Database: Dissertations
Title: The effects of using multiple
representations on tudents' knowledge and perspectives of basic
algebraic concepts
Author(s): Hail, Christopher Jason
Degree: Ed.D.
Year: 2000
Pages: 00303
Institution:
University of Kentucky; 0102
Advisor: Director
William S. Bush
Source: DAI, 61, no.
07A (2000): p. 2636
Standard No: ISBN:
0599870028
Abstract: The purpose of this study was to
describe the effects of using multiple forms of representation
(experiencedbased scripts, spoken language, manipulatives,
graphs, tables, and written symbols) on students' understanding
and perspectives of basic algebraic concepts. Specifically, the
study attempted to answer the following questions:
(1) Do students'
knowledge and perspectives of variables and equations change
during introductory algebra lessons? If so, how?
(2) Do multiple
representations aid students in developing knowledge and
perspectives of introductory algebraic concepts? If so, how?
(3) Do students'
perspectives of functions change during introductory algebra
lessons? If so, how?
(4) Do students prefer
one form of representation over others when solving algebraic
problems? If so, why?
(5)
Are students able to move flexibly among
representations and perspectives when solving problems? If not,
why not?
A fourweek teaching experiment was conducted
with twentynine students in a prealgebra class. Multiple forms
of representation were used to introduce algebra from a
structural functional approach.
The purpose was to help students understand
variables, equations, and Solving equations. Students were
expected to learn to view functions as operations (operational
perspective) and as objects (structural perspective).
The findings showed that students used graphs
and manipulatives to attach meaning to operations on symbols.
Students also used these representations to explain symbolic
procedures and errors. The manipulatives also helped students
learn to solve equations. Graphs helped students view variables
as more than an abbreviation and as representing a range of
variables. Finally, graphs and manipulatives helped students
view the equal sign as a comparison symbol.
Students preferred working with graphs by the
end of the study, although many students did not develop
understanding of the graphical representation. Five students
developed a structural perspective of functions and only two
students demonstrated flexibility in solving problems. Some
students, however, could change perspectives and representations
in order to solve problems very early in their algebra
experience.

Title:THE EFFECTS OF USING CUISENAIRE RODS ON
THE MATH ACHIEVEMENT OF SECOND GRADE STUDENTS
Author(s): EGAN, DONNA LEE
Degree: ED.SPEC.
Year: 1990
Pages: 00049
Institution: CENTRAL MISSOURI STATE UNIVERSITY;
0958
Advisor: Chair: SHARON LAMSON
Source: MAI, 29, no. 03, (1990): 0365
Abstract: The purpose of this study was to
determine the effects of using Cuisenaire rods on the math
achievement of second grade students in the Marshall Public
Schools. This research was designed to show findings that would
help administrators and educators determine the effectiveness of
using Cuisenaire rods with second grade students. The
introduction stresses the importance of using an appropriate
curriculum for young children and summarizes difficulties
students and teachers face when using a manipulative curriculum.
A review of the literature showed the importance of making math
meaningful by using math manipulatives. Researchers agreed that
the primary focus of the curriculum should be the development of
mathematical understandings and relationships. The control group
was instructed by using a traditional math textbook. The
experimental group was instructed using Cuisenaire rods. A
chapter on methodology provides a description of the Cuisenaire
program. Results and recommendations for further study are
provided. The results of the study indicate that using
Cuisenaire rods for math instruction was no more effective than
teaching math with the use of a traditional textbook

Title: AN INVESTIGATION OF TWO INSTRUCTIONAL
METHODS FOR
TEACHING SELECTED PREALGEBRA CONCEPTS TO
MINORITY
ATRISK
SEVENTHGRADE MATHEMATICS STUDENTS
Author(s): RODGERS, CYRUS EUGENE
Degree: ED.D.
Year: 1995
Pages: 00101
Institution:
UNIVERSITY OF MISSOURI  SAINT LOUIS; 0481
Source: DAI, 56, no. 08A, (1995): 3042
Abstract: This research investigates two
instructional methods for teaching prealgebra concepts to
minority seventh grade mathematics students. The research
utilizing the two methods termed concept based instruction and
symbolic instruction took place in a suburban middle school
located just outside a large midwestern metropolitan city.
Subjects for the investigation were divided into three groups
and comprised two categories of students, traditional algebra
students and non traditional algebra students. Subjects in the
traditional algebra group were those students whose standardized
mathematics test scores on the Missouri Mastery and Achievement
Test, were in the top three percent. Non traditional algebra
students' scores were in average range, but not remedial, as
defined by the school districts' standards. Each of the three
groups, denoted as Accelerated Symbolic Instruction (ASI),
Concept Based Instruction (CBI), and Symbolic Instruction (SI),
were administered a pretest followed by fourteen days of
instruction involving the same content material and utilizing
the method assigned to its group. All groups were administered a
posttest on the fifteenth day. The prealgebra concepts studied
were limited to perimeter, area, and volume of solids. Assessing
these two methods supported the hypothesis that instruction
utilizing manipulatives and a concept based approach is a
superior method of instruction.
The SI treatment consisted of symbolic
instruction, predominately symbol manipulation, with an emphasis
on algorithmic approaches. The ASI (traditional algebra
students) group received the same type of instruction covering
the same content in normal classroom situations on an
accelerated basis. The CBI group covering the same content,
received instruction utilizing manipulatives, discourse, and a
concept based approach. All groups utilized journal writing as a
part of the instruction and were administered pretest/posttest
attitude surveys.
Each of five
dependent measures were utilized to assess the outcomes and
included scores on content tests of perimeter, area and volume
as well as mathematics attitude scores and journal rubric
results. The results of this investigation showed that as
hypothesized, the Concept Based Instruction resulted in
significantly higher achievement in all areas except volume of
solids.

Title: AN INVESTIGATION OF THE USE OF THE
ALGEBRA MANIPULATIVES WITH COMMUNITY COLLEGE STUDENTS
Author(s): DYER, LAURA A.
Degree: ED.D.
Year: 1996
Pages: 00130
Institution: UNIVERSITY OF MISSOURI  SAINT
LOUIS; 0481
Source: DAI, 57, no. 05A, (1996): 1985
Abstract : This investigation focused on the use
of an algebra manipulative called algebra tiles in the community
College mathematics classroom. Manipulatives are regularly used
in the elementary and secondary level mathematics classrooms and
the argument is made that manipulatives have a role in the
postsecondary classroom as well.
Ninety students at Belleville Area College
served as subjects. These students were registered in one of
four Intermediate Algebra courses. Two of the courses were
randomly selected to be taught sing algebra manipulatives. The
remaining two courses were taught using traditional symbolic
instruction. Pretests and Posttests in the content areas of
polynomial addition, polynomial multiplication, and polynomial
factoring were administered. Pretests and posttests in the
form of two different attitude surveys were also administered.
The attitude surveys used were the Dutton Test
and the Semantic Differential Test.
A quasiexperimental research design was used.
The statistical analysis was a three factor Repeated Measures
Analysis of Variance. The factors were treatment condition
(treatment, control), instructor (A,B) and time of testing
(pretest, posttest). All interactions and main effects were
analyzed.
Results of the analysis revealed significant
differences in the mean performances between the students in the
manipulative instruction classroom and the students in the
traditional symbolic instruction classroom on the polynomial
multiplication content test.
Content learning of polynomial multiplication
increased significantly for community college students who
received manipulative instruction.
_______________________________________________________________
Title: EXPERIENTIAL LEARNING OF ALGEBRA: USING
MANIPULATIVES IN
A COLLEGE DEVELOPMENTAL ALGEBRA PROGRAM
Author(s): KINARD, AMELIA SPETH
Degree: PH.D.
Year: 1996
Pages: 00141
Institution: UNIVERSITY OF SOUTH CAROLINA; 0202
Advisor: Major Professor: JACK M. OTT
Source: DAI, 57, no. 11A, (1996): 4678
Abstract: Experiential learning in mathematics
has been advocated
at all grade levels by the National Council of
Teachers of Mathematics since 1989. Mathematics philosophers and
theorists have touted its importance since the times of the
early Greeks. This study was conducted to examine the effects of
manipulatives in the teaching of developmental algebra on
college students. In particular the study concentrated on the
use of algebraic manipulatives in the teaching and learning of
geometricrich and algebraicrich topics.
Four topics were selected from a developmental
algebra course, two geometricrich and two algebraicrich. The
control group was taught algebra as an analogy to arithmetic and
the experimental group was taught using manipulatives. At the
end of each of the four lessons, students in both groups were
given a one hour quiz consisting of twenty questions which
tested the material covered. At the end of the course in which
the control and experimental groups participated, the students
were each given a comprehensive final examination. Analysis of
student data showed no significant differences between the
classes receiving instruction with the manipulatives and those
that did not.
No significant differences between treatment
groups and no significant interaction between treatment groups
and achievement were found. Although the data showed no
differences, there is anecdotal evidence that manipulatives did
aid conceptualization in the two geometricrich topics. The
researcher believes that there were many factors that influenced
these results. Conclusions and recommendations for further study
are included in the final chapter.

Title: EFFECTS OF USING MANIPULATIVE MATERIALS
TO TEACH REMEDIAL
ALGEBRA TO COMMUNITY COLLEGE STUDENTS ON
ACHIEVEMENT AND
ATTITUDES
TOWARDS MATHEMATICS
Author(s): MARTELLY, DIANA I.
Degree: ED.D.
Year: 1998
Pages: 00209
Institution: FLORIDA INTERNATIONAL UNIVERSITY;
1023
Advisor: Major
Professor: BARRY GREENBERG
Source: DAI, 59, no.
03A, (1998): 0706
Abstract: This dissertation derived hypotheses
from the theories of Piaget, Bruner and Dienes regarding the
effects of using Algebra Tiles and other manipulative materials
to teach remedial algebra to community college students. The
dependent variables measured were achievement and attitude
towards mathematics. The Piagetian cognitive level of the
students in the study was measured and used as a concomitant
factor in the study.
The population for the study was comprised of
remedial algebra students at a large urban community college.
The sample for the study consisted of 253 students enrolled in
10 sections of remedial algebra at three of the six campuses of
the college. Pretests included administration of an achievement
premeasure, Aiken's Mathematics Attitude Inventory (MAI),and
the Group Assessment of Logical Thinking (GALT).Posttest
measures included a course final exam and a second
administration of the MAI.
The results of the GALT test revealed that 161
students (63.6%) were concrete operational, 65 (25.7%) were
transitional, and 27 (10.7%)
were formal operational. For the purpose of
analyzing the data, the transitional and formal operational
students were grouped together.
Univariate factorial analyses of covariance
($„alpha$ =.05) were performed on the posttest of achievement
(covariate =achievement pretest) and the MAI posttest (covariate
= MAI pretest).
The factors used in the analysis were method of
teaching (manipulative
vs. traditional) and cognitive level (concrete operational vs.
transitional/formal operational).
The analyses for achievement revealed a
significant difference in favor of the manipulatives groups in
the computations by campus. Significant differences were not
noted in the analysis by individual instructors.
The results for attitude towards mathematics
showed a significant difference in favor of the manipulatives
groups for the collegewide analysis and for one campus. The
analysis by individual instructor was not significant. In
addition, the collegewide analysis
was significant in favor of the
transitional/formal operational stage of cognitive development.
However, support for this conclusion was not obtained in the
analyses by campus or individual instructor.
Title: The use of manipulatives in middle school
algebra: An application of Dienes' variability principles
Author(s): Gningue, Serigne Mbaye
Degree: Ph.D.
Year: 2000
Pages: 00290
Institution: Columbia University; 0054
Advisor: Sponsor Bruce Vogeli
Source: DAI, 60, no. 12A (2000): p. 4356
Abstract: The purpose of this study was to
determine the effects of the use of Dienes' variability
principles implemented With manipulatives on middle school
students' ability to perform the algebraic processes of
(1) simplifying
algebraic expressions,
(2) solving linear
equations,
(3) multiplying
algebraic expressions, primarily binomials, and
(4) identifying
multiple representations of a linear function. The differences
in the effects that are related to factors including ability
level, achievement in prior mathematics courses, age, and gender
were studied. Interviews with selected students were also
conducted.
The subjects, 53 sixth graders who completed
only the two units on algebraic expressions and equations, and
53 seventh graders who did all four topics of the study, were
aged 11 and 12. The standardized test results from school
records were used to measure the mathematics ability of each
student prior to the experiment, and to classify each subject as
being belowaverage, low, medium or high in mathematics ability.
The investigator identified two “perceptual
variates” and a different number of “mathematical
variates” for each topic. “Perceptual variates”
simply represented the materials used in the experiment, while “mathematical
variates” described the many irrelevant attributes whose
variations do not change the general mathematical concept. A
preliminary unit on integers was also implemented with all
students so they could form mental representations they used
later when studying the chosen units. An investigatordesigned
test administered after each topic, had questions classified
each as belonging to a certain difficulty level determined by
the number of mathematical variates identified in the question.
Results showed that the application of Dienes'
variability principles for all four topics was successful for
almost all students. The study found no genderrelated
differences in either age group. In both groups however, there
were significant differences favoring the high achievement
groups, and none between the medium and low groups. While
agerelated differences favored the 12yearolds in the
equationsolving topic, 11yearolds performed almost as well as
12yearolds in the algebraic expression topic. In general,
performance on all topics was described as satisfying,
considering that, questions students solved in the study, were
usually well beyond middle school expectations.
Database: Dissertations
Title: An investigation of the use of algebra
tiles to introduce
the basic algebraic concepts of integers,
equations, and
polynomials in community college developmental
algebra classes
Author(s): Dell'Isola, Ida McCalip
Degree: Ph.D.
Year: 1999
Pages: 00123
Institution: The University of Tennessee; 0226
Advisor: Major Professor John R. Ray
Source: DAI, 61, no. 01A (1999): p. 119
Abstract: The purpose of this study was to
examine the effectiveness of using activities involving algebra
tiles to introduce algebra concepts. The study was conducted
with 119students enrolled in six elementary algebra classes at
Walters State Community College. Three instructors each taught a
control and a manipulativebased class. The study was designed
to compare the mathematical achievement, retention, anxiety, and
attitude of students based on treatment groups and past algebra
experience groups.
In this quasiexperimental study, equivalency of
treatment groups was determined by independent samples ttests
on pretest scores. Analysis of variance on preand posttests
with treatment,instructor, and past algebra experience as
factors and pretest and posttest as repeated factors was used
to analyze differences in achievement, attitude, and anxiety. To
measure retention, the posttest was used as a covariate in an
analysis of variance on the final exam.
The most significant finding (p = .045) was that
students with no recent experience in the manipulativebased
group experienced greater retention than did students with no
recent experience in the control group. The mean achievement
gain of all norecentexperience students was significantly
higher than the mean gain of recentexperience students (t =
−4.884,df = 117,and p < .001).Additionally, the mean
attitude of all norecentexperience students was better than
that of recentexperience students (p < .001). There was a
significant time by treatment group interaction for attitude ( p
= .034). At
pretest the control group attitude score appeared to be higher
than the manipulative group score. Over
time a trend appeared to be developing with the manipulative
group's attitude improving so that the gap was narrowing. There
was no significant difference in the anxiety of students in the
treatment and control groups. Both groups experienced a
significant lessening of anxiety.
The researcher believes it is significant that
the previous experience group with the better attitude
experienced a significant difference in retention when
instructed with manipulatives. With the attitude of the
manipulative group showing a trend toward improvement, there is
reason to believe that the treatment might have produced greater
differences in a longer study.
Database: Dissertations
Title: AN INVESTIGATION OF THE USE OF THE
ALGEBRA MANIPULATIVES
WITH COMMUNITY COLLEGE STUDENTS
Author(s): DYER, LAURA A.
Degree: ED.D.
Year: 1996
Pages: 00130
Institution: UNIVERSITY OF MISSOURI  SAINT
LOUIS; 0481
Source: DAI, 57, no. 05A, (1996): 1985
Abstract: This investigation focused on the use
of an algebra manipulative called algebra tiles in the community
college mathematics classroom. Manipulatives are
regularly used in the elementary and secondary level mathematics
classrooms and the argument is made that manipulatives have a
role in the postsecondary classroom as well.
Ninety students at Belleville Area College
served as subjects. These students were registered in one of
four Intermediate Algebra courses. Two of the courses were
randomly selected to be aught using algebra manipulatives. The
remaining two courses were taught using traditional symbolic
instruction. Pretests and Posttests in the content areas of
polynomial addition, polynomial multiplication, and polynomial
factoring were administered. Pretests and posttests in the
form of two different attitude surveys were also administered.
The attitude surveys used were the Dutton Test
and the Semantic Differential Test.
A quasiexperimental research design was used.
The statistical analysis was a three factor Repeated Measures
Analysis of Variance. The factors were treatment condition
(treatment, control), instructor (A,B) and time of testing
(pretest, posttest). All interactions and main effects were
analyzed.
Results of the analysis revealed significant
differences in the mean performances between the students in the
manipulative instruction classroom and the students in the
traditional symbolic instruction classroom on the polynomial
multiplication content test.
Content learning of polynomial multiplication
increased significantly for community college students who
received manipulative instruction
Database: Dissertations
Title: THE EFFECT OF ALGEBRA TILE USE ON THE
POLYNOMIAL
FACTORING ABILITY OF ALGEBRA I STUDENTS
Author(s): GOLDSBY,
DIANNE SIMPSON
Degree: PH.D.
Year: 1994
Pages: 00236
Institution:
UNIVERSITY OF NEW ORLEANS; 0108
Source: DAI, 57, no. 08A, (1994): 3434
Abstract: This study investigated the
differential effect of manipulative use in the form of algebra
tiles with
teacher explanation on the polynomial factoring
ability of Algebra I students and the traditional method of
teacher explanation alone. The study involved 247 students in 6
schools in a suburban public school system. The data, using a
multivariate analysis of covariance yielded a significant
difference in the posttest, total facets competency, and
proficiency in factoring scores when considered together. The
univariate tests indicated statistically significant differences
in the adjusted means for the Facets total and Proficiency
scores, with the manipulative group having higher adjusted
means. On the six facets of factoring, the MANCOVA showed
statistical significance and the ANCOVAS yielded statistically
significant differences for 4 of the 6 facets. These facets
dealt with common factors, the correct selection and placement
of signs, the exponents, and the application to equations.
Instruction using algebra tiles on a Algebra I factoring of
polynomials unit resulted in higher scores than traditional
instruction using teacher explanation alone on these facets. The
use of manipulatives resulted in higher scores than teacher
explanation alone across grades nine and ten and achievement
levels.
