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 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
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 100th 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.
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.
Some kindergarten children can also solve simple multiplication (repeated addition) and simple division (repeated subtraction) problems. For example,
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)