Algebraic Skills Research Articles

Combined development of middle school children's conceptual understanding in momentum conservation, procedural skills and epistemological awareness in a constructionist learning environment

1D collisions are part of the curriculum for 15–16 year olds in the educational system of Cyprus. The curriculum activities in this domain commonly rely on a mathematical approach and are constrained by prerequisite algebraic skills. Conservation laws have the broadest possible application of all laws in physics and are thus considered by many scientists to be the most fundamental laws in our representations of nature. However, they are currently presented in the educational system in a very didactic manner with the result that students identify them as meaningless algorithms to be implemented in working out a solution as to the outcome of a collision. In this paper, we focus on the design of inquiry based curriculum that aims to help 13 and 14 year olds to foster conceptual understanding and modelling skills in an integrated manner in the domain of 1D collisions. Through the model-based curriculum students are also expected to appreciate the important role of conservation laws in science as a fundamental principle in analysing process end states. We will also present some preliminary pre-test results related to our threefold objective.

Short comments on technical note—The EOQ and EPQ models with shortages derived without derivatives

We study the paper of Ronald, Yang and Chu that was recently published in the International Journal of Production Economics to find the optimal solution for the EOQ and EPQ models without derivatives. We will offer a simple algebraic method to replace their sophisticated algebraic skill, and then we point out a possible direction for further research.

Visual Salience of Algebraic Transformations

Information processing researchers have assumed that algebra symbol skills depend on mastery of the abstract rules presented in the curriculum (Matz, 1980; Sleeman, 1986). Thus, students' ubiquitous algebra errors have been taken as indicating the need to embed algebra in rich contextual settings (Kaput, 1995; National Council of Teachers of Mathematics [NCTM] Algebra Working Group, 1998). This study explored a nonrepresentational account of symbolic algebra skills as feature correlation within the visual field. We present evidence that algebra students respond spontaneously to the visual patterns of the notational display apart from engagement with the declarative content of the rules. Thus, persistent algebra errors may reflect disengagement from declarative content rather than an inability to deal with it. We sketch a Lexical Support System designed to sustain students' engagement with the declarative content of algebraic rules and processes, thus complementing the exciting curricular possibilities being developed for referentially rich algebra.

Technical note: The EOQ and EPQ models with shortages derived without derivatives

Grubbström and Erdem (Int. J. Prod. Econom. 59 (1999) 529) derived the EOQ model with shortages and complete backlogging without derivatives, and they predicted that their method could be introduced to younger students in school without the knowledge of calculus. However, their algebraic procedure is too sophisticated to be absorbed by ordinary readers. Recently, Cárdenas-Barrón (Int. J. Prod. Econom. 70 (2001) 289) extended the algebraic method to solve the EPQ model with shortages and complete backlogging. We will point out that his method has the same problem as Grubbström and Erdem's. In this note, our method applies basic algebraic skill to derive the optimal solution for EOQ and EPQ models with shortages and complete backlogging.

IMPLEMENTATION AND ASSESSMENT OF ON-LINE GATEWAY TESTING

ABSTRACT Gateway tests provide a means of assuring that students in reformed precalculus and calculus courses acquire the algebraic and computational skills needed in courses following these, while allowing the focus of the course to be on the conceptual understanding intrinsic to the reformed courses. However, logistical difficulties plague the pencil-and-paper administration of these tests, significantly decreasing their usefulness. To address these difficulties we have implemented an on-line version of the gateway tests which allows students to practice the skills on their own and take the test for a grade in a proctored environment. We report on the details of our implementation and assessment of our system, including our administration procedures and lab use. Our assessment included evaluation of students' perceptions of the effectiveness of the program and their actual skill acquisition, and showed that the students' skills improved in the course of the on-line gateway testing.

The relationship between mathematics preparation and conceptual learning gains in physics: A possible “hidden variable” in diagnostic pretest scores

There have been many investigations into the factors that underlie variations in individual student performance in college physics courses. Numerous studies report a positive correlation between students’ mathematical skills and their exam grades in college physics. However, few studies have examined students’ learning gain resulting from physics instruction, particularly with regard to qualitative, conceptual understanding. We report on the results of our investigation into some of the factors, including mathematical skill, that might be associated with variations in students’ ability to achieve conceptual learning gains in a physics course that employs interactive-engagement methods. It was found that students’ normalized learning gains are not significantly correlated with their pretest scores on a physics concept test. In contrast, in three of the four sample populations studied it was found that there is a significant correlation between normalized learning gain and students’ preinstruction mathematics skill. In two of the samples, both males and females independently exhibited the correlation between learning gain and mathematics skill. These results suggest that students’ initial level of physics concept knowledge might be largely unrelated to their ability to make learning gains in an interactive-engagement course; students’ preinstruction algebra skills might be associated with their facility at acquiring physics conceptual knowledge in such a course; and between-class differences in normalized learning gain may reflect not only differences in instructional method, but student population differences (“hidden variables”) as well.

Revitalizing Algebra: the effect of the use of a cognitive tutor in a remedial course

This study investigated the effect of instruction with a cognitive tutoring software system in a remedial algebra course. The performance on algebra tasks of students who attended the experimental (cognitive tutor) and a control class was compared. The results indicated that the two groups of students were equally proficient with respect to algebraic manipulation skills. However, students who attended the experimental algebra section performed significantly better in problem solving than students in the control section. This finding suggested that the use of the cognitive tutor (a) improved students' problem‐solving abilities; (b) fostered student development of richer concepts of variable and function; and (c) improved students' procedural abilities in approaching and carrying through mathematical analyses of relatively complex situations.

The Mathematics Background of Undergraduate Engineers

A major problem for engineering undergraduates is a lack of basic skills in number and algebra. While we seek middle-term solutions to these problems we must do something for the immediate term. It is suggested that the other engineering subjects be taught in the first semester in a qualitative way.

The Effects of Curriculum on Achievement in Second-Year Algebra: The Example of the University of Chicago School Mathematics Project

We examine the performance of 8 pairs of 2nd-year algebra classes that had been matched on pretest scores. One class in each pair used the UCSMP Advanced Algebra curriculum, and the other used the 2nd-year-algebra text in place at the school. Achievement was measured by a multiple-choice posttest and a free-response posttest. Opportunity-to-learn (OTL) measures were used to ensure that items were fair to both groups of students. UCSMP students generally outperformed comparison students on multistep problems and problems involving applications or graphical representations. Both groups performed comparably on items testing algebraic skills. Hence, concerns that students studying from a Standards-oriented curriculum will achieve less than students studying from a traditional curriculum are not substantiated in this instance.

Using Integral Transforms to Estimate Higher Order Derivatives

1. INTRODUCTION. When doing error analysis for numerical quadrature, achieving good uniform bounds on higher order derivatives of the integrand is paramount. As undergraduates become increasingly adept with programmable calculators, numerical integration schemes such as Simpson's Rule and the Trapezoidal Rule take on a new relevance. Although it may be a rare calculus class that dwells overmuch on error bounds for such schemes, this may be due as much to the perceived paucity of interesting examples for which decent error bounds are readily achievable as to the general weakness in algebraic skills necessary for the requisite understanding of inequalities. The purpose of this article, therefore, is to offer some interesting, non-trivial examples for which the error analysis, if not elegant, is at least simple enough to carry out in the classroom.

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.

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

The Effects of a Graphing-Approach Intermediate Algebra Curriculum on Students' Understanding of Function

In this study, we extended O'Callaghan's computer-intensive algebra study by using his component competencies and the process-object framework to investigate the effects of a graphingapproach curriculum employing the TI-82 graphing calculator. We found that students in the graphing-approach classes demonstrated significantly better understanding of functions on all 4 subcomponents of O'Callaghan's Function Test, including the reification component, than did students in the traditional-approach classes. Additionally, no significant differences were found between the graphing-approach and traditional classes either on a final examination of traditional algebra skills or on an assessment of mathematics attitude.

Purpose in school algebra

Algebra is commonly regarded as a problem area for many students, which has led teachers in schools to question its necessity, whereas lectures of tertiary-level courses continue to complain of incoming students' lack of algebraic skill. In considering how this situation should be addressed, I need first to clarify the aims and objectives of the school algebra course. I then review the research evidence on student's performance and proceed to some suggestions for curriculum modifications that might be helpful.

PLACEMENT IN FIRST-YEAR COLLEGE MATHEMATICS COURSES USING SCORES ON SAT-MATH AND A TEST OF ALGEBRA SKILLS

ABSTRACT A test of algebra skills was developed and used in conjunction with math SAT scores to place students in freshman math courses. Correlations with final grades indicated that most students would be correctly placed, far more than would be the case using SAT scores alone. The test was used during the next college year with similarly good results.

The Descriptive Tests of Mathematics Skills:Predictive Validity for an Elementary Mathematics Concepts and Structures Course

The Descriptive Tests of Mathematics Skills (DTMS) were designed for a wide range of uses at the lower division level. Two DTMS tests, Arithmetic Skills and Elementary Algebra Skills, were used to set a minimum standard for transferring juniors majoring in elementary education to enroll in an advanced mathematics concepts and structures course at a midwestern upper-division university. For the 60 students passing both tests and earning a grade in the class, significant predictive validity coefficients were found. When multivariate analysis was applied, it was found that the model could be simplified to only the Elementary Algebra Skills test. Nevertheless, DTMS can be used effectively to place upper-division elementary education majors into an advanced mathematics concepts and structures course.

An Analysis of the Validity of SAT Entrance Scores for Accelerated Classes

This study examined the validity of SAT cutoff levels used for entrance into fast-paced summer classes. Thirty-eight junior-high aged students whose SAT scores were below the required cutoff (SAT-V≥430 or SAT-M≥500) were admitted into fast-paced classes in literature, a teacher-paced course, and algebra, a self-paced course. Each below-score student was matched to an above-score student in his class on age, grade, sex, race, and family income. There was a small but statistically significant difference in achievement on standardized tests for the literature students, with above-score students scoring higher. There was no difference between above and below scorers in mathematics on achievement measures, including standardized tests, and in the number of chapters completed during the course. For either class, below and above score students did not differ on previous exposure to the content area; there were some differences between the groups on study skills and self-concept. Further, when individual profiles for below-score students were examined, below-score students who were most successful did not differ from other less successful students on any of the variables examined with the exception of study skills in algebra. There was a tendency for below-score students in algebra who were the most successful in the course to have higher scores on a study skills assessment than below-score students who had smaller gains. The results generally support the use of SAT cutoff scores in a teacher-paced course but suggest that cutoff scores might be less critical for motivated students in self-paced classes.

Learning Problem-Solving Skills in Algebra

In this article we describe aspects of our intelligent tutor for basic algebra. A main goal of the project is to develop a computer tutoring system whose skills and knowledge approximate those of a high-quality human tutor. We are particularly interested in exploring novel learning opportunities that can be made available to students for the first time by exploiting the reactive capabilities of such intelligent tutors. In this context, we focus here on the role of an algebra expert system embedded in the tutor. We discuss how it can be used to help students learn several nontraditional types of skill and knowledge in the context of algebra, including goal-directed reasoning skills, and debugging techniques.

Descriptive Tests of Mathematics Skills: A Follow-up of Performance of Older Master's Level Students

The academic performances of older Master's level business administration students in calculus and statistics were compared for students passing the algebra tests of the Descriptive Tests of Mathematics Skills and for those required to remediate their basic algebra skills deficiency. Of the 606 students entering the Master's program for the last 4 yr., 118 enrolled in calculus and 128 enrolled in statistics. For both courses, students passing the Descriptive Tests earned higher grades than students remediating their deficiency. The difference was statistically significant for the calculus course but not the statistics course. These findings with the exception of the nonsignificant difference in the statistics course grade were consistent with the literature regarding older junior level business administration students.

Factors Affecting Achievement in the First Course in Calculus

Three groups of students at Illinois State University (of respective sizes 235, 157, and 397) were used as subjects to determine which factors were significant predictors of success in the first course in calculus. The second and third groups were used to provide replications of the initial study. Academic independent variables considered were: ACT scores, high school rank, high school GPA, high school algebra grades, and the score from an algebra pretest. Biographical independent variables considered were: sex, birth order, family size, and high school size. The dependent variable was a function of the student’s course grade in the first semester of calculus. The use of stepwise and all-subsets regression procedures on the three groups revealed in each case that the best combination of predictors consisted of the algebra pretest and high school rank. From this result, the investigators concluded that the combination of algebraic skills, as represented by the score on the algebra pretest, and long-term pe...