ERIC Identifier: ED446826 Publication Date: 2000-10-00
Author: Richardson, Kathy Source: ERIC Clearinghouse on
Elementary and Early Childhood Education Champaign IL.
Mathematics Standards for Pre-Kindergarten through Grade 2.
Concern about the mathematics achievement of America's youth has reached a
new level. It is clear in this increasingly technological and global society
that achievement in mathematics will have a major impact on students' "career
aspirations, their role in society, and even their sense of personal
fulfillment" (Malcom, 1999). This high level of concern has resulted in a
growing appreciation for the impact that early mathematics learning could have
on the life course of young children. Traditionally, mathematics has received
little attention in preschools and in many K-2 classrooms (Johnson, 1999). This
situation is changing as mathematics learning is recognized as critical to
The National Council of Teachers of Mathematics (NCTM, 2000) recently
published Principles and Standards for School Mathematics, building on its 1989
Curriculum and Evaluation Standards for School Mathematics. The current version
includes pre-kindergarten standards for the first time and outlines the
mathematics that children should learn as they progress through school. While
some observers question the appropriateness of any standards for this age group,
other critics argue that the current standards may be less develop-mentally
appropriate than the previous standards, moving in the direction of content
knowledge and product orientation. This Digest discusses the latest mathematics
standards for young children and how teachers can use developmentally
appropriate practices to help children meet these standards.
WHAT MATHEMATICS SHOULD PRE-K THROUGH GRADE 2 STUDENTS BE LEARNING?
Many early childhood educators approach teaching
mathematics with feelings of anxiety. However, the mathematics presented in
Principles and Standards for School Mathematics provides a broad view of what
mathematics is and can be for young children--a view that early childhood
educators implementing developmentally appropriate practices can use.
Mathematics can provide children with ways to understand and appreciate the
world around them and enrich rather than narrow children's experiences.
Principles and Standards for School Mathematics identifies both content and
Content Standards. The content standards are organized into several areas:
(1) number and operations, (2) geometry, (3) measurement, (4) data analysis and
probability, and (5) algebra. Mathematics in the early years is not just a
simpler version of mathematics that children will learn later. Rather, teaching
about mathematics in early childhood classrooms provides foundational concepts
that are key to understanding more formal and abstract ideas. To be truly
prepared for later math, young children need to develop flexibility in thinking
about numbers (NCTM, 2000). For example, they need to know that 5 is 1 more than
4 and 2 less than 7. They need to know that 5 objects can be arranged in
different ways: as 3 and 2 or 4 and 1, and also as 2 and 2 and 1. They need to
be able to solve problems by using relationships such as 3 + 3 = 6, so 3 + 4
must be 7 (Richardson, 1999a; Althouse, 1994).
To understand measurement, children first need to be aware of what can be
measured. They need to line things up, to cover spaces with blocks that fit
together, and to pour sand or water from one container to another. If children
are going to understand geometric principles, they first must put together
blocks to make new shapes and to recognize the difference between a triangle and
a rectangle. In short, children need to experience the applications of
mathematics in their everyday lives.
Process Standards. As stated in Principles and Standards for School
Mathematics, "Learning with understanding is essential to enable students to
solve the new kinds of problems they will inevitably face in the future" (NCTM,
2000, p. 21).
The process standards set out in Principles are congruent with
developmentally appropriate practice and include (1) problem solving, (2)
reasoning and proof, (3) communication, (4) connections, and (5) representation.
The standards suggest that young children should be encouraged to solve
problems, investigate, and use mathematics to find out things they don't already
know. Children can be encouraged to reason, to conjecture about the way things
are, and to check those conjectures. The emphasis is on children thinking for
themselves, rather than repeating what the teacher wants. Children will want to
communicate, listen, and clarify their own thinking in the process of
communicating with others.
WHAT QUESTIONING TECHNIQUES CAN TEACHERS USE?
questioning techniques--including those intended to help children understand
concepts, hypothesize, and generate interesting questions--can help children
appreciate the mathematics that surrounds them.
Rather than engaging in rote counting and numeral recognition, children can
be encouraged to ask: How many are there? Can we find out without counting them
all? How many do we need? Do we have enough? Who has the most? Are there any
extras? What happens when we take numbers apart or put them together?
Rather than simply learning the names of basic shapes, children can discover:
How are these shapes alike? How are they different? Which ones fit together?
Which ones leave spaces? What can we build with these? What other shapes can we
make using these shapes?
Rather than learning how to use a ruler, children can determine: Which is
bigger? More? Heavier? Longer? Shorter? How can we find out?
Children can prepare to represent data in charts and graphs by sorting and
organizing objects into groups to find out which group has more or less. "Do we
have more red apples or more green apples?"
Rather than using symbols to stand for various amounts, children can work
with ideas related to generalization and predictability through the exploration
of patterns. What comes next? How do you know?
Mathematics also helps children understand, organize, and analyze their
science experiences. They can experience the connections between math and music
when exploring rhythm and patterns, and between math and art when working with
symmetry and design.
HOW CAN WE HELP ALL CHILDREN MEET THE STANDARDS?
suggests that children's math concepts are often more sophisticated than
traditionally assumed (Gelman, 1999). However, in our eagerness to help children
meet the standards, we must take care not to use instructional methods that give
the appearance of high math achievement but that, in fact, interfere with real
growth in understanding. Children are much more capable and confident when they
are allowed to make sense of things instead of trying to follow someone else's
thinking. Elkind (1999) reminds us that, as we seek to determine what is
possible for children, "the only way to understand how children learn a concept
is to observe them in the process of acquiring it."
Teachers, parents, and administrators will want to keep in mind that meeting
the standards is a long process. For example, one state lists in its Math
Checklist for Kindergarten, "Identify, name and draw a circle, rectangle,
triangle, hexagon, rhombus, trapezoid, and square" (Arkansas, 2000). Before
children can meet this standard with understanding, they must be able to
recognize a variety of triangle shapes and distinguish these from a variety of
hexagon shapes. They must be able to draw straight and diagonal lines of similar
lengths and connect them appropriately. In that same checklist is the standard
"Counts and keeps track of up to 20 objects." Even this seemingly
straightforward skill is highly complex and understood in different ways as
children develop competence over time (Richardson, 1997b, 1999a). Clearly, early
childhood educators will need to continue to explore ways to individualize the
mathematics curriculum for all children.
The most effective way to meet standards is to
work toward them by beginning wherever the child is. Any other strategy simply
wastes the child's time and prevents the development of the essential
foundational understandings and skills needed for future success. It is
important to be in tune with both the accomplishments and the still undeveloped
ideas that are a part of the child's growing understanding. It is exciting and
even inspiring to think that we can provide more mathematics for children than
we may have done in the past. We do not have to fear raised expectations as long
as we look first to the child with respect for wherever he or she is on the
journey toward deeper understanding of mathematics.
FOR MORE INFORMATION
Althouse, R. (1994). INVESTIGATING
MATHEMATICS WITH YOUNG CHILDREN. New York: Teachers College Press.
American Association for the Advancement of Science. (1999). DIALOGUE ON
EARLY CHILDHOOD SCIENCE, MATHEMATICS, AND TECHNOLOGY EDUCATION. Washington, DC:
Author. ED 427 877.
Arkansas Department of Education Web site. (2000). Available:
Bredekamp, S., & Copple, C. (Eds.). (1997). DEVELOPMENTALLY APPROPRIATE
PRACTICE IN EARLY CHILDHOOD PROGRAMS(Rev. ed.). Washington, DC: NAEYC. ED 403
Bredekamp, S., & Rosegrant, T. (Eds.). (1995). REACHING POTENTIALS:
TRANSFORMING EARLY CHILDHOOD CURRICULUM AND ASSESSMENT(Vol. 2). Washington, DC:
National Association for the Education of Young Children. ED 391 598.
Burke, M. J., & Curcio, F. R. (Eds.). (2000). LEARNING MATHEMATICS FOR A
NEW CENTURY. Reston, VA: National Council of Teachers of Mathematics.
Copley, J. V. (Ed.). (1999). MATHEMATICS IN THE EARLY YEARS. Reston, VA:
National Council of Teachers of Mathematics.
Elkind, D. (1999). Educating young children in math, science, and technology.
In American Association for the Advancement of Science, Dialogue on early
childhood science, mathematics, And technology education. Washington, DC: AAAS.
ED 427 877.
Gelman S. A. (1999). Concept development in preschool children. In American
Association for the Advancement of Science, DIALOGUE ON EARLY CHILDHOOD SCIENCE,
MATHEMATICS, AND TECHNOLOGY EDUCATION. Washington, DC: AAAS. ED 427 877.
Johnson, J. R. (1999). The forum on early childhood science, mathematics, and
technology education. In American Association for the Advancement of Science,
Dialogue on early childhood Science, mathematics, and technology education.
Washington, DC: AAAS. ED 427 877.
Malcom, S. (1999). Making sense of the world. In American Association for the
Advancement of Science, DIALOGUE ON EARLY CHILDHOOD SCIENCE, MATHEMATICS, AND
TECHNOLOGY EDUCATION. Washington, DC: AAAS. ED 427 877.
National Council of Teachers of Mathematics. (2000). PRINCIPLES AND STANDARDS
FOR SCHOOL MATHEMATICS. Reston, VA: Author.
Richardson, K. (1997a). MATH TIME: THE LEARNING ENVIRONMENT. Norman, OK:
Richardson, K. (1997b). Too easy for kindergarten and just right for first
grade. TEACHING CHILDREN MATHEMATICS, 3(8), 432-437. EJ 543 623.
Richardson, K. (1999a). DEVELOPING NUMBER CONCEPTS. White Plains, NY: Dale
Richardson, K. (1999b). UNDERSTANDING GEOMETRY. Bellingham, WA: Lummi Bay
Rowan, T., & Bourne, B. (1994). THINKING LIKE MATHEMATICIANS. Portsmouth,
NH: Heinemann. ED 408 183.
Wolf, D. P., & Neugebauer, B. (Eds.). (1996). MORE THAN NUMBERS:
MATHEMATICAL THINKING IN THE EARLY YEARS. Redmond, WA: Child Care Information
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