Algebra Instruction Research Articles

Online Algebraic Tools for Teaching

Many free online tools exist to complement algebraic instruction at the middle school level. This article presents findings that analyzed the features of algebraic tools to support learning. The findings can help teachers select appropriate tools to facilitate specific topics.

Twin Texts and Technology: Enhancing Algebra Instruction for Elementary Teachers

Twin Texts and Technology: Enhancing Algebra Instruction for Elementary Teachers

Strategies for Teaching Algebra to Students With Learning Disabilities: Making Research to Practice Connections

To improve student success in mathematics, the use of research-based interventions is necessary to help secondary students with learning disabilities (LD) access the algebra curriculum. The authors provide an overview of the following research-based approaches: explicit instruction, graduated instructional sequence, technology, and graphic organizers. For each approach, a summary of the research is provided along with suggestions for how the approach can be used by classroom teachers to support algebra instruction.

Prospective teachers’ views on the use of calculators with Computer Algebra System in algebra instruction

Although growing numbers of secondary school mathematics teachers and students use calculators to study graphs, they mainly rely on paper-and-pencil when manipulating algebraic symbols. However, the Computer Algebra Systems (CAS) on computers or handheld calculators create new possibilities for teaching and learning algebraic manipulation. This study investigated the views of Turkish prospective secondary mathematics teachers on the use of advanced calculators with CAS in algebra instruction. An open-ended questionnaire and group interviews revealed prospective teachers’ views and beliefs about when and why they prefer three possible uses of CAS—black box, white box, or Symbolic Math Guide (SMG). The results showed that participants mainly preferred the white box methods and especially SMG to the black box method. They suggested that while the black box method could be used after students mastered the skills, the general white box method and SMG could be used to teach symbolic manipulation.

Graphic Organizers Applied to Secondary Algebra Instruction for Students with Learning Disorders

Students who have particular difficulty in mathematics are a growing concern for educators. Graphic organizers have been shown to improve reading comprehension and may be applied to upper level secondary mathematics content. In two systematic replications, one randomly assigned group was taught to solve systems of linear equations through direct instruction and strategy instruction. The other group was taught with the same methods with the addition of a graphic organizer. Students who received instruction with the graphic organizers outperformed those who received instruction without the organizers. They also better understood the related concepts as measured by immediate posttests in both replications. The difference in understanding concepts was maintained on a 2–3 week posttest.

The case of bruce: A teacher’s model of his students’ algebraic thinking about equivalent expressions

The purpose of this article is to describe a middle school mathematics teacher’s model of his students’ responses to algebraic tasks involving equivalent expressions and the distributive property. The teacher engaged in two model-eliciting activities designed for teachers by creating a library of his students’ work and an accompanying “Ways of Thinking”[WOT] sheet (Doerr & Lesh, 2003). These activities were designed to help reveal the teachers’ models of students’ algebraic thinking and to promote the development of that model. Results of the analysis showed that the teacher developed a clearer understanding of the role of a variable in algebraic instruction. The teacher employed visual strategies for the first time and began to perceive their usefulness in helping students understand the equivalence of two expressions.

Teaching Algebra to Students with Learning Disabilities

In many states, students with learning disabilities are required to take algebra in high school. These students are usually served in inclusive settings (the general education classroom), where they often struggle with algebra because the content is so abstract. Although limited research and literature are available on algebra instruction for students with special needs, recognized strategies that promote students' learning can help make the algebra experience more enjoyable and successful for them and for the other students in the class, as well. This article highlights characteristics of students with learning disabilities, reviews current literature on algebra and students with learning disabilities, summarizes some of the recommendations, and describes how they can be put into practice.

The MATCH Program: A Preparatory Chemistry and Intermediate Algebra Curriculum

Since 1994 we have taught a program that combines instruction in intermediate algebra and preparatory chemistry. These are two critical courses for students interested in science, mathematics, engineering, and the health professions. Students receive instruction that coordinates the order and treatment of topics between the two segments. They also receive their instruction in the same classrooms, a tactic that, along with cooperative learning methods in the discussion classroom, fosters strong peer-to-peer connections. The program succeeds, especially in preparing students for general chemistry courses.

Teachers' and Researchers' Beliefs about the Development of Algebraic Reasoning

Mathematics teachers and educational researchers ordered arithmetic and algebra problems according to their predicted problem-solving difficulty for students. Predictions deviated systematically from algebra students’ performances but closely matched a view implicit in textbooks. Analysis of students’ problem-solving strategies indicates specific ways that students’ algebraic reasoning differs from that predicted by most teachers and researchers in the sample and portrayed in common textbooks. The Symbol Precedence Model of development of algebraic reasoning, in which symbolic problem solving precedes verbal problem solving and arithmetic skills strictly precede algebraic skills, was contrasted with the Verbal Precedence Model of development, which provided a better quantitative fit of students’ performance data. Implications of the findings for student and teacher cognition and for algebra instruction are discussed.

Grade 6 Students' Preinstructional Use of Equations to Describe and Represent Problem Situations

The purpose of this study was to investigate 6th-grade students' use of equations to describe and represent problem situations prior to formal instruction in algebra. Ten students were presented with a series of similar tasks in 6 different problem contexts representing linear and nonlinear situations. The students in this study showed a remarkable ability to generalize the problem situations and to write equations using variables, often in nonstandard form. Although students were often able to write equations, they rarely used their equations to solve related problems. We describe students' preinstructional uses of equations to generalize problem situations and raise questions about the most appropriate curriculum for building on students' intuitive knowledge of algebra.

Introductory algebra: Four approaches or one?

This edited volume is a worthy attempt to draw together recent debate and research regarding the traditionally difficult problem of presenting algebra to those not enculturated into the world of algebraic process and thought. The early "errors" orientation of research into children's ability to work in an algebraic mode, undertaken by writers such as Kiichemann (1981) and Booth (1984), seemed to indicate that the focus on decontextualised operations led to many misconceptions and an unwillingness or inability to transfer the skills to problem­ solving situations. Since then many attempts have been made to identify alternative approaches to algebraic instruction with the aim of widening the availability of algebraic skills in the community. This book is an attempt to synthesise the progress thus far. Those readers who have followed the work of the Algebra special interest group at the international conferences of the International Group for the Psychology of Mathematics Education (PME) in the 1990s would be familiar with most of the writers and the nature of the debate, if not with some of

Algebra Instruction for Students with Learning Disabilities: Implications from a Research Review

A review of six published studies on algebra interventions for students with learning disabilities (LD) revealed that successful interventions included instruction on domain-specific knowledge, as well as general problem-solving and self-regulation strategies. Promising practices include sequential introduction of word problems through concrete, semi-concrete, and abstract phases, and peer feedback. Use of selected computer-assisted programs also may hold promise. These findings are discussed in the context of the requirements of successful algebra performance, developmental problem-solving literature (Mayer, 1989; Schoenfeld, 1992), and characteristics of students with LD.

SOFTWARE ACTIVITIES FOR LINEAR ALGEBRA: CONCEPTS AND CAVEATS

ABSTRACT This article describes some uses of software for undergraduate linear algebra. The examples illustrate some innovations that enhance linear algebra instruction using geometric visualization. In addition, we present some seemingly benign examples that lead to interesting caveats. These caveats encourage the student to be mindful of limitations of software for numerical and symbolic calculations.

A Comparison of the Effectiveness of Videodisc and Traditional Algebra Instruction for College-age Students with Learning Disabilities

The purpose of this study was to compare the effectiveness of a videodisc program, mastering equations, roots and exponents (system impact, inc., 1989) with a traditional text-based program. this study was conducted as part of a transition program during the summer before these students first entered college. the control group (n = 13) received instruction from a typical intermediate-level college algebra text, whereas the treatment group (n = 13) received instruction from the mastering equations, roots and exponents videodisc program. data analysis indicated that the treatment group receiving videodisc instruction significantly outperformed the control group on two posttest measures of algebra skill. the videodisc-instructed group also earned significantly higher grades in their first algebra class in the fall semester following the treatment phase of the study. the authors attributed the effectiveness of the videodisc instruction to the proven elements of effective instruction embedded in the program's lesson presentation format and supporting materials.

Computer‐assisted instruction and the library: A case study—in disappointment

A project was undertaken to determine the appropriateness of providing subject‐based courseware in an academic library's software center or microcomputer lab. The courseware was intended to provide remedial instructional support to re‐entry students in selected subjects. For this project, college algebra became the chosen subject because there appeared to be widespread local agreement that a number of adult students needed remedial instruction in college algebra. The question of the appropriateness of CAI in the library remains open. This service seems to be a viable one for academic libraries to offer. Success would be dependent on wide ranging cooperation involving the library, teaching faculty, computing staff, and instructional technologists.

Research issues in the learning and teaching of algebra

Contents: C. Kieran, S. Wagner, The Research Agenda Conference on Algebra: Background and Issues. Part I:Past Research and Current Issues. J.A. Thorpe, Algebra: What Should We Teach and How Should We Teach It? J.R. Leitzel, Critical Considerations for the Future of Algebra Instruction. C. Kieran, The Early Learning of Algebra: A Structural Perspective. L.R. Booth, A Question of Structure. N. Herscovics, Cognitive Obstacles Encountered in the Learning of Algebra. D. Tall, Different Cognitive Obstacles in a Technological Paradigm. S. Chaiklin, Cognitive Studies of Algebra Problem Solving and Learning. R.B. Davis, Three Ways of Improving Cognitive Studies in Algebra. J.H. Larkin, Robust Performance in Algebra: The Role of the Problem Representation. P.W. Thompson, Artificial Intelligence, Advanced Technology, and Learning and Teaching Algebra. M.W. Lewis, Intelligent Tutoring Systems: First Steps and Future Directions. J.J. Kaput, Linking Representations in the Symbol Systems of Algebra. D. Kirshner, Critical Issues in Current Representation System Theory. J.T. Fey, School Algebra for the Year 2000. S.L. Senk, Toward School Algebra in the Year 2000. Part II:A Research Agenda. S. Wagner, C. Kieran, An Agenda for Research on the Learning and Teaching of Algebra. L.R. Booth, The Research Agenda in Algebra: A Mathematics Education Perspective. M.W. Lewis, The Research Agenda in Algebra: A Cognitive Science Perspective. S.L. Rachlin, The Research Agenda in Algebra: A Curriculum Development Perspective. Part III:Theoretical Considerations. R.B. Davis, Research Studies in How Humans Think About Algebra. J.H. Larkin, Eight Reasons for Explicit Theories in Mathematics Education. D. Wheeler, Contexts for Research on the Teaching and Learning of Algebra.

A Way to Individualize Instruction in First-Year Algebra

Here is a tried and proved variation on individualized instruction.

Main Features of the New Mathematics Syllabus for 1st Grade

After September 1, 1969, all 1st grades in schools of the RSFSR will be using a new mathematics syllabus. This syllabus maintains a certain continuity with the old one, but quite a number of problems are treated in a fundamentally new way. The basic core of the course specified for grades 1 to 3 remains arithmetic, but elements of geometry and preliminary instruction in algebra are integrated around this core. While continuing to devote the most serious attention to the task of developing conscious and lasting computational skills in pupils, the syllabus orients the teacher, in the study of numeration within the ranges of 10 and 100, toward acquainting children with several characteristics of the natural order of numbers and, in the study of addition and subtraction, with the characteristics of arithmetical operations and with the connections between these operations.

Experimental Programs: Programmed instruction in ninth-grade algebra

There is little doubt that the ideal learning situation is the one where there is one teacher for each student. Although this ideal is obviously impossible in today's schools, it is evident that educators are aware that individual differences among students do exist. Ability grouping, team teaching, special seminars, accelerated and enriched programs, and large-scale guidance plans are among the current attempts to meet the needs of the individual learner.

Individualized Instruction in Algebra

It has been said, time after time, that the most neglected pupil in our high school is the brilliant boy or girl. There seems to be a leveling down of the classwork to meet the “tempo” of the average child and partially, if not wholly, neglect the superior students upon whom our future leadership will depend.