Teaching Young Children Mathematics

Janice Minetola, Robert G. Ziegenfuss, J. Kent Chrisman, Robert Ziegenfuss

• Publisher: Routledge
• Year: 2013
• Print ISBN: 9780415641630
• eBook ISBN: 9781136175435
• Topic: Education
• Subtopic: Education General

Teaching Young Children Mathematics provides a comprehensive overview of mathematics instruction in the early childhood classroom. Taking into account family differences, language barriers, and the presence of special needs students in many classrooms throughout the U.S., this textbook situates best practices for mathematics instruction within the larger frameworks of federal and state standards as well as contemporary understandings of child development.

Key topics covered include: developmental information of conceptual understanding in mathematics from birth through 3rd grade, use of national and state standards in math, including the new Common Core State Standards, information for adapting ideas to meet special needs and English Language Learners, literacy connections in each chapter, 'real-world' connections to the content, and information for family connections to the content.

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Nurturing Young Children’s Mathematical Development

How Do Young Children Learn Mathematics?

This question has been the focus of many approaches, theories, and practices regarding the growth, development, and learning of young children. While children have similar feelings as adults concerning fear, pain, sadness, and joy, they are not merely miniaturesized adults; their learning needs and patterns are different (Taylor, 1999). Learning mathematics for young children should be grounded in experiences that are developmentally appropriate for a specific age group as well as the individual child, for even though a group of children may be similar in age, their developmental levels—physical, emotional, social, and cognitive—may be quite varied. Because of the variance of developmental levels in children of similar ages, educators look at general characteristics of children at different ages in planning learning experiences. They know that not all children are at the same developmental levels at the same time. Effective teachers get to know their students well and enact developmentally appropriate practices in their classrooms that help individual children reach goals that are both challenging and attainable, while at the same time, being responsive to the social and cultural context in which their students live (Copple & Bredekamp, 2009).

Theories vs. Assumptions

Children from birth through the primary grades in school pass through the first three stages of intellectual development. They begin and continue to learn through their five senses—sight, hearing, taste, touch, and smell. They learn to control their body movements and continue to develop physical or body skills. They question, imitate, and practice and learn best through concrete, hands-on experiences. Although quite home-based and self-centered at first, they expand beyond these characteristics to interact with others and to explore other environments (Taylor, 1999).

Best practices in classrooms are not based on assumptions or beliefs; they are based on knowledge of how children learn and grow. Early childhood educators draw on the work of those whose theories are anchored in developmentally appropriate practice. Some of these theorists include Jean Piaget, Erik Erikson, John Dewey, and Lev Vygotsky. A brief overview of each theorist’s work follows; however, you are encouraged to read more extensively in order to gain a greater understanding of the theories and how they relate to the mathematical development of young children. This section is included in this chapter to link your understanding of child development to children’s mathematical thinking.

Cognitive or Intellectual Development

Jean Piaget (1896–1980) was a Swiss epistemologist whose work is described as a cognitive-development theory. He studied the cognitive or intellectual development of children and formulated a model with four stages of development. His model of intellectual development includes the following categories (Taylor, 1999) (implications for mathematical understanding in each stage are also included):

Piagetian Stages (Taylor, 1999, p. 3)

Stage 1: Sensorimotor (0–24 months of age)

Implications for Mathematical Understanding: Children in the sensorimotor stage learn through their senses. They repeat number words they hear from media such as television and from people in their environment. Some children learn to count numbers, typically from one to ten and begin to recognize some number symbols. They learn terminology related to mathematics, such as big and small. By the end of this stage children are developing the concept of object permanence, or the idea that objects exist even when they are out of sight (Charlesworth, 2005).

Stage 2: Preoperational (two–seven years of age)

Implications for Mathematical Understanding: Children in the preoperational stage are the focus of early childhood education. Although they still use their senses to learn about their world, they begin to learn through concrete, hands-on experiences. Mathematics manipulatives, or concrete objects that children can handle, become an important part of their learning experiences. For example, children need to have access to various objects in their classrooms to count, add together, and subtract from one another. They need to examine shapes, use various measuring tools, and use real-life objects such as clocks, a cash register, money, and calendars in their dramatic play areas. As children’s language development continues, they begin to use more terminology related to mathematics in referring to size, weight, shape, time, and length. As children reach the later end of this stage, they use symbolic behaviors in their play. For example, a stick could represent a fishing rod, or a block could represent a toy car or train car. They begin working with classifying objects (putting things in logical groups) and seriation (putting objects in a logical sequence) (Charlesworth, 2005).

Stage 3: Concrete Operational (seven–11 years of age)

Implications for Mathematical Understanding: Children in the concrete operational stage are beginning to work with symbolic activities—the beginning of logical thought, according to Piaget (Geist, 2009). They are able to represent objects using other objects or symbols. For example, when drawing a picture representation to aid in solving a subtraction problem involving ducks on a pond, children would be able to draw Xs or some other symbol to represent the ducks rather than draw pictures of ducks. Also, children at this stage do not need to see five objects to know what the number five represents.

Stage 4: Formal Operational (11 years of age on)

Implications for Mathematical Understanding: Children in the formal operational stage are capable of solving mathematics problems in a logical and systematic way. They begin to understand abstract mathematics concepts and solve abstract problems (Charlesworth, 2005). For example, children in this stage of cognitive development no longer need concrete manipulatives or picture representations to solve math problems. They understand that numerals represent quantities and can solve problems (add, subtract, multiply, and divide) using standard algorithms.

Social-Emotional Development

Erik Erikson’s work (1902–1994) focused on eight stages of psychosocial development, in which each stage is characterized by a positive and negative attribute (Bickart, Jablon, & Dodge, 1999). The four stages listed below relate to young children’s psychosocial development. Implications for mathematical understanding are included for each stage.

Erikson Stages (Taylor, 1999, p. 4)

Birth–one year: Trust vs. Mistrust

Implications for Mathematical Understanding: In any academic area, the ability to trust oneself is important, and this is especially true in mathematical thinking. Trust or mistrust has the potential of affecting their attitudes towards adults, including their teachers, as they grow and develop (Taylor, 1999).

One–three years: Autonomy vs. Shame and Doubt

Implications for Mathematical Understanding: This is the stage when children are learning to become self-sufficient. They are learning to walk, feed themselves, use the toilet, and engage in other adult-like activities. Depending on the attitudes of caregivers towards children’s development in these areas, children can learn to feel good about themselves or doubt their abilities (Taylor, 1999).

Three–six years: Intuitive vs. Guilt

Implications for Mathematical Understanding: During this stage, children are engaged in adult-like activities as they are experimenting with their world (Taylor, 1999). Attitudes and feedback they receive from caregivers can directly relate to children’s feelings of confidence in their ability to work with mathematical concepts such as numbers, shapes, and patterns, or they learn to feel guilty about their inabilities.

Seven–11 years: Industry vs. Inferiority

Implications for Mathematical Understanding: Children in this stage are product-oriented and want to do well and feel confident in their abilities. If children feel confident in learning mathematics, they will take risks, problem solve, and take on mathematical challenges. This in turn helps them learn and grow mathematically. Children in this stage who do not feel confident in their abilities to do mathematics shy away from risk-taking and problem solving. They learn to dislike mathematics because they have feelings of inferiority and the attitude of “I can’t do it” (Bickart et al., 1999).

In working with young children, it is important for teachers to make daily decisions that help children meet success and progress in a healthy manner so positive attributes are developed as opposed to negative ones. Children need to learn to trust, develop confidence and adequacy in their abilities, and to become self-sufficient as they engage in adult-like activities. These abilities are critical learning, practicing, and using mathematics throughout the child’s life.

Inquiry-Based Learning

John Dewey (1859–1952) has had a tremendous influence on education in the United States. He believed that schools should teach children to think by engaging them in hands-on, inquiry-based learning experiences. He believed that education should involve community experiences and also involve physical activity and free play (Ryan & Cooper, 2007; Taylor, 1999). Examples of hands-on projects and inquiry-based math learning experiences will be included in each of the following chapters.

Socio-Constructivism

Lev Vygotsky (1896–1934) was a contemporary of Piaget, whose work was unpublished until after his death. He stressed the importance of language and social and cultural context in helping children develop thinking skills. His work emphasizes the social nature of learning and the importance of scaffolding children’s learning through teacher–child and child–child relationships. He wrote about the Zone of Proximal Development (ZPD) in emphasizing these relationships—the zone being the “distance between the child’s ability to solve problems independently compared to his ability to solve them with the assistance of someone more competent than he” (Taylor, 1999, p. 5). Vygotsky also believed that learning was at the heart of a child’s development; in other words, learning leads development.

Two implications for mathematical learning with young children from Vygotsky’s theory are that children can learn mathematical concepts and operations from other children if the daily activities are planned for such interactions, and mathematical activities must be planned just ahead of the child’s developmental level. These require ongoing assessment and careful planning. Activities and projects will be included in later chapters that highlight these two implications.

Developmentally Appropriate Practice (DAP)

The National Association for the Education of Young Children (NAEYC) published a position statement titled Developmentally Appropriate Practice in Early Childhood Programs Serving Children from Birth through Age 8, adopted in 2009, that provides a framework for best practices in educating young children that promotes optimal development and learning. The position statement is grounded in research, theories of growth and development of children, and knowledge concerning effective education. The NAEYC position statement reflects a commitment to excellence and equity in educating all children.

Early childhood practitioners make many daily decisions concerning goals for children’s learning and how to help them achieve these goals. In making sound decisions, practitioners need to consider knowledge about general, age-related characteristics of children, knowledge about individual children, and knowledge about the cultural and social context in which individual children live. These pieces of knowledge are essential in guiding early childhood practitioners in planning appropriate learning experiences for individual children that will support them as they progress in learning and growing (Copple & Bredekamp, 2009).

The NAEYC position statement identifies 12 well-supported principles for educating young children. These include:

1. All the domains of development and learning—physical, social and emotion, and cognitive—are important, and they are closely interrelated. Children’s development and learning in one domain influence, and are influenced by, what takes place in other domains.

2. Many aspects of children’s learning and development follow well-documented sequences, with later abilities, skills, and knowledge building on those already acquired.

3. Development and learning proceed at varying rates from child to child, as well as at uneven rates across different areas of a child’s individual functioning.

4. Development and learning result from a dynamic and continuous interaction of biological maturation and experience.

5. Early experiences have profound effects, both cumulative and delayed, on a child’s development and learning; and optimal periods exist for certain types of development and learning to occur.

6. Development proceeds toward greater complexity, self-regulation, and symbolic or representational capacit...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Dedication
• Table of Contents
• List of Figures
• List of Tables
• Chapter 1. Nurturing Young Children’s Mathematical Development
• Chapter 2. Mathematics Curricula for Young Children, Standards, and Assessment
• Chapter 3. Numbers and Operations
• Chapter 4. Patterns, Functions, and Algebraic Thinking
• Chapter 5. Geometry
• Chapter 6. Measurement
• Chapter 7. Data Analysis and Probability
• Index