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Table of Contents

• Manipulatives Are Not Enough: Necessary Conditions for Math/Science Concept Development
• Total Quality Education
• Implications of Understanding the Brain
• Whole Language, Literacy Development, and Kindergarten
• Adult Interactions
• New Jersey Transition Project

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Manipulatives Are Not Enough:
Necessary Conditions for Math/Science Concept Development

David Wright, Ed.D.

For some time educational writers have been concerned about many of the concepts taught in both math and science and the developmental appropriateness of them. Underhill has stated, "Evidence is accumulating that implies strongly that we are teaching mathematical concepts and skills that many learners in many classrooms cannot understand."' Kovalik argues that "a large percentage of the concepts identified for elementary students is, in fact, beyond what students can understand at those ages; they may be able to memorize and parrot them back but that does not indicate a real understanding."' Unfortunately, the early introduction of complex, perhaps inappropriate, concepts in math and science, during the early childhood years, has been accepted by many educators as long as they were introduced with concrete, manipulative objects.

Using manipulative materials to introduce mathematical or scientific concepts is typically based on the learning theories of Maria Montessori and Jean Piaget who suggested that children construct knowledge from their actions on the environment. And such actions require objects to manipulate.' It is also based on an understanding of how concepts are learned as expressed by Beattie:

"...it is generally agreed that an optimal learning sequence of mathematical concepts moves from concrete to semiconcrete (iconic) to abstract learning experiences. Such a sequence involves objects and their manipulation, pictures or models of real objects, and finally, the use of abstract symbols."⁴

Similarly, Jack Ott and others have stated that "familiar, concrete experience actual or recalled should be a first step in the development of new abstract concepts and their symbolization."⁵

As a result of these understandings about learning, teachers energetically use an approach to mathematics and science instruction that emphasizes manipulation. The manipulation activities generally go well and the children seem to be learning. Encouraged by the results, teachers move the activities to the symbolic level and find, to their dismay, that many of the children are not able to deal with the concept in a symbolic way. In frustration many teachers resort to the old methodology of telling the children the rules necessary for completing the symbolic tasks and forget the concern for concept development.

What is incorrectly assumed by many teachers is that there is some sort of "burst of understanding" that occurs for all children as a result of manipulative activities. It is felt that the connectedness from concrete to semiconcrete to abstract happens magically, as if the materials themselves contain the understanding. If the children just get enough "hands on experiences," the understanding will be absorbed through the pores of the hands. The materials, of course, do not contain the understanding; the child does. These misunderstandings, based on a widely accepted theory of learning, need changing.

Another View

While it is true that Piaget felt that children construct their own knowledge, he also believed that cognitive development occurs in stages along a continuum and that the use of concrete materials is important at each of the stages in order for the child to build mental images.' In addition, it seems that the child's readiness for concrete experiences is based on a complicated, not well understood, combination of prior learnings and current level of cognitive functioning. This means that in order to understand a mathematical or scientific concept, the child must be at the appropriate stage of cognitive development for that particular concept, even when using manipulatives.

For instance, Piaget suggests that primary grade children cannot think meaningfully about things which are not real. To the young child who is in the primary grades, reality is that which one can see, hear, taste, touch or smell, yet we sometimes try to teach these children concepts that have no basis in reality. An example of this is arithmetic regrouping. Regrouping numerical values is not part of the primary child's reality. Regrouping is an abstract concept that was developed in the mind and is not physically manifested. Only when the child has reached the stage of concrete thought (usually around age eight) is it possible to meaningfully learn about abstract ideas using concrete experiences. With the onset of concrete thought the manipulation of a variety of materials is appropriate for developing a beginning understanding of arithmetic regrouping.

Science too has concepts that appear to be inappropriate for many young children. Brain researchers and Seven Intelligence advocates have expressed concern about the early introduction of inappropriate concepts in math and science. Kovalik states that

"This is particularly true in science. For example, solar system as a subject for second or third graders is wildly age-inappropriate. The concepts are highly abstract and not experienceable - the ground on which we stand is spinning at hundreds of miles an hour, and the distances between planets are computed in millions of miles or even light years, a measurement that most adults can't relate to. "⁷

The understanding of mathematical and scientific concepts, then, is dependent on two factors that are closely related: experiences at the concrete, semiconcrete, and abstract level and the child's stage of cognitive development. The child's level of cognitive development cannot be rushed and changes slowly over a period of time. It also takes time and a sequence of experiences from the concrete to abstract in order to build mental images in the mind of the child who is cognitively ready.

While Piaget's stages do have ages attached to them, it is clear that not all seven or eight year olds are concrete thinkers and, conversely, some six year olds are. How a teacher decides what is a developmentally appropriate concept is difficult and will not always "hit the mark". Two strategies that are typically taken are considering the "age appropriateness" and the "individual appropriateness" of each concept.

Age appropriate math and science concepts are those concepts that one would typically expect children of that age to understand, given good teaching techniques such as the use of manipulatives. This knowledge comes from experience with children of this age and understanding of preoperational and concrete thought. Using the example of mathematical regrouping, one would expect that most third graders should be able to deal with this concept.

Individually appropriate math and science concepts are more difficult to identify. This requires careful observation and knowledge of individual children in the classroom. Those children who seem to be able to grasp somewhat difficult concepts might benefit from a concrete introduction to an idea like regrouping even if they aren't seven or eight years of age.

Initially, however, it is probably best for the teacher to use the age appropriate guidelines to determine math and science, concepts early on in the year, and then, as time passes and knowledge of individuals increases, one can substitute more individually appropriate concepts.

When a person can understand and successfully think through a math or science problem, a sense of competence ensues. A curriculum that most closely matches a child's level of thinking, and therefore makes sense, helps that child feel competent to understand and use what is learned. On the other hand, when something does not make sense and a solution appears out of reach, a person feels incompetent. Developmentally inappropriate math or science expectations such as arithmetic regrouping foster, in many children, a sense of incompetence.

When a young child is presented with concepts not appropriate for his or her level of cognitive development, the child becomes frustrated. The result can be devastating to the child's developing sense of self as mathematician or scientist-as a person capable of understanding and using our mathematical and scientific symbol system. Such children may come to view mathematics and science as mysterious, incomprehensible systems.

To avoid this situation, it is recommended that each teacher keep in mind not only the need to introduce math and science concepts through manipulatives, but also the developmental stages of the children in the class. That is, each teacher should try to teach those concepts which are developmentally appropriate.

¹ Underhill, B. (1985). One Point of View Let's Diagnose the Curriculum. ARITHMETIC TEACHER, 33 (4) 1.

² Kovalik, S. and Olsen (1993). ITI: THE MODEL INTEGRATED THEMATIC INSTRUCTION (2nd Edition): Susan Kovalik and Associates, 6.

³ Piaget, J. (1964). THREE LECTURES: PIAGET REDISCOVERED, Ithaca, New York: Cornell University Press.

⁴ Beattie, 1. D. (1986). The Number Names: An Aid to Understanding Place Value. ARITHMETIC TEACHER, 33 (5) 24.

⁵ Ott, J., Snook and Gibson (1991). Understanding Partitive Division of Fractions. ARITHMETIC TEACHER, 39 (2) 7.

⁶ Ibid.

⁷ Op. cit.

David Wright is Professor, Elementary and Early Childhood Education, Western Oregon State College. He may be contacted at Western Oregon State College, Monmouth, Oregon 97361, (503) 838-8330. This article appeared in the Volume 30, Number 4 (Summer 1995) issue of the OAEYC Bulletin, the quarterly publication of the Oregon Association for the Education of Young Children, and is reprinted here with the express permission of Dr. Wright and OAEYC.

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Total Quality Education

The Woods Quality Center has produced a 17-minute video entitled Total Quality at Prairie View: "A New Way of Learning". Using the Prairie View Elementary School in Cedar Rapids, Iowa, as a resource, the tape focuses on the critical roles children can play in their own education.

Highlights include observing students while they conduct self-assessments, lead parent conferences and classroom meetings, work in teams, and plan with a systems approach. Children solve problems, set goals, and c art their progress in an environment where teachers become "supporters rather than judges, coaches rather than lecturers, partners with students and parents rather than isolated within the walls of each classroom."

The Woods Quality Center is a coordinated effort by area private and public organizations to integrate continuous quality improvement into communities in Iowa. The Center was formed in 1993 as the result of a shared vision of a Total Quality environment for the entire community and a belief that all organizations benefit from a commitment to such quality improvement: businesses, government, labor, manufacturers, nonprofits, and schools.

The videotape may be ordered for $89, plus $5 per tape for shipping and handling, by contacting The Woods Quality Center, 4401 Sixth Street, SW, Cedar Rapids, Iowa, 52404-4499, (319) 399-6798; the director of the Center is Robert McNiel. Persons wishing to learn more about Prairie View Elementary may contact its principal, Marilyn Miller, at (319) 848-5260.

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Implications of Understanding the Brain

A collaborative relationship should exist between scientists and educators, writes Philip Cohen in "Understanding the Brain: Educators Seek to Apply Brain Research". To emphasize the importance of such collaboration and how it can directly affect education, Cohen reviews Robert Sylwester's book, A Celebration of Neurons: An Educator's Guide to the Human Brain, and finds that in it Sylwester provides examples of recent discoveries which have specific implications for primary-grade teachers and other educators.

When learning to read, for example, a child has to develop new connections between the regions in the brain that process oral and written language. These connections "may grow to be more efficient," according to research, if early reading instruction adjusts to more normal processing rates, instead of maintaining slower rates which appear to be easier for the child.

During the early childhood years, children develop strategies for solving arithmetic problems by using their fingers and/or other manipulatives. They learn a new strategy most efficiently when adults present them with problems that can be solved using that strategy, rather than allow the children to become frustrated with the current strategies they are employing. The implications of brain research are, according to Sylwester, that teachers should be providing information about a variety of arithmetic strategies and facilitating children's learning to self-monitor the use of these strategies in problem solving.

Cohen writes that such suggestions are based on an understanding of the brain's plasticity, its ability to grow and adapt in response to environmental stimuli. This complexity of the brain means that a method of instruction which is based on complex experiences is a more effective means than one which narrowly focuses on subject disciplines and the accumulation of facts and skills. Cohen quotes Renate Caine, co-author of Making Connections: Teaching and the Human Brain, who champions the complex experience method: "That means that the children have hands-on experience. They have the potential to engage in dialogue with other people. They have the opportunity to express something orally, the opportunity to express something in written form, the opportunity to touch, to recreate."

Research on brain functioning can also serve to identify inappropriate educational practices. Cohen cites Susan Kovalik, an education consultant and developer of the Integrated Thematic Instruction model, who has written that "what has to go are the curriculum pieces we've clung to for centuries. Disciplines have to go; the textbooks have to go; the worksheets have to go--because they have nothing to do with how the brain works."

Philip Cohen's article, "Understanding the Brain: Educators Seek to Apply Brain Research," appeared in the September 1995 issue of ASCD's Education Update (Volume 37, Number 7). Robert Sylwester's and Renate Caine's books are both available from the Association for Supervision and Curriculum Development, 1250 N. Pitt Street, Alexandria, Virginia 22314, (703) 549-9110.

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Whole Language, Literacy Development, and Kindergarten

An article by Scott Willis entitled "Whole Language: Finding the Surest Way to Literacy" comprises the entire Fall 1995 issue of Curriculum Update. He writes that the philosophy of whole language instruction has "swept the primary grades" and that although most teachers still use a basal reader, they have also tended to incorporate aspects of whole language into their reading curriculum. The use of whole language, like other educational reforms, is currently being re-examined, and Willis explores the arguments on both sides of the debate.

Willis references many of the leading authorities on literacy development in his provocative exploration. Gay Fawcett, director of curriculum and instruction for the Summit County (Ohio) Education Service Center provides a framework for the debate by indicating that "everyone - even the most hardcore whole language advocates - acknowledges that kids need to learn letter-sound relationships."

Jerome Harste, professor of language education at Indiana University-Bloomington, declares that "whole language is the best phonics program there is," and Constance Weaver, professor of English at Western Michigan University recommends strategies for teaching phonics in the context of whole language passages.

Phonetic teaching in such a manner is only "incidental", and thus inadequate, according to Jeanne Chall, professor emerita of the Harvard Graduate School of Education. "Sometimes you have to take things out of context" in order to teach them effectively, she writes." God doesn't say everything has to be in context."

Steve Stahl, professor of reading at the University of Georgia, is concerned about the elimination of the direct teaching of phonic skills, and about teachers' waiting for teachable moments. "What if that moment arises when the teacher is working with another kid? The teacher can't be everywhere."

In a similar vein, Bill Honig, the former California Superintendent of Schools and currently a professor at San Francisco State University, believes that the state's literature-based, language arts framework was "disastrous for huge numbers of kids in California.... We took for granted that teaching phonics would happen."

"Whole language has gone way too far in under emphasizing word attack and phonetic skills," asserts Bob Slavin, codirector of the Center for Research on the Education of Students Placed At Risk, at Johns Hopkins University. Yet Yvetta Goodman of the University of Arizona contends that "If given rich experiences, kids will find ways to come to literacy" and that the recommendation that children, who have not had rich early literacy experiences, receive lots of direct skills instruction "blows my mind."

In the midst of the arguments, Marilyn Jager Adams, professor of cognitive and linguistic sciences at Brown University, says that "the best bet is to invest in kindergarten" to help children who lack a literacy background. Scott Willis writes that All young children benefit if engaged in activities (such as games and rhymes) that direct their attention to the sounds of language. By finding ways to engage kindergartners with print, stories, and the sounds and structure of language, teachers can make sure young children have "phonemic awareness"-an understanding of the logic of written language. This understanding lays the groundwork for later learning. Adams further states that teaching phonics to children who don't have such phonemic awareness is "a big waste of time. The very kids who need it the most are just not responding."

A reprint of this issue of Curriculum Update (Fall 1995) is available for $1.00 plus a , stamped, self-addressed envelope from the Association for Supervision and Curriculum Development, 1250 N. Pitt Street, Alexandria, Virginia 22314, (703) 549-9110. When ordering, refer to stock number 1-95232.

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Adult Interactions

The subject of what constitutes appropriate adult interactions in primary-grade classrooms and other early childhood settings is frequently examined. The topic has, in fact, been explored in past issues of Of Primary Interest. In the Colorado Quality Standards for Early Childhood Care and Education Services, a new section has been developed which focuses on "The Role and Function of the Teaching Team" (Spring 1995/Volume 2, Number 2); and Elena Bodrova and Deborah Leong are championing the role of the primary-grade teacher in raising the level of a child's assisted performance, while facilitating the practice of what she/he can do independently (Fall 1995/Volume 2, Number 4).

In the Fall 1995 issue of the High/Scope Resource, Mary Hohmann and David P. Weikart provide information about quality interactions in a preschool environment, which apply equally to the primary grades. They identify five key strategies adults should use: Provide a variety of materials for children to work with, in order to assure that there are sufficient opportunities for children to make choices and to manipulate the materials.

• Provide space and time for children to use materials by arranging and equipping play areas and planning a consistent daily routine.
• Seek out children's intentions, thereby strengthening their sense of initiative and control.
• Listen for and encourage children's thinking through relaxed conversations which repeat, amplify, and build on what the child says.
• Encourage children to do things for themselves, by allowing them to think of and practice ways of solving the everyday problems they encounter, and by referring children to one another for ideas, assistance, and conversation.

Hohmann and Weikart have developed a list of ingredients which should be present in the active learning of an early childhood environment: materials, manipulation, choice, language from the child, and adult support. They caution teachers that once the environment is prepared, adults should "continue to be active and involved, observing children and supporting their initiatives throughout the day."

The High/Scope Resource and other publications are available from the High/Scope Educational Research Foundation, 600 North River Street, Ypsilanti, Michigan 48197, (800) 407-7377.

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New Jersey Transition Project

CORN Associates, in partnership with Wayne General Hospital Child Care Center, Wayne, New Jersey, has received a $50,000 grant to develop a project promoting ways for early childhood teachers in public and private schools, parents, and administrators to work together to ensure successful educational transitions from preschool to third grade.

The grant is part of a $1.3 million fund jointly administered by Bell Atlantic, the International Brotherhood of Electrical Workers (IBEW), and Communication Workers of America (CWA).

Ten informational workshops will be presented on issues related to educational continuity. They will be attended by representatives from five preschools, 35 public schools and their "feeder" districts, as well as other interested state educators. Each workshop will be videotaped and will include informational handouts. For more specifics about the project, interested individuals may call Kathryn Arabia, Director of the Wayne General Hospital Child Care Center, at (201) 904-0202.

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This page last updated: December 13, 2007