Mathematics and Young Children (page 4)
During the early childhood years, children develop cognitive understandings that are foundational to learning the mathematical content described in the NCTM standards. One such understanding is the ability to classify. Putting objects or ideas with similar characteristics into groups demonstrates classification competence.
Although this cognitive ability seems simple for us as adults, children require considerable practice and time to understand classification. For example, a 3-year-old child given a set of colored blocks and asked to “put blocks together that are the same” may playfully organize and reorganize the blocks, not really using any logical thinking in creating his groupings.
Classification skills are fundamental to many mathematical concepts. For example, writing the numeral 43 requires an understanding of the “tens place” and “ones place” as different groupings. In addition, the study of algebraic functions places a heavy emphasis on the ability to classify, for example, “Consider n to be the set of all integers greater than zero.”
Teachers and caregivers can provide many opportunities for practicing an understanding of classification. Mary Baratta-Lorton (1976), in her classic book Mathematics Their Way, provides many good ideas for simple materials that can be used for sorting and classifying tasks:
- People in the classroom
- Buttons for grouping
- Old bottle caps
- Natural materials for sorting such as acorns, leaves, rocks, and shells
- Nuts and bolts
- Teacher-directed activities using geoboards (square board with 25 regularly spaced pegs over which rubber bands can be stretched)
Ordering objects from smallest to largest is referred to as seriation. This sequencing can be based on height, weight, shades of color, or any other characteristic. This is another important cognitive task for young children to master. It is essential to an understanding of the number system.
Many opportunities to practice seriation are necessary for children to truly make sense of it. Although parts of this cognitive understanding are seen in many children at age 3 or 4, the full development of this concept is often reached as late as age 8 or 9. Piaget spent considerable time studying the growth of this developmental task (Flavell, 1963).
Many excellent commercial materials are available to give children practice with seriation. One well-known example is Montessori’s cylinder block. Each rectangular block has several wooden cylinders that fit into holes ordered from smallest to largest in the block. Children practice their sequencing skills by finding the right cylinder for each hole. Another set of materials that can be used for seriation activities are Cuisenaire Rods. These multicolored rods begin with a small cube as the basic unit and grow step by step to the longest rod, which is 10 units in length.
Being able to recognize and create visual, auditory, spatial, and numerical patterns is another important mathematical understanding. The discipline of mathematics is logical and based on patterns of all sorts. The number system, for example, with groupings of 10 has a clear pattern that children must recognize to truly understand its complexities.
Students must also master patterns in arithmetic, algebra, and geometry. Teachers can provide young children with many meaningful opportunities to engage in patterning activities. Some examples of appropriate materials and activities include:
- Stringing beads in patterns
- Constructing designs with pattern blocks
- Repeating clapping patterns
- Listening to musical patterns
- Building with Unifix cubes (plastic cubes that can be snapped together to form patterns)
- Playing with Cuisenaire Rods
Children’s understandings of number concepts develop rapidly during the early childhood years. While a 3-year-old often is just beginning to understand that “1” is a small number and others are larger, 5-year-olds have typically mastered basic number concepts through “9” (Murray & Mayer, 1988). During the primary years, children develop the ability to count forward and backward, skip count (counting by twos, fives, tens, etc.), and understand numbers into the hundreds (Charlesworth, 2005).
Much of the preschool child’s understanding of numbers comes from repeated counting experiences. Many songs and finger plays (e.g., “Five Little Speckled Frogs” and “Ten Little Monkeys”) give children enjoyable opportunities for rote counting experiences. The day-to-day life of the classroom and home provide many other meaningful opportunities to count and understand numbers. Discussing the calendar at group time, counting crackers in the snack bowl, and finding out how many ladybugs were caught on the playground are examples of these natural opportunities for counting.
It is important to remember that primary-age children often use counting as an aid in solving addition and subtraction problems. Although this strategy eventually becomes cumbersome and slow, it helps many children make the transition to more mature arithmetic skills. Using concrete materials as an aid in counting is still a necessity for many primary-age children and should be made available for those who need them.
During the primary years, children develop the ability to understand addition, subtraction, and multiplication. However, part of the problem with the traditional approach to teaching these skills is that not all children are cognitively ready to begin when the teacher starts this instruction. Some may need more counting experience first, while others should spend additional time developing basic number concepts. When the teacher uses a constructivist approach and allows children to manipulate materials and discover arithmetic understandings as they work on real-world problems, children are much more able to set their own pace and conquer this task.
Another important mathematical understanding to be emphasized during the early years is the ability to quantify materials in the world. Finding the height, weight, volume, and dimensions of objects are examples of measurement. Piaget’s work tells us that, until children have reached the stage of concrete operations (around age 7 or 8), they have difficulty measuring using standardized units such as inches, pounds, and liters (Flavell, 1963). Younger children, however, can learn much when given the opportunity to measure with nonstandard units.
The preschool and kindergarten child can engage in the following types of informal measurement activities:
- Use blocks to measure tables, floor space, and other elements of the classroom environment.
- Find out how many plastic cups of sand or water it takes to fill containers in the sand or water play area.
- Use a balance to compare weights of different objects.
- Trace full-body silhouettes of children and have them compare heights.
Children in the primary grades can engage in many of these same activities but with the use of standardized units of measurement. Children this age can use and understand rulers, weight scales, and one-cup containers. Measurement activities should remain meaningful and relevant to the children’s lives. To achieve this goal, consider weighing the class guinea pig, measuring the dimensions of playground equipment, and discovering how many liters of water are needed to fill the classroom sink.
The study of two- and three-dimensional shapes and how they are related to one another is called geometry. Although this topic is often thought of as part of the high school mathematics curriculum, it is highly applicable in the early childhood classroom. The young child’s world is filled with interesting shapes to explore and understand.
Young children develop geometric understandings from playing with materials such as unit blocks, pattern blocks, tangrams, and paper for origami. In addition, teachers can help children identify the many shapes that exist in the classroom and outdoors on the playground by casually pointing them out.
© ______ 2009, Merrill, an imprint of Pearson Education Inc. Used by permission. All rights reserved. The reproduction, duplication, or distribution of this material by any means including but not limited to email and blogs is strictly prohibited without the explicit permission of the publisher.
- Kindergarten Sight Words List
- Coats and Car Seats: A Lethal Combination?
- Signs Your Child Might Have Asperger's Syndrome
- Child Development Theories
- GED Math Practice Test 1
- Social Cognitive Theory
- The Homework Debate
- 10 Fun Activities for Children with Autism
- First Grade Sight Words List
- Graduation Inspiration: Top 10 Graduation Quotes