دراسات علمية حول اليدويات

Abstract: The purpose of this study was to describe the effects of using multiple forms of representation (experienced-based 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 four-week teaching experiment was conducted with twenty-nine students in a pre-algebra 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.

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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

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Title: AN INVESTIGATION OF TWO INSTRUCTIONAL METHODS FOR

TEACHING SELECTED PRE-ALGEBRA CONCEPTS TO MINORITY

AT-RISK

SEVENTH-GRADE 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 pre-algebra 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 mid-western 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 post-test on the fifteenth day. The pre-algebra 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 pre-test/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.

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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 post-secondary 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. Pre-tests and Post-tests in the content areas of polynomial addition, polynomial multiplication, and polynomial factoring were administered. Pre-tests and post-tests 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 quasi-experimental 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 (pre-test, post-test). 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.

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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 geometric-rich and algebraic-rich topics.

Four topics were selected from a developmental algebra course, two geometric-rich and two algebraic-rich. 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 geometric-rich 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.

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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 pre-measure, 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 college-wide analysis and for one campus. The analysis by individual instructor was not significant. In addition, the college-wide 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 below-average, 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 investigator-designed 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 gender-related 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 age-related differences favored the 12-year-olds in the equation-solving topic, 11-year-olds performed almost as well as 12-year-olds 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 manipulative-based 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 quasi-experimental study, equivalency of treatment groups was determined by independent samples t-tests on pre-test scores. Analysis of variance on pre-and post-tests with treatment,instructor, and past algebra experience as factors and pre-test and post-test as repeated factors was used to analyze differences in achievement, attitude, and anxiety. To measure retention, the post-test 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 manipulative-based group experienced greater retention than did students with no recent experience in the control group. The mean achievement gain of all no-recent-experience students was significantly higher than the mean gain of recent-experience students (t = −4.884,df = 117,and p < .001).Additionally, the mean attitude of all no-recent-experience students was better than that of recent-experience students (p < .001). There was a significant time by treatment group interaction for attitude ( p = .034). At pre-test 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 post-secondary 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. Pre-tests and Post-tests in the content areas of polynomial addition, polynomial multiplication, and polynomial factoring were administered. Pre-tests and post-tests in the form of two different attitude surveys were also administered.

The attitude surveys used were the Dutton Test and the Semantic Differential 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.