The foundation for children’s mathematical development is established in the earliest years. Mathematics learning builds on the curiosity and enthusiasm of children and grows naturally from their experiences. Mathematics at this age, if appropriately connected to a child’s world, is more than “getting ready” for school or accelerating them into elementary arithmetic. Appropriate mathematical experiences challenge young children to explore ideas related to patterns, shapes, numbers, and space with increasing sophistication.—(NCTM, 2000, p. 73)

Mathematics is a particular way of thinking and all children everywhere do it quite naturally. From their earliest encounters, children explore the abstractions of mathematics. Parallel to the development of language skills is the development of concepts related to basic areas of mathematics. We can follow the development of mathematical concepts as we look at infants and toddlers. The basic mathematical concepts addressed here are pattern, sequence, seriation, spatial relationships, object permanence, sorting, comparing, classifying, and one-to-one correspondence.

### Pattern

An important mathematical concept that infants develop is pattern. Pattern is the underlying theme of all mathematics and science. It is our ability to discover and recognize patterns that helps us understand how our world works in logical and predictable ways. Experiences with observing and making sense of patterns are what helps young children become logical thinkers who can reason and think critically.

As infants are cared for in predictable ways, they experience the idea of patterns. They easily begin to recognize and anticipate the rhythm or pattern of their care. As they experience this daily routine, infants come to anticipate the sequence of events. Such experiences are important to the development of recognizing the logical patterns that will be discovered later in their mathematical and scientific experiences. As babies approach their first birthday, they anticipate sequences and patterns in games that include patty-cake, peek-a-boo, singing, dancing, touching of the nose and toes, and feeling different textures. These rich experiences help children to develop the ability to predict and anticipate events.

### Sequence

Sequence, like pattern, is a mathematics concept that children internalize early in life. Sequence refers to the organization and order of successive events and experiences. Recognizing sequences helps young children’s developing sense of order, logic, and reason. They begin to recognize the sequencing of their day and are able to predict what may happen next. They may also observe the sequence of seeds growing into plants, the sun rising and setting, the melting of snow, or the leaves falling from the trees. Each of these events involves sequences in nature. As children become more sophisticated observers, they can discover sequences in daily activities and involvements.

Before the age of 2 years, children tend to involve themselves in activities that require sequencing, such as taking turns, following a certain order when doing a task, or learning how to get dressed. Children may also use sequencing in play; for example, they may push all the blocks off the table and then one by one pick them up and put them back on the table, only to knock them all off again. Young children often repeat a sequence of events numerous times, because the predictability of these actions is enjoyable.

As children listen to stories either from books or other people, they begin to build concepts for sequence: what comes first in the story, next, and how the story logically unfolds. They often like to predict what may happen next. This type of activity helps to build a mind-set for rational and logical thinking, which again are important skills for budding mathematicians and scientists.

When children play in a sandbox they are constructing. When they use building blocks they set goals for what they want to build. Playing with dolls and figures requires that the children develop story lines about what the dolls are doing. Playing with racing cars promotes decisions about which car comes in first. Each of these activities designed by children requires an understanding of the concept of order and sequence. We can evaluate children’s true understanding of sequence better by watching their actions than we can by listening to their verbalization. A child might be using sequence skills in many activities, but because of their developing language skills, may not be able to describe to you the process they are doing through play.

Soon after children start to make sentences, they give great detail about processes they use to make play-doh cookies, paint a picture, or create roads in the sandbox. They tell about how they will plant seeds in the garden or in the flower box.

Children will act out sequences of events with toy figures that may represent family members—Mom-mom, Poppy, Daddy, Mommy, Aunt Rachel, and cats Shadow and Sunshine. This type of play illustrates how a child makes sense out of events. Such active play fosters personal meaning (Isenberg & Quisenberry, 2002). Children can also be encouraged to put photographs of the family vacation at the seashore in order according to what happened first, second, and so forth. Such observations are valuable in evaluating the developmental level of children.

### Seriation

Seriation is a mathematics concept that involves organizing or ordering things in a logical way. Consider toys that can be manipulated, such as different-sized stacking rings or blocks. Early in the use of these types of toys, children do not attend to the seriated relationship, of which ring or block goes on first. Over a period of time, however, the child will try to put the largest item on the bottom, as the rings are seriated by size. Exploring and discovering this seriated set of rings is important for logical mathematical thinking. In addition, these types of investigations are interesting, engaging, and motivational.

Other types of seriated toys and tools include cookie cutters in different sizes and pie plates in varying diameters. These toys can be explored for seriation of their nesting attributes. Such explorations can prompt children to tell stories relating to the seriated sizes of the toys, with encouragement from teachers. For example, a young child playing at a play-doh center with a seriated set of bunny cookie-cutter shapes could be asked about the “baby bunny” and the “mommy bunny,” and eventually she might tell a story about them. Children’s self-directed play often develops and utilizes seriation skills. Seriation becomes more natural as children enter school as kindergarteners and continues to become more sophisticated through the primary grades.

### Spatial Relationships

Another important concept developed in mathematics is spatial relationships. The games and interactions that comprise play in infancy also help babies become aware of their body parts and develop a sense of their physical self. Such exploration helps them to know where they are in relation to their world. As toddlers become more skilled with moving about their world, a concept called navigation, they experience spatial relationships firsthand. They navigate themselves through a play tunnel or space fort, they begin to climb on play structures and equipment. These experiences will be the foundation for more mathematics concepts to follow involving directionality and position in space, such as the concepts of up, down, over, next to, under, above, beside, in between, first, and last.

### Object Permanence

Babies can discover the important concept of object permanence in a simple game of peek-a-boo. For very young babies, this is a fun game because it elicits an element of surprise when someone appears and then disappears. Learning that the person or object does not actually disappear is a major accomplishment for a little one. Such a discovery is important for the mathematics concepts that will follow. Once babies know, by about 9 months of age (Piaget, 1963), that something is still there even when it is hidden, they will begin to be more observant and notice similarities and differences among the objects themselves. Such observations and experiences of objects lead to sorting and classifying. Piaget explains that knowledge arises neither from objects nor the child, but from interactions between the child and those objects.

### Sorting

Sorting occurs when things with like attributes are grouped together. As Poole (1998) reports, if you give an 18-month-old five blocks and one ball, the child will handle and examine the ball (the different object) for a longer time than the blocks. This activity suggests that the child feels the difference between the blocks and the ball and wants to explore the different one longer to make sense of the difference. As such, the concept and process of sorting begins to be evident. When young children put the blocks in the block corner, the books on the book shelf, their socks in the big box, or the toy animals in the wooden barn, they are sorting.

Toddlers are likely to group similar objects together quite easily, whereas seriating or sequencing objects by a specific characteristic is more difficult. For example, if you give 4-year-olds a group of stuffed teddy bears and ask them to arrange them by size, they will focus on the big bears and the little bears without considering the seriation or the ordering by size.

### Comparing

When toddlers sort items and put them into two groups, such as the big teddy bears and the little teddy bears, they are demonstrating the concept of comparing. In comparing, children identify and examine specific properties of different objects or ideas and then make judgments about how they are similar and how they are different. Comparing causes a person to look at details and specifics instead of generalities, to observe and study more carefully. Noting that some things are big and some things are little requires a judgment about attributes or qualities of things.

Having children make comparisons is valuable because it requires them to actively make observations related to specific items. They must look for divergent ideas, to go beyond the obvious. Beginning activities that involve comparing tend to be of objects common to the children’s environment. They can compare cars and trucks, dogs and cats, apples and oranges, or cookies and crackers. These activities are important for children in that they promote the use of the five senses.

After a child has developed and used the skill of comparing concrete objects, teachers can extend this to comparing ideas. Children can compare sunrises and sunsets, seasons, a pumpkin and an apple, a carrot and cucumber, story lines from Eric Carle books, or songs and poetry. Charting their ideas about the similarities and differences of two things or events is helpful in developing expressive language.

### Classifying

Classifying is a way of comparing. Classification refers to putting like things together and naming the group, such as big bears, little bears; shiny shells, dull shells; round buttons, square buttons; or smooth rocks, rough rocks. Classification schemes are important for young children to construct, as they are central to scientific thinking. Rocks, seashells, birds, seeds, and just about everything in nature has a classification system defining it.

More specifically, classifying is the division of items into groups by identifying a specific attribute that we recognize. We identify specific attributes or pertinent discrete characteristics that distinguish items within the collection. We can classify cars and trucks, dogs and cats, meats and vegetables.

Classifying is a natural activity for young children. They love to collect items from nature. Rocks, leaves, acorns, seashells, and pinecones become important collections.

Children separate their treasures by texture, color, shape, size, and favorites. Attributes are inherent characteristics of objects. Classification is often explored with commercial manipulatives called attribute blocks. Typical attribute block sets classify by color (red, blue, yellow), size (big, small), shape (square, circle, rectangle, triangle), and thickness (thick and thin). Attribute blocks are valuable for initiating classification development because the attributes are discernable.

It is also good to use more natural objects than attribute blocks, objects that are real aspects of the child’s world. As mentioned, young children can gather, play with, and sort items such as seashells, leaves, seeds, rocks, and other natural objects from their backyard, nearby park, or other natural area.

Classification is an extension of sorting. In a classification, the collection of items continues to be divided into subgroups until each item is unique. Children can design classification of common objects around them. Classifications tend to be based on more obvious characteristics so that all can agree on how the items fit in the classification.

### One-to-One Correspondence

As toddlers have experiences with authentic hands-on learning, and with counting and sharing, they begin to develop the concept of one-to-one correspondence. When distributing a snack, for example, toddlers can give each child one mini muffin or one orange section. In this way, they are learning to relate to the notion of one for each person: “One for you, one for you, one for you, and one for me.” Teachers can provide many opportunities to distribute “one” to “each child” and model this concept.

One-to-one correspondence extends as toddlers count a collection of items. Toddlers learn the counting sequence of numbers but are not always consistent in their ability to name the numbers in order. Toddlers seem to enjoy practice counting and know “how” to count although the actual sequence of naming numbers is not always followed. Teen numbers are often confused or left out. Teachers can model the stable order principle and keep track of counted items when counting a collection of things (e.g., 1, 2, 3, 4,...).

Learning to count is a similar process to learning the alphabet. Wanting to show what they know, children take great pride counting aloud. They seem to enjoy reciting the counting numbers and will attempt to count objects in their environment. They may not have one-to-one correspondence or stable order, but practicing these skills is important. We must recognize, however, that children’s verbal counting does not indicate real conceptual numeric understanding, only that they can sequence particular sounds. We may ask toddlers how many blocks they have, and they may give a number that seems to be selected at random. By about age 4 or 5 years, however, they are able to understand the logical concepts for numbers under 10.

As their skills develop, children relate the rote counting sequence to rational counting. They might have a stack of blocks in front of them and start pointing to particular blocks and, at the same time, start the counting sequence. However, close observation indicates that they might count at a different rate than that at which they point. It is also common for them to point to the same items more than once.

For children to be able to count rationally, they need to demonstrate one- to-one correspondence. This is demonstrated when a child actually relates the counting to specific individual items. Touching each item only one time as the child recites the counting sequence illustrates this concept. “This is car 1. This is car 2. This is car 3.” Children can confirm their one-to-one correspondence by keeping track of each item counted.

Relating the concept of one-to-one correspondence to rational counting is a complex skill. Young children must be able to keep track while reciting a stable order of numerals to their one-to-one counting. This skill often does not occur until the kindergarten years.

The mathematical concepts highlighted here are important components of rational and logical thinking. As children interact with their environment and with people in their world, they begin to see order in and make sense of their world.