37?/67531/metadc331046/...use in grades K-2 and 3-5 with similar results (Kloosterman, P. & Harty, HI., 1987). The teachers at the K-2 level reported more frequent use of manipulatives

37? JH&Jd

Sio. 3/3B

FIRST-YEAR TEACHER USAGE OF MANIPULATIVES IN MATHEMATICS

INSTRUCTION: A CASE STUDY

DISSERTATION

Presented to the Graduate Council of the

University of North Texas in Partial

Fulfillment of the Requirements

For the Degree of

DOCTOR OF PHILOSOPHY

By

Barbara N. Sylvester, B. S., M. S.

Denton, Texas

December, 1989

Sylvester, Barbara N., First-Year Teacher Usage of Manipulatives in

Mathematics Instruction: A Case Study. Doctor of Philosophy (Elementary

Education), July, 1989,186 pp., 9 tables, bibliography, 63 titles.

This qualitative case study examined the use of manipulatives in

mathematics instruction by six first-year intermediate teachers in a north

Texas school district. Their preparation for, access to, and perceptions

about manipulatives were examined. Specific content associated with

manipulative usage was identified.

The following conclusions were drawn from the study. The number of

manipulatives the subjects were exposed to in college varied widely.

Teachers exposed to the most manipulatives used more and taught a wider

variety of topics with manipulatives than those who had seen fewer

materials. A second important factor was the teacher's perceptions

concerning availability of the manipulatives. When subjects felt materials

were readily available, they chose to use them more often than when they

were perceived to be less accessible. The six teachers used many

manipulatives when teaching measurement, time and money, and basic

fractional concepts and relatively few when developing computational

algorithms for the four operations. Teachers' editions of textbooks were

the main resource used to plan lessons. The district's curriculum guide,

which was full of lessons using manipulatives, was rarely utilized.

The following recommendations were made from the study. Better

communication of the district's goals for including concrete objects in

/6o /*s

mathematics instruction was needed. Provision of inventories for each

building and the district's Instructional Center was suggested so that

teachers would know what was available. Since some materials were new

to some teachers, voluntary minisessions in their use prior to student

instruction was suggested. Since the college preparation of the teachers

played a direct role in their use of manipulatives, mathematics educators

should conduct additional research to determine if the wide disparity of

exposure to materials found in this group was an anomaly, or whether it

exists on a wide-scale basis. Issues including content covered, aspects of

manipulative usage, and implementation of the levels of concept

attainment are examples of possible research interest.

TABLE OF CONTENTS

Page

LIST OF TABLES v

Chapter

I. INTRODUCTION 1

Purpose of the Study

Research Questions

Significance of the Study

Definition of Terms

Limitations

Assumptions

Chapter Bibliography

II. REVIEW OF RELATED LITERATURE . .8

Historical Perspective

Cognitive Psychology Influences

Brain Research

Research Supporting Manipulative Use in Classrooms

Manipulative Use in Classrooms

Teacher Education Programs

Summary


III. PROCEDURES 31

Research Approach

Population

Procedures for Data Collection

Data Analysis


IV. PRESENTATION AND ANALYSIS OF DATA 39

Data From the Four Elementary Buildings

Data From the Central Office


V. FINDINGS, CONCLUSIONS, AND RECOMMENDATIONS . . . . 135

Findings

Conclusions

Recommendations


APPENDICES 156

BIBLIOGRAPHY 180

IV

LIST OF TABLES

Table Page

1. Manipulatives Listed in Anne's Lesson Plans 53

2. Manipulatives Listed in Alice's Lesson Plans 64

3. Manipulatives Listed in Beth's Lesson Plans 83

4. Manipulatives Listed in Brenda's Lesson Plans 96

5. Manipulatives Listed in Cathy's Lesson Plans 109

6. Manipulatives Listed in Dena's Lesson Plans 125

7. Average Perceived Level of Need for

Mathematics (Content and Application) 129

8. Average Perceived Level of Need for

Mathematics (Manipulatives) 130

9. Comparison of Exposure to Manipulatives

with Usage 150

CHAPTER I

INTRODUCTION

The use of concrete objects, referred to as manipulatives, as a means

of introducing mathematical concepts and skills has been advocated by the

National Council of Teachers of Mathematics (NCTM) for nearly half a

century (Worth, 1986). Numerous articles have appeared during that time

in Arithmetic Teacher (Beattie, 1986; Young, 1983), Mathematics Teacher

(Bright & Harvey, 1988) and various NCTM Yearbooks (Lindquist &

Shulte,1987) explaining the rationale and demonstrating the uses for

these manipulatives in elementary and secondary classrooms.

Support has also grown for use of manipulatives for mathematical

concept building through research conducted about learning itself from a

variety of sources. Piaget's (1963) four stages of human development

theory and Skemp's (1971) two-stage learning model both rely on the use

of manipulatives for successful concept attainment. Currently the

findings of research about the functioning of the brain, the place learning

occurs, reveals the importance of continued physical manipulation not

just during the early elementary years, but into adolescence (Hart, 1987;

Mann & Sabatino, 1985).

Suydam (1984) indicated that teachers believed manipulatives should

be a part of mathematics instruction. She also reported findings from

studies which indicated the use of manipulatives increased achievement

for a variety of topics for grades K-8 no matter the achievement level or

ability level of the students. Parham (1983) examined sixty-four research

studies and concluded that students who had used manipulatives outscored

those who did not.

Studies have been conducted examining the extent to which

manipulatives are used in elementary classrooms. First grade teachers

reportedly use manipulatives for mathematics instruction most often with

a continued decrease in manipulative use from grades two throughout the

elementary years (Suydam, 1984). Another study compared manipulative

use in grades K-2 and 3-5 with similar results (Kloosterman, P. & Harty,

HI., 1987). The teachers at the K-2 level reported more frequent use of

manipulatives than did the 3-5 teachers. Scott (1983) suiveyed teachers

of grades kindergarten through five in an urban area and found that only a

limited number of them used manipulatives over five times during an

entire year.

Teacher preparation courses include readings, discussions, and

demonstrations of these techniques. Very little is known, however, about

what occurs within the first year teacher's mathematics classroom in

terms of manipulative use. A comprehensive study which investigates this

phenomenon would be beneficial to mathematics educators. In addition,

school district personnel who hire first-year teachers may be assisted in

their staff development programs with this information.

Purpose of the Study

The purpose of this study was to determine the extent to which

manipulatives were used in the instruction of mathematics by first-year

intermediate grade teachers.

Research Questions

The purpose of this study was to determine:

1. Were teachers familiar with manipulative materials which could be

used to facilitate the learning of mathematical skills and concepts?

2. What types of manipulatives were available to the teachers? Who

decided their selection?

3. What manipulatives, if any, did the teachers choose to use in the

instruction of mathematics at the intermediate grade levels?

4. For which topics in mathematics did the teachers choose to use

manipulatives as a vehicle of instruction?

5. Why did the teachers match particular topics and manipulatives?

6. What were the teachers' perceptions about the use of manipulatives

in mathematics?

7. What methods of evaluation, if any, did the teachers use to

determine the effectiveness of manipulative use?

Significance of the Study

Shulman (1987) discussed a need for educational case studies to be

included in case books which would be beneficial for preservice, novice and

inservice education. Good and Biddle (1988) stated, "We contend that the

expansion of observational research can yield better theories for

understanding the learning of mathematics and other subjects and can

produce more adequate models for improving teaching" (p. 114). Cooney,

Grouws and Jones (1988) cited the lack of observational information to be

a "missing link" in the knowledge of teaching mathematics. They felt

observations allowed for perspective development and identification of

possible important educational variables.

This research provides a case study on the use of concrete

manipulatives during mathematics instruction by first-year intermediate

grade teachers.

Definition of Terms

The following terms have restricted meaning and are thus defined for

this study:

1. Manipulatives were defined as concrete objects used to enhance the

learning of mathematical concepts. These could be commercially

purchased materials such as Cuisenaire rods and base ten blocks or

teacher-made materials such as paper pattern blocks or base ten

models made from beans and tongue depressors.

2. A first-year teacher was defined as a teacher who had completed a

teacher education program but had no prior teaching experience

other than student teaching.

Limitations

This study provided a thick description of the use of manipulatives by

intermediate grade teachers in one school district. Due to the qualitative

nature of this study, generalizations to other settings are inappropriate.

Data was gathered through observation during mathematics instruction,

interviews with teachers and administrators, and examination of

appropriate written materials such as lesson plans, textbooks and

curriculum guides, then analyzed qualitatively. Observations of classroom

instruction in mathematics were followed by interviews with the

teachers. In studies of this nature, both observer bias and subject

self-reports can produce error so results should be considered with

caution.

Assumption?

The major assumption underlying this study was:

The first-year teacher makes final decisions concerning content and

methodology in terms of the mathematics classroom.

CHAPTER BIBLIOGRAPHY

Beattie, I. D. (1986). Modeling operations and algorithms. Arithmetic

Teacher. 23(6), 23-28.

Bright, G. W., & Harvey, J. G. (1988). Games, geometry, and teaching.

Mathematics Teacher. £1(4), 250-259.

Cooney, T. J., Grouws, D. A., & Jones, D. (1988). An agenda for research on

teaching mathematics. In T. J. Cooney, D. A. Grouws, & D. Jones (Eds.),

Effective mathematics teaching (pp. 253-261). Reston, VA: National

Council of Teachers of Mathematics.

Good, T. L., & Biddle, B. J. (1988). Research and the improvement of

mathematics instruction: The need for observational resources. In

T. J. Cooney, D. A. Grouws, & D. Jones (Eds.), Effective mathematics

teaching (pp. 114-142). Reston, VA: National Council of Teachers of

Mathematics.

Hart, L. A. (Speaker). (1987). The brain approach to learning (Cassette

Recording No. 1987-03). Reston, VA: Association of Teacher

Educators.

Kloosterman, P. & Harty, H. (1987). Current teaching practices in science

and mathematics in Indiana (Report 143V Indianapolis: Indiana State

Department of Education.

Lindquist, M. M., & Shulte, A. P. (Eds.). (1987). Learning and teaching

geometry. K-12 (1987 Yearbook). Reston, VA: National Council of

Teachers of Mathematics.

Mann, L., & Sabatino, D. A. (1985). Foundations of cognitive process in

remedial and special education. Rockville, MD: Aspen Systems

Corporation.

Parham, J. L. (1983). A meta-analysis of the use of manipulative

materials and student achievement in elementary school

mathematics. Dissertations Abstracts International. 44A. 96.

Piaget, J. (1963). The attainment of invariants and reversible operations

in the development of thinking. Social Research. 3Q, 283-299.

Scott, P. B. (1983). A survey of perceived use of mathematics materials

by elementary teachers in large urban school district. School Science

and Mathematics, S3(1), 61-68.

Shulman, L. (Speaker). (1987). A vision for teacher education (Cassette

Recording No. 1987-02). Reston, VA: Association of Teacher

Educators.

Skemp, R. S. (1971). The psychology of learning mathematics.

Hammondsworth, England: Penguin Books.

Suydam, M. N. (1984). Manipulative materials. Arithmetic Teacher. 31 (5).

27.

Worth, J. (1986). By way of introduction. Arithmetic Teacher. 33(6), 2-3.

Young, S. L. (1983). Teacher education: How teacher educators can use

manipulative materials with preservice teachers. Arithmetic

Teacher, 31(4), 12-13.

CHAPTER II

REVIEW OF RELATED LITERATURE

This literature review presents information describing manipulatives

in mathematics education. It begins with an historical perspective.

Rationales are given for the use of concrete manipulatives in the

mathematics classroom stemming from both brain research and the study

of human development. Implications of that research are then examined.

Studies examining the actual use of manipulatives in elementary

classrooms are included.

Manipulative Usage in Mathematics Education

Historical Perspective

The first half of the twentieth century marked a change in the rationale

for choosing content and methodology in mathematics classrooms

(Brownell, 1986,1954). At the beginning of the century, learning occurred

almost wholly at the rote level and content was quite difficult. This

regimented method of instruction was based on the doctrine of Formal

Discipline which stated that the mind was divided into separate pieces

called faculties which required strenuous training. In 1935 William

Brownell set the stage for the next several decades by proposing the idea

that in order for learning of mathematics to be permanent, it must be

8

meaningful. Included in Brownell's plan was a greater emphasis on

experiential learning, the use of concrete objects for children to

manipulate as a means of understanding new skills and concepts, and a

change in teacher role from telling students about mathematics to

facilitating discovery of principles by students (Marks, Hiatt & Neufeld,

1985).

The above ideas fit in well with the progressive mood in education in

vogue at the time. John Dewey, identified as one of the foremost

progressive education advocates, felt that the materials used for learning

were of utmost importance in building the experiences needed for

successful learning to take place (McNeil, 1985). A description of

Brownell's (1935) proposal is found in the tenth yearbook published by the

National Council of Teachers of Mathematics (NCTM).

Cognitive Psychology Influences

In 1953 the NCTM published their eighteenth yearbook entitled

Multi-Sensorv Aids in the Teaching of Mathematics (Worth. 1986). It was

during this time that the writings of Jean Piaget were beginning to have

profound effects on the views the educational community held toward child

development. Piaget (1963) defined four periods of development through

which he felt all individuals passed: the sensorimotor period, the

preoperational period, the concrete operations period, and the formal

operations period. Piaget believed that it was not until individuals

progressed to the formal operations period that formal thought processes

such as formal analysis of situations and abstracting hypotheses were

10

possible. Students in the middle elementary grades are most often found

in the concrete operations period where "...thought is still bound to the

concrete and tied to perception" (Wadsworth, 1978, p. 19). Kennedy and

Tipps (1988) described students at this level as needing "...experiences

with many concrete objects in order to represent abstract ideas and the

operations involving those ideas" (p. 15).

While Piaget's model described learning in general, Zoltan P. Dienes

looked specifically at the learning of mathematics (Post, 1988). The four

principles which marked Dienes' (1960) theory of learning mathematics

were based on the assumption that students must be actively involved in

the learning process. In his "Dynamic Principle," Dienes stated that

complete understanding of a new idea is a process which is composed of

stages which occur over time. The beginning stage involves the learner in

unstructured play with the object or objects to be used in the second stage

where structured experiences are provided by the teacher. The third part

of this principle involves the gradual convergence upon the concept to be

taught and opportunities for application of the concept in real world

situations.

The next two principles Dienes (1960) suggested are known as the

"Perceptual Variability Principle" and the "Mathematical Variability

Principle" which work together to help students develop mathematical

concepts. Dienes stated in the first principle that a variety of experiences

with different materials is needed for students to eventually abstract a

concept. He explained in the second principle that students need to be

given experiences with examples of concepts where critical attributes

11

remain the same and noncritical attributes are varied. This allows the

child to generalize the concept.

The last principle Dienes (1960) put forth, the "Constructivity

Principle," dealt with identifying students as either constructive or

analytical thinkers. These stages are analogous to Piaget's concrete

operational and formal operational periods. The principle was based upon

the idea that students must experience concepts constructively before

analysis was possible.

Skemp's (1971) learning model also involved two stages. The first

stage allowed for both in-school and out-of-school manipulation of

objects which provided a base on which abstractions were later built. The

internalization of these experiences must occur before the student is able

to apply the concept in problem situations. Adequate time must be given

students for this to occur.

Another advocate of manipulative use for concept introduction has been

Jerome Bruner (Reys, Suydam & Lindquist, 1989). He described three levels

of developmental learning: enactive, iconic, and symbolic. The enactive

level requires the handling of "real-world" objects to allow direct

interaction with the physical world. The iconic level relies on various

types of visual images to represent objects from the physical world. The

symbolic level is the most abstract of the three. Symbols (i.e."+" or "23")

are abstractions of reality.

Bruner's work has been extended further by Lesh (1979). A diagram of

the model appears in Appendix A (Lesh & Zawojewski, 1988, p. 62). There

are five components of this model. Three of these components correspond

12

to Bruner's model. Lesh's "Manipulative Models" corresponds to Bruner's

enactive level, the "Static Pictures" relates to the iconic level, and the

"Written Symbols" is analogous to the symbolic level. The "Spoken

Language" and "Real Scripts" are added aspects to this model not found in

Bruner's. The arrows represent the translations from one mode to another

and within a mode. Post (1988) stated that, "Stressing the various

translations within and among these modes of representation is the most

important contribution of this model" (p. 14). An example of a

within-mode translation would be asking a student to use two different

manipulative aids to show the same concept. When students use base ten

blocks to work a problem, then draw a picture representing that problem,

they have accomplished a translation from one mode to another.

Brain Research

The theories set forth by Brownell in the thirties and Piaget and Bruner

in the fifties and sixties have become the cornerstone for the current

activity-based mathematics education programs (Kroll, 1989). These

theories are being given further credence by the psychobiologists of today.

Restak (1979) uses the term psychobiology to describe the research field

which attempts to make sense of our actions within our world by

combining brain research and psychology.

Early information about the human brain has come from studies on

people with brain injuries or people who have had surgery performed in

hopes of curing an illness. Due to today's technology, noninvasive

techniques allow examination of the brain from different viewpoints (Mann

13

& Sabatino, 1985). The first part of this section describes the techniques

used to gain information about the brain. Models of brain action follow

with implications for education given.

The oldest of these techniques is Electroencephalography more

commonly called EEG. This technology dates back to 1924 when Hans

Berger, a German psychiatrist, hypothesized that brain activity could be

measured by placing electrodes on the patient's head. This technique has

allowed researchers to measure different patterns of brain activity. With

the advent of computers, this method was enhanced even more. Using the

average evoked response method (EMR), visual, auditory, or tactile stimuli

were repeatedly shown to the patient while attached electrodes recorded

responses from specific brain locations. Currently color images are

generated through a technique labeled BEAM or brain electrical activity

mapping. These new forms hold promise for further study of cognition.

A second type of technology which has given information about the

brain is the computerized axial tomography or CAT scan. This machine

allows X-rays to be sent throughout the body and then generates

three-dimensional images for examination. Differences in one brain from

another can be examined through this visual output. Nuclear magnetic

resonance (NMR) also may hold promise for providing visual images for

investigation. Images have been produced using a combination of magnetic

fields and radio waves. One advantage of this system over the CAT has

been the fact that the NMR is better able to penetrate bone allowing for

more precise readings. A second advantage of the NMR has been related to

improved patient safety due to the fact that X-rays are not used.

14

A final piece of equipment which may hold promise is the positron

emission transaxial tomography or PETT. It also has provided color images

of the brain during cognition. Radioactive glucose has been injected into

the subject intravenously. During the test this glucose has appeared in

larger amounts in the parts of the brain where the greater activity has

occurred. This has allowed scientists to learn more of the brain's

physiology.

Information from this technology has allowed models to be

hypothesized which can have great implications for educators, particularly

in terms of content scope and sequence and teaching methodology (Hart,

1987; Mann & Sabatino, 1985). To fully understand these ideas, it is first

necessary to examine two different models used to represent information

about the brain.

In the first model, Dr. Paul D. MacLean, Director of the Laboratory of

Brain Evolution and Behavior of the National Institute of Mental Health,

hypothesized the existence of the human brain not being a single unit, but a

triune body (Restak, 1979). MacLean stated, "The three brains amount to

three interconnected biological computers, each having its own

intelligence, its own subjectivity, its own sense of time and space, and its

own memory and other functions" (Restak, 1979, p. 52).

In MacLean's model the three brain parts are layered. The innermost,

smallest and most primitive brain is the R-complex or "reptilian" brain

which manages basic functions and is thought to encourage ritualistic,

compulsive behaviors (Restak, 1979; Hart, 1982). In addition, awe for

authority, "a proclivity for prejudice and deception," and "a tendency

15

toward imitation" stem from this part of the human brain.

The second layer is called the paleomammalian brairi or limbic system.

It surrounds the reptilian brain and accounts for almost twenty percent of

the total brain mass. This part of the brain has been under investigation

for over twenty-five years. During that time it has been demonstrated

that the limbic system affected formation of memory, thermal

equilibrium, hormones, and drives. Our emotions stem from this area.

The outermost layer is called the neomammalian braiin or cerebrum. It

is the largest of the three, by far, occupying about seventy-five percent of

the total brain mass. It's complexity allows for much storage, but this

also makes it much slower than its two counterparts (Hart, 1982). "The

cerebrum or newest brain handles language and all we think of as

education" (p. 199).

When discussing the brain, Restak (1979) hypothesized a relationship

between psychobiology and cognitive theory as proposed by Piaget. Restak

(1979) wrote:

I am introducing Piaget at this point because I think his theories of

psychobiological development and intelligence fit best with recent

psychobiological discoveries about consciousness. Although Piaget

does not refer to brain processes as explanatory concepts (even now

such correlations are only beginning to be drawn), his experiments are

rooted in the soundest possible psychological methods: systematic

study of how the brain evolves from a reflex machine (sensory motor

period) to the level where consciousness first appears, (p. 261)

The second hypothesized model is one which deals primarily with the

16

cerebral cortex or cerebrum (Mann & Sabatino, 1985). It is made up of two

hemispheres which are connected by a band of nerves called the corpus

callosum which allows communication to continually occur between the

two. Each of the hemispheres has been divided into five lobes for purposes

of locating motor and sensory functions. Research has been inconclusive

about more complex processes such as memory, problem-solving and

perception. Injuries to or removal through surgery of some areas of the

brain associated with certain cognitive abilities may result in either a

temporary or permanent loss of those abilities. A variety of theories exist

about the brain's ability to transfer or relocate.

Cerebral dominance, a theory proposed by Samuel Orton in 1928, has

been perpetuated throughout the last sixty years. Orton believed that one

hemisphere must be more dominant for normal cognitive thought to be

possible. If one hemisphere did not take charge, confusion reigned. This

theory still retains credibility as Silver and Hagin (1976) hypothesized

some learning disabilities were due to a lack of dominance.

Much of the early brain research consisted of examining brains that had

been injured or changed through surgery. Due to these experiences, early

conceptions of brain functioning models insinuated that the two

hemispheres could be thought of in terms of two separate brains (Restak,

1982).

Not to be ignored is the sequence of the development of the brain itself.

Brain growth can be examined in two stages (Restak, 1979). The first

stage begins at conception and continues through the second trimester of

pregnancy. It has been commonly accepted that most of our neurons are

17

created at this time. The second stage begins during the third trimester

and continues until the infant is about six months of age. It is during this

time that the glia and the dendrites, cells which support the nervous

system, are formed. The glia form on the neural branches and the dendrites

form synapses or connections between neurons.

Myelin, a special form of glia, has been said to be of special importance

to educators (Hart, 1987). Its function is to cover the axon of neurons

allowing for quicker travel of nerve impulses. What has been said to be

special about the myelination process is the length of time it takes and the

order in which it occurs (Denman, 1988). The myelination or coating of

axons takes several years to complete; in fact the process takes twenty

years on the average (Hart, 1987). The process occurs in a sequenced

fashion with the prefrontal lobes being the last completed. Since these

prefrontal lobes are thought to be responsible for "higher mental activity"

including long-term planning and goal-centered behavior (Restak, 1979),

the educational implications, particularly for young children, are

tremendous.

Levy (1982) stated that while the hemispheres were different, the fact

that they are connected by the corpus callosum was significant. The

corpus callosum's function is "to process information and to derive

perceptions at the same time" (p.180). It follows then that information

about a student's dominant hemisphere clues the educator how to introduce

material for a given student, not how it must be totally taught.

Brennan (1982) advocated the teaching of lessons to the whole brain.

She suggested that this be accomplished by presenting each lesson two

18

ways: inductively for the left-dominant student and deductively for the

right-dominant one. Gregorc (1982) also pointed out the need for teachers

to balance their instructional strategies. In doing so, the student will

have a better working relationship between hemispheres.

Hart (1982,1987) proposed his Proster Theory, a brain-based theory of

human learning based heavily on MacLean's triune-brain model. The major

emphasis of the model was for the learner to acquire programs deemed

useful by the learner. This model was based upon the following brain

functions:

1. Housekeeping. The brain controls the many subsystems of the

body that keep us alive, such as blood pressure, digestion, salt

level, temperature, etc., as well as balance and muscle

coordination.

2. Adjustment. The brain adjusts these systems to meet

conditions and needs. The major adjustments involve what we

call "emotions." The prime concerns here are individual and

species survival.

3. The brain receives and processes informational input and

extracts patterns from it. So to speak, it "recognizes what is

happening."

4. The brain builds and stores programs for all kind of activities.

(Hart, 1982, p. 198)

It is easy to see the correlation between the "reptilian brain" with

"Housekeeping," the "limbic systems" with "Adjustment" and the

"neomammalian brain" with Hart's third and fourth points dealing with

19

patterns and programs. These patterns and programs, known as learning,

are stored in the neomammalian brain.

Another element of Proster Theory dealt with the concept of

downshifting. Hart (1987) maintained that when humans encountered

situations which were unpleasant or endangering, the reptilian and/or

limbic system became the dominant brain force. When students perceive

school as an unpleasant experience, downshifting occurs. The desired

learning is unable to occur when this happened. Therefore, to successfully

implement brain-compatible education, schools must be restructured to

allow the brain to function at its learning level.

Research Supporting Manipulative Use in Classrooms

The theories of cognitive development put forth by Piaget, Dienes,

Bruner and Lesh have captured the educational community's attention. The

topic of manipulative use in mathematics instruction has been the focus of

many studies. Suydam and Higgins (1977) noted in a review of research on

the topic that in over half the studies, the children who used manipulatives

scored higher on achievement tests than did their counterparts. In the

great majority of the remainder of the studies, the two groups compared

were about equal in achievement.

Parham's (1983) analysis of sixty-four research studies showed even

stronger results. When tested for achievement, the average score for the

students who had used manipulative materials during instruction was

approximately the eighty-fifth percentile while those who did not use

manipulatives scored at the fiftieth percentile.

20

Specific content areas in mathematics have also been examined in

terms of manipulative use. Driscoll (1984), when summarizing research on

rational numbers, wrote, "Research has shown that the sensible use of

concrete materials is effective in teaching rational numbers" and

"Concrete materials are useful beyond the primary grades and, indeed,

beyond elementary school" (p. 35). Hunting (1984) concurred with these

findings.

Canny (1984) investigated the use of manipulatives with fourth graders

in terms of problem solving skills. Her conclusion stated that when

manipulatives are used to introduce content, the students scores are

especially higher.

The National Council of Teachers of Mathematics (1980) published "An

Agenda for Action," a concisely written document consisting of eight

recommendations. The fourth of these recommendations stated, "Stringent

standards of both effectiveness and efficiency must be applied to the

teaching of mathematics" (p.11). There were four subrecommendations

accompanying this; one stated that a variety of materials, strategies, and

resources should be used for instruction including "the use of

manipulatives, where suited, to illustrate or develop a concept or skill" (p.

12).

Each year the February issue of the NCTM's elementary journal,

Arithmetic Teacher, is completely devoted to one topic. In 1986, the topic

was manipulatives (NCTM, 1986). Rationales, research, exemplary

programs, and selection criteria for manipulative use were presented.

The NCTM (1989) published a document entitled Curriculum and

21

Evaluation Standards for School Mathematics. In the section which

described the K-4 mathematics curriculum, one of the underlying

assumptions stated that children must be actively involved when doing

mathematics. One of the implications of this assumption was the need for

teachers "to make extensive and thoughtful use of physical materials to

foster the learning of abstract ideas" (p. 17). The publication was

described by Willoughby (1988) as a "liberating document" which contained

ideas about the importance of all children studying mathematics at

advanced levels, the use of technology in mathematics classrooms, and

recommended a radical change in the evaluation system used to determine

success of the program at all levels.

Phi Delta Kappa (Kroll, 1987) published a collection of articles about

mathematics in the Exemplary Practice Series. One of the eight areas

discussed in the publication was entitled, "Overview: Provide A Concrete

Foundation By Using Manipulatives." In this section of the book there were

six articles discussing the use of concrete manipulatives. The articles

included a philosophical discussion about manipulative use, research

findings about mathematics and achievement, and descriptions of programs

currently in use which rely heavily on manipulatives.

State and local education agencies have also been advocating the use of

manipulatives in classrooms. The Mathematics Section of the Texas

Education Agency (Peavler, DeValcourt, Montalto & Hopkins,1987) included

the recommendation that, "Concrete materials should be used to introduce

new concepts; attention should be paid to developmental sequencing from

concrete to pictorial to abstract, with deliberate attention to transitions

22

from one level to the next" (p. 5). The topic "manipulative resources" was

included in a report published for the New York City Board of Education

(1986). The report was used to organize staff development sessions for

teachers in the district.

Manipulative Use in Classrooms

Manipulative use is advocated by many groups. Cognitive psychologists,

psychobiologists, professional associations and state education agencies

agree on this point. A natural inquiry then seems to be to examine what

extent manipulatives are actually used in mathematics classrooms. Few

studies were found that examine this question.

One survey concluded that thirty-seven percent of classroom teachers

used manipulatives less than once a week (Fey, 1979). The same study

stated that in nine percent of the elementary classrooms manipulatives

were never present.

Wiebe (1981) sent a questionnaire asking first-grade teachers what

manipulatives were available and to estimate the amount of time

manipulatives were used for four different groupings of activities.

Included were the amount of time a) students used manipulatives, b)

students had manipulatives available, c) manipulatives were used by

teachers or students to model instruction, and d) manipulatives were used

for free exploration, drill and practice, introduction and development of

concepts and solving "real" problems. Observations were then made in the

classrooms to attempt to validate the questionnaire. The teachers were

fairly accurate in reporting the various manipulatives available in their

23

school settings. The teachers usually overestimated the amount of time of

manipulative usage.

Scott (1983) also surveyed teachers about manipulative use. He found

manipulative usage was not prevalent, and the textbook was the main

material source used for instruction. Few teachers used manipulatives

over five times a year. The amount of manipulatives used seemed to be a

linear function of grade level with the most usage at the lower elementary

grades and the least taking place in the upper grades. Even at the

first-grade level, however, less than sixty percent of the teachers used

manipulatives. Most of the teachers surveyed indicated they would like

access to more materials.

Kloosterman and Harty (1987) sent questionnaires to elementary

principals throughout the state of Indiana asking them to provide

information about both science and mathematics education within their

buildings. The results paralleled that of Scott (1983). The students in

grades K-2 used mathematics manipulatives more often than those in

grades 3-5. The inverse held true for science manipulatives.

Another study examined the use of four specific manipulatives: unifix

cubes, Cuisenaire rods, abacuses and base ten blocks (Perry & Grossnickle,

1987). A survey was sent to teachers of grades K through 3 inquiring about

availability and usage of the manipulatives. Cuisenaire rods and unifix

cubes were available in more than seventy percent of the schools while

less than forty-six percent had access to abacuses and base ten blocks.

When the categories "some" and "few" were combined, the unifix cubes

were used by seventy-five percent of the responders, forty percent of them

24

used Cuisenaire rods, twenty-nine percent used base ten blocks and

nineteen percent used abacuses.

Teacher Education Programs

Dossey (1981) summarized findings of a survey conducted to determine

information about teacher preparation programs in elementary education.

The three major topics on the questionnaire dealt with university

mathematics requirements, mathematics methods requirements, and

pre-student teaching experiences. Ninety percent of the schools surveyed

had at least one methods course in mathematics. Ninety -one percent of the

institutions teaching a methods course reported an emphasis on the use of

manipulatives in class.

With the emphasis today from research, it is a fair assumption to make

that the majority of teachers who have been graduated since Dossey's

(1981) study was completed have had exposure to manipulatives. The

emphasis placed on manipulatives can be approximated by examining the

textbooks written for methods courses. Appendix B contains a chart

showing the comparison of several current mathematics education

textbooks. Chapter topics and manipulative usage are evaluated.

Summary

The use of concrete manipulatives in mathematics instruction at the

elementary level has broad research-based support. National, state, and

local policies advocate the use of these manipulatives in mathematics

classrooms. Reported research shows that manipulative use in classroom

25

settings is limited. Teacher education programs report inclusion of

instruction with manipulatives in mathematics education methods courses.

Little is known, however, about the decisions first-year teacher teachers

make in implementing their mathematics programs.


Brennan, P. K. (1982). Teaching to the whole brain. In Student learning

styles and brain behavior (DP. 212-213). Reston, VA: National

Association of Secondary School Principals.

Brownell, W. A. (1935). Psychological considerations in the learning and

the teaching of arithmetic. In Teaching of Arithmetic. Tenth Yearbook

of the National Council of Teachers of Mathematics. Reston, VA:

National Council of Teachers of Mathematics.

Brownell, W. A. (1954). The revolution in arithmetic. Arithmetic

Teacher. £(2).

Brownell, W. A. (1986). AT classic: The revolution in arithmetic.

Arithmetic Teacher. 21(2), 38-42.

Canny, M. E. (1984). The relationship of manipulative materials to

achievement in three areas of fourth-grade mathematics:

Computation, concept development and problem-solving.

Dissertations Abstracts International. 45A, 775-776.

Denman, T. I. (1988, November). Whole-brain development and the

mathematics classroom. Paper presented at the meeting of the

National Council of Teachers of Mathematics, Baton Rouge, LA.

Dienes, Z. P. (1960). Building UP mathematics. London: Hutchison

Education.

26

27

Dossey, J. A. (1981). The current status of preservice elementary

teacher-education programs. Arithmetic Teacher. 2S(1), 24-26.

Driscoll, M. J. (1984). What research says. Arithmetic Teacher. 31(6),

34-35,46.

Fey, J. T. (1979). Mathematics teaching today: Perspectives from three

national surveys. Arithmetic Teacher. 2Z(2), 10-14.

Gregorc, A. F. (1982). Learning style/brain research: Harbinger of an

emerging psychology. In Student learning styles and brain behavior (pp.

3-10). Reston, VA: National Association of Secondary School

Principles.


Recording No. 1987-03). Reston, VA: Association of Teacher Educators.

Hart, L. A. (1982). Brain-compatible education. In Student learning styles

and brain behavior (pp. 199-202). Reston, VA: National Association of

Secondary School Principals.

Hunting, R. P. (1984). Understanding equivalent fractions. Journal of

Science and Mathematics Education in Southeast Asia. Z(), 266-33.

Kennedy, L. M., & Tipps, S. (1988). Guiding children's learning of

mathematics (5th ed.). Belmont, CA: Wadsworth Publishing Co.


and mathematics in Indiana (Report 143). Indianapolis: Indiana State


28

Kroll, D. L. (1989). Connections between psychological learning theories

and the elementary mathematics curriculum. In P. R. Trafton (Ed.), New

Directions for Elementary School Mathematics (pp. 199-211). Reston,

VA: National Council of Teachers of Mathematics.

Kroll, D. L. (Ed.). (1987). Mathematics. Bloomington, IN: Phi Delta Kappa.

Lesh, R. & Zawojewski, J. S. (1988). Problem solving. In T. R. Post (Ed.),

Teaching Mathematics in Grades K-8 (pp. 40-77). Boston: Allyn and

Bacon.

Lesh, R. (1979). Mathematical learning disabilities: Considerations for

identification, diagnosis, and remediation. In R. Lesh, D. Mierkiewicz, &

M. B. Kantowski (Eds.), Applied mathematical problem solving.

Columbus, OH: ERIC/SMEAR.

Levy, J. (1982). Children think with whole brain: Myth and reality. In

Student learning styles and brain behavior (pp. 173-184). Reston, VA:

National Association of Secondary School Principals.



Corporation.

Marks, J. L., Hiatt, A. A., & Neufeld, E. M. (1985). Teaching elementary

school mathematics for understanding (5th ed.). New York:

McGraw-Hill Book Company.

McNeil, J. D. (1985). Curriculum: A comprehensive introduction (3rd ed.V

Boston: Little, Brown and Co.

National Council of Teachers of Mathematics. (1989). Curriculum and

evaluation standards for school mathematics. Reston, VA: Author.

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National Council of Teachers of Mathematics. (1986). Focus issue:

Manipulatives. Arithmetic Teacher. 22(6).

National Council of Teachers of Mathematics. (1980). An agenda for

action. Reston, VA: Author.

New York City Board of Education. (1987). Mathematics instruction grades

4 & 5. staff development. (Report No. ISBN-88315-904-X). New York

City: Author. (ERIC Document Reproduction Service No. ED 290 632)

Parham, J. L. (1983). A meta-analysis of the use of manipulative materials

and student achievement in elementary school mathematics.

Dissertations Abstracts International. 44& 96.

Peavler, C. S., DeValcourt, R. J., Montalto, B. & Hopkins, B. (1987, August).

The state of the state: Curriculum recommendations for the rest of the

twentieth century. Paper presented at the meeting of the Conference

for the Advancement of Mathematics Teaching, Austin, TX.

Perry, L. M. & Grossnickle, F. E. (1987). Using selected manipulative

materials in teaching mathematics in the primary grades. CA: (ERIC

Document Reproduction Service No. ED 250 155)


in the development of thinking. Social Research. 2Q, 283-299.

Post, T. R. (1988). Some notes on the nature of mathematics learning. In

T. R. Post (Ed.), Teaching mathematics in grades K-8 (p. 1-19). Boston:

Allyn and Bacon, Inc.

Reys, R. E., Suydam, M. N., & Lindquist, M. M. (1989). Helping children learn

mathematics (2nd ed.). Englewood Cliffs, NJ: Prentice Hall.

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Restak, R. M. (1982). The brain. In Student learning styles and brain

behavior (pp. 159-172). Reston, VA: National Association of Secondary

School Principles.

Restak, R. M. (1979). The brain: The last frontier. New York: Warner

Books.



and Mathematics. 83m. 61-68.

Silver A. A., & Hagin, R. A. (1976). Search. New York: Walker.



Suydam, M. N. (1984). Manipulative materials. Arithmetic Teacher. 31(5).

27.

Suydam, M. N. & Higgins, J. L. (1977). Acitivitv-based learning in

elementary school mathematics: Recommendations from research.

Columbus, OH: ERIC/SMEAC.

Wadsworth, B.J. (1978). Piaget for the classroom teacher. New York:

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Wiebe, J. H. (1981). The use of manipulative materials in first grade

mathematics: A preliminary investigation. School Science and

Mathematics. Sl(5), 388-390.

Willoughby, S. S. (1988). Liberating standards for mathematics from

NCTM. Educational Leadership. 4£(2), 83.

Worth, J. (1986). By way of introduction. Arithmetic Teacher. 33(6k 2-3.

CHAPTER III

PROCEDURES

Research Approach

A qualitative case study method was selected for this research. Four

types of data are most often associated with qualitative research:

participant observation, ethnographic interviewing, artifact: collection,

and researcher introspection (Eisenhart, 1988). "When ethnography is

underway, all four of these methods are often employed together. Each is

useful for providing a different perspective on the topic of interest" (p.

106).

Participant observation is the most common method of data collection

(Goetz & LeCompte, 1984). Spradley (1980) lists five categories of

participant observation commonly used in qualitative research:

nonparticipation, passive participation, moderate participation, active

participation, and complete participation. Each research setting dictates

the degree of participation and observation the researcher chooses to do.

Denizen (1978) describes three forms of interviews: the scheduled

standardized interview, the nonscheduled standardized interview, and the

nonstandardized interview. Regardless of the type employed, an interview

allows the researcher "to gain a deeper understanding of how the

participants interpret a situation or phenomenon than can be gained

through observation alone" (Stainback & Stainback, 1988, p. 52).

31

32

Artifact collection includes written and graphic materials related to

the topic (Eisenhart, 1988). In this category Goetz and LeCompte (1984)

also include objects such as materials found in classrooms. Non-symbolic

materials collection is also referred to as collection of physical traces.

Researcher fieldnotes play an important role in data collection.

Fieldnotes consist of detailed information of the setting and happenings

occurring during an observation and researcher introspection or reflection

(Bogdan & Biklen, 1982). This second part of fieldnotes is a necessary part

of the data collection process as it helps to remind the researcher of his

role in the process.

Corroboration, or triangulation as it is commonly called, plays an

important role in qualitative research (Stainback & Stainback, 1988). They

state, "The aim is not to determine the truth about some social

phenomenon or cancel out bias in any one research method or data source.

Rather the purpose of triangulation is to increase one's understanding of

whatever is being investigated" (p. 71). Using a variety of data gathering

techniques including surveys and the like, as well as the four previously

discussed help increase that understanding.

Finally a rigorous, in-depth analysis of one or a small number of

instances of a setting or subject constitutes a case study (Bogdan &

Biklen, 1982). The consideration of first-year teacher usage of concrete

manipulatives is such a topic making a case study ideal. Examination of

teacher and administrator attitudes and perceptions about manipulative

use will also be included in the study. A qualitative approach is the

appropriate research methodology to deal with this data.

33

A qualitative approach was chosen for this research topic for the

following reasons:

1) The objective of this study was to describe the extent to which

concrete manipulatives were used by first-year intermediate grade

teachers during mathematics instruction. The goal of the data

analysis was a more complete understanding of the underlying

concepts affecting the decisions made by first-year teachers.

2) A case study was an appropriate model choice due to the detailed

information which was collected.

3) Data included fieldnotes from observations, transcripts of

interviews with teachers, principals and/or assistant principals,

the elementary mathematics curriculum coordinator, and the

administrator in charge of all elementary programs. In addition,

textbooks, state mandates, curriculum guides, and lesson plans

were analyzed. These required qualitative methods of analysis.

4) Due to the single district location of this study, it would be

inappropriate to make generalizations from the research findings.

5) Since data were collected and analyzed using a grounded theory

approach (Glaser & Strauss, 1967), specification of an a priori

hypothesis was inappropriate.

Population

The six subjects of this study were all first-year intermediate level

teachers employed in a single northeast Texas suburban school district.

They were all in their early to mid-twenties. All had attended universities

34

in the state of Texas. Three had gone to different large state universities.

One had attended a large, private school while the other two went to the

same small, private institution although they were graduated a year apart.

The teachers were located in four different elementary buildings. All

participants remained in their positions throughout the data collection

period.

Each building had a principal and an assistant principal. Three

principals and one assistant principal were interviewed during the course

of the study. The elementary mathematics coordinator and the

administrator in charge of all elementary programs for the entire district

were also interviewed.

Procedures for Collection of Data

This study was conducted in a large suburban school district in north

Texas. Permission was obtained for the researcher to observe six

first-year teachers between January and May of 1989. The teachers and

building administrators were told that the researcher was interested in

observing first-year teachers during mathematics instruction. The

administrator in charge of all elementary programs was given detailed

information about study objectives. He agreed not to share these details

with anyone within the district.

Preliminary contact was made in January, 1989, with each teacher. At

that time, general background information about the teachers was

gathered. This information included personal data and information about

their educational background. Each teacher was observed five times. The

35

first two observations were scheduled; the remainder were unannounced.

The observations took place during the regularly scheduled time for

mathematics instruction.

The researcher took fieldnotes during the classroom observations.

These fieldnotes consisted of two types of material. The first type dealt

with a complete description of the setting. The second type of material

found in fieldnotes was reflections the researcher made while the

observations were occurring and usually contained a concluding

commentary which was reflective at the end of each observation. As soon

as possible following each of the first three observations, a taped

interview between the teacher and researcher was held. Observations four

and five were conducted before the final interview was done. With the

exception of one participant, Anne, five observations and four interviews

occurred. Due to technical difficulty, Anne's third interview was not

recorded so could not be transcribed for analysis. Therefore only three

interviews were included in the data relating to Anne.

Taped interviews with building principals and/or assistant principals,

the elementary mathematics coordinator, and the administrator in charge

of all elementary programs were also done. No administrators were

interviewed until all teachers had been observed a minimum of two times.

The administrator interviews generated data about their observations of

the teachers during mathematics instruction, information about materials

or resources within their building and information about purchasing

policies which had an effect on manipulative use. In addition, documents

which may have had some influence on the teachers' choices such as the

36

textbooks and teacher's editions used, elementary mathematics guidelines

from the Texas Education Agency, and local curriculum guides were

examined. The mathematics lesson plans for each teacher were also

inspected. Through the use of observations, interviews of both teachers

and administrators, and inspection of documents pertaining to

mathematics instruction, the process of triangulation provided a check for

internal validity.

Data Analysis

The constant comparison method of data analysis was used (Stainback

& Stainback, 1988; Bogdan & Biklen, 1982). The data analysis took place in

two phases. The initial phase began immediately with the first contact

and continued through the data collection period. The second phase began

when all data were gathered.

In the first phase, the content of fieldnotes and transcripts of

interviews with the teachers were analyzed to determine units of analysis.

Once units of analysis began to appear, the data were compared and sorted

so that relationships were identified. This cycle of constant comparison

was repeated throughout the data collection period. Information already

gathered should and did have an effect on further data collection sessions

as the ideas generated during analysis form substantive theory. The same

type of iterative process was used to analyze the central office

administrators' interviews, then again on the written material pertinent to

the study.

37

When all of the data were collected, the second phase began. First the

data on individual teachers was compared for similarities and differences.

All other data were then added so comparisons could be made to identify

relationships which indicated explanations of first-year teacher's

decision- making behavior concerning mathematics manipulatives.


Bogdan, R. C., & Biklen, S. K. (1982). Qualitative research for education:

An introduction to theory and methods. Boston: Allyn and Bacon, Inc.

Denizen, K. (1978). The research act tend ed.). New York: McGraw Hill.

Eisenhart, M. A. (1988). The ethnographic research tradition and

mathematics education research. Journal for Research in Mathematics

Education. 12(2), pp. 99-114.

Goetz, J. P. & LeCompte, M. D. (1984). Ethnography and qualitative design

in education research. Orlando: Academic Press, Inc.

Spradley, J. P. (1980). Participant observation. New York: Holt, Rhinehart

& Winston.

Stainback, S., & Stainback, W. (1988). Understanding arid conducting

qualitative research. Dubuque, IA: Kendal/Hunt Publishing Company.

38

CHAPTER IV

PRESENTATION AND ANALYSIS OF DATA

The results of the study are presented in this chapter in two sections.

The first section describes data collected at each of the four buildings

from both the teachers and the building administrators. Each teacher's set

of observations and interviews are described as they occurred

chronologically. Teacher lesson plans were also examined.

The final teacher interviews were conducted after all classroom

observations had been completed. At the beginning of eaich final interview,

the teachers were told that sufficient data had been collected about most

aspects of teaching mathematics, but information on manipulative

materials was sparse. They were then asked to fill in the checklist found

in Appendix C. The list contained thirty-six manipulatives taken from the

state recommended lists, textbooks, and curriculum guides. The subjects

provided information about their preteaching experience with

manipulatives, their perceptions of the availability of the materials

within their rooms and buildings, and which materials had been used during

the year in mathematics instruction.

Building administrators, principals or assistant principals, were

interviewed, also. The administrator interviews took place after a

minimum of two observations had been conducted. Administrators

provided information about the building philosophy about mathematics, the

39

40

teachers, the availability of manipulative materials, and the budgeting

process within the building.

The second section contains information gathered from the two

central office administrators. It also reports data from the district

mathematics curriculum guides for and the teachers' editions of the

mathematics textbooks.

Data From the Four Elementary Buildings

There were four elementary buildings visited during the study. The

buildings will be referred to as Buildings A, B, C, and D henceforth.

Buildings A and B each had two first-year teachers observed in the study.

Buildings C and D each had one first-year teacher. Both observations and

interviews conducted will be discussed. Interviews with principals in

buildings A, C, and D were conducted. The principal of Building B felt that

the assistant principal would be a better source of information, so the

administrator interviewed in Building B was the assistant principal. In

building D, both the principal and the assistant principal were questioned.

Building A

Building A contained grades K through 5. In addition, the district's

elementary gifted program, a one-day-a-week pullout program for grades

three through five was housed in the facility. The students were

ethnically mixed and came from a wide span of the socioeconomic strata.

The classes observed in this building were held in self-contained rooms.

41

The building had undergone complete remodeling the previous summer and

had carpeting in all areas and was newly decorated.

Teacher A1 - Anne

Anne was a tall, slender blonde in her early twenties. She was very

energetic both during interviews and when observed teaching her students.

Anne was graduated from a large state college in December a year before

and had spent a semester substitute teaching in a mid-Texas district prior

to her coming to her current job. She had student taught iin second grade,

two grade levels below her present position.

Anne had a degree in elementary education with specializations in

reading and mathematics. She exhibited much confidence during

discussions about teaching mathematics. She felt she had received an

excellent preparation for teaching in general and mathematics in

particular. When asked to describe her mathematics education course, one

of the first things she stated was that the instructor stressed the use of

lots of manipulatives because, "the more hands-on, the better for the

students." During that initial meeting, she listed various manipulatives

she had been taught to use in that course.

Several topics and activities were seen during the five observations.

The students were always actively involved in part or all of the lessons in

this classroom.

Observation One

The class was involved in using menus in a problem solving format

42

during the first observation. The students were initially seated on the

floor or in chairs in a group near the teacher in the front of the classroom.

A word problem involving the use of the menu which the teacher was

holding was written on a large, dry-erasable board. In the problem, the

students were asked to plan a meal that "Hungry Roger Rambo" could eat

with his last $20.

The teacher first reviewed the different parts of a menu with the

students, then asked them to recall what they should do when they saw a

problem like the one on the board. Anne read a variety of items from the

menu for the students to hear, then asked them to plan Roger Rambo's

menu. The students listed several menu items, then figured the total.

When the total was too large, they eliminated items and refigured the bill.

Once the final list of items was decided upon, the class figured the total

bill including tax and tip.

The students were given individual menus from a different restaurant

as an independent assignment. Problems were written on the overhead

projector and the students were asked to find the correct answers using

the menus. The problems on the overhead were much more specific and

required simple addition and multiplication to find the answers.

Interview One

The first interview was held two days after the observation. Anne

gave a general description of her students at this time. The bulk of the

discussion, however, revolved around the resources Anne used to plan

lessons, in particular the observed lesson, and the materials available to

43

help teach mathematics lessons in general. She discussed the fact that the

only things she found in the room at the beginning of the year were the

textbooks, the district curriculum guides, and one class set of small

cardboard clocks. The menus used in the lesson had been collected by Anne

a short time previous to their use.

The discussion included references in Anne's lesson plans to materials

she had included to be used in a lesson later that week. The lesson topic

was multiplication of amounts of money and she had made reference to the

fact that she would use plastic pieces of money. When asked about this,

she explained that another teacher at that grade level had been teaching

for a long time and had accumulated a great number of manipulatives. The

teachers were free to borrow materials from her and that was where Anne

planned to get the manipulatives for the lesson.

Mention was also made of the money the district allotted each teacher

for math and science manipulatives. At that time Anne had not received

the catalogs from the building administrators or the grade level lead

teacher so had not an opportunity to order anything with her budgeted

money.

Observation Two

Two activities took place during the second observation. The students

worked together under teacher direction on another problem situation

involving menus as seen in observation one. The students each had a menu

and were involved through teacher questioning strategies. The second

activity involved the introduction of one-digit division using the

44

traditional division algorithm. This lesson was chosen for observation by

the researcher because Anne's lesson plans said that "tactile objects"

would be used to illustrate the concept.

The lesson was introduced with the following problem:

There are 27 bicycle tires in a store. How many bicycles

can get a set of new tires and how many tires will be left

over?

A picture of a bicycle was then drawn on the board. Anne asked the

students to figure out what they were trying to solve. Students

volunteered that there were two tires per bicycle and that the problem

could be solved by using division. Anne then introduced the steps of the

traditional division algorithm giving the students the saying, "Daddy

divides, Mommy multiplies, Sister subtracts, Brother brings down and

Rover gets the remainder," to assist them in remembering the steps used

to describe the algorithm. Several problems were then worked using the

algorithm, but manipulative objects were not present.

Interview Two

The second interview immediately followed the second observation.

Anne discussed the use of the "Touch Math" program she had recently

learned about from one of the building's special education teachers. She

had introduced it to her entire class knowing that certain students had not

been able to master addition, subtraction, multiplication and division any

other way. Since division was the topic currently under discussion and

division required a basic knowledge of the other operations, Anne felt the

45

students needed additional alternatives to figure out correct responses to

problems.

When asked about the "Family" story that helped students remember

the division algorithm, Anne said she had asked Alice, another first-year

teacher who had previously introduced division to her students, what had

helped the children learn the division process. Alice, who was also a

subject in the study, had already taught the material to her students and

was able to suggest that strategy to Anne.

Although over a month had passed, when Anne was asked about

ordering her materials she replied she had not been able to do so yet. Upon

further questioning, she intimated that she felt it was a grade level

holdup, but admitted she was not at all familiar with the entire budget

process.

Observation 3

Anne spent the first ten minutes of class reviewing two-digit divisor

problems where the divisors could all be rounded down to the nearest ten

by putting five problems (eg. 593 + 83) on the board and asking the

students to work them at their desks.

After the board activity was completed, Anne asked Jonathan to be her

"Trivial Pursuit" person. Jonathan went over to the counter and pulled a

slip of paper from a jar. The paper read $8.20 + 4. Jonathan went to the

board and worked the problem successfully.

The students then opened their textbooks and worked together on word

problems which required them to choose number sentence from a list of

46

three choices. The first problem dealt with the difference in the length of

two dinosaurs measured in meters. To make the unit more meaningful,

Anne got out a meter stick and held it up for the students to see. After the

first problem was worked, the class had to go to their regularly scheduled

library period.

While the students were lining up to leave the room, Anne explained

that the "Math Trivia" was the method by which she was reviewing

previously covered topics. Each day one student was asked to draw a

problem from the jar, work it on the board, and explain it to the class.

Observation 4

The class began with Anne's version of review, "Math Trivia." On this

day Brenda was asked to get a problem from the review jar. The problem

was 902 - 367. Brenda worked the problem correctly, then Anne asked how

she would check the work. Brenda was able to describe the process for the

class.

Anne's students were learning how to use what she termed

"guesstimation" skills to estimate the answer to problems with two-digit

divisors and three-digit dividends. As suggested in her teacher's edition,

she used problems where the divisors all were to be rounded up. She began

the lesson with three problems on the board and worked with the class

using a great deal of choral response from the students during each step of

the problems. The entire class time was spent using this method of

working these problems.

47

Observation 5

Anne's lesson plans stated that her students would be studying a

variety of topics dealing with time on the day chosen for the fifth

observation. In her lesson plans, she mentioned she would be using

individual clocks for each student.

The class again began with "Math Trivia." Jarod drew the following

problem from the jar:

Mr. Cane has 37 tomatoes. He puts them into three packages.

How many tomatoes are in each package? How many are left

over?

At first Jarod was unable to begin to work the problem. Anne prompted

him by asking him to make a drawing on the board to illustrate the

problem. Jarod then drew three large rectangles and began to draw circles

representing the tomatoes. He drew a circle in the first rectangle, then

one in the second rectangle followed by one in the third rectangle, then

repeated the process counting the tomatoes as he went. When he was

finished Anne asked the class what operation would simplify that for them

and most of the students replied chorally, "Division." The problem was

then worked in the traditional format using 37 +• 3.

Next Anne asked her students to take out their textbooks and turn to a

page concerning minutes, hours, and days. While the students were doing

this, Anne passed out small, cardboard clocks to each student. An oral

review was then given which included identifying the hour, minute, and

second hands. A discussion of the relationship between each of the hands

followed.

48

The first activity involved the teacher giving a specific time such as

"half past four" or "thirty till five," then the students were asked to move

the minute and hour hands on the clock to illustrate the specified time.

Once Anne observed that the students were successful with this, she

moved to problems like, "It is now 11:37. What time is it in five minutes?"

The students responded by moving the hands of the clocks to the correct

positions. Anne walked around the classroom checking individual's

answers to the problems she was asking.

Questions such as, "Let's say we begin science at 2:20 and end it at

2:45. How long is science class today?" were also posed so the students

could figure out how much time had elapsed. All during the questioning

done by Anne, the students used the clocks to deduce their answers. Some

students diligently used the clocks each time while others did not,

preferring to work the answers in their heads. Several problems from the

books were worked, then an assignment was made from a worksheet which

accompanied the textbook.

Interview 3

Anne was exposed to more manipulatives in her college preparation for

teaching mathematics than any of the other subjects. She checked

twenty-six items out of the thirty-two on the list as being included in her

college mathematics education course as shown in Appendix C. She

indicated that sixteen items were located in her classroom with an

additional five somewhere else in her building. Of the thirty-two items,

she had used nineteen in her instruction between the beginning of the

49

school year and the middle of May.

Anne was then asked to explain some of the things she had done with

the manipulatives she had indicated that had been used during the year.

She cited using the various fraction models and the Cuisenaire rods to help

students understand fraction names and develop the concept of equivalent

fractions. She mentioned using overhead materials while the students

worked in small groups. At the time of the interview, those were the only

concepts that had been taught concerning fractions.

At the beginning of the year, Anne used place value mats and chips

with numbers on them to identify place value and to compare the value of

numbers represented in this manner. The spinners and the die were

primarily used to generate random numbers to be used in word problems.

The play money was used to develop and refine money concepts,

particularly the menu math.

Anne stated that the metric measurement devices had been used for

both science and mathematics so her students had covered this material

twice. The standard measurement tools had been used only with

mathematics. Base ten blocks had been used to demonstrate volume

concepts.

Anne reported that she had ordered materials and they had been

delivered. She stated that there had been some confusion about the

ordering procedure which had kept her from ordering sooner. She was also

asked what she would do differently with math instruction the following

year. She replied,

Lots more manipulatives. I'll have a lot better idea of what's

50

available to me. I'll have the materials I've ordered; they will be

in my room now. I'm aware of what other teachers ordered, those

teachers who cooperate around here. I'm welcome to use their

things and they're welcome to use mine. I'm going to make it a

point at least once a week that we're using manipulatives in the

math lesson, at least once a week. It does take extra time, but I

think its really worth it, it makes the lesson more meaningful. It

puts a lot more meat into it.

She also discussed moving math instruction from the afternoon to the

morning, including more problem solving activities, allowing the students

to work in cooperative groups more often, and allowing the students to use

both calculators and computers because they were the "tools of the

nineties."

Lesson Plans

Further evidence of Anne's use of manipulatives was found in her

lesson plans as indicated in Table 1. She planned to use play money for

two separate lessons in multiplication of money and listed using real

money to introduce division of money. Measurement tools like metric

rulers were specifically listed for learning about measuring centimeters

and finding perimeter of polygons.

Anne was not consistent about listing the manipulatives used in her

lesson plans or about using materials she had listed. The Cuisenaire rods

she used to work with equivalent fractions the week previous to the

interview were not written in her plans. She also had written that she

51

Table 1

Manipulatives Listed in Anne's Lesson Plans

Topic Manipulative^

Problem solving with money

Multiplying money values

Multiplying dollars < $10

Introduction of one-digit division

Division of money

Introduction to measurement

Perimeter of polygons

Area of surface

Volume

Reading Celsius thermometers

Time topics

Equivalent fractions using models

Menus

Play money

Play money

Tactile objects

Real money

Centimeter rulers

Measurement tools

Graph paper & crayons

Base ten blocks

Thermometers

Individual clocks

Manipulatives divided into

parts

See Appendix C for an additional chart on Anne.

planned to use place value models in her introductory lesson on the

division algorithm in her lesson plans, but no models were used when the

lesson was observed.

52

Teacher A2 - Aline

Alice was a tall, slender brunette. She was very soft spoken both

when teaching and during interviews. Alice originally majored in

psychology, then decided she wanted to teach. She returned to college and

took more than sixty additional hours to obtain elementary certification.

She also had attended a large state university.

Alice had student taught within the district, but in a different

building. She had spent eight weeks teaching second graders then eight

weeks with fourth graders the previous spring. When questioned about

which grade level she liked best, she explained that she had thought she

would like the younger students better, but found she really enjoyed

working with the intermediate grades better.

Alice was a little nervous at the introductory meeting with the

researcher. She apologized over the "clutter" in the room. The room itself

was always neatly decorated with both student work and teacher-made

articles. Alice said that she liked to draw bulletin board characters

herself as was evident in the decoration of her room.

A discussion of her college math background ensued. She said that

different methods for teaching had been discussed. The class talked about

adapting a mathematics program for gifted students and got many

practical suggestions. Her professor had required that they make several

manipulatives and she recalled making a number line, place value charts

and a fishing game where the students "fished" for problems to solve. She

made the comment that she did not use manipulatives every day, but they

were used to introduce a concept or when the principal came into the room.

53

Observation 1

The first lesson observed by the researcher dealt with multiplying

two- and three-digit numbers by an even tens factor. Alice carried her

teacher's edition around during the entire lesson, relying on it to give

sample problems. First she asked the students to open their books to the

correct page, then she wrote the problem 30 X 128 on the chalkboard in

vertical format. She placed a zero in the ones place in the quotient, then

had a student work the remainder of the problem. After this was

accomplished, she reviewed the procedure for working problems with zeros

in one of the factors.

Six additional problems were worked on the board. When that was

finished, she had a student pass out a worksheet which accompanied the

textbook page for the students to work at their seats. The sheet contained

thirty computation problems and two word problems requiring the same

multiplication skill taught in the lesson. Alice then proceeded to walk

around the room for twenty-five minutes checking the students' mastery of

the material as indicated by their independent work.

Interview 1

The interview occurred two days after the observation. Alice

discussed topics she had previously covered during the school year. She

had spent the majority of her time working on computation skills including

addition, subtraction, and multiplication and division facts. They had most

recently completed multiplication by one-digit factors.

Alice found no materials except the textbook and curriculum guides in

54

her room when she began teaching. When asked about having an opportunity

to order things, she brought up the $100 each teacher was budgeted for

math and science equipment. She had not ordered anything at the time and

had not even seen a catalog. The reason she cited for not ordering

materials yet was, "There is a grade level shelf in the grade chairman's

room and I'm not familiar with what's there. There's a lot and I don't want

to reorder something we have."

When asked if she had used anything from the shelf, she said she had

used counters with the students when multiplication and division facts

were being developed. She had grouped the students for those activities.

Alice discussed both the textbook and the curriculum guide as teaching

resources. She said that she probably used the textbook a little each day.

She had followed some of the methods suggested to develop the

multiplication algorithm from the textbook, but felt that the students

were unnecessarily confused so she would do it a different way the next

year. She was currently using the book itself for examples and the guided

practice portion of her lesson. The practice masters which accompanied

the book were used as the independent student work.

The district's curriculum guide provided a content time line. Alice

felt she was following the time table suggested. Other than using the time

line as a gauge, she stated she did not use the math curriculum guide. She

commented, "I don't feel like I have enough time to go to the curriculum

guides."

55

Observation 2

The class began with Alice reviewing division facts. The new material

for the lesson was problems with two-digit dividends, one-digit divisors

and two-digit quotients. She told a story about a family to help the

students remember the steps of the algorithm. The family consisted of "a

Daddy, a Mother, a Sister, a Brother and the dog, Rover." She told the

students that the "Daddy divides, Mother multiplies, Sister subtracts,

Brother brings down and Rover gets the remainder."

Alice worked several problems relating each step to the algorithm.

She added an additional item after "Sister subtracts" when she presented

the steps as she worked the initial problem. She said that after the sister

subtracted, she had to check to see if the house was clean so the brother

could bring down the next visitor. Alice then had the students come up one

at a time to work problems at the board. This occupied the remainder of

the lesson.

interview 2

This interview was held on the next school day after the observation.

Alice explained that the story she used to help teach the algorithm was

something she had learned from her cooperating teacher during student

teaching. She had not had an opportunity to use it previously, though, as

the students had completed division before she entered the fourth grade

classroom.

Alice discussed the difficulties found in teaching division to her

students. She felt that they did not have sufficient mastery of the

56

multiplication and division facts. When asked how she was able to make

the concepts clear for the students, she replied that she had used

materials like beans or small pieces of paper which the students used at

their desks to count out groups like three groups of eight.

The availability of certain manipulatives was also discussed. When

Alice was asked about using base ten blocks instead of counters, she said

she was familiar with the materials because she had used them during her

second grade student teaching experience to work with addition and

subtraction concepts. In this building, however, the base ten blocks were

located in another teacher's room. This was a problem because she did not

always remember to get them. She summed up her feelings about the

matter by saying, nl like having manipulatives, but it's just kind of hard

when they're not right here."

A variety of metric measurement devices were sitting on the table in

the back of Alice's room. Metric measurement was a topic to be taught in

both science and mathematics. When asked about these manipulatives, she

said that they were currently being used in her science class, but she had

not yet taught any measurement in mathematics. Furthermore, she had not

checked to see the similarities and differences which existed between the

two measurement units.

Alice had referred to the science curriculum guide for activities for

the unit on metric measurement, but felt that many of them were

unrealistic. Several required students to use charts that made no sense

while others requested students to measure a large number of items which

would be unrealistic in terms of time and resources. She summed it up by

57

saying, Hl don't have enough supplies for every child to have one so if they

have to rotate, it takes way too long. They were to measure ten things. It

takes them a long time to measure one."

She showed support for the use of manipulatives by the comment, "If I

had the money I would invest in enough supplies for twenty kids to do

everything alone because I think it's so neat when they can manipulate it

themselves." Alice said that there were enough rulers for each student to

be able to complete the length measurement assignments. She stated that

the mathematics unit on measurement would not be affected by the fact

that the material had already been studied in science. She planned to

"review it (measurement topics) like they've never seen it before."

Approximately a month had passed between the first and second

interview. During that time, Anne had received the materials with which

to order the manipulatives budgeted for mathematics and science. Her

unfamiliarity with available materials was evident when she commented,

I want to be careful what I get because I want it to be something

that I'll really need, not something that looks neat. I almost hate

to order anything this whole year in a way, because I'd rather wait

till the end of the year and look back and say, "That would have

been neat to have and that, but I'm glad I didn't get that thing

because I never would have used it."

Observation 3

Alice began the class period by handing out pieces of paper to the

students. The papers were about 3" by 11" and the boys received blue while

the girls got pink. When all of the pieces of paper had been handed out,

58

Alice told the students to fold the paper in half the long way while she

demonstrated the process for the students. After the students folded their

papers, they were told to open them back up and smooth them out. The

students were then asked to shade in one-half of their paper. Alice used

this time to introduce writing the number name for one-half explaining

that the numerator represented the shaded part and the denominator the

total number of parts. The process was repeated twice more with the

students folding the paper into fourths and then sixteenths. Throughout

the time Alice was giving oral instructions, she was walking around the

room observing the students working. If she found someone who had not

understood the directions, she stopped to clarify them for the student.

Worksheets were then passed out which showed various shapes divided

into equal parts. Different numbers of the parts were shaded for each

shape, and the students were asked to name the fraction which described

the shaded parts. The remainder of the lesson was spent on this activity.

Alice commented at the end of the class about how much more concrete

fractions were compared to division.

Interview 3

The interview took place the day after the observation. Alice was

asked to provide further explanation of her comments of the previous day

about fractions being more concrete than division. She stated that while

the concept of division and the division facts could be illustrated

concretely, larger problems like 936 21 were too cumbersome to easily

illustrate with manipulatives such as counters or money.

59

When asked about the use of the paper folding activity, Alice stated

that the idea was in the teacher's manual but she had heard of it

previously. This led to a discussion of the philosophical reasons behind the

use of manipulative materials. Alice expressed the belief that through the

use of manipulatives, students have a better initial understanding of

concepts. She believed that "you get a quicker understanding of a new

concept if it is presented in some tangible way." She stated that she had

thought of several additional manipulatives such as graham crackers, M &

M candies, and chocolate bars that break into pieces, to use when

developing fraction concepts.

Alice also reported at this time that she had placed her order for

manipulative materials. She chose to order many overhead materials such

as coins, powers of ten, fraction strips, fraction circles, a geoboard, and

spinners. In addition, a classroom set of rulers, a metric pan balance set,

and a metric beaker set were also requested. She picked the overhead

materials because they could be used to demonstrate concepts for the

students or the students could come to the overhead and manipulate the

materials themselves. She stated that she could borrow sets of the

materials from the grade level shelf if she wanted class sets to use with

the students.

Observation 4

This class period began with students grading homework assignments.

The topic covered on the assignment being graded was changing an

improper fractions to mixed numerals. All of the problems had answers

60

that were whole numbers, an example being 24/4. Alice asked the students

to clear their desks when the homework grading was completed and

immediately began the new lesson.

She used the sentence, "Some of you have asked what happens if the

top number can't be divided by the bottom number evenly," to introduce the

new material. Then she proceeded to draw two circles on the board

dividing each into fourths. She shaded one complete circle and talked

about how that was 1. She asked the class, "How many fourths make a

whole?" and was answered correctly.

Alice then shaded one-fourth of the other circle and asked, "How many

fourths are shaded altogether now?" A student answered 5 correctly. The

teacher then used the traditional division format to show that 5 + 4 = 1

1/4. The students had previously written division in this manner, but had

not taken remainders and expressed them in fraction form.

A teacher-made overhead transparency was then used to go through

eight more problems consisting of different shapes with the class working

together. The students were given a worksheet with a few problems

similar to the ones worked in class and a few sets of blank objects which

were to be done with a partner. One person was to shade some of the

objects and the other was to determine the mixed numeral that

represented the shaded objects.

Observation 5

This observation occurred approximately three months after

observation number two. During interview two, a discussion was held

61

about measurement supplies observed at the back of the classroom dealing

with the metric measurement then being studied in science. The topic

being presented to the students during observation 5 was length

measurement using the rulers measuring centimeters. Alice passed out the

rulers which showed both metric and standard scales. She had ordered a

class set to ensure consistency in measurement by the students. Students

verbally identified the metric side and when asked, explained the steps

they should follow to accurately measure something that was flat. These

steps were summarized on the chalkboard by the teacher.

A worksheet which accompanied the book was passed out containing

pictures of eight objects to be measured and five lengths (eg. 6 cm) to be

drawn below the length. The class worked together to determine the

lengths with each student doing his own measurement. When the

worksheet was completed, Alice told the students that their book

assignment was similar to the worksheet. The two assigned pages in the

book directed the students to measure thirteen drawn objects to the

nearest centimeter and draw five specified lengths. The final part of the

assignment asked the students to take five common objects found at the

desk such as their pencil and their book and estimate their lengths, then

measure them, and find the difference between the two.

Interview 4

Alice and one other subject, Beth, tied for least number of

manipulatives used in their college mathematics education class. Alice

was only exposed to five manipulatives: the meter stick, tangrams,

62

geoboards, an abacus and place value mats. When asked about the lack of

checks in the column, she explained that her instructor had verbally

stressed the use of manipulatives in mathematics instruction, but had

brought none to class himself. The students were required to make five

manipulatives for class, but only one was allowed to be purchased. They

were then to bring them to class to share. She summed up her preteaching

experiences by saying, "My only exposure to manipulatives was what other

students made and brought in." Alice did, however, have some experience

with base ten blocks during her second-grade student teaching experience.

Twice the number of the manipulative materials were used by Alice

throughout the year than she had actually been exposed to in her

mathematics education class. The place value mats she had made as one of

her manipulatives in college were used, but the tangrams, geoboards, and

the abacus had not been part of her own classroom instruction. She used a

variety of fraction models as well as an abundance of measurement tools.

At the time of the final interview, she had yet to cover standard measure,

so none were indicated on her checklist.

Alice chose to use only the materials that were found in her room to

teach mathematics. She was aware of other materials in the building, but

did not indicate that she had used any of them. On several occasions she

expressed that the present location of the materials outside her room was

a deterring factor from their use. She did not want to interrupt another

class to get the materials she could use. When asked if the manipulatives

were located in the library or some other easily accessible location, would

she use them more, she replied she would.

63

A lack of knowledge of what was available within the building was

also expressed at this time. If a written inventory of materials existed,

Alice was unaware of it.

When asked what changes she would make in mathematics instruction

next year, she replied, "I'll probably just have to get more aggressive or

assertive about the manipulatives I want and just go get them and not

worry about anything. That's probably the biggest change I'll make."

Lesson Plans

During the first three months of the four and one-half month research

period, no manipulatives were listed in Alice's lesson plans. Table 2

summarizes the topics and manipulatives found in the final six weeks of

the project. All of the manipulatives listed in her plans were used to

develop fraction concepts. This may be somewhat misleading, however, as

the measurement tools found used in instruction from the checklist or

observed by the researcher were not in the lesson plans.

Principal From Building A - Mr. Anderson

Mr. Anderson was an experienced principal near retirement. He had

been at Building A for many years. When asked to describe the building's

mathematics program, he stated that his classrooms were all

self-contained with no team teaching occurring. The teachers were

responsible for all instruction within their rooms and were to teach the

minimum learning objectives presented in the district's curriculum guide.

He explained that the learning objectives existed before the state

64

Table 2

Manipulatives Listed in Alice's Lesson Plans

Topic Manipulative^

Writing fractions from parts of a whole

Writing fractions for part of a group

Finding fractional parts of groups

Writing equivalent fractions using models

Writing equivalent fractions by

multiplying

Comparing fractions with like

denominators

Comparing fractions with like

denominators (again)

Introduction of mixed numbers models

Writing fractions as whole or mixed

numbers

Adding like fractions

Fraction manipulatives

Use student groups as

examples

M & M candies

Teddy Grahams

Folded paper circles

Fraction models

Fraction models

Overhead with fraction

Overhead with fraction

models

Fraction models

See Appendix C for an additional chart on Alice.

mandated essential element lists were written, so the district's learning

objectives were modified so that the essential elements were included.

Mr. Anderson stated that the district's learning objectives were more

65

comprehensive than the state essential element lists.

Mr. Anderson sent several of his grade level chairman to the state

mathematics conference during the summer, but did not recall whether the

new teachers had been given any type of inservice geared directly toward

mathematics at the beginning of the school year. He explained that the

inservices are planned on the perceived needs of the teachers through the

use of a questionnaire sent out the preceding spring.

The ordering process for the manipulatives was explained by Mr.

Anderson. He provided the catalogs and the suggested grade level

manipulative lists both from the state and the district for the teachers

use. Inservice on manipulatives was sometimes also held. This consisted

of people from the different commercial companies making presentations

of materials available to the staff. Then the teachers were free to make

the final decision as to what they wanted to order with their $100. The

teachers might choose to pool their funds and purchase larger items as a

grade level or to place their orders individually. Mr. Anderson felt that all

of his teachers took advantage of the monies at their disposal to order

items for their classrooms.

No special meetings were held in Building A for first-year teachers

concerning the ordering process. The first-year teachers were encouraged

to seek assistance from the grade-level chairperson. This assistance

could consist of help with making lists of particular manipulatives

compatible with the grade level's learning objectives or might simply be

help locating the catalogs or filling out the forms.

66

Mr. Anderson did not recall observing either Anne or Alice during

mathematics instruction. His general policy was to schedule first-year

teacher observations while they were teaching in the language arts block.

He did, however, say that during observations of mathematics instruction

by other teachers that manipulatives were being used. He recalled seeing

instruction with base ten blocks and interlocking cubes.

A discussion of the necessity to develop mathematics concepts from

the concrete level first through the abstract level ensued. Mr. Anderson

stated that manipulatives needed to be used in instruction "to help make

the concepts clear." He also felt that in the past mathematics was

perceived as being difficult and boring. To support his hypothesis, he

stated, "This emphasis on manipulatives is a step in the direction of

making it a lot more meaningful and making difficult concepts a lot more

understandable." In his opinion, the district was really stressing the use

of manipulatives by providing the money for teachers to purchase

materials.

Building B

Grades K-5 were taught in Building B. The students were ethnically

mixed and came from a wide span of the socioeconomic strata. Many of the

students in the two classes observed were international students who had

come to the United States within the past year. The classes observed in

this building were held in self-contained rooms. The rooms were quite

large with windows along one wall. The floors were tiled, not carpeted,

and not much storage area was found in the room. The building was

67

scheduled to be completely remodeled during the next summer.

Teacher B1 - Beth

Beth was a brunette of medium height and average build. She had

attended a large, private university where she was graduated the previous

spring. She majored in elementary education with a reading specialization.

She student taught for a semester at the fourth-grade level, but she was

only there half-days. This meant that she saw little to no mathematics

instruction and only taught mathematics for two weeks when she covered

geometry with her students.

Her classroom was the most multicultural room observed. She

commented at one point that out of her fluctuating population of 20 to 22

students, she had seven different countries other than the United States

represented in her classroom during the year. Several of the students

spoke only limited English when they entered her room. She expressed the

thought that she had enjoyed the experience and would miss the variety if

her next class was more homogeneous.

Beth readily admitted that mathematics was not her favorite subject.

She confessed to having enough difficulty with college algebra that she had

to take the course more than once. She described the content of her

mathematics education course as "a lot of worksheets, a few

manipulatives and no explanation."

Beth's mathematics instruction took place in the early afternoon

immediately following her class* scheduled lunch time. In her eyes, her

students were quite interested in mathematics. They were a bright group

68

that generally caught on to concepts easily. She voiced having had

apprehensions about teaching mathematics due to her lack of knowledge,

but said her students made teaching the subject quite enjoyable.

Observation One

Beth began the lesson by having the students open their textbooks to

the page that introduced the day's objective. The topic was division

problems with one-digit divisors and three-digit quotients. The students

had previously had problems with two-digit quotients. Beth used the

suggestions in the teacher's edition to begin the lesson. She wrote the

problems 2 + 2,20 + 2, 200 + 2 and 2000 + 2 on the chalkboard and

proceeded to work the problems starting with the simplest and moving to

the most complex. She questioned the students throughout the problems as

to the steps that needed to be followed to arrive at the correct solution.

The next problem written on the board was 684 + 5. The students were

asked to work the problem at their seats. Once most of the class had

calculated the quotient, Beth orally discussed the problem again as it was

worked relying on questioning techniques to actively involve the students.

These steps were followed seven more times with a different student

coming up and explaining the problem after the class had an opportunity to

work it. When the class grew restless, Beth assigned eight division

problems which required the students to practice the day's objective and

eighteen multiplication problems for purposes of review.

69

Interview One

The first interview was held two days after the observation. Beth

explained that thus far she had only covered computation topics with her

class. They had completed addition and subtraction and had finished

multiplication problems with a one-digit number multiplying a four-digit

number. They were now nearing completion of one-digit division. When

asked if topics like geometry, measurement or fractions had been covered,

Beth replied that the teachers from the next grade level where TEAMS

tests were given had told them not to worry about those topics. They

wanted them to be competent with their multiplication and division skills.

Lesson planning was also discussed. Beth said that she relied heavily

on her book. She stated, "That's the only resource I have right now besides

just experiences I had in being taught how to do it." She later mentioned

checking the district's enabling objectives found in the curriculum guide to

make sure she was completing them in good time.

Beth referred to the fact that she had gone to her assistant principal

for help when she was planning her division unit. She felt that she did not

know how to teach division effectively and the teacher's edition of the

textbook had not provided enough information. A third grade teacher's

edition was given to her. She explained, "I got that to try to get it straight

in my head - different ways that I could explain it (division) to them."

Beth commented on the fact that the assistant principal and the four

teachers at the grade level who had previously been in the building were

valuable resources for both Brenda and her, the other first-year teacher at

the grade level also involved in the study.

70

No materials except for textbooks and curriculum guides were found in

the room when Beth arrived in the fall. The building had previously had

four teachers at the grade level and had increased that number to six at the

beginning of the current year. Beth had been given the opportunity to place

her order for manipulative materials during the first semester. They

arrived the week before Christmas break. Beth explained that the

assistant principal had given the teachers each a list and let them pick out

what they wanted to order.

At this point, she walked over to the shelf and carried the materials to

the table. She had ordered dominoes, a scale, and one set each of

Cuisenaire rods, geoblocks, three dimensional shapes, fraction models, and

base ten blocks. She had not had an opportunity to use any of them,

however. Beth felt she did not have enough experience to successfully

incorporate the materials into her lesson and planned to use the teacher

work day at the end of that week to "explore" the manipulatives. Beth also

mentioned the individual slates that she and another teacher shared. When

asked about other materials that were available to her within the building,

she replied, "That's a good question. I don't know."

None of the inservice that Beth attended had covered any topics in

mathematics. Several of her colleagues had attended workshops during the

summer, but there had been no time for them to share the ideas with Beth.

She expressed a desire to attend summer workshops in mathematics during

the coming summer.

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Observation Two

The learning objective taught in the second lesson was rounding a

two-digit divisor up so that quotients could be more readily estimated.

The students had previously worked with two-digit divisors that had to be

rounded down. Beth put the problems 242 37 and 545 + 59 on the

chalkboard. She asked Angela to explain the first step of division. Angela

replied that first the divisor had to be rounded, then the number in the ones

place had to be covered so they could estimate.

Once this was done, Beth referred to the phrase, "Drive My Super Cool

Buggy," as a way to make sure that all the steps had been included in the

division algorithm. The students were to remember that the "Drive stood

for Divide, My for Multiply, Super for Subtract, Cool for Check and Buggy

for Bring Down." This was obviously something they were) accustomed to

using while working division problems.

The students were then asked to open their textbooks to a page which

contained thirty division problems which required them to round the

divisor up before estimating. One more problem was worked together with

the teacher emphasizing each of the steps of the division algorithm. Beth

assigned three problems for the students to work as she walked around the

room checking progress, then the three were discussed on the board. The

students were given an assignment from the page.

Interview Two

When asked about the origin of the phrase "Drive My Super Cool Buggy,"

Beth explained that one of the fifth grade teachers had shared it with her

72

during lunch time. Beth had been expressing her frustration due to her

students' confusion about division, so the other teacher told her what she

had found helpful.

On the wall next to her desk, Beth had hung a sign which read:

Tell me, I forget.

Show me, I remember.

Involve me, I understand.

Ancient Chinese Proverb

When asked why it was there, she responded, "I put it there so I'd

remember not only to tell kids things, which is what I've done a lot, but to

involve them. Let them have some manipulatives and things, not just with

math but with other subjects as well." She added that when the students

asked her why it was there, she told them it was to help remind her of the

best way to teach them. She went on to say that it was difficult to do

with mathematics. The sign itself had been handed out at an inservice.

In terms of mathematics, Beth called herself "a book taught person."

She commented that using the manipulatives was an improvement. She

described her background further:

I think we should use manipulatives a lot more because we just skip

the concrete level and just go straight to the abstract. I think that's

one reason I didn't understand it, because I need to touch it a lot of

times and do different things and experiment with it. I find it

easier to do math in my head now that I'm older, but I couldn't do it

then and if I couldn't do it in my head then, I had a hard time getting

it onto the paper. If I could have started with my hands and let it

73

get into my brain first, I think I would have done it a lot better.

When Beth referred to the terms "concrete" and "abstract," the researcher

asked what meaning those words held for her. She responded that she had

learned them in college and could apply them to her own learning. She

cited an experience she had as a child when she had a difficult time

understanding whether 1/3 cup or 1/4 cup was larger. She said that she

knew if someone had just brought in a set of measuring cups it would have

been much more clear.

Beth credited many of her college professors with not only advocating

the use of manipulatives, but demonstrating it in their own lessons. She

conceded that the one thing she had gained in her mathematics education

class was a more clear understanding of fractions. The teacher presented

fractions using Cuisenaire rods and Beth summed up the experience by

stating, "That's one thing that I did get out of that math class -- the only

thing. I think it was because we did use the manipulatives." Most of the

time, though, she said that she had only her own learning experiences in

mathematics to draw upon "until someone tells me a better way to do it."

Observation 3

Beth began the lesson by asking her students to get out their books and

turning to the first page of the chapter on measurement. She explained

that part of the chapter, the metric portion, would be a review of some of

the things they had already studied in science. At this point, she reviewed

some of the different types of measurement they had already experienced.

Included in the list were length, weight, and liquid volume. She told the

74

class that they would be learning about standard units in the chapter, but

the first lesson would be a review of centimeters.

Two girls were asked to give each student a paper clip. All of the

students were asked to take out the pencil sharpeners from their pencil

boxes. The class was then directed to measure one side of their desk

twice, once with the paper clip and once with the pencil sharpener. The

students began measuring. Beth wrote the column titles "Paper Clips" and

"Pencil Sharpeners" on the board. She called upon several students to give

their measurements, then followed with a discussion of nonstandard

versus standard units. This idea was mentioned in the teacher's edition of

the textbook.

There was not a class set of rulers, so Beth asked the students who

had rulers to take them out. Because there were students without rulers,

she told the class that some of them might have to work with partners.

Several objects were drawn on the pages in the book. The students were

asked to measure them to the nearest centimeter. Beth allowed time for

each item to be measured, then she called on students to answer. Thirteen

objects were measured in all.

A chart appeared at the bottom of the second page of the textbook. It

listed five objects: "Your pencil, Your book, Desk top, Crayon and Longest

finger." The directions told the students to first estimate the length of

each object, measure it using a ruler, then find the difference between the

estimates and the actual lengths. Beth assigned this chart for homework.

She verbally gave directions as to how to construct the chart while

drawing a model on the board. She also reviewed the abbreviated label for

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centimeter and what it meant to find the difference between two things

before allowing the students to begin working.

Interview 3

This interview, originally scheduled for a few days after the

observation, was delayed a week due to inclement weather. The building

had held an open house for parents the night before the interview and the

room was elaborately decorated. Beth was noticeably tired. She said that

this was the first day that was somewhat normal in quite a while as they

had been busily preparing for the open house during the previous week.

A brief discussion of the observed lesson took place. Beth replied that

the paper clips used by the students during the nonstandard measurement

practice were all alike when questioned about their size. She stated that

the lesson was a review of what had been covered in science and the lesson

they were to do in the book that day on kilometers was also a review. Her

class had not yet covered meters in mathematics class, but she did have at

least ten meter sticks to use when they studied the topic. The meter

sticks had arrived after metric length had been covered in science, so the

students had not yet had an opportunity to use them.

While Beth was describing an activity she had done with her student

teaching class which involved group work, the researcher asked if she had

used that technique much during mathematics instruction during the

present year. She said that she had tried it when working with money

because there weren't enough materials for every child to have their own.

Beth also stated that even with the new materials she got, there weren't

76

enough things for all students to really have an opportunity for much

hands-on access to things. If she had enough materials, she probably would

have used some of the activities described in the book for Cuisenaire rods

or base ten blocks for "carrying and things that were in the book, but there

weren't enough materials to do it." She then admitted that she was not all

that familiar with how to use the manipulatives, but wanted to remedy

that over the summer by attending a course somewhere.

Later, when asked about which materials she had used from her order,

she said just the meter sticks and the dominoes. She had used the

dominoes to help Sherry, a slow learner, understand division. Beth would

set up an array with the dominoes and Sherry would break the array into

groups the size of the divisor. She did comment, however, that when she

got to fractions, she planned to use the overhead fraction models she had

purchased.

The TEAMS tests had been given during the time period between the

second and third interviews. Beth mentioned that she really wanted to

teach fractions well because the teachers from the next grade level had

reported that there were many problems on the test concerning fractions.

Beth felt the students were going to need quite a bit of work with the

topic because she had tried to use terms like one-third and three-fourths

during her measurement unit which confused the students. She had already

decided that the basic concept of a fraction needed to be developed before

measurement was taught next year.

As a first-year teacher, Beth felt her mathematics teaching was about

average. When asked what suggestions she would have for administrators

77

to help new teachers have a better year, she immediately replied, "Provide

the materials at the beginning of the year and teach us how to use them.

They're not doing me any good if I don't know how to use them." She did not

think that Brenda, the other new teacher on her grade level, had used the

manipulative materials she had ordered very much either. Another

suggestion she had was to pair each new teacher with an experienced

teacher so the new people could observe the materials in use. Even though

she felt her principal would have allowed this to occur, the only other

class to have the same mathematics period was Brenda, and both were

first year teachers.

Observation 4

Beth stood in front of the class holding a bucket of learning links at

the beginning of the lesson. She asked the students if they were ever

scared or anxious when they were about to try something new. She then

told them she was having those feelings about the lesson because they

were going to be using some manipulatives which were ntjw not just to

them, but for herself as well. She broke the class into groups of two and

handed out about twelve links per group.

The class was told they were going to practice making fractions with

the links. Beth proceeded to give the students directions as to which links

to use. The first problem was, "Four red links, what fraction describes

this? The students correctly answered 4/4 or 1. The next instructions

were to use six links and have one-half of them green and one-half of them

red. As the students completed their chains, they held them up and Beth

78

walked around the room to check their progress. She continued this

pattern to observe the students' work throughout the lesson.

Beth also asked questions like, "Five links with 4 green and 1 red.

What fraction describes the red part? the green part?" When one student

was called on for an answer, Beth told the class to put their thumbs up if

they agreed. Several more of the same type problems were given the

students. In all of the exercises with the links, the students were naming

fractions.

Next the students were directed to open their books to a page entitled,

"Finding Parts of a Group." The instructions in the teacher's edition said to

use counters to begin the lesson. The problem "1/2 of 8" was to be

illustrated for the students concretely. The process was described in

great detail. Eight softballs appeared at the top of the student's page with

verbal instructions that one-fourth of them belonged to a girl named

Candice. The students were asked to find out how many of the balls were

in one-fourth of eight. In the middle of the page, the problem was written

"1/4 of 8 = ?" with an explanation requiring the students to divide eight by

4 to obtain the answer 2.

Beth did not use the learning links or any other manipulative to

introduce the lesson. She immediately directed the students to the

algorithm as shown in the textbook. The students were quite confused as

to what they were to do. The questions they asked indicated they did not

understand the concept at all. Beth was unable to clear the confusion. At

this point, the counselor arrived to give a presentation to the class, so

mathematics was finished.

79

Since the counselor was with the class, Beth was free to talk

immediately after the lesson. She expressed her frustration at not being

able to clearly present the lesson concepts. She said that she had recently

heard a news report that claimed mathematics instruction was not

concrete enough for students. She knew how true that was, but was not

always able to make lessons like the one that day concrete. She then

shared that her principal was sending her to a two-week seminar in June

held at a university at the district's expense to gain a better understanding

about mathematics, particularly hands-on mathematics. Beth was very

excited about the opportunity because she knew this was an area she

needed to learn more about in order to grow as a teacher.

Observation 5

Beth spent about ten minutes reviewing various aspects of adding and

subtracting fractions containing the same denominators. She began with

addition of like fractions where the sum was less than one, then reviewed

the additional steps that needed to be followed when the sum was an

improper fraction and needed to be changed to a mixed numeral. She

showed examples of subtraction problems with two proper fractions, then

finally reviewed addition and subtraction of mixed numerals with like

denominators. During the review, Beth referred to various pages in the

textbook for the examples presented.

After the review of problems with like denominators, Beth began the

new material. The focus of the lesson was addition of fractions with

unlike denominators. She first reviewed the steps necessary to change a

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given fraction to an equivalent form where the new denominator was

provided, an example being "1/2 = ?/8." Beth told the students they would

need to know how to do this in order to be able to add unlike fractions.

Next Beth used the following word problem to set up a situation

requiring the addition of two unlike fractions:

Rosita is on the swim team. Each day she swims the backstroke for

1/3 mile and the butterfly stroke for 1/4 mile. How far does she

swim each day? (Harcourt Brace Jovanovich, Inc., 1985, p. 274)

The students were told by the teacher that they would not be able to use

the method they had previously learned for adding fractions because the

denominators were different. She proceeded to explain the process in very

abstract terms asking questions like, "What is the first number three and

four can go into evenly?" Being led by questions asked by Beth through the

process, the students came up with the correct sum, 7/12. Beth then asked

the students to read the section in the book that explained the rule, "When

denominators are unlike, find equivalent fractions so that both fractions

have the same denominator" (Harcourt, Brace, Jovanovich, Inc., 1985,

p. 274)

One more problem was worked with Beth again leading the students

through the process by questioning. It became obvious rather quickly that

the students had not made the connection between the equivalent forms

and the original values. Beth stated, "Let's think back to when we got

equivalent fractions. Obviously I'm moving too fast." The remainder of the

class time was spent reviewing the method of changing given fractions to

81

their equivalents. This was also all done at the abstract level. The

homework assigned provided more practice of this skill.

Interview 4

As show in Appendix C, Beth tied with Alice for exposure to the least

number of manipulatives in their mathematics education classes. Both

teachers had seen only five of the thirty-six materials found on the list.

Beth, however, had eighteen of the items in her classroom and listed

twenty-five within her building. Of the materials available, she had only

used fourteen during the school year. With the exception of the play money

which had been used during the first semester to teach various concepts

like making change, the only manipulatives Beth used were for either the

fraction or the measurement units.

Beth did not use any manipulatives to work with the computation

concepts that represented the majority of the instructional time in

mathematics during the year. She admitted several times that she did not

know how to use materials like base ten blocks or Cuisenaire rods to show

these computational concepts. The other reason she gave for not including

more manipulatives in her mathematics instruction was the lack of enough

materials to allow students to work individually or in pairs with them.

She felt that the groups of four or five students which she tried were

counterproductive in that the students fought over materials rather than

learning the concepts.

When using materials like the measurement tools or the learning links,

Beth s only form of evaluation was to walk around the classroom and

82

observe the students as they worked. She had never thought about

assigning some sort of grade which would be figured into their average or

using the manipulatives in a testing situation

Beth again discussed the helpfulness she had experienced from the

assistant principal, Mrs. Barber. Beth felt that Mrs. Barber had particularly

aided her during her quandary about teaching division by researching the

subject and bring materials to her attention as well as soliciting help

from experienced teachers for Beth.

A discussion of the lesson on addition of unlike fractions revealed that

the reteaching the following day had gone much better. Beth felt that she

had done well by simply stopping instruction when she realized her

students did not have a sufficient background in prerequisite material, in

this case equivalent fractions. She also mentioned that later on in the day

of observation five she had placed fraction materials on the table for

students to "play with." Several students took advantage of the

opportunity and later expressed surprise that they hadn't done better with

the material the first time around.

Beth expressed the belief that fractions and division were the two

most difficult topics taught during the year in mathematics. She stated,

"Students have no basis for the fractions at all." She had already decided

that during the next year she would incorporate some cooking, which

required the use of fractions, into her mathematics curriculum.

Lesson Plans

Only one entry was found in Beth's lesson plans concerning materials

83

or manipulatives which would be used during mathematics instruction. As

shown in Table 3, the day she planned to introduce metric length using

centimeters, she listed metric rulers. Different manipulatives were

observed during mathematics instruction, and Beth referred to a variety of

other manipulatives on her checklist. Her lesson plans did not reflect this

manipulative usage.

Teacher B2 - Brenda

Brenda was a petite blonde. She attended a mid-sized state school

where she majored in elementary education and minored in early childhood.

Both she and her husband were first-year teachers within the district. Her

husband taught on the secondary level. Originally he had been placed in a

middle school, but at the beginning of the second semester had been moved

to a high school assignment.

Brenda had student taught for a half semester each on two grade

levels, kindergarten and third grade. She described her mathematics

education program as being of limited value. The majority of the time, the

students had worked on writing units of instruction. Her unit had been on

Table 3

Manipulatives Listed in Beth's Lesson Plans

Topic Manipulative^)

Introduction of measurement - centimeters Metric rulers

See Appendix C for an additional chart on Beth.

84

place value. The instructor had not required any sort of textbook, so she

had none to use as a reference.

The student enrollment varied little in Brenda's room. The number of

students remained between nineteen and twenty throughout the entire

research project. There were almost twice as many girls as boys in her

classroom. When asked to tell about her students, she described them as

being all on-level with only one slow learner and no resource or ESL

students. She identified three students as needing enrichment although

none of her students took part in the district's one-day-a-week pullout

program for the gifted.

Observation 1

Upon their return from lunch, the students went to their seats and took

out their mathematics books. The book page number was listed on the

board by the teacher and the students followed directions. She began the

lesson with a review of the material covered the previous day, one-digit

multiplication with regrouping of the ones occurring (eg. 3 X 26). The

objective for the day's lesson was to carry that one step further so the

ones and tens places both were regrouped.

Brenda had a set of Cuisenaire rods in the front of the room with her.

Using the problem listed at the top of the teacher's manual, 4 X 32, she

demonstrated the regrouping of twelve longs and eight singles into a

bundle of ten longs, two separated longs and eight singles. She followed

the steps suggested in the teachers edition, but the students sitting in

back of the front row had difficulty seeing the process as she had the

85

blocks on a desk in the front of the room. Brenda went through the process

very quickly one time, then put the manipulates away.

Her next problem was 6 X 27 which she explained step by step on the

board at the abstract level only. In this problem, both the ones and the

tens columns involved regrouping. After asking if there were questions

and not receiving any responses, Brenda passed out individual chalkboards,

chalk and a paper towel to each student. She explained that they would be

using these new materials for the first time so that all of the students

would have a chance to work the problems while she walked around and

checked their progress.

The students worked several problems of the same kind on their

boards. When asked if they wanted to continue practicing, they responded

in unison that they did. Brenda then said she wanted to challenge them.

The next two problems were (3 + 6) X 16 and 7 X (16 + 10). Once a correct

answer had been given, Brenda demonstrated the process of calculating the

answer for those who had not been able to compute it. The last problem

was given orally in story form. "A group of 26 students wrote a report

about arts and crafts. Each student wrote 8 pages. How long was the

report?" The students were given an opportunity to calculate the solution.

The assigned homework was a xeroxed master which accompanied the

textbook page used for problems during instruction.

Interview 1

When asked about the manipulative demonstration at the beginning of

the lesson, Brenda explained that the directions had been in the teacher's

86

edition of the textbook. She stated, "I tried to show them with concrete

things how these numbers can be represented by something else. We've

already gone through all of the place value stuff. Sometimes they have

trouble with it and that's why I was reviewing it." The manipulatives used

in the lesson had not been ordered until after the initial instruction in

place value had been completed.

Brenda's room had no materials except textbooks arid curriculum

guides when she arrived. She had been given a number of things including a

set of base ten blocks from her mother who taught fifth grade in a district

in the southern part of the state. Brenda said she had been able to order

her materials in October and had received the majority of the order by the

end of November. When asked what she had been able to use from the order,

she stated, "Just showing how they represent the numbers. That's the only

way I could think of really."

The chalkboards had been in the building less than a week before the

first observation. The students had used them for the first time. The

boards were to be shared between two classrooms so they were not

available on a daily basis, although the other classroom did not have

mathematics during Brenda's scheduled time.

Brenda had attended one place value workshop during inservice. Her

mother had given her copies of inservice materials in mathematics she had

received in addition. Brenda commented that she wished she had more

ideas about things to do with her students in mathematics, but she just

didn't know what else to do.

87

Observation 2

The second lesson observed began with a concrete demonstration of

multiplication of money. Brenda explained to her students that they were

going to learn how to multiply amounts of money, which was very similar

to the multiplication they had been doing in class. She was holding

pennies, dimes, $1-bills, and $10-bills in the form of play money in the

front of the room. She questioned the students about the different

exchanges between the monetary pieces.

To introduce multiplication of monetary values, Brenda used the

suggestions found in teacher's edition for the topic taught. She chose

three students, who all happened to be girls, to come up and had each hold

three dimes and seven pennies. The following was then written on the

board:

3 dimes 7 pennies

3 dimes 7 pennies

3 dimes 7 pennies

9 dimes 21 pennies

The girls were instructed to put all the same type of currency into piles,

so one pile held 9 dimes and the other 21 pennies. Next Brenda

demonstrated exchanging two groups of ten pennies each for two dimes.

Her piles then contained 11 dimes and 1 penny. Then she exchanged 10

dimes for a $1 -bill. There was one $1 -bill, one dime, and one penny

totaling $1.11.

Following the concrete demonstration, Brenda wrote the problem

3 X $ .37 on the chalkboard in vertical form and proceeded to explain the

88

algorithm to the students. She explained that the only new part involved in

the problem was the placement of the decimal point and dollar sign. Two

more problems were worked on the chalkboard.

Brenda lifted her screen which revealed a menu she had written on the

board. The menu and directions looked like this:

Menu Orders

Cheeseburger $ 1.65 Cheeseburgers

Steak sandwich $2.25 Steak sandwiches

French fries $ .45 French fries

Brenda asked the students to copy the order and fill in the blanks with

numbers between 2 and 9. The students were then matched in pairs. Each

student was asked to calculate his own bill and that of the other person's

order. The individual chalkboards were passed out for the students to use

for the assignment. The class worked quickly and quietly as a whole.

The last problem Brenda gave the class for practice was 7 X $ 52.04.

She told the students to work the problem, show it to her, then hand in

their chalkboards. Once this was completed, Brenda referred the students

to a problem in the textbooks which was starred to indicate a higher

difficulty level. It read as follows

Team jackets cost $24.50 each. Six pairs of shorts cost

$53.79. How much does it cost to buy jackets and shorts

for 6 team members? (Harcourt, Brace, Jovanovich, Inc.,

1985, p. 159)

The first five people who completed the work were to receive treats which

consisted of small pieces of hard candy. When finished, the students were

89

to check with the teacher who sat perched on a stool in the front of the

room. Several students came to Brenda with incorrect solutions before

anyone had it right. Her responses varied from a simple, "No," to "You need

to look at this (referring to some specific thing) again." The first correct

answer took approximately three minutes. The researcher stayed ten

minutes after the problem had been assigned. During that time many

students had shown work to Brenda, but only three had answered the

problem completely correctly.

Interview 2

The interview began with a discussion of the problem at the end of the

observed lesson. Brenda said she hadn't anticipated that the problem would

be so difficult for the students. It had taken about twenty-five minutes

for five students to obtain the correct answer and although the first three

had figured the solution without hints, she had worked through part of the

problem before the last two could come up with the answer.

When asked if the play money was part of the materials she had

ordered, Brenda replied that the bills were located in her room, but the

coins had been borrowed from another teacher on the same grade level. She

went on to explain that each of the teachers had materials in their room,

and you could borrow it if you knew it was there. She cited the example

that when she wanted to use a class set of Cuisenaire rods she had some in

her room and could go to the room next door to get the rest.

Brenda explained that she had finished multiplying by one-digit

numbers and was ready to begin one-digit divisor division. Her lesson

90

plans said that she would be using place value models to explain a problem.

When asked about it, she said the problem was again listed at the top of

the teacher's edition. She felt she generally followed the directions on the

pages and described herself as "a by the book person."

Observation 3

When the class returned from lunch, Brenda placed a transparency on

the overhead which contained four multiplication problems like 5 X 36 as a

review. The students spent five minutes settling down and working the

problems. Four students were then chosen to work the problems for the

class on the overhead. A second transparency sheet was placed on the

overhead which contained four division problems with one-digit divisors,

two-digit dividends and two-digit quotients. The students were told to

check their answers by multiplying.

Brenda introduced the new materials with the problem 120 + 8. She

wrote this in traditional format, then explained to the students that even

though there was a hundreds place in the dividend, the quotient may have

either two or three digits. She demonstrated the process by covering up

the 20 in 120 and asking how many times eight went into one. Since there

were no groups of eight in one, she placed an 'X' straight above the

hundreds place. They worked through the problem, but several students

were unable to quickly answer questions concerning multiplication facts.

Brenda got very upset with the students and spent a couple of minutes

verbally chastising them for their lax attitude.

91

A third row of problems was then placed on the overhead which

contained the problems: 248 + 4,876 + 3 and 644 •+• 7. The students worked

the problems at their seats, then Brenda had them worked on the overhead

again. She was visibly upset with their progress and told them they would

do some more of the same kind of work the next day.

Interview 3

Brenda was quite agitated throughout the third interview. She was

trying to get the room ready for the school's spring open house which she

had heard was an extravaganza. In addition, she felt her students were not

putting forth the effort needed to master some of the required material

and the students who most needed parental assistance with school work

were getting the least amount.

During the interview, Brenda expressed frustration with the fact that

several of her students did not yet know their multiplication tables. She

was upset with two students in particular who were not even able to use

strategies of any sort to make the calculations. Brenda had assisted

Alyson, one of the students having difficulty, with 7X7. She had told

Alyson to begin with the fives and then figure out 7X8. This same

technique had been one that Brenda had used as a student and found

successful when she couldn't remember a particular fact. When asked if

the student's lack of fact mastery irritated her, she replied, "I am very

irritated by it. They're lazy, it's just laziness."

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Observation 4

The students first checked their homework papers from the day before.

They had been working on a review of metric length and their assignment

had four separate parts: 1) to tell whether stated objects were more than

a meter, 2) to decide whether the correct unit of measure to use was

centimeter, meter or kilometer, 3) to work conversions between meters

and centimeters and 4) when given two different measures (eg. 4 m and

472 cm), to find the difference between them. This worksheet took about

fifteen minutes to grade.

Brenda then introduced a lesson focusing on perimeter. She had rulers,

a measuring tape, a yardstick and a meter stick, any of which could be used

to determine the perimeter of a shape. A rectangle was drawn on the board

to represent a desktop. Brenda asked the students to take the rulers from

their desks and measure the four sides of their own desktops centimeters.

Janice was asked to give her desktop's measurements which were 58, 58,

44 and 44. Brenda wrote these numbers on the chalkboard in column

addition form. Then she asked the students if there was any other way the

total distance around the desktop could be calculated. Another student, a

boy, suggested that both 58 and 44 could be multiplied by 2, then the

products added together. Brenda explained that when there were sides

which were the same length, this method could be used.

The students were asked to turn in their books to two pages which

redefined perimeter and contained several shapes drawn to scale with

lengths provided. There were also two problems where the shapes were

noticeably larger and the students were required to do the measurement of

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the sides themselves. The last section was three word problems requiring

the students to figure out the perimeter for purposes of fencing or framing

areas. Brenda used several of the shapes as further examples encouraging

the students to find more than one way to calculate the correct answer

when applicable. The assigned independent practice was a worksheet with

activities similar to the textbook pages used for guided practice.

Observation 5

Brenda reviewed the meanings of perimeter and area with the

students, then introduced the concept of volume. The base ten blocks were

used first by the teacher, then by the students to build structures pictured

in their textbooks. The blocks were counted to determine the number of

cubic units each structure possessed. At first, one student would come to

the front of the room and construct the three dimensional model of the

structure pictured in the textbook. The class would determine whether or

not the structure had been built correctly. If so, the blocks would be

counted to determine the volume in cubic centimeters.

This process was repeated several times, then the students were

divided into groups of two, given a set of base ten blocks, and instructed to

build the remaining models, determine the volume, and record their

answers. Brenda cautioned the groups not to mix up the sets of blocks

because several of them had been borrowed from another room. She also

encouraged the students to try to estimate the volume before constructing

the shapes. Once the students had completed the examples in the textbook,

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they were assigned a worksheet which also contained drawings of

structures where the volume was to be determined.

The class reaction to the assignment was mixed. Most students were

able to construct the simple structures, but many groups had trouble with

those which did not have even, smooth sides. The students became

restless when they reached those problems because Brenda was not able to

get around to answer the questions quickly.

Interview 4

Brenda was exposed to a few more materials than either Beth or Alice

in her college mathematics education course, as show in Appendix C, but

fell short of the remainder of the subjects. She and Beth reported

identical numbers of manipulative materials available in their classrooms

and nearly the same in the building. This was not surprising as the

assistant principal had provided assistance to the new teachers when

ordering materials. Brenda reported that she had used fewer materials

than she had available in her classroom. Some of the materials she ordered

arrived after the topic for which they were purchased had been covered and

other materials she did not feel comfortable using yet.

A variety of materials was used by Brenda as she taught various

topics. Manipulatives were used for fractions, measurement, and place

value concepts. The majority of the manipulatives she used, ten of the

fifteen, had been used to demonstrate measurement concepts.

A strong belief in the importance of using manipulative materials was

expressed by Brenda when she stated, "If I'm just talking about it

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(mathematics) or reading about it they just stare at me. When they look at

it or they get to play with it, no matter how old you are, you want it

hands-on. If they're able to play with something, they can see it." She felt

that this conviction came mainly from the courses she had taken in early

childhood rather than her elementary education work.

Brenda was in the process of planning her summer at the time the final

interview occurred. She was one of three teachers from her building being

sent to the state mathematics conference and was quite excited because

she had been able to sign up for a manipulative workshop on computation

skills. She reported that when she looked through the program she spotted

a course entitled something like, "Division With Manipulatives," and the

first thought that went through her head was, "Oh, I didn't know you could

use those with division." She was very excited to be given the opportunity

to broaden her mathematics background.

Lesson Plans

Brenda's lesson plans showed a slow, steady use of manipulatives

throughout the semester. As seen in Table 4, manipulative materials were

used for computational skills as well as concept introduction for fractions

and measurement. Brenda's lesson plans were only available through the

time the observations occurred. She taught her fractions unit after that,

so no entries appear on the chart for the fraction models she is reported to

have used.

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Table 4

Manipulates Listed in Brenda's Lesson Plans

Topic

Multiplication of money

Introduction of two-digit divisors

Division of money

Measurement - centimeters

Measurement - meters

Volume

Weight - grams & kilograms

Measurement - inches

Manipulative^

Play money

Place value model

Play money

Metric rulers

Meter sticks

Cuisenaire rods (singles)

Scale

Standard rulers

See Appendix C for an additional chart on Brenda.

Principal From Building B - Mr. Baxter

Mr. Baxter, the building principal, stated that he would be happy to

speak with the researcher, but he felt that the assistant principal, Mrs.

Barber, would be able to provide more information. Mr. Baxter had been

spending a great amount of time meeting with central administrators and

architects preparing for the complete renovation of the building. He

explained that Mrs. Barber had been assisting teachers with curriculum

during this time.

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Assistant Principal From Building B - Mrs. Barber

Mrs. Barber was a medium-height woman in her early forties. She had

been a teacher within the district for some time and was the full-time

assistant principal for the first time during the current school year. She

had previously been a facilitator within the district being responsible for

teacher appraisal at two buildings including Building B.

When asked to describe the mathematics program in Building B, Mrs.

Barber explained that the students were taught in self-contained

classrooms. Within classrooms, homogeneous grouping occurred to meet

student needs. She explained that there was an emphasis in the building as

well as the district on manipulative use. To emphasize this she stated,

"We're buying lots of manipulatives and encouraging all of the teachers to

use them." She discussed teacher inservices held both during the school

year and in the summer which helped promote familiarity with the

materials.

Mrs. Barber was the person responsible for coordinating the ordering

of mathematics materials in Building B. She held meetings for each grade

level to explain the process and distribute materials which might help

teachers become better informed as to what was available from different

commercial distributors. Included in the materials were catalogs from

various approved vendors and the recommended list of manipulatives from

both the local curriculum guides and the Texas Education Agency's

Mathematics Framework. The teachers were then allowed to decide what

materials would be most helpful for their rooms. Mrs. Barber recalled

spending time with both Beth and Brenda during this ordering process.

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When the individual teachers had completed their purchase requests,

Mrs. Barber compiled a building purchase order for each vendor. This

process took place early in the fall, so the great majority of manipulatives

had been received by the individual teachers by the end of November of the

current school year.

Although mathematics was not her area of specialization, it was

evident that Mrs. Barber had devoted much time to mathematics instruction

in order to assist the teachers in the building. The majority of her

teaching experience had been at the primary level, so she had little actual

classroom experience with concepts such as division and fractions. She

had, however, researched instructional strategies for teaching those

topics and shared the information with her staff. Mrs. Barber took

seriously her role as building instructional leader. She was aware of the

lack of knowledge expressed by both Beth and Brenda in terms of using

manipulatives to demonstrate computation concepts. She had been

instrumental in informing them about the summer opportunities they would

be attending at the district's expense in order for them to become more

knowledgeable about manipulatives and mathematics education.

Another function Mrs. Barber performed in Building B was to be in

charge of the materials inventory. She shared a copy of the fourth grade

mathematics materials inventory with the researcher. Included on the

computerized inventory were the materials names, publishers, and current

locations within the building. When asked if the teachers had a copy of the

inventory, she stated that she did not think they did.

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Building Q

Grades K-5 were taught in Building C. In addition, several special

education programs were housed in the building. The student population

varied ethnically and socioeconomically. Building C, like Building B, was in

the planning process for renovation. The rooms were to be carpeted,

painted, and have more storage built in during the coming summer months.

Building C was the only school where homogeneous grade level groups

were taught for mathematics and language arts. The students had been

tested at the beginning of the year. During language arts and mathematics

instructional periods the students, the students took their class supplies

and went from their homerooms to the room occupied by their assigned

teacher. Cathy, the first year teacher observed from building C, had been

given her grade level's top achievers in mathematics.

Teacher C1 - Cathv

Cathy was a tall, large-boned woman in her mid-twenties. She had

been graduated from college over a year before beginning her teaching

career. She had spent the previous year working for a large

communications company. Cathy had attended a small, private university

located in the immediate area. She had majored in elementary education

with a specialization in chemistry. During her undergraduate program, she

had also taken several of the courses necessary for secondary

certification, but had not completed the process.

Cathy's student teaching experience had been with third graders in a

very wealthy district. She had student taught half-days for a semester, so

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had riot had an opportunity to teach much mathematics or science. Even so,

she expressed being very comfortable with both science and mathematics

instruction. She felt this was due to her university preparation where she

had learned how to use manipulatives to teach both math and science

concepts. She also mentioned using the textbook from her mathematics

education class to help her plan lessons for her class.

Cathy's scheduled mathematics instruction time was 12:45 -1:45 p.m.

each day. Due to the fact that she had students from each grade level

homeroom, this schedule rarely varied. Her mathematics class had

twenty-five students. She described the group as being "quick learners."

The students were not only able to master all of the district's enabling

objectives within the prescribed time, but were often able to participate

in enrichment activities such as studying personal finances including

organizing and using checkbooks, collecting standard and metric data from

a weather station, and writing word problems to accompany the various

computational units studied.

Observation One

Cathy informed her students they were going to start a chapter on

geometry at the onset of the first observation. She then asked the

students to define "geometry." Troy volunteered that geometry meant

shapes. Cathy then asked the class to name the shapes with which they

were familiar. Virtually all hands were raised with over a dozen

responses including two- and three-dimensional shape names. As the

students called out the shape names, Cathy wrote them on the chalkboard.

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Once the students were unable to name more shapes, Cathy wrote three

column titles on the board: "2-Dimensional," "3-Dimensional" and

"Polygons (5+)." She inquired as to the difference between two- and

three-dimensional objects. A student responded, then Cathy erased all but

the title "3-Dimensional." She explained that they would be studying

three-dimensional objects that day, then proceeded to name specifically

the cone, cylinder, cube, sphere, and rectangular prism.

Next the students were asked to open their textbooks to a two-page

spread which contained pictures of the shapes themselves and pictures of

everyday objects which exemplified those shapes. In addition, Cathy had a

tub which contained each of the shapes made from wood. She asked the

students to examine both the objects she was holding and the page in the

book as they discussed characteristics and everyday examples found in

their school or homes of each.

The second part of the instruction dealt with the introduction of the

geometric terms "faces, flat surfaces, edges and corners." Cathy held a

rectangular prism using parts of it as examples when she defined each of

the terms. Next the cone and cylinder were used to demonstrate the term

"curved surface." To practice these new definitions, Cathy drew a chart on

the board which the students copied. The chart contained the name of each

shape on the right and the geometric terms newly introduced across the

top. The students were to determine how many of each term each shape

studied possessed. She then broke them into groups of four asking that

each group turn in only one chart. There were five groups and each group

got one shape. When they were finished, they passed their shape to the

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next group of students and either worked on the one that had been given to

them or talked quietly until the next shape arrived. The students spent the

remainder of the allotted class time working on this assignment.

Two minutes before the students were to change classes again, Cathy

told them to return to their seats. Several students had not been able to

complete the assignment because not all of the shapes had been to their

group. When asked what they should do, Cathy responded that they would

have to use the pictures in the book to finish the rest of the assignment.

Interview One

Cathy explained that the lesson observed had been the first lesson

from the book on geometry, but she had previously discussed plane figures

with her students. She apologized that the lesson had not gone as smoothly

as she had hoped, but when she went to the library to get the six buckets of

shapes she had planned to use with her students, only one bucket was left

on the shelf. Later she explained that in Building C all materials were

checked out through the library. Checkout came on a first come, first

serve basis and she did not think she could reserve materials for a specific

date.

When asked about materials found in her room at the beginning of the

year, Cathy responded that she had only the textbooks and curriculum

guides from the district. She did use her textbook from her mathematics

education course entitled Mathematics. A Good Beginning by Troutman and

Lichtenberg (1984) when she planned an unfamiliar topic.

103

At the time of the first interview, she had not received any

information about ordering manipulatives, but was aware that a budget

existed. She discussed borrowing Cuisenaire rods from the first grade and

had checked materials out from the district's Instructional Center (IC).

She remembered going to the IC for sets of money, fraction bars and

additional Cuisenaire rods so her students could all use them.

Cathy had not attended any mathematics inservices. She did

remember, however, receiving a writeup from the district elementary

mathematics coordinator on writing word problems. She looked through a

set of papers to locate the item. She admitted she had not had an

opportunity to read it although she said the stack it had been in was part

of the material she wanted to go through on the following Friday which

was a teacher work day.

Observation Two

The second observation began with Cathy placing a transparency on the

overhead projector which contained problems written within the district

to help students review for their upcoming TEAMS test. Due to the poor

quality of the transparency, Cathy had to read each question to the class.

She eventually rewrote several of the problems on the board so the

students could read them. Finally she became so frustrated with the

process that she told the students to put their papers away; they would do

something else.

The next several minutes were spent reviewing the material already

covered from the geometry unit. Cathy asked the class questions, allowed

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time for thought, then randomly called on students to answer. When she

had completed the review, she told the students they would be discussing

angles that day. Once the term angle was defined, she had them locate

angles throughout the classroom. Several examples were given. She then

had them open their textbooks to the page that introduced the concept

"right angle." The students discussed why certain examples were or were

not right angles. To check their answers, Cathy had the students use the

corner of a piece of notebook paper.

The previous day Cathy had asked the students to bring their glue with

them when they came to class. When the examples in the book were

completed, she asked them to take out their glue. She had student helpers

pass out sheets of construction paper while she handed out toothpicks to

the students. The students were instructed to experiment making designs

which contained right angles with the toothpicks on their paper. When they

were satisfied with their design, they could use the glue to stick the

toothpicks to the paper. The remainder of the class time was spent on this

activity.

Interview Two

During the second interview, Cathy expressed frustration with the

TEAMS review materials. She felt that the transparency quality was not

sufficient. Part of the problem, she admitted, was the location of the

screen and plugs within her classroom. She expressed the hope that during

the remodeling process that would be corrected.

105

Cathy explained that the topic "Angles" was strictly an enrichment

part of her textbook. The idea to use toothpicks with the students

originated with the teacher's edition. She planned to hang the pictures

along the hallway outside her classroom where space was provided for

work to be displayed. Cathy commented, "You can walk down the halls and

you've got reading and stories and art, but you never see anything math

oriented."

Cathy discussed the importance of introducing concepts at the

concrete level and moving toward the abstract level in mathematics

instruction at this time. When asked why she felt this was important, she

replied, "Because it works. I tried introducing things-here's the book,

let's see what's here. It's like speaking Greek, it goes right over their

heads." She related the experience of trying to teach the concept of

regrouping in subtraction with her top-level group. She said they were

having a very difficult time when the concept was presented abstractly, so

she brought in base ten blocks to help develop the concept. Cathy

demonstrated and led the lesson with the overhead models while the

students each had access to a set. The students then quickly learned the

concept. The materials were obtained from both the library and first grade

classrooms.

When asked about the availability of any materials she wanted to use,

Cathy commented, "Somebody has them either at the instructional center

or the first and second grades have a lot of things." The only thing she had

wanted but had difficulty obtaining was the computer which had finally

arrived in the classroom.

106

Observation 3

Each student was given twelve red squares made from construction

paper about 1 1/2" X 1 1/2". Cathy explained to the students that they

would be learning about division. The students were asked to pretend that

each square was a book. Next they were to figure out how many books

would be in each row if the books were divided into four rows. This

exercise was repeated for six rows, three rows and two rows. Cathy

walked around the room the entire time the students were working with

the squares of paper observing student progress. Each time the answer

was calculated through the movement of manipulatives, a written summary

was placed on the board until the four notations appeared:

4 groups of 3 • 12

3 groups of 4 = 12

6 groups of 2 = 12

2 groups of 6 = 12

After the students stacked the counters, they discussed the data. Several

of the students immediately saw that the words "groups of" could be

replaced by a multiplication sign.

Instructions were then given for the students to open their books to

the pages which contained introductory material concerning division

concepts. Cathy followed the sequencing established in the book to

continue the lesson. She introduced the relationship between

multiplication and division, the division number sentence, and the correct

terminology naming parts of a division equation. The class worked through

several problems where a given number of objects (dividend) was split into

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groups with the same number in each group (divisor) and the students were

asked to determine how many groups were made (quotient). Under each

picture the problem was written in the form "15 + 3 = An assignment

of twelve problems which required the same process was given as

homework.

Interview 3

Cathy discussed using counters for a variety of lessons on

computational skills throughout the year. She said that using them as a

way to explain the concept of division was a natural extension of their

explanations for addition, multiplication and division. There had been

several types used: pennies, straws, rulers, pencils and pieces of papers.

Cathy described several of these activities.

When asked about ordering mathematics and science materials, Cathy

replied that the principal was checking to see just how much money was

available before the orders were able to be placed.

Observation 4

Expansion of the division concept was the focus of the next observed

lesson. Cathy began by placing three review problems on the board, 6 X 5, 6

X 50 and 6 X 500. The products were calculated and the pattern discussed.

She then wrote the problem 48 + 6 using the division box and asked, "How

many sixes fit in 48?" Next she changed the dividend to 480. Through

questioning strategies, the students were able to compute the answer.

Students looked at 6 + 3 then 60 +• 3. Cathy then showed the class how to

108

determine where the first digit of the quotient should be placed. She

reviewed the steps of the algorithm. The class worked several problems

together, then an assignment practicing the skill was given. No

manipulative materials were observed during the lesson.

Observation 5

Students spent the majority of the time during this observation

working on division worksheets. The problems on the sheet consisted of

one-digit divisors with two-digit quotients. Cathy's lesson plans had

indicated that the concept would be introduced on that particular day, but

when the researcher arrived, the concept introduction had taken place the

previous day so the students were simply practicing the algorithm.

Interview 4

Cathy was exposed to a variety of materials in her mathematics

education course. The manipulatives checked could be used to demonstrate

computational concepts as well as measurement, geometric and fractional

ones. There were fewer materials found in Cathy's classroom than in any

of the other rooms, but she had access to twenty-four of the thirty-six

manipulatives on the checklists in her building. As found in Appendix C,

Cathy used fifteen different items during the course of the year.

Cathy was the only teacher who mentioned obtaining mathematics

materials from the district's Instructional Center. Cathy explained that

she had gone one afternoon after school to visit the facility to see what

109

was available. She had checked materials out from the center several

times during the school year.

The curriculum guide for Cathy's grade level listed nine separate

topics to be studied during the year: numeration and place value, addition,

subtraction, multiplication, division, measurement, fractions, geometry,

and time and money. During the final interview, when asked to describe

how each manipulative checked was used, Cathy cited each of these topics

in connection with manipulative usage.

Although satisfied with her first year's mathematics instruction,

Cathy felt frustrated about not using certain manipulatives such as

tangrams. She stated, "I just didn't get the chance to use them this year. I

love using them because I think they are interesting and I think they'd get a

lot out of using them."

Cathy expressed frustration that she did not always think of things far

enough in advance in order to assure that the materials necessary would be

available. When asked if that was partially a function of being a first year

teacher, she responded, "I think so. I think about these great ideas the day

I want to use them and they're (the manipulatives) not necessarily there."

Cathy felt that a comprehensive list which indicated exactly which

materials were available within the building and within the district's

Instructional Center would be of assistance to teachers.

Lesson Plans

Cathy's lesson plans were quite sparse. The great majority of the

110

time, only the objective arid the page numbers appeared. Only three entries

made any reference to manipulatives, as shown in Table 5.

Principal From Building C - Dr. Connors

Dr. Connors was a young, highly articulate woman. She had come to the

district three years earlier to accept the position as principal of Building

C. She had experience as a classroom teacher and as an assistant principal

and had earned a Ph. D. in education before joining the district.

The mathematics program at Building C was different from the other

buildings in that in grades three through five, the students were

homogeneously grouped for instruction according to "skill level." Dr.

Connors emphasized the fact that the groups were quite flexible so the

students moved up or down according to the skills studied. In addition,

experimental programs using Touch Math and cooperative learning were

also found in the building.

Table 5

Manipulatives Listed in Cathy's Lesson Plans

M e Manipulative^

Identification of solid geometric shapes Tubs of shapes

Review of geometric shapes Geometric shapes

Introduction of concept of division Counters for students

See Appendix C for an additional chart on Cathy.

111

Dr. Connors was able to converse about current practices in

mathematics instruction and the manipulatives themselves. She was quite

excited about a new purchase she had recently made for her building,

Mortenson Mathematics Manipulatives. She described these new learning

aids as a cross between base ten blocks and Cuisenaire rods. They could be

used to demonstrate numeration and place value concepts as well as the

four basic operations with whole numbers.

" Building C has always been a place where principals spend money on

instruction," stated Dr. Connors when asked about the large supply of

materials in the school library. She said that the first year she was

principal, she had quite a bit of input into the teachers' purchases. The

second year she allotted the money on a grade level basis. Due to the large

amount of materials already in the building, some of the teachers did not

spend all of the allocated funds. Dr. Connors stated that at least

eight-five percent of the teachers in the building used the materials for

mathematics instruction.

Building D

Building D had recently been through remodeling. In addition to the

internal updating of the existing rooms, an entire new wing had been added.

Renovation of the building had taken longer than expected in the fall, so

teachers had not been able to enter their assigned rooms during the

previous summer.

The student population had been steadily growing for a number of

years. Many new teachers had been added to the faculty in grades K-5.

112

Homogeneous grade level groupings occurred in Building D for the language

arts. The students returned to their assigned homerooms for the remainder

of instruction including mathematics.

Teacher D1 - Dena

Dena had short, auburn hair and was of medium height and average

build. She was in her early twenties and had been graduated a year later

than Cathy from the same small, private university. She majored in

elementary education with a specialization in English.

Dena student taught in second grade in a wealthy suburban school

district. She taught half-days for a full semester and was not able to

observe or teach mathematics. Realizing this to be a disadvantage, Dena

enrolled in an additional mathematics practicum during the spring

semester of her senior year. She was placed in a fourth grade classroom in

Building D. Mrs. Dobbs, the assistant principal at that time, observed Dena

teaching and brought her to the principal's attention. Mrs. Danvers, the

principal, hired Dena for her present position.

During her college preparation, Dena used two texts for her

mathematics education class. The texts were Guiding Children's Learning

of Mathematics (Kennedy & Tipps, 1984) and Helping Children Learn

Mathematics (Reys, Suydam & Lindquist, 1984). The first thing Dena

mentioned when asked about her preparation for teaching mathematics was

the "concrete to pictorial to abstract presentation of concepts." She added

the instructor had also stressed modeling of all of the levels and the

importance of understanding the process steps involved.

113

The language arts period began early in the day, so mathematics

instruction could not begin until the students had returned to their

homerooms. Dena's scheduled time was 11:00 -11:45 a.m. on Mondays,

Wednesdays and Fridays and 12:15 -1:15 p.m. on Tuesdays and Thursdays.

Three days a week the students had mathematics before lunch and two days

after lunch.

Dena's class consisted of twelve boys and 7 girls. During the

observation period, she had one female join the class and she lost one

male. Dena described her mathematics class as having average ability

although she felt they did better overall in mathematics than in subjects

requiring more language arts skills. One of her students participated in

the district's gifted and talented program, but she felt that three or four

could qualify if space were available.

Observation One

The class was in the middle of a unit on measurement when the

observations began. Dena began the first lesson by reviewing the concepts

of perimeter and area. She then held up a tissue box and asked the students

to tell her how much the box could hold. One student volunteered, "Weigh

it?" and another said, "Volume." Dena explained they would be using cubic

units to measure volume.

Dena had a box of sugar cubes beside her in the front of the room. As

she constructed the object, she described the figure which was two cubes

by three cubes by one cube. She then counted the six cubes that

constructed the solid shape and explained the volume was equal to 6 cm3.

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When the figure was completed, Sarah, a student in the front of the room,

commented, "Oh, it's just like area!" Dena then explained that area talked

about flat objects like things drawn on pieces of paper.

The same process was repeated, this time having a figure with 2 cubes

by 3 cubes by 2 cubes. The cubes were again counted to figure the volume

which was 12 cm3. Dena constructed two more figures, then counted the

cubic units in each. All shapes were constructed at the front of the room.

The students did not have an opportunity at this time to make any of the

figures themselves.

Next she had the students turn in their books to a two-page

presentation of volume. The students were to look at the pictures of the

shapes, then count how many cubic centimeter blocks each contained. The

first six shapes drawn were rectangular prisms. The last two problems,

however, were rectangular prisms with one to three cubes removed leaving

uneven figures. The textbook pages were used as practice for the class.

An independent assignment was placed on the overhead projector. There

were nine problems on the transparency with two being irregularly shaped.

At first, the students were told to count the number of cubes needed to

construct each shape from the pictures. When several students had

problems with the assignment, Dena told the students they could go to the

back of the room and build the shapes so they could determine the number

of cubic units.

Interview One

This interview took place immediately after the observation. Dena

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walked her students to the lunchroom, then returned to the room. When

asked what she took into consideration as she planned the lesson on

volume, Dena replied, "Well I knew that I had to do something with the

concrete and I didn't know what. At the last minute I thought of the sugar

cubes, but I wish that I had used them differently." She went on to explain

the changes she would have made in the lesson which included a more

precise method of investigating height, width and depth of the figures. In

addition, she wished she had passed the cubes to the students so they

would have been able to manipulate the materials themselves. Since

volume was to be continued the following day, Dena planned to incorporate

the changes into the next lesson.

The topics measurement of length, perimeter and area which preceded

the lesson on volume were also discussed. Dena explained how she

presented all of these concepts using identifiable examples for her

students. She related centimeters to the width of their index fingers, used

the meter sticks to measure a variety of lengths, related perimeter to

bulletin board border and discussed the area of new carpeting that had just

been placed in the room.

Observation Two

During the second observation, Dena conducted a review of the two

measurement systems the students had been studying. Initially she asked

questions like, "If we had to weigh this trash can (holding up the can for

the students to see), what units of measurement could we use?" Next she

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asked the students to think about units that could be filled in the following

chart:

Length Weight Volume

Metric:

Standard:

As the students would name a unit, for example meter, Dena would ask

what they could think of that would be about the same as that unit. The

student responded with everyday objects that were about the same length,

weight, or volume. In addition, they had to tell where the unit belonged on

the chart.

In the next part of the review, Dena orally named objects and

specified length, weight, or volume and a measurement system, then asked

the students to tell her which unit in that system was most appropriate.

The final portion of the review dealt with perimeter and area. Dena gave

scenarios to the students, then asked them which would be the appropriate

calculation, perimeter or area.

About half of the allotted mathematics time remained when the

review had been completed. Dena asked the students to have only paper and

pencil on their desks as they were going to begin a new kind of thinking

problem. She had written five pieces of information about different

students' grades, then drew a Venn Diagram which she called "circle logic"

for the solution of the problem. Dena labeled each circle A, B or C and

discussed the places where the circles overlapped. The class worked

together to determine where each of the five fictitious students were to

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be placed within the Venn Diagram. Once the problem was solved, Dena

reviewed each of the placements to see if students could verbalize the

information. A second problem was then put on the board which dealt with

three types of reading material. The same procedure was followed as the

class found the correct placement solutions.

Interview Two

Once her students had been taken to the cafeteria, Dena returned to

the room and got two transparencies from her desk. She explained that

these were examples of the types of review problems the students were

doing as they entered the classroom in the morning to help review the

skills taught for TEAMS. She felt she was not teaching specifically for the

TEAMS test, but that the material on the transparencies represented a

quick way to review the topics taught thus far during the year.

Dena expressed her belief again during this time about using concrete

objects with students. When asked why her students had named objects

like raisins and thumb joints when discussing measurement she stated, "I

know that grounding something in a physical thing will help that child

learn it forever, better than if I just said, This line right here is this.1 I

knew that multisensory thing would give it to them."

At the beginning of the year, Dena decided she would try to devote

Fridays to "problem solving." Most of the problem solving which had

occurred consisted of word problems and logic problems. In addition to the

Venn Diagrams which had been observed during the lesson, she had given

the students a great many grid logic problems throughout the year. She had

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also begun to use nonroutine problems with her students. The example she

cited was the problem which asked the students to determine which order

a farmer had to transport his grain, chicken and fox so that the three

objects made it across a river safely. She had found several problem

solving strategies in her mathematics textbook, but had been exposed to

many in her college mathematics education course which she incorporated

into her mathematics curriculum.

Dena was the only teacher to have materials left from a previous

teacher in her classroom. She had found a class set of calculators,

individual chalkboards, a few games and some flashcards in her room as

well as the textbooks and district curriculum guides. She had placed her

order for the manipulatives, but the order had not yet arrived. Included in

her order was a class set of base ten blocks and three books. Of the three

books, two were computational drill worksheets. The third book was The I

Hate Mathematics! Book by Marilyn Burns which contained ideas for

activities for a variety of mathematics topics.

Observation 3

The lesson began with a review of what they had done the previous

day. The students had used calculators to explore the concept of

multiplication. They had determined that multiplication was a quick way

to add the same value over and over. In addition, they had discovered that

when two numbers were to be multiplied, the order in which they were

entered into the calculator made no difference on the final product. As

Dena reviewed this knowledge, she used mathematically correct

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terminology to describe factors, products and the commutative property.

Dena explained that they would be using a deck of cards in their

lesson. She had divided the cards into groups of four of the same value.

The first time she held up three cards which each showed twos. She talked

to the students about three groups of two, then had them count the total

number of characters on the three cards. On the board she wrote, "3 groups

of 2," to describe the set. Beside this she wrote "3X2 = 6." She

demonstrated the multiplication concept several times using similar

examples.

Next the class was divided into groups of two. Each group of students

was given a set of four cards. Dena asked the students to write the

different combinations they could make with the given card in word form.

They wrote both the word format and the number sentence format. The

students then shared their findings with the class.

To culminate the lesson, Dena brought out her tape recorder. She

played a tape entitled "Rap With the Facts" for the students. The

multiplication facts for ones and twos were set to music and worded in a

catchy manner.

Interview 3

Immediately after the observation, the interview took place. Dena's

first comments were about how relieved she was to have the TEAMS

testing over for the year. When asked how long it would be before results

were back, she said that she had looked over the exams and her students

had done well.

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She expressed her excitement about beginning multiplication with her

students. To prepare for the unit, she had gone back to the textbooks she

had used in college and read the chapters that dealt with multiplication. It

refreshed her memory about using the correct terminology, but had not

been especially helpful with ideas for introducing the concepts as Dena did

not have access to the materials suggested. When asked if she had

discussed the unit with other teachers on her grade level, she explained

that they had told her about the race track which appeared on the wall

above the windows which would be used to chart student mastery of fact

families. She had not discussed actual instruction of the multiplication

with them.

Dena described the lesson involving calculators in detail. She had

enough calculators so only two students had to share during the lesson.

She explained that they began by exploring with the calculator itself.

When the students were comfortable with the keys, they began the lesson

itself. First she would ask the students to enter an addition problem like 4

+ 4 + 4 and calculate the sum. Next the students were asked to enter 3 X 4 ,

or the multiplication sentence which coincided with the repeated addition.

The students quickly surmised that multiplication was repeated addition.

Because some of them had entered the factors in the multiplication

sentence in different order, they also discovered the commutative property

of multiplication. That was not part of the planned lesson, however. Dena

said they related it back to the commutative property of addition also.

Next Dena discussed the observed lesson. She said that at first while

she was planning the lesson, she did not know what type of manipulative to

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use with her students. She considered beans, but felt thai they would be

too messy. Finally she came up with the idea of using decks of cards.

When asked to name the objective or objectives of the lesson, she said her

goal had been to familiarize the students with the correct vocabulary and

give them a general understanding of what it meant to multiply. She felt

those students who could already call out their multiplication tables had

learned them on the rote level alone. They did not understand the concept

of multiplication. She went on to say that even though they worked with

the ones, twos, threes, and fours with the cards, she didn't expect any king

of fact mastery from the lesson, only an understanding of the concept.

Dena had planned to introduce the ones, twos, and threes during the

first week of instruction on multiplication. She said that it was a

tentative timetable because she had no idea how long it would take. She

made a point of reiterating that there was a difference in knowing the

facts by memory and having the mental tools to figure out one that was not

committed to memory. Her goal was to work toward memory, but do it by

teaching the children the process of multiplication so they could solve one

they did not know by heart.

Since Dena had previously stressed the importance of teaching

concepts at the concrete level, the researcher asked how she would

continue doing that when she got to the higher numbers like eights or

nines. Dena confidently replied, "I don't know, but I'll find something. I

always do. I've got my beans back there that I can always get."

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Observation 4

Several short activities were witnessed during the fourth

observation. The desks had been arranged in pairs. As the students entered

the classroom, one person in each pair got out flashcards and proceeded to

drill the other person. When a timer went off, the partners switched rolls

and the drill continued. When the timer sounded again, the drill stopped.

The second activity involved the review of the concept perimeter. One

student supplied the definition. Others volunteered examples of things for

which the perimeter could be calculated. One student named the clock,

which was round. Dena gave a brief explanation of the concept

"circumference."

The final activity was a problem solving exercise Dena found

described in The I Hate Mathematics! Book by Marilyn Burns. Dena

presented the problem of six friends meeting on a street corner and all

shaking hands with every other person once and only once. The students

were first asked to estimate how many handshakes had occurred. Several

of them volunteered answers such as 36,12 and 54. Some of the students

suggested using themselves as the groups of six. Dena then placed six

volunteers in the front of the room. The students at their seats were

notetakers who were to keep a record of the number of handshakes.

Observation 5

Dena began the lesson by drawing pictures of two of the same sized

circles on the board. One of the circles she divided into two parts shading

one part; the other was divided into four parts with two parts shaded. She

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reviewed the concept of equivalent fractions with the students. She

summed up the idea by saying, "So far we've been talking about pieces of

one thing."

Next she introduced the concept of mixed numerals by having two

circles each divided into four parts drawn on the board. Dena told the

students that these were mini-pizzas. She shaded one entire circle and

one out of the four parts of the second circle. Under the completely shaded

circle she wrote the numeral "1" and under the partially shaded circle she

wrote "1/4." She told the students that if they ate all the shaded parts,

they would have eaten one whole mini-pizza and one-fouirth more of

another mini-pizza or one and one-fourth mini-pizzas. She wrote "1 1/4"

on the chalkboard and explained a mixed numeral had a whole number part

and a fractional part.

Dena then used another model to demonstrate mixed numerals. She

had a package of graham crackers where each cracker was divided into four

pieces. She held up an unbroken cracker and referred to it as " 1 S h e then

broke one of the other crackers into four pieces. She used the crackers to

model mixed numeral values asking that the students tell what mixed

numeral each set of crackers represented. During this time, Dena was the

only person to manipulate the crackers.

The remainder of the lesson was from the textbook. The students

looked at pictures representing different mixed number values and had to

come up with the correct name. This same skill was practiced in the

independent assignment.

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Interview 4

Dena's manipulative checklist responses are found iin Appendix C. She

had been exposed to a large variety of manipulatives in her mathematics

education program. She indicated that twenty-five of the thirty-six listed

had been used in class. She had access to twelve within her own room, the

majority being related to measurement. There were twenty-three found in

her building. Dena indicated use of sixteen of the manipulatives on the list

during the year, but observations and interviews showed she used many

noncommercial items as manipulatives within her classroom such as sugar

cubes, tissue boxes, bundled coffee stirrers and decks of cards.

When talking about manipulative usage for specific topics, Dena

mentioned all nine of the topics listed in the grade level's district

curriculum guide. She discussed using manipulatives in connection with

numeration and place value, all four whole number operations,

measurement, fractions, geometry and time and money.

Even though manipulatives had been used during the fraction unit,

Dena stated that when she taught it the next year some changes needed to

be made. She felt like she had done too much of the demonstrating. She

described it this way,".. .most of it was pictorial rather than holding on to

it when I worked with fractions." She believed that too many concepts had

been presented into too short a time, therefore the students did not do as

well as they should have with the content.

Dena was looking forward to a workshop she would be attending when

school was out on numeration and place value. A great deal of work with

manipulatives was included in the course and Dena felt she needed to

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improve in that area. She commented, "I hope that next year my year will

be much more manipulative." She felt that during the current year she had

used too many paper-only activities and wanted to actively involve her

students in mathematics instruction more.

lesson Plans

Dena listed some manipulatives in her lesson plans. Table 6

enumerates all of these instances within the period of observation.

Measurement activities, fractional, geometric and multiplication concepts

were included on the chart. Other units including computational

operations,

numeration and place value, and time and money were not topics taught

during the observed time period.

Principal From Building P - Mrs, Dflnvers

Mrs. Danvers had been principal of Building D for many years. She had

seen much population growth in the school during that time. Due to this

growth, many teachers had been added to the faculty. The building had been

renovated and had an addition put on the previous summer. Six more

classrooms were to be added the following summer.

When asked to describe the math program in her building, she stated

that a great deal of time was spent on mathematics. The teachers

followed the district's mathematics curriculum including the time table

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Table 6

Manipulatives Listed in Dena's Lesson Plans

Topic Manipulative^

Volume Sugar cubes

Milliliter & liter Models of milliliter & liter

Gram & kilogram Scales

Measuring inches Rulers

Fraction review Fraction models

Introduction of multiplication concept Calculators

More multiplication concepts Playing cards

Commutative property of multiplication Beans

Equivalent fractions Paper folding

Mixed numerals Graham crackers

See Appendix C for an additional chart on Dena.

provided for them in the curriculum guide. She added, "We feel very

strongly about using manipulatives, hands-on experiences so the children

can internalize these concepts."

Mrs. Danvers explained that in her building the teachers were given a

set amount of money to buy materials and allowed to make their own

choices. If the teachers felt that the science curriculum was more needy

than the mathematics curriculum in any given year, perhaps due to a new

textbook adoption, the teachers could exercise their own prerogative and

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order what they felt best. No special funds were allotted to furnish brand

new classrooms with manipulatives past second grade.

The grade level chairman were considered by Mrs. Danvers to be a very

important link in the communications chain for first-year teachers. She

said she depended upon them to share information like the district's

recommended list of manipulatives. Mrs. Danvers also stated that the new

assistant principal, Mrs. Doe, had taken on quite a large part of the

responsibility of instructional leadership within the building.

Mrs. Danvers felt the number of teachers using manipulatives on a

regular basis was quite high, but expressed the desire to use the newly

acquired computer system to begin to catalog the materials within the

building. She wanted teachers to be able to locate resources for specific

topics. She felt the sharing of ideas and materials would benefit the

school.

Data From the Central Office

The two central office administrators involved in the research project

were the elementary mathematics consultant, Mrs. Evans, and the

Supervisor of Elementary Programs, Mr. Edwards. In addition, the

district's curriculum guides and the teacher's editions of the textbooks

were examined for data concerning manipulative usage.

Elementary Mathematics Consultant - Mrs. Evans

Mrs. Evans was one of four elementary consultants within the district.

It was her first year in her current position. Each consultant was assigned

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to four elementary buildings as a generalist who assisted teachers in any

way possible. Another part of the job was to help plan inservice at the

district level. In addition, she was given responsibility for elementary

mathematics instruction throughout the district.

The first thing mentioned by Mrs. Evans when she was asked to tell

about the district's elementary mathematics program was, "We have a

philosophy in the district that math manipulatives are very, very

important." This commitment was more than just verbiage as monies had

been budgeted over the past four years so that each elementary teacher

could order manipulatives. In addition to the budget for all elementary

teachers, class sets of manipulatives like Cuisenaire rods had been

provided at the first and second grade levels for a few years and she would

like to see that extended to the third grade. Mrs. Evans felt like the

commitment to mathematics education in general came directly from her

boss, Mr. Edwards, who originally was a mathematics teacher.

Mrs. Evans was in charge of disseminating information about the

district-wide ordering of manipulatives. She explained that she sent a

memo to each school supplying data regarding the ordering process.

Included in the memo was a list of the state recommended manipulatives

with additional ones added at the district level. She supplied current price

lists and recommended vendors on the form, too. This information was

given to the principal and assistant principal at each building, but Mrs.

Evans did not believe the teachers were automatically sent copies. It was

left to the building administrators to do that.

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The district also provided many opportunities for teachers to attend

inservice geared specifically toward mathematics. They had offered

several summer courses where teachers could attend various portions of

the Texas Education Agency's Mathematics Training Modules. Several

teachers had been sent to commercial workshops, workshops at the

regional service center, and many of the district's teachers attended the

state mathematics conference held each August at district expense. Since

most of these inservice opportunities had occurred the previous summer,

few first-year teachers had been able to participate in them. Several of

the new teachers, however, had been given the opportunity to do so

immediately following the completion of their first year. Beth, Brenda and

Dena had all mentioned district supported mathematics workshops when

discussing their summer plans.

Mrs. Evans explained that the planning for the following year's

inservice was underway. A survey form had been sent to each teacher and

administrator within the district asking them to rank their perceived level

of need for over sixty items. The two items on the questionaire that dealt

with mathematics were "Mathematics (Content and Application)" and

"Mathematics (Manipulatives)." A five point scale was give the teachers

with a 1 perceived as "Low" need and a 5 perceived as "High" need. Tables 7

and 8 indicate the order rank and the average level of need for the two

mathematics related classifications within the district.

Mathematics manipulatives was ranked much higher than mathematics

content and applications in all cases. In both grade levels where

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Table 7

Average Perceived Level of Need for Mathematics (Content and Application^

Personnel/Grade Level Order (1 - 64) Average level Qf Need

Teachers 48 2.3

Administrators 41 2.9

Pre-K 6 3.4

K 29 2.5

1 50 2.3

2 37 2.7

3 53 2.5

4 25 2.8

5 48 2.4

Composite 46 2.3

observations occurred, the need for inservice with mathematics

manipulatives was ranked in the top ten out of sixty-four choices given.

Supervisor of Elementary Programs - Mr. Edwards

Mr. Edwards, the Supervisor of Elementary Programs, had been with the

district for many years. He had previously been a teacher, a principal, and

a consultant in the system. His teaching field had been mathematics

education and it was obvious he had kept pace with the area. Mr. Edwards

was the only person interviewed in the study who specifically knew the

focus of the study was teacher usage of manipulatives rather than

mathematics education in general.

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Table 8

Average Perceived Level of Need for Mathematics (ManipulativesV

Personnel/Grade Level Order M - 64) Average Level of Need

All Teachers 15 2.3

All Administrators 5 2.9

Pre-K 1 3.4

K 8 2.5

1 20 2.3

2 12 2.7

3 7 2.5

4 3 2.8

5 9 2.4

Composite 18 2.8

According to Mr. Edwards, the district's mathematics program was

built around the state mandated essential elements. At the time the

elements were first required, the district incorporated them into the lists

of enabling objectives for each grade level. These served as a minimum or

base guideline with the teachers being urged to go beyond when possible.

Flexible grouping of students was supported by the administration if it

proved to be the vehicle that best met student needs.

Mr. Edwards voiced his opinion of the district's commitment to

manipulatives by saying:

We encourage, not only verbally but through the budget process,

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teachers to have manipulatives to use in the classroom in order to

teach concepts to children so that we're just not trying to introduce

them to something that's theoretical or abstract and expect them to

understand it and be able to use it and retain it over a long period of

time. We budget $100 per year for every teacher to buy mathematics

and science manipulatives. We've done that now for the past four

years. We think that is the best way to teach mathematics.

He explained the money was distributed on a campus by campus basis with

spending left to the discretion of the administrators for distribution. The

building consultants worked with the principal and staff in the buildings to

help determine what they had and what materials would be available for

purchase.

Purchasing the manipulatives was just half the battle according to Mr.

Edwards. Further concern was expressed about the use of manipulatives

within classrooms when he stated:

We need not only to purchase materials, but there needs to be good

training on how you utilize the materials, too. Just to take money,

buy something and put it in a classroom, you've run half the race. If

you don't know what to do with it, you're limited. But if you have it

to work with and don't know what to do with it or don't know all the

different things that can be done with it, you're still limited.

To support the training, the school district also supplied funds to send

teachers to classes and conferences like the state mathematics conference

held each summer. In addition both the regional service center and the

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district itself offered workshops to help teachers better learn to use

materials.

Mr. Edwards stated that there were four days of general inservice each

year and four additional days for all new teachers. The new teachers had

to come the week before the others and hear about topics such as

discipline, the appraisal system, and the district's curriculum. Of the four

general days, about half of the time was spent in district-wide inservice

and the other half devoted to building inservice. Mr. Edwards explained

that the new teacher inservices and the district-wide inservices were

planned by the central office personnel including the consultants and

himself. One of the days the past fall had been devoted to curriculum and

instruction and operational procedures in the classrooms, but nothing

pertaining strictly to mathematics had been offered.

Textbooks and Curriculum Guides

Both the district curriculum guides and the mathematics textbooks

used in the classroom provided suggestions for teaching lessons. The

curriculum guide suggestions were keyed into the district's mathematics

enabling objectives which were the core of the district curriculum. In the

textbook, proposed items including materials were written in the teacher's

editions for introducing the lesson concepts and using the page or pages to

be taught. Charts listing the manipulative materials from both curriculum

guides and textbooks for each grade level observed are found in Appendix D.


Burns, M. (1975). The I hate mathematics! book. Boston: Little, Brown

and Company.

Harcourt Brace Jovanovich. (1985). Mathematics today, level 4. Orlando,

FL: Author.

Kennedy, L. M. & Tipps, S. (1984). Guiding children's learning of

mathematics (4th ed.). Belmont, CA: Wadsworth, Inc.

Reys, R. E., Suydam, M. N. & Lindquist, M. M. (1984). Helping children learn

mathematics. Englewood Cliffs, NJ: Prentice Hall.

Troutman, A. P. & Lichtenberg, B. K. (1982). Mathematics: A good

beginning (2nd ed.). Monterey, CA: Brooks/Cole Publishing Co.

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CHAPTER V

FINDINGS, CONCLUSIONS AND RECOMMENDATIONS

Due to the qualitative nature of this study, generalizations can not be

made from the results. The research does, however, paint a narrative

picture of the background experiences, instructional planning, and lesson

presentations of six first-year intermediate grade level teachers in a

north Texas district as they teach mathematics. The study focuses

specifically on the teachers' usage of manipulative materials. It examines

the teachers' familiarity with the materials, the availability of those

materials, the utilization of manipulatives, and teacher perceptions of

manipulative usage.

Findings

District Commitment

The commitment to the use of concrete manipulatives in the

instruction of mathematics held by the Texas school district where the

research was conducted was evident in every phase of this research. Only

the Supervisor of Elementary Programs, Mr. Edwards, knew that the

objective of the researcher was to examine manipulative usage within

elementary mathematics classrooms, yet the elementary mathematics

consultant, all building administrators, and written materials such as the

district's curriculum guides exemplified this intent. All building

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136

administrators, when asked to describe their school's mathematics

program, spent time explaining the role of manipulatives in mathematics

instruction. The elementary mathematics consultant, Mrs. Evans, talked at

great length about various aspects of manipulative usage within the

district as she described the district's elementary mathematics program.

In addition to verbally supporting the inclusion of manipulatives in

mathematics instruction, the district's actions demonstrated this

commitment. For the past four years funds had been budgeted on a

district-wide basis at the rate of $100 per teacher per year for

mathematics and science manipulatives. The district had offered summer

inservices and workshops through the regional educational service center

for teachers to learn more about the use of the materials. The district had

also set aside funds for teachers to have the opportunity to take special

courses at universities or attend the annual state mathematics conference

held each summer.

Teacher Familiarity with Manipulatives

A large discrepancy between the six first-year teachers was noted

when information was gathered about their familiarity with manipulatives.

At the beginning of their final interviews after all observations had been

completed, the teachers were asked to indicate on a checklist which

manipulatives they had seen during their college preparation programs.

The thirty-six item checklist was comprised mainly of manipulatives

found on the Texas Education Agency's recommended minimum classroom

manipulative materials list with other additions taken from the district's

curriculum guides.

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As indicated in Appendix E, Alice and Beth had been exposed to the

least number of materials with only five each. Brenda had seen eight,

Cathy fifteen, Dena twenty-five and Anne twenty-six. When discussing

their college preparation, Anne, Cathy and Dena all mentioned the

consistent reference to and use of manipulatives by their professors. Beth

stated that only a few materials had been demonstrated in her class, but

they spent a great deal of time doing problems on worksheets. Alice

recalled that the only manipulative materials found in her class were the

ones the students themselves brought to class. Writing units of

instruction occupied the majority of time spent in mathematics education

class for Brenda.

The subjects had diverse experiences with mathematics during their

student teaching experiences as well. Three of them, Dena, Cathy and Beth,

had student taught only half-days, so they missed mathematics instruction

the majority of the time. Cathy had planned and taught a unit on geometry

so that she could do some mathematics. After Dena had completed her

student teaching, she enrolled in a one-hour mathematics practicum course

to give herself some experience teaching mathematics in an actual

classroom.

Availability of Manipulatives

Of the six first-year teachers, four walked into rooms containing only

textbooks and curriculum guides. Anne stated that she found one class set

of small clocks. Dena and another new teacher on her grade level split

materials left by a former teacher who had retired. Dena got class sets of

calculators and individual chalkboards, some flash cards, and a few games

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that could be used in conjunction with mathematics. Brenda had been given

a set of base ten blocks and a few other materials by her mother who was

a fifth grade teacher in a south Texas school. Each of the teachers was

allowed to place an order, however, for the materials budgeted by the

district sometime between October and March. All teachers ordered at

least some mathematics manipulatives.

The teachers indicated on a checklist the number of materials found in

their rooms and within their buildings at the time of the final interviews.

The results appear in Appendix F. While relatively few differences existed

between the amount of materials reportedly found at the building level, a

large difference existed between the number of items reported within

classrooms.

The two teachers from Building A indicated a slight differences in

both the number of items reportedly found within their classrooms and

building. Alice listed thirteen of the items in her room while Anne checked

sixteen. Anne indicated that twenty-one items were located in the

building while Alice reported only seventeen. The Building B teachers,

Beth and Brenda, reported twenty-four and twenty-five items within the

building respectively. Both teachers indicated that eighteen items were

located in their classrooms, but the lists were not identical. Cathy

indicated only four items in her classroom, but stated that twenty-four

manipulatives could be found in Building C. Dena checked twelve items in

her classroom and twenty in Building D.

Manipulatives were stored mainly in individual classrooms in

Buildings A, B and D. Building A teachers reported that a veteran team

member kept most of the grade level's mathematics materials in her

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classroom closet. Both teachers reported a hesitancy to "bother" the

teacher in order to obtain the materials. Alice admitted she had not even

felt comfortable enough to explore all of the materials located there.

In Buildings B and D, the manipulatives were spread evenly throughout

the classrooms. The teachers of building B described frequent sharing of

materials both between themselves and the other four teachers at the

grade level. Dena reported less of that type of sharing in Building D. It

may be noted, however, that Dena used more everyday objects. She brought

in sugar cubes to represent cubic units and shoe boxes, soup cans and balls

to demonstrate different geometric solids. During her review of

measurement units, her students were able to compare various units to

common objects.

In Building C, most of the materials were kept in the library and

checked out to teachers at all levels. Cathy reported that while there were

large numbers of materials available to be checked out, this was done on a

first come, first serve basis which had been a problem in that materials

she wanted to use were not always there when needed. She was not aware

of any way she could request materials for a specific day and time.

Although a trip to the district Instructional Center (IC) had been part

of the inservice conducted for new teachers at the beginning of the year,

only Cathy had checked out mathematics materials from the IC. She had

gone back one day after school to investigate more specifically what was

available. She stated that the reservation process used at the IC made

obtaining materials easy.

None of the teachers were aware of printed lists that told them

exactly what materials were available within their buildings or the IC.

140

Two of the schools, Buildings C and D, were in the process of

computerizing their inventories according to their principals. Mrs. Barber,

the assistant principal of Building B, shared a printout with the researcher

listing all materials found in the fourth grade area, but the teachers were

unaware of its existence.

Manipulative Usage bv Teachers

A variety of manipulatives were used by each of the first-year

teachers. Appendix G shows a listing of the manipulatives each teacher

reported using. Seven items, the interlocking cubes, tangrams, mirrors,

geoboards, attribute blocks, decimal squares and the abacus, had not been

used at all. Cathy was the only teacher to incorporate pattern blocks and

two-dimensional geometric models in lessons while only Anne had used

number cubes and Dena the calculators. Three items were used by all

teachers: the circular fraction models, the metric rulers arid the meter

stick. Standard measurement tools, thermometers, base ten blocks,

Cuisenaire rods, and other types of fraction models were used by the

majority of the first-year teachers.

The district's curriculum guides listed numeration/place value, whole

number computation, fractions, geometry and measurement as topics to be

taught at both grade levels observed. In addition, the third grade guide

listed a separate unit on time and money. Three teachers, Alice, Beth and

Brenda had not covered geometry when the research period was concluded.

Two additional fourth grade units, graphing and decimals, came at the end

of the year and had not been taught by any of the teachers when the

research project ended. When examining the relationship between the

141

topics taught and the manipulatives used, several interesting points

became apparent.

Manipulatives were used for instruction in all topics (see Appendix H).

Each teacher used at least one manipulative to teach each topic with the

exception of Beth who did not teach numeration/place value with

manipulatives. Her limited experiences in mathematics education and her

lack of familiarity with place value models may have accounted for this.

The teachers all expressed comfort with the manipulatives used to

teach measurement, time and money, and fraction concepts. The same was

not true for the whole number operations, even though the teacher's

editions of the textbooks and the district's curriculum guides presented

these computational concepts and skills using both counters and place

value models. Anne, Cathy and Dena had used both models during

instruction. Alice, Beth, and Brenda felt comfortable presenting the

concepts of addition, subtraction, multiplication and division with

counters, but did not know how to work with place value models such as

base ten blocks.

Advocates of manipulatives stress that the learners themselves must

manipulate the materials in order for ideas to eventually be abstracted

from the concrete to the pictorial, then to the symbolic levels (Cruikshank

& Sheffield, 1988). Data derived from observations in classrooms of the

first-year teachers demonstrated that the teachers placed the

manipulatives in the hands of the students only part of the time.

Brenda used manipulative materials in three of the five lessons

observed. During one class, she modeled the regrouping process with the

Cuisenaire rods for the students during a lesson where they were learning

142

to multiply a one-digit number by a two-digit number. The multiplication

itself was not demonstrated with the place value models, just the

regrouping. The students did not touch the manipulatives at any time. The

students were then shown the standard, abstract algorithm and expected to

work the problems. During a later interview Brenda explained that until

she received the program for a summer workshop, she did not know that

manipulatives could be used to show multiplication or division.

In another lesson Brenda was teaching, the students were to multiply

amounts of money. The teacher's edition of the textbook suggested using

play money to demonstrate the problem as repeated addition. When the

coins were combined, the twenty-one pennies were exchanged for two

dimes and one penny and then the eleven dimes were exchanged for a

$1 -bill and one dime. Again, no students used the coins to help build

concepts for themselves.

During the first observation in Dena's classroom, she used sugar cubes

to construct three-dimensional figures to teach her students about volume.

Several different figures were constructed, but Dena did all of the work

while the students watched. When the students went to their textbooks to

do the assignment, they had problems calculating the correct answers. In

the interview that followed the observation, Dena said that: she had made

an error by not allowing the students to construct the figures themselves.

She planned to reteach the lesson the next day allowing the students to

have twelve sugar cubes each so they could build the objects.

Teacher Perceptions About Manipulatives

All six subjects spoke in favor of the use of manipulatives during

143

mathematics instruction. Cathy, Anne, Alice, and Dena used the phrase

"concrete to abstract" when referring to the sequence in which they knew

they should present new concepts and computational skills. Alice felt her

lack of exposure to techniques with manipulatives in her mathematics

education class hindered her being able to accomplish that task. The other

three, however, felt confident with the materials. All six of the teachers

expressed the desire to incorporate more manipulatives in their

mathematics instruction during their second year of teaching.

Dena, Brenda, Anne, and Beth made references to their own

mathematics backgrounds in the interviews. All four had little or no

experience with manipulatives as students. Brenda expressed the feeling

that students benefited greatly from hands-on opportunities. They all

believed they would have been much better mathematics students

themselves if they had been given a chance to explore mathematical

concepts and skills concretely.

Beth had used relatively few manipulatives until she began the

measurement unit which came toward the end of the year. Virtually all of

the hands-on materials she used during the year were for measurement and

fractions. During her final interview, she commented that she found that

the students not only enjoyed the materials, but some would not have been

able to master certain concepts without them.

Brenda verbally supported the use of manipulatives. During

observations, however, she most often used the materials in

demonstrations rather than allowing the students to handle them. When

teaching division, she expressed much frustration with her students' lack

of mastery of multiplication facts and felt it was due to their laziness

144

rather than any lack of concept mastery. She expressed the belief that the

responsibility to learn rested wholly with the students.

Evaluation of Classroom Use of Manipulatives

The only method of evaluation of instruction involving manipulatives

seen or discussed by the six teachers was observation. Each of the

subjects was seen walking around their classrooms observing students

with manipulatives. Brenda summed up the perceived teacher's role when

she stated, "I just watch and see if they're doing what I was doing. If they

look strange I go over and show them. I just watch their faces and listen

to how they talk to each other."

The two most common places within a lesson cycle that manipulatives

were observed occurred during the introduction of the material and the

students' guided practice. Occasionally the students were allowed to

continue to use the materials to assist them with their independent work.

None of the teachers used manipulatives as part of any formal evaluation

process. Beth cited lack of materials in great enough numbers to be able to

use them to formally evaluate. She also felt like a formal evaluation

setting required silence, and that would not occur if the students were

using manipulatives. Dena explained that she was not certain just how to

incorporate manipulatives into the evaluation process. She hoped to

acquire this skill at the mathematics workshop she had registered to

attend during the summer.

Conclusions

This north Texas school district embraced the assumption espoused by

145

the National Council of Teachers of Mathematics (1989) that physical

materials aided in the acquiring of abstract mathematical concepts and

skills. All administrators verbally expressed that opinion when asked

about their school's or district's mathematics programs. In addition, large

amounts of funds were budgeted over a period of time for the acquisition

of materials and for the continuing education of teachers in the use of

those objects.

The six first-year intermediate grade level teachers understood on a

theoretical level the need for students to use concrete objects when

learning new mathematics skills and concepts. The phrase "concrete to

abstract" was used by over half of the teachers to describe the desired

sequence of content presentation. Several teachers referred to the

developmental levels proposed by Piaget (1963), which were further

supported by the research findings of psychobiologists (Restak, 1982,

1979), when explaining the importance of beginning at the concrete level

with intermediate level students.

They were not always able, however, to put the that theory into

practice. Some of the teachers did an admirable job incorporating the

materials into their curriculum; others were not able to do as well.

Several different factors accounted for this.

Familiarity With Materials

Just as Dossey (1981) had stated in his research findings,

manipulatives were used in all college mathematics education classes.

There was, however a great discrepancy noted in preteaching experiences

with manipulative materials between the subjects. This was one reason

146

first-year teachers may or may not succeed in using manipulatives

effectively in their own classrooms. When one has never seen an object,

much less learned to teach with it, it becomes virtually impossible to

successfully include that object in one's instructional repertoire.

The teachers who had been exposed to the most manipulatives during

their mathematics education programs were the teachers who used the

most manipulatives during their first year of teaching. Cathy, Dena and

Anne not only used more different types of materials than the others, they

also introduced a wider variety of topics with manipulatives than the

other three teachers. Dena provided many noncommercial materials to

represent mathematical concepts on a concrete level.

Availability

There were two aspects of availability noted by the researcher. The

first aspect dealt with physical availability of the manipulative materials.

Since most of the teachers entered virtually empty rooms, the acquisition

of manipulatives played an important part in their use. At the end of the

research period, all teachers had ordered and received at least some

mathematics manipulatives.

When asked to give information about the specific manipulatives found

in their classrooms and building, Alice and Anne reported quite different

numbers (see Appendix F). In both instances, Alice listed fewer

manipulatives than Anne. Alice had been exposed to twenty-one fewer

manipulatives than Anne in her preservice education experience (see

Appendix E). This lack of familiarity with materials may account for

some of the discrepancies. Another factor contributing to the differences

147

found in Building A may have been the method by which materials were

ordered. The teachers were simply given catalogs and asked to fill out

order forms. In Building B, Mrs. Barber, the assistant principal, met with

the new teachers and helped them order their materials. Even though

neither Beth nor Brenda had been exposed to many manipulatives in their

preteaching background as seen in Appendix E, they listed the highest

numbers of manipulatives found both in their classrooms and their building

(see Appendix F). The assistance provided by the building administrator

appeared to make a large difference in material availability for these

first-year teachers.

Cathy showed the least number of materials within her room, but this

did not mean she was not able to obtain manipulatives. With the materials

centrally located in Building C's library, Cathy had ready access to class

sets of many items which the other teachers would have had to borrow

from more than one or two locations in their own buildings.

The perceived availability of the mathematics manipulatives was a

second factor which affected the use of the materials. When the subjects

felt the materials were readily available, they chose to use them more

often than when they were perceived to be less accessible. Three distinct

methods of storing manipulatives existed within the schools examined

which had a direct effect on the perceived availability of those concrete

objects.

Cathy, the one participant who stated that any type of materials could

be found if a person was willing to look, taught in Building C which had a

great many materials centrally located in the school library. Anne and

Alice, the two teachers from Building A, both stated that they were

148

hesitant to use the manipulatives belonging to the grade level because they

were stored in another teacher's room. They did not want to "bother" their

team member. Beth and Brenda of Building B, where materials were kept in

each teacher's room, discussed the sharing of materials within their grade

level. Building D's Dena mainly used the materials found in her room. In

addition, she described many instances where she had used noncommercial

materials as manipulatives.

During the first three interviews, several teachers commented that

they were unaware of which materials were available to them outside

their rooms, grade levels or schools. They did not have access to

inventories which would have given them information about both what was

available and where the manipulatives were housed within the district.

Mrs. Barber, the assistant principal at Building B, had shown the

researcher the inventory for materials in her building, but the teachers had

not had access to the the list. Mrs. Danvers expressed the desire to create

an inventory of materials and ideas for the teachers to share and Dr.

Connors discussed the compilation of a list for her building. Both felt it

would be possible since a computer system had been recently installed in

the library at each building, but neither was completed at the end of the

research period. If the teachers had possession of materials inventories

for their buildings and the Instructional Center, they would have been more

aware of what materials were available and might have been more inclined

to use the manipulatives.

Manipulative Usage in Instruction

When planning lessons, all teachers referred frequently to the

149

teacher's edition of the district adopted textbook. This concurs with part

of Scott's (1983) findings that the textbook was the main resource used by

teachers. Although curriculum guides were available in every room, little

use of them was made for mathematics. The teachers did not take full

advantage of district prepared materials which advocated the use of

manipulatives through suggested activities.

All teachers used manipulative materials at least once when observed.

When data was analyzed from observations, lesson plans, and interviews,

it was concluded that while some used many more manipulatives than

others, they all included manipulatives in their lessons more than the five

times a year average reported by Scott (1983).

During all observed lessons including manipulatives, the materials

either introduced the lesson content itself or were used by students to

practice skills or concepts under teacher direction. Seldom were students

seen using the materials to work on assigned independent practice. When

the manipulatives were used to introduce lesson content, they were not

always made available for the students to handle. Oftentimes the teachers

merely demonstrated the concept, then proceeded to continue teaching for

mastery at the abstract level. Two reasons emerged for this pattern:

teachers were not familiar enough with the process of using the materials

with an entire class or there were not enough materials perceived

available for a whole group of students to handle.

An interesting relationship became evident when the number of

manipulatives seen in mathematics education courses was compared to the

number the teachers reported using during their first year in the

classroom. As shown on Table 9, teachers who had seen more materials in

150

college used more manipulatives in their own classrooms. The three

teachers who had seen the least number of materials in college all

incorporated more into their teaching. The great majority of these

materials were used to teach fractions or measurement. Anne and Dena,

the two who had seen the most manipulatives in college, used the most in

their classrooms. In addition, Dena was observed using many

noncommercial materials to illustrate mathematical concepts.

Another aspect of manipulative usage found interesting by the

researcher was the decisions made by the teachers concerning which

manipulatives best illustrated given concepts. All subjects chose suitable

manipulatives for the following topics: measurement, time and money, and

fractions. The most obvious example where appropriate models were not

used by several teachers was the lack of place value models for building

concepts about whole number operations, particularly multiplication and

Table 9

Comparison of Exposure to Manipulatives with Usage

Number of Manipulatives

Teacher Mathematics Education Reported Usage

Anne 26 19

Dena 25 16

Cathy 15 15

Brenda 8 15

Beth 5 14

Alice 5 10

151

division. Both the district's curriculum guides and the teacher's editions

of the textbooks advocated the use of place value models in the sections on

multiplication and division. These manipulatives were used by most

subjects to demonstrate numeration and place value concepts, but were

seldom incorporated in instruction of multiplication and division.

The teachers themselves demonstrated varying degrees of

understanding about place value models. Anne, Cathy and Dena expressed

comfort concerning their use. Alice knew place value models could be used

to present the concepts, but was unsure how to teach with them. Both

Brenda and Beth expressed being surprised when they had discovered that

multiplication and division could be concretely shown.

Recommendations

Recommendations for the District

To complement the verbal and financial commitment to the use of

manipulative materials for mathematics and science instruction found in

this district, efforts should be made to relay that message to teachers,

particularly those newly employed with the district. This might be

accomplished during the week of inservice for new employees, at the

regular inservice, or during after school minisessions scheduled early in

the year. A variety of topics could be presented which would assist

teachers in the planning and teaching of mathematics. Information

concerning the use of textbooks and curriculum guides as references, the

location and types of manipulative materials available, sample lessons

demonstrating the use of manipulatives with specified topics, and methods

of evaluating manipulative use in mathematics might be included.

152

Since the teachers were not very familiar with the district prepared

curriculum guides which contained many fine teaching ideas and activities,

the district should provide more time introducing the curriculum guides. If

the teachers perceived the guides to be of assistance in planning lessons,

they would be more likely to be used.

With the acquisition of computers in the school libraries, the means by

which inventories can easily be generated exists. These inventories can be

used to make teachers aware of the great numbers of manipulatives within

the building or the district's Instructional Center. Awareness is the first

step needed by teachers when they begin to plan a unit of instruction.

Efforts should be made to place these inventories in teachers' hands,

perhaps as addenda to the curriculum guides or the building handbooks.

As shown in the district-wide inservice needs assessment in Table 8,

the third and fourth grade teachers as a whole within the district felt the

need to learn more about mathematics manipulatives. The third grade

teachers ranked the item seventh out of sixty-four while the fourth grade

teachers voted it third. Therefore the fact that many of the subjects in

the study discussed being unfamiliar with one or more of the

manipulatives presented to them on the checklist was not a surprise.

To assist both new and former teachers, the district could do a needs

assessment to determine the manipulatives and topics with which

teachers felt least familiar. A series of optional minisessions devoted to

the use of those specific topics could then be offered. The classes could

present an overview of the topic, how to effectively introduce the topic

with a variety of manipulatives, and provide assistance to teachers on the

evaluation of activities that enhance the instruction of the material.

153

Since the teachers all follow the same time line established by the

district, the minisessions could be presented shortly before instruction on

a topic was anticipated to begin.

Further research might be conducted at the district level to gain

information about the perceived availability of materials. Three separate

storage systems existed within the buildings: 1) the materials were

located in a central location for all teachers to check out, 2) the materials

were stored in a specified location for the grade level like a teacher's

room, and 3) the materials were distributed evenly between all classes on

a grade level. A study to determine if a difference existed between three

methods by which the manipulatives were stored might prove valuable.

A followup study of the subjects as they continue their teaching

careers over the next two years might be useful to determine their growth.

Since at least three planned to attend mathematics related classes or

workshops during the summer, it would be anticipated that changes in

teaching would occur.

Recommendations for Mathematics Educators

Teacher educators may want to focus research on the content of

mathematics education courses. The impact the subjects' preservice

experiences had on their mathematics teaching was tremendous. The great

disparity noted in various aspects of the college preparation of this small

group might be just an anomaly. However, if this amount of difference

does exist in the preparation of future elementary teachers on a

wide-scale basis, the educational community must know. Issues including

154

topics covered, aspects of manipulative usage, and implementation of the

levels of concept attainment are examples of possible research interest.


Cruikshank, D. E. & Sheffield, L. J. (1988). Teaching mathematics to

elementary school children. Columbus, OH: Merrill Publishing

Company.


teacher-education programs. Arithmetic Teacher. 22(1), 24-26.


in the development of thinking. Social Research. 20, 283-299.


behavior (pp. 159-172). Reston, VA: National Association of

Secondary Principals.


Books.



and Mathematics. £2(1), 61-68.

155

APPENDICES

156

APPENDIX A

LESH'S MODEL FOR MODE TRANSLATIONS

157

158

Lesh's Model for Mode Translations

Static Pictures

y Written Symbols

Manipulative Models

Real Scripts

Spoken Language

APPENDIX B

MATHEMATICS EDUCATION TEXTBOOK COMPARISONS

159

160

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APPENDIX C

TEACHER SUMMARY DATA ON MANIPULATES

161

Anne's Manipulative Summary Data

162

College In In Used Math Ed Classroom Building In Class

Counters Interlocking cubes Individual counters

Base ten blocks Pattern blocks Mirrors x Fraction models

Circles Squares Bars

Measurement tools Rulers - inch Rulers - metric Yardstick Meter stick Scale for student weight Scale for grams Thermometers Clocks Metric volume Standard volume

Play money Geometry models

Two-dimensional Three-dimensional

Tangrams Geoboards Learning links Abacus Attribute blocks Number cubes (dice) Spinners Place value mats Decimal squares Cuisenaire rods Calculators

x x x X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

Column totals: 26 16 21 19

Alice's Manipulative Summary Data

163

Manipulative College In In Used Math Ed Classroom Building In Class


Base ten blocks Pattern blocks Mirrors Fraction models






x x x x

x x x X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

Column totals: 13 17 10

Beth's Manipulative Summary Data

164

College In In Used Math Ed Classroom Building In Qlm








x x x

x x x

x x x x

x x

x x

x x

x x x

x x x

x x X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X


Brenda's Manipulative Summary Data

165









x x x x

x x x

x x x

x x x x

x x

x x

x x x x

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X


Cathy's Manipulative Summary Data

166

Manipulative College In

Math Ed In

Classroom Used

Building In Class Counters

Interlocking cubes Individual counters







x x

x x X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X


Dena's Manipulative Summary Data

167



Base ten blocks Pattern blocks Mirrors x Fraction models






x x x

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

Column totals: 25 12 23 16

APPENDIX D

CURRICULUM GUIDE AND TEXTBOOK MANIPULATIVE LISTS

169

Curriculum Guide and Textbook Manipulative Lists Third Grade

Curriculum Guide Texfoppk

Balance scale Base ten blocks Base ten stamps Beans Birthday candles Centimeter ruler Fahrenheit thermometer Fundamath Geoboards Meter stick Model clock Pattern blocks Play money Pegboard Popscicle sticks Rubber bands Ruler Straws String or yarn Unifix cubes

Abacus Blocks (both 1" & 1 cm) Calculators Calendars Clocks Counters Egg cartons Fraction models Graph paper Magazines Meter & yardsticks Metric volume containers Newspapers Paper cups Paper plates Paper strips Place value models Plane figures Play money Rubber bands Rulers (cm & inch) Scales Sheets of paper Solid geometric figures Standard volume containers Straws Thermometers

170

Curriculum Guide and Textbook Manipulative Lists Fourth Grade

Curriculum Guide Textbook

1" cube blocks Abacus Balance scales Base ten blocks Base ten rubber stamps Bean bags Bingo markers or quiet counters Colored counting cubes Cuisenaire rods Dominoes Flannel board Fraction bars Fraction block kit Fraction disks Fraction models Fundamath Geoboards Geostrips Interlocking cenimeter cubes Interlocking cubes or chains Mirror/Geometric shapes Model containers Number cubes & spinners Paper plate fractions Place value charts Place value mats Play money Rulers Solid geometric shapes Tang rams Thermometers

Abacus Balance scale Calculators Calendars Clocks Counters Crackers Fraction models Graph paper Magazines Metric & Yardsticks Metric volume materials Newspapers Number cubes (dice) Objects for nonstandard measure Objects to group (for

multilication and division) Paper clips Paper squares for area Paper strips Place value charts Place value models Plane figures Play money Rubber bands Rulers (cm & inch) Solid geometric shapes Standard volume materials String Thermometers Unit cubes for volume

APPENDIX E

EXPOSURE TO MANIPULATIVES IN MATHEMATICS

EDUCATION CLASSES

171

Exposure to Manipulatives in Mathematics Education Classes

172

Bwlding JB_ D Manipulative Anne Alice Beth Brenda Cathy Dpna Counters

Interlocking cubes Individual counters







x x x x x

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X


APPENDIX F

MANIPULATIVE AVAILABILITY

173

1 7 4

Manipulative Availability in Classrooms

Building A B C D Manipulative Anne Alice Beth Brenda Cathv Dena Counters

Interlocking cubes X X

Individual counters X X X X

Base ten blocks X X X X

Pattern blocks Mirrors Fraction models

Circles X X X X

Squares X X X

Bars X X X X

Measurement tools Rulers - inch X X X X X X

Rulers - metric X X X X X X

Yardstick X X X X X X

Meter stick X X X X X

Scale for student weight X

Scale for grams X X X X

Thermometers X X X X X

Clocks X X X

Metric volume X X X X

Standard volume X X

Play money X X X

Geometry models Two-dimensional X

Three-dimensional X X

Tangrams X

Geoboards Learning links Abacus Attribute blocks Number cubes (dice) X

Spinners X X

Place value mats X X

Decimal squares X X

Cuisenaire rods X X X

Calculators X

Column totals: 16 13 18 18 4 12

Manipulative Availability in Buildings

175

Buildine A B C D Manipulative Anne Alice Beth Brenda Cathv Dena Counters

Interlocking cubes X X X X

Individual counters X X X X X X

Base ten blocks X X X X X

Pattern blocks X X X

Mirrors Fraction models

Circles X X X X X X

Squares X X X X X

Bars X X X X X X

Measurement tools Rulers - inch X X X X X X


Yardstick X X X X X X

Meter stick X X X X X X

Scale for student weight X X X X X X

Scale for grams X X X X X

Thermometers X X X X X X

Clocks X X X X X

Metric volume X X X X X

Standard volume X X X

Play money X X X X X

Geometry models Two-dimensional X X X

Three-dimensional X X X X

Tangrams X X

Geoboards X X

Learning links X X X

Abacus X X X

Attribute blocks X X

Number cubes (dice) X X X X X

Spinners X X X

Place value mats X X X X

Decimal squares X X

Cuisenaire rods X X X X X X

Calculators X

Column totals: 21 17 24 25 24 23

APPENDIX G

MANIPULATIVES USED THROUGHOUT THE YEAR

176

Manipulatives Used Throughout the Year

177

Building A B C D Manipulative Anne Alice Beth Bnenda Cathv Dena Counters

Interlocking cubes Individual counters X X X

Base ten blocks X X X X

Pattern blocks X

Mirrors Fraction models

Circles X X X X X X

Squares X X X X X

Bars X X X X X X

Measurement tools Rulers - inch X X X X X


Yardstick X X X X X

Meter stick X X X X X X

Scale for student weight X X

Scale for grams X X X X

Thermometers X X X X X

Clocks X X X X

Metric volume X X X X X

Standard volume X X X X

Play money X X X

Geometry models Two-dimensional X

Three-dimensional X X

Tangrams Geoboards Learning links X X

Abacus Attribute blocks Number cubes (dice) X

Spinners X X

Place value mats X X

Decimal squares Cuisenaire rods X X X X

Calculators X

Column totals: 19 10 14 15 15 16

APPENDIX H

MANIPULATIVES CHOSEN FOR SPECIFIC CONTENT

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Documents

37?/67531/metadc331046/...use in grades K-2 and 3-5 with similar results (Kloosterman, P. & Harty, HI., 1987). The teachers at the K-2 level reported more frequent use of manipulatives