Math is a way of thinking — and children do it naturally, from discovering that hidden things still exist, to grasping the concept of numbers, and then learning to classify and count.

By Carla Poole, Susan Miller, Ed.D, and Ellen Booth Church

Ages 0 to 2: Poole

When a baby looks around, she sees lots of confusing things. As she grows, she'll learn to make sense of the world — and the care you give her plays a crucial role. A nurtured baby with predictable daily routines will develop the thinking skills that lead to mathematical thinking.

The natural rhythms in thoughtful caregiving — where a cuddle and story time always precede a nap, for instance — set the stage for an understanding of patterns. Even a 3-month-old baby anticipates regular events in her daily routine. By nine months, she'll anticipate the final tickle in a familiar game with a giggle or squeal. Your loving care makes life feel orderly and logical — important feelings that help the baby understand what she sees and hears.

The wonderful games in which we count baby's toes, touch her nose, and clap hands, help her develop an awareness of her body and learn where her body parts are. The baby develops a physical sense of where she is in relation to the world — the necessary groundwork for all spatial relationships.

Hidden, But Still Here

During late infancy, babies grasp an important developmental concept — object constancy. Through playful games like peek-a-boo, the baby gradually learns that people and things exist even when she can't see them. This realization is key for all learning and is especially crucial for developing thinking and number skills.

Definitely Different

As the child comes to firmly grasp object constancy, she starts to notice differences between objects. Give an 18-month-old five blocks and one ball, and she'll handle and examine the ball — the different object — longer than the others.

Soon the toddler begins to sort objects by type, size, and color. Give three containers to a 2-year-old playing with cars, table blocks, and toy figures, for example, and she'll most likely group the similar objects together and put each set in a separate container. The concept of classification is beginning to take hold.

Each One Is One

As the child moves through her twos, she begins to develop one-to-one correspondence. She knows, for instance, that when she counts her cookies, each is given one number. The exact order of those numbers, however, is still pretty much a mystery.

Up, Under and Forward

Now that the child can get up and go, she can experience spatial relationships hands-on. The physical experience of pushing a step stool over to a high window and climbing up for a better look gives real meaning to the words up, on top, and through.

By her third birthday, the remarkable toddler has acquired many important skills. Now she can discriminate simple geometric forms, do two-piece puzzles, and sort objects by size and color. The child has an impressive number of mathematical thinking skills, all of which lead to the more abstract concepts she'll learn in the coming years.

What You Can Do

Playful interactions and an interesting environment help infants and toddlers begin a pleasurable journey toward math skills. You can help make it a fun ride!

• Recite predictable rhymes to help children understand visual patterns. Infants and toddlers are attracted to the rhythmic word patterns found in finger games and nursery rhymes.
• Provide simple puzzles and shape sorters with two or three shapes. A favorite toddler puzzle has only one shape, a circle, but the circles are of different colors and can be matched to the colors on the puzzle's board.
• Offer lots of water and sand play. These activities help children grasp abstract concepts like weight and mass. Provide plenty of shovels, cups, bowls, and funnels for fun exploration.

### At a Glance

• An infant who experiences predictable routines now will be better able to understand logical patterns later on.
• Infants learn about their physical relationship to others when you count their toes, tickle their tummies, and clap their hands.
• Babies can learn big concepts like object constancy through simple games such as peek-a-boo.
• Once toddlers grasp object constancy, sorting by size and color is almost within reach.

Ages 3 to 4: Miller

Fascinated by birthdays and the concept of age, Jessica delights in holding up three fingers as she proudly tells everyone she meets, "I'm 3 years old!" Having made this exciting realization, Jessica likes requesting three crackers at snack time and has learned to count to three out loud. The concept of three is very important to her.

Preschoolers are mastering one-to-one correspondence and the concept of numbers. But they're still in the pre-operational phase and are more concrete than logical. Threes and fours tend to base their judgments on how things look — they're sure that the tall pile of blocks is bigger than the short one even if both have six blocks. And if objects aren't side by side or in a line, it's far more difficult for preschoolers to perform the mental actions of comparing or counting.

Type by Type

Preschoolers learn to classify according to one characteristic at a time — such as size, shape, or color. When focusing on just one quality, threes and fours can make a variety of classifications. However, placing items in a series is difficult because it requires that they consider a specific characteristic across a large number of objects.

When 4-year-old Tim is asked to arrange a series of stuffed animals by size, he selects two animals and picks the larger of the pair and then does the same with two more animals — without considering them in relation to the previous two. It's difficult for him to do more than find the smallest and largest of a pair. However, he can sort them by size, placing all of the large animals in the doll carriage and all the small ones in the baby crib.

By the Numbers

By 3-and-a-half, children can understand the principle of cardinality and realize that the last number in a counting sequence is the quantity of the objects in the set. When 4-year-old Josh picks up the Lincoln Logs and finishes counting "1,2,3,4," he knows that he is holding four logs.

While some preschoolers can be taught to count by rote to as high as 20, the rote recitation of numbers in long sequences isn't really meaningful. It's far more appropriate for threes and fours to connect actual items to the corresponding number. For example, 4-year-old Julio shows an early understanding of addition by counting the number of blocks in two piles — three blocks and three blocks — to determine that he has six blocks to build with.

What You Can Do

The best way for preschoolers to develop mathematical concepts is through hands-on explorations like these:

Explore the math in everyday experiences. Children can classify groceries by size or shape in the dramatic-play area, count out napkins at snack time, and sort the balls by how high they bounce in the playground. Reinforce one-to-one correspondence by asking children to match paint containers with paintbrushes at the art table.

Offer interesting materials for children to sort and classify. There are lots of attractive everyday items children can use to classify and place in a series — buttons, large plastic lids, plastic utensils, and so on. Legos, wooden blocks, or unifix cubes are also good for counting, sorting, and matching.

Document activities and real-life experiences and places. After cooking, create a bar graph showing how many children worked on each step. Or take a photo when you visit a construction site, and sort and count the equipment afterward.

Make counting meaningful. Instead of providing inappropriate number ditto sheets and flash-card activities, offer children blank paper, markers, and classroom items such as toy trucks to count. Give children index cards and markers so they can create their own number cards.

### At a Glance

• Preschoolers understand one-to-one correspondence but still base their judgments on appearance, not logic.
• Threes and fours learn to classify items by one characteristic at a time.
• Threes and fours understand that when they count items, the last number is how many they have.
• Although learning to recite numbers by rote isn't meaningful, preschoolers can count and begin to add a few real objects.

Ages 5 to 6: Church

By the time children reach kindergarten, they're using mathematical thinking skills throughout the day, every day. You can hear it in their conversations: "Look, I'm the same height as the climber." "I have more crayons than you." "This is the smallest shell on the beach!" Kindergartners are thinking about what they see and applying abstract concepts to compare, quantify, and evaluate the world around them.

Concrete Concepts

Five- and 6-year-olds approach the world from a very practical level, using mathematical thinking and reasoning in daily experiences. Kindergartners are using the mathematical concepts (such as shape, size, and amount) they acquired in their preschool years and applying them to a variety of real-life situations. They can not only name shapes but also sort, classify, and graph them into meaningful categories.

Kindergartners use their thinking skills to understand the things they see and interact with in hands-on explorations. They do more than count and name numerals; they compare which has more or less, estimate how many, and use superlative words such as biggest, longest, and tallest.

More or Less

Fives and sixes are learning the underlying structures of mathematical thinking. They're becoming aware of the patterns in the world around them. Knowing how many is progressing into knowing how many more or less. As nonstandard measurement skills are developing, children can estimate the length or height of an object using anything from apples to unifix cubes.

Although they have developed a variety of mathematical thinking skills, most kindergartners aren't ready to grasp formal addition and subtraction. Research has shown that 5- and 6-year-olds can learn the rote skill of adding and subtracting columns of numbers, but they often do not understand what they're doing. Learning to compute column addition isn't meaningful if the child has no conception of what the numbers represent.

What You Can Do

Kindergartners think and talk about mathematical concepts all the time. You can help them explore math even more.

Focus on spatial concepts and terms. Invite children to notice similarities and differences and compare sizes and amounts in their everyday experiences. Provide pairs of shirts, toy cars, books, and other items and ask children to name all the similarities and differences.

Provide opportunities for estimation and prediction. Ask children how many more blocks they'll need and how tall their structure might get. At first, they may make wild estimations of amounts, but over time, and with experience, they'll make more educated guesses and closer approximations.

Cut down on workbook use. Kindergartners learn new skills best when they see their usefulness to everyday life. Count the number of children at group time, estimate how many cups are needed for snack time, and sort the blocks and toys at cleanup time. Use pencil-and-paper activities only as a follow-up to these and other concrete experiences.

### At a Glance

• Kindergartners use mathematical concepts to compare and quantify all the time, every day.
• Fives and sixes are using the abstract concepts they've gained to understand concrete situations.
• Kindergartners notice patterns and make estimations and predictions.
• Formal addition and subtraction can be taught by rote, but most kindergartners don't have the abstract thinking skills to understand what they're doing.

Carla Poole is an advisor in the infant-parent development program at Bank Street College of Education in New York, and a child development specialist at Bellevue Hospital.

Susan A. Miller, Ed.D., a veteran teacher and director, is a professor of early childhood education at Kutztown University of Pennsylvania, and author of the Problem Solving Safari series of teaching guides.

Ellen Booth Church
is an early childhood consultant for the New York State Department of Education and for programs across the country.