A group of three-year-olds sits in a circle, eagerly awaiting the story, *Puppy
Says 1-2-3* (Singer 1993). They have not heard the tale of the puppy that
counts and squeaks as their teacher, Miss Lily, squeezes his tummy. The class
counts along with puppy. “Puppy looks up and what does he see? ONE little
squirrel climbing up ONE big tree. Puppy says ONE.”

Then the children play a game with three rubber cows, a mat of green “grass,”
and a bowl turned into a barn. One child says, “Let’s put one
cow on the grass. Let’s put another cow on the grass. Now we have *two*
cows.” The cows go *into* the barn, *around* the barn,
and *on top of* the barn. The teacher gives them makeshift cardboard
“bridges.” The cows go *over* the bridge and *under*
the bridge.

The teacher prepares many centers that are found throughout the room. Besides housekeeping, a picture-book center, a computer center, and a puzzle center, there are many centers devoted to early mathematical experiences. After much repetition, the puppy book will go in the picture-book center to be read and reread. While the teacher knows that some of the children have been able to count to 10 since the age of two, she realizes that very young children need a foundation with small sets, 1-2-3, emphasizing one-to-one correspondence. At a later time Miss Lily will introduce numbers up to five or 10.

During their music time, the class sings very simple songs with much repetition. Today they sing “Row, Row, Row Your Boat.” The sound of the music and the lyrics reinforce the concept of pattern. Later in the year the children will construct simple patterns in artwork and with manipulatives.

The children choose a center to explore. The matching center contains bins of socks, mittens, identical farm animals, and zoo animals. The children put identical pairs together by type, not color. In the comparing center, a child sorts stuffed animals into “big” piles and “little” piles with the teacher’s help. Next week, the comparing center will feature a different pair of words.

Later in the week, the class gathers in a circle to practice sorting. They concentrate on things they know, such as food, toys, clothes, and ways to travel. They sort themselves by boy/girl, hair clips/no hair clips, buttons/no buttons, and so on. Miss Lily avoids categories that might cause hard feelings, such as a certain brand of tennis shoe versus a dress shoe.

In another session, Miss Lily introduces nesting toys that illustrate ordering. She asks the children to find the biggest one. The children point to the biggest cup. She takes it out of the line, places it near her, and asks, “Now which one is the biggest?” Miss Lily gathers various nested sets of measuring cups, kitchen bowls, plastic glasses, and commercial nesting toys for her ordering center. Throughout the week, children choose to visit this center, and they try to put the items in order of size.

In the pouring center, a child fills plastic containers of many sizes with
scoops of rice. The teacher helps with words like *empty/full* and
*little/big*. The child fills the cup to the *top*. Later in
the year, the class will discuss “which jar has *more*, and which
jar has* less*.” A sturdy balance sits on the counter nearby.
Children take turns weighing fruit, feathers, and small items such as erasers,
chalk, markers, crayons, and toy cars. Each child tells Miss Lily which items
are *heavy* and which are *light.*

The block-building center is a major center and the cornerstone of a mathematically rich environment. Blocks are essential tools for creativity, dramatic play, and geometry—for girls as well as boys. The teacher rotates groups of children to give everyone a chance to use the building blocks. At first, a child might pile the blocks to make a tower, then make a simple enclosure, and eventually master the challenge of roofing or bridging the space between two walls. With time and practice, children may build elaborate structures that have evidence of symmetry, sound construction, and aesthetics.

Miss Lily understands that developing concepts about space is a natural part of growing up. She recognizes that children need opportunities to study the relationships between objects, places, and events (the study of topology) more than they need the ability to draw common shapes such as a circle or a square.

Miss Lily creates opportunities to explore *proximity*, asking questions
such as “Where am I?” or “Where is it?” *Separation*
refers to the ability to see the whole object as comprised of individual parts.
Puzzles and model building encourage this ability. The nesting-center toys
promote *order*, including reversing one’s thinking. Miss Lily
also talks about last week’s events as well as what is happening today.

*Enclosure* refers to being surrounded or boxed in by the surrounding
objects. The points on either side can enclose a point on a line. In three-dimensional
space, a fence can enclose the animals, or a canister with a lid can enclose
the cereal. The teacher helps by saying, “Is the lid closed so the beads
won’t spill out?” or “Open the closed door so we can hang
up our coats.”

All of these activities contribute to the overall development of a child’s
*spatial* sense. In its 1989 *Curriculum and Evaluation Standards
for School Mathematics*, the National Council of Teachers of Mathematics
(NCTM) defines spatial sense this way:

Spatial sense is an intuitive feel for one’s surroundings and the objects
in them. To develop spatial sense, children must have many experiences that
focus on geometric relationships: the direction, orientation, and perspectives
of objects in space, the relative shape and sizes of figures and objects,
and how a change in shape relates to a change in size. (page 49)

Spatial sense contributes to the study of geometry, and is an integral part of the preschool curriculum.

*Number sense* is using common sense based on the way numbers and tools
work within a given culture. It involves an appreciation for the reasonableness
of an answer and the level of accuracy needed to solve a particular problem.
It is a complex set of interrelated concepts (Smith 1997), including

- reading numerals (For example, “It’s a three.”)
- writing numerals, a visual-motor task.
- matching a number to a set, or the principle of cardinality. (A child counts five beans and answers the question, “How many?”)
- having an intuitive feel for how big a number is. (“Is 15 closer to 10 or closer to 50?”)
- being able to make reasonable guesses using numbers. (The small jar could not hold more than 100 goldfish crackers.)
- seeing part-whole relationships using sight or abstract thinking, not counting. (“I have two green bottle caps and three purple bottle caps.”)

The teacher facilitates the development of number sense and spatial sense throughout the preschool years.

Most experts believe that young children possess a substantial amount of informal knowledge about mathematics. The teacher’s role is to create a link between children’s ability to use informal math and the ability to understand the more formal math found in grade school (Ginsberg 1996).

Teachers must help children construct and elaborate upon what they already know, so they can “re-invent” mathematics for themselves. A reflective teacher helps the child discover and communicate ideas that would have not occurred spontaneously to the child without the adult’s help (Vygotsky 1978). As children mature, they find patterns and solve problems far beyond what is typically found in the preschool-kindergarten curriculum (Resnick et al. 1991; Carpenter et. al. 1993).

Mr. Toby has a kindergarten class of 15 students. At the beginning of each day, they place a picture of themselves that has been glued to a magnetized orange juice lid on the attendance chart. The children count and decide how many people are in class today and how many are absent. They decide if there are more boys than girls or vice versa, and they figure out the difference. They chart the weather for the day by placing a magnetized counter under the category chosen: sun, clouds, rain, snow. They know that they can only choose one at a certain time in the morning.

They sequence the day’s activities in a pocket chart. Mr. Toby knows
that young children cannot comprehend the traditional calendar, i.e., a five-row
and seven-column matrix, with both ordinal and cardinal numbers (Schwartz
1994). He will gradually use a weekly schedule and then a two-week schedule
before introducing a more comprehensive calendar. The children keep track
of how many days they have been in school by putting a straw for each day
in a container labeled the “1’s cup.” When there are 10
straws in the cup, they bundle them and move them to the “10’s
cup.” Sometime in February there will be 10 groups of 10 straws to move
to the “100’s cup,” and the class will celebrate the 100^{th}
day of school. They will decorate a cake with 100 candles and have a party
with a “GORP” mix, which consists of small snack items (raisins,
cereal, chocolate chips, etc.) that the students bring from home and sort
into groups of 100. They will enjoy a day filled with many activities that
use 100 items.

The circus is a popular kindergarten theme. Over several days the class will participate in many creative art, creative movement, science, dramatic play, and cooking activities. A teacher plans a number of math activities, some of which are described in the text that follows.

- In pairs, the children decide how to line the edges of a small table with peanuts. They pay close attention to covering the edge and having the peanuts touch.
- After they finish lining the perimeter of the table, the children remove the peanuts while counting them with the teacher. Kindergarten children enjoy counting to 100 and beyond. (It may take more than 100 peanuts to line the perimeter of some tables.)

- Read and reread the story of Mirette, and highlight the events in sequence:
- Mirette lives in a boardinghouse.
- A new tenant, a retired high-wire performer, arrives.
- He teaches Mirette to walk the high wire.
- He returns to the stage.

- Have the class act out and retell this story in sequence.

- Research the weight of a baby elephant. Compare it to the weight of a newborn person.
- Make a more weight/less weight chart, with pictures of things that might weigh more or less than a baby elephant.

- Make designs with two kinds of nuts, so each design uses six nuts.
- Fill the whole table with designs. Tell the teacher about your combinations, for example, “This one has two pecans and four peanuts. It looks like a star.”

(For additional math activities, see Smith 1997.)

Mr. Toby concentrates on pattern-work and part-part-whole designs with each number from 4 to 12. Later in the year, he will introduce simple story problems, following the Cognitively Guided Instruction Approach (Carpenter and Moser 1983; Carpenter and Moser 1984; Carpenter et al. 1990; Peterson et al. 1989). Many kindergartners are able to solve the following types of problems by using counters or their fingers or by drawing.

*The circus ring had three clowns. Four more clowns join them. Now how many clowns are in the ring?**The clown had nine pieces of candy. He gave away four pieces. How many pieces does the clown have left?*

Some kindergarten children can also solve simple multiplication (repeated addition) and simple division (repeated subtraction) problems. For example,

*The clown had three bags of candy. There were five pieces of candy in each bag. How many pieces did the clown have?**The clown had 12 pieces of candy. He gave three pieces to each child. How many children received candy?*

Mr. Toby’s classroom provides the time and structure needed by children to explore significant mathematics. He encourages his students by respecting and valuing their ideas and validating their ways of thinking. He challenges the class to take intellectual risks by posing interesting questions to the group. They learn to support their responses with mathematical ideas. Finally, he encourages all students to participate, so they gain confidence in their ideas.

All preschool and kindergarten teachers must pay attention to the key ingredients for success: a well-prepared environment, a developmentally appropriate math curriculum, and an awareness of the teacher’s role. The process of learning is never over, but the journey is worth taking.

Carpenter, T.P., and Moser, J.M. (1983). *The acquisition of addition and
subtraction concepts. In The acquisition of mathematical concepts and processes*,
eds. R. Lesh and M. Landau, 7–44. New York: Academic Press.

Carpenter, T.P, and Moser, J.M. (1984). The acquisition of addition and subtraction
concepts in grades one through three. *Journal for Research in Mathematics
Education*, 15: 179–202.

Carpenter, T.P., Ansell, E., Franke, M.C., Fennema, E., and Weisbeck, L. (1993).
Models of problem solving: A study of kindergarten children’s problem-solving
processes. *Journal for Research in Mathematics Education*, 24(5):
427–440.

Carpenter, T.P., Carey, D., and Kouba, U. (1990). *A problem solving approach
to the operations. In Mathematics for the young child*, ed. J.N. Payne,
111–131. Reston, VA: National Council of Teachers of Mathematics.

Ginsberg, H.P. (1996). *Toby’s math. In The nature of mathematical
thinking*, eds. R.J. Sternberg and T. Ben-Zeev, 175–202. Hillsdale,
NJ: Lawrence Erlbaum Associates.

McCully, E.A. (1992). *Mirette on the high wire*. New York: G.P. Putnam
& Sons.

National Council of Teachers of Mathematics. (1989). *Curriculum and evaluation
standards for school mathematics*. Reston, VA: Author.

Peterson, P., Fennema, E., and Carpenter, T. (1989). Using knowledge of how
students think about mathematics. *Educational Leadership*, 46(4):
42–46.

Resnick, L., Bill, V., Lesgold, S., and Leer, N. (1991). Thinking in arithmetic
class. In *Teaching advanced skills to at-risk students*, eds. B. Means,
C. Chelmer, and M. Knapp. San Francisco: Jossey-Bass.

Schwartz, L.L. (1994). Calendar reading: A tradition that begs remodeling.
*Teaching Children Mathematics*, 1: 104–109.

Singer, M. (1993). *Puppy says 1,2,3*. Hong Kong: Reader’s Digest
Young Families, Inc.

Smith, S.S. (1997). *Early childhood mathematics*. Needham Heights,
MA: Allyn & Bacon.

Vygotsky, L.S. (1978).* Mind in society: The development of higher psychological
processes.* Cambridge, MA: Harvard University Press.

*Susan Speery Smith is an associate professor in the College of Education
at Cardinal Stritch University in Milwaukee, Wisconsin.*

Copyright © 1999 by the American Association for the Advancement of Science (AAAS)