stephen Johnson
  • Fish Eyes , by Lois Ehlert

5-year-olds – See this age across all domains Preschool

What educators might observe

• Quickly names the number of objects in groups of five without counting
• Tells what number comes after any number between one and 20
• Explains that numbers tell “how many” of something
• Recognizes and writes numerals 1 through 10
• Divides up to 20 objects between four or five people by dealing out an object to each person
• Uses the words “first,” “second,” “third,” up to and including “tenth” (e.g., says, “I came in fourth” when describing a race with classmates
• Counts on their fingers
• Keeps track of objects that have and have not been counted (e.g., slides objects they have already counted to the side)
• Counts backward from 10 by taking away objects

Ways educators might support the child's development

• Provide bags of objects to count (10 acorns, 14 keys, 19 cubes) and ask children to count and draw the objects in them
• Allow children to freely play with math manipulatives before using them to solve problems
• S et up the dramatic play area as a store, and invite children to use number cards to “purchase” objects
• Analyze the mathematical thinking in children’s errors
• Provide games with dice and dominoes so that children can practice recognizing the number of items in a small set without counting
• Find math in the body, and play finger math; for example, hold up three fingers on one hand and ask children how many fingers are folded down

6-year-olds – See this age across all domains Early elementary

What educators might observe

• Understands that 10 is always 10, whether it is made by adding 6+4 or 7+3
• Starts counting with numbers other than one (e.g., asked to count from four to seven, counts four, five, six, seven)
• Answers questions about quantity such as, “How many more?” “How many fewer?” by counting up or down (e.g., when asked how many is three more than six? The child responds, “Six, seven [puts up a finger], eight [puts up another finger], nine [puts up a third finger])
• Understands place value; understands value of a digit according to the place of the digit within a number
• Makes whole objects from parts when counting (Given three whole plastic eggs and four plastic egg halves, knows they have five whole eggs)

Ways educators might support the child's development

• Create a number line adding one number each day for the whole year, and use it to practice counting by ones, twos, fives, and tens
• Offer structures like a 100’s chart or number line sections for organizing number systems
• Play games in which children roll a dice or spin a spinner and move a piece along a path
• Provide strong context for solving problems with numbers (With 23 children in the class, does everyone have a snack if you have 18 snacks? How many more we need or how many extra do we have?)

7-year-olds – See this age across all domains Early elementary

What educators might observe

• Realizes that the number is unchanged, even when the arrangement looks different (e.g., when counters are spread out, says that there are the same number of counters as when they were together)
• Understands the reversibility of operations (e.g., subtraction is the reverse of addition)
• Uses groups, skip counting, and place value to quickly identify sets shown quickly (“I saw three sets of ten and three sets of two, so the answer is 36.”)
• Uses a mental number line to compare numbers
• Counts forward and backward by ones and using skip counting
• Solves all types of single digit addition and subtraction problems, with flexible strategies and known combinations

Ways educators might support the child's development

• Play more games on a number line (If you start at 12, how many to get to 15? How do you get to 100 from 64?)
• Keep locating math in the body; ask questions like how many fingers in the class?
• Provide base 10 materials to model numbers
• Play “I’m Thinking of a Number” in which children seek clues of what numbers are smaller or larger than the one in mind

8-year-olds – See this age across all domains Early elementary

What educators might observe

• Uses groups, multiplication, and place value to visually recognize sets shown briefly (“I saw groups of 10s and twos, so I thought six tens is 60 and four twos is eight, so 68.”)
• Rounds numbers to the nearest 10 or
• Builds skills to multiply and divide up to 10 × 10 accurately
• Solves word (story) problems using addition, subtraction, multiplication and division

Ways educators might support the child's development

• Engage children in a cooking activity that calls for ingredients in ½, ¼, ¾, or ¹∕₃ measures
• Ask children to write word problems that fit number sentences for their classmates to solve
• Practice addition, subtraction, multiplication, and division skills using Jeopardy-type games