Early acceleration of students in mathematics: Does it promote growth and stability of growth in achievement across mathematical areas?

Using data from the Longitudinal Study of American Youth (LSAY), the present study examined whether early acceleration of students into formal algebra at the beginning of middle school promoted evident growth in different mathematical areas (basic skills, algebra, geometry, and quantitative literacy) and stable growth across these mathematical areas. Results of multivariate multilevel analyses showed that low achieving students who were accelerated into formal algebra grew faster than not only low achieving students who were not accelerated but also high achieving students who were not accelerated. The rates of growth of accelerated low achieving students were even comparable to those of accelerated high achieving students. All low achieving students showed the same potential to take advantage of early acceleration regardless of their individual, family, and school characteristics. Early acceleration also promoted stability of growth across mathematical areas, and this stability was not dependent on student and school characteristics.     

Teaching Algebra to Students with Learning Difficulties: An Investigation of an Explicit Instruction Model

Abstract. Thirty‐four matched pairs of sixth‐ and seventh‐grade students were selected from 358 participants in a comparison of an explicit concrete‐to‐representational‐to‐abstract (CRA) sequence of instruction with traditional instruction for teaching algebraic transformation equations. Each pair of students had been previously labeled with a specific learning disability or as at risk for difficulties in algebra. Students were matched according to achievement score, age, pretest score, and class performance. The same math teacher taught both members of each matched pair, but in different classes. All students were taught in inclusive settings under the instruction of a middle school mathematics teacher. Results indicated that students who learned how to solve algebra transformation equations through CRA outperformed peers receiving traditional instruction on both postinstruction and follow‐up tests. Additionally, error pattern analysis indicated that students who used the CRA sequence of instruction performed fewer procedural errors when solving for variables.     

运用化归数学思想,把握初中代数基本建构

化归思想是初中代数学习的重要思想,有助于学生完成代数基本建构。基于此,本文在分析化归数学思想内涵的基础上,结合代数问题解答例子从化归思想理解、运用和拓展三方面提出了初中代数教学运用化归思想的方法,为关注这一话题的人们提供参考。     

The roles of reification and reflective abstraction in the development of abstract thought: Transitions from arithmetic to algebra

This study utilized a psychological constructivist perspective to examine the transitions that students make from arithmetic to algebra in the context of problems, that from the expert's perspective, involve the concept of linear inequality. Unstructured interviews were used to gather data that were used to develop an explanation concerning student understanding. Thirteen college students were interviewed individually and asked to solve nine related tasks. The interviews were videotaped and the protocols were analyzed to document student conceptions. Five case studies were used to develop and substantiate an explanation regarding students' transitions from arithmetic to algebra. Cifarelli's (1988) levels of reflective abstraction and Sfard and Linchevski's (1994) theory of reification provided a framework for this explanation. This paper discusses an integration of Cifarelli and Sfard's constructs. Students who completed a transition to algebra operated at higher levels of reflective abstraction than students who were unable to complete such a transition. Operating at higher levels of reflective abstraction enabled these students to consider concepts as both processes and abstract objects. Developing this ability was found to be critical in achieving a transition to using algebraic methods.     

Structures of Constant Rate Word Problems: A functional approach analysis

Many recent projects have lately explored function-based approaches to algebra. For the last five years we developed and tested such an approach to algebra, the 'Visual Mathematics'¹ curriculum, in grades seven to nine. While teaching this curriculum we learned from our students' work that while functions form an important basis for solving various sorts of algebra word problems, they introduce some tension in dealing with other sorts of word problems. The data accumulated during our experiments led us to conjecture that viewing algebra rate problems through a function lens offers new ways to look at similarities and differences among word problems. This article presents a theoretical framework for a systematic investigation of the structures of algebra word problems and their relationship to function-based solutions. We think that for those who take a function approach to algebra, this paper suggests new insights into the categorization of word problems. This revised version was published online in July 2006 with corrections to the Cover Date.     

College Algebra: Past,Present, and Future

Issues within college algebra are well-documented. Over the past two decades, reform has brought about significant modifications in the course, yet to date there has been no synthesis of the efficacy of these efforts. This paper situates college algebra within a historical context, and synthesizes research on reform. I argue that, notwithstanding the successes brought about by change, the course is still not in a position to fulfill quantitative literacy requirements; that is, there exists nontrivial components of quantitative literacy that a college algebra course, by its nature, tends not to provide. I conclude with realistic steps forward for administrators and instructors.     

The Struggle to Pass Algebra: Online vs. Face-to-Face Credit Recovery for At-Risk Urban Students

Students who fail algebra are significantly less likely to graduate on time, and algebra failure rates are consistently high in urban districts. Identifying effective credit recovery strategies is critical for getting students back on track. Online courses are now widely used for credit recovery, yet there is no rigorous evidence about the relative efficacy of online versus face-to-face credit recovery courses. To address this gap, this study randomly assigned 1,224 ninth graders who failed algebra in 17 Chicago public high schools to take an online or face-to-face algebra credit recovery course. Compared to students in face-to-face credit recovery, students in online credit recovery reported that the course was more difficult, were less likely to recover credit, and scored lower on an algebra posttest. There were no statistically significant differences by condition on any outcomes measured during the second year of high school (standardized mathematics test and algebra subtest scores, likelihood of passing subsequent math classes, cumulative math credits, or on-track rates). The benefits and challenges of online learning for credit recovery are discussed in light of the findings to date.     

A structural model of algebra achievement: computational fluency and spatial visualisation as mediators of the effect of working memory on algebra achievement

The goal of this study was to develop and evaluate a structural model of the relations among cognitive abilities and arithmetic skills and college students’ algebra achievement. The model of algebra achievement was compared to a model of performance on the Scholastic Assessment in Mathematics (SAT‐M) to determine whether the pattern of relations is similar for different types of higher level maths achievement. Structural equation modelling was used to test the effects of working memory, 3D spatial ability, and computational fluency on both types of higher order maths achievement. Computational fluency had the strongest effect on algebra achievement, with 3D spatial ability and working memory showing moderate effects. In contrast, 3D spatial ability had a stronger effect on SAT‐M scores than did computational fluency. Computational fluency and 3D spatial ability completely mediated the effect of working memory for both algebra and SAT‐M achievement.     

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 effects of high school math curriculum on college attendance: Evidence from the NLSY97

Using a sample of youth who graduated from high school in the late 1990s and early 2000s, this paper examines the impact of high school math curriculum on the decision to go to college. Results that control for unobserved differences between students and their families suggest that a more rigorous high school math curriculum is associated with a higher probability of attending college and of attending a 4-year college. The household fixed effect results imply that students who take an advanced academic math curriculum in high school (algebra II or precalculus, trigonometry, or calculus) are about 17 percentage points more likely to go to college and 20 percentage points more likely to start college at a 4-year school by age 21 compared to those students whose highest math class was algebra I or geometry.     

谈谈如何学好高等代数

高等代数是高等院校里数学专业的一门专业基础课程,该课程概念多,抽象,且逻辑性强。不少学生感觉基本概念难以理解,习题难做,基本方法难以掌握。针对这些困难,提出学好该课程的系统的学习方法与建议。     

代数应用题图式研究概述

代数应用题图式是学生在代数应用题学习过程中,对学习材料概括的基础上形成的、存储在长时记忆中的、具有一定框架结构的陈述性知识.模板图式、家族图式、概念图式和类别图式是心理学家研究数学学习心理常用的4种图式类型,也是代数应用题学习环境的主要设计元素.当今代数应用题图式的一个重要应用就是基于图式的代数应用题学习环境设计.     

Premarital Sex, Social Support, Submissive Behaviors, and Loneliness Among Turkish University Students

The purpose of this study was to investigate whether differences in social support, submissive behaviors, and loneliness existed among Turkish university students who had had premarital sexual intercourse and those who had not. Using self-reported questionnaires, students who had experienced sexual intercourse were contrasted with those who had not. Data were gathered on 420 university students in Ankara, Turkey. MANOVA statistics were used in the data analysis. Results indicated that more than 75?% of the total respondents reported that they had never had sexual intercourse; the frequency of sexual intercourse was higher among males than females. It was found that students who had had sexual intercourse reported less perceived support from family. The two groups were not significantly different, however, in perceived support from friends, submissive behaviors and loneliness. Possible interpretations of these data and implications for future study are suggested.     

EXAMINING THE RELATIONSHIPS AMONG CLASSROOM CLIMATE, SELF-EFFICACY, AND ACHIEVEMENT IN UNDERGRADUATE MATHEMATICS: A MULTI-LEVEL ANALYSIS

For nearly 50 years, leaders in American industry, military, education, and politics have focused considerable attention on STEM (science, technology, engineering, and mathematics) education. Given the increased societal demand for STEM careers, the relationships among classroom climate, self-efficacy, and achievement in undergraduate mathematics needed to be examined. A purposeful sample of college algebra instructors (n = 15), employed at public 4-year universities in various states (n = 10) across the nation, was administered the Principles of Adult Learning Scale at the beginning of the semester to assess classroom climate. At the end of the course semester, their college algebra students (n = 326) were administered the Mathematics Self-Efficacy Scale-Revised and final college algebra examinations. The results of the multi-level analysis indicated: (a) students having higher mathematics self-efficacy also had higher mathematics achievement, (b) teacher-centered classroom climates had greater mathematics self-efficacy levels, (c) classroom climate was not a significant predictor of mathematics achievement, (c) classroom climate did not moderate the relationship between mathematics self-efficacy and achievement, and (d) although boys reported higher mathematics self-efficacy than girls, gender differences were not found to exist in regard to mathematics achievement.     

Correlations between teaching practices and class achievement in introductory algebra

Certain teaching practices were found to be linked to the educational achievement of students learning algebra for the first time. Collective teaching was observed with an instrument based on one developed by Stallings (1977). Student achievement appeared related, to some extent, to academic discourse. For this to be the case, this discourse had to come mainly from the teacher who was dispensing instruction, lecturing in a large-group context with or without teaching materials, asking questions, helping the student unable to give a correct answer, or giving feedback. The quantity of such discourse was also important. It appeared that it must be plentiful, but not excessive. Correlations with student achievement of large-group instruction and of instruction where the teacher monitors student work are the main focus of this paper. Student discourse in general, and non-academic discourse in particular, appear to have negative effects on achievement if they become frequent.     

Online Tutoring Research Study for Remedial Algebra

In 2012, the City University of New York undertook to examine whether online tutoring would be helpful to remedial algebra students. The research study was done in the spring 2012 semester at the Borough of Manhattan Community College (BMCC). The research question was two-fold: (a) Does online tutoring help improve pass rates among remedial algebra students? and (b) Do remedial algebra students find that online tutoring is helpful? A random sample of eight sections of Elementary Algebra was chosen from the 112 sections of that course offered in the spring semester. This amounted to a total of 195 students in the research sections, with 2,521 remaining students in the control (nontreatment) sections. All students in the eight treatment sections were given access to the Brainfuse online tutoring system. Midterm scores, final exam scores, and opinion surveys were collected. An overwhelming proportion of those who used the service found it helpful (94.7%) and would recommend it to a friend (100.0%). However, the pass rates in the study were nearly the same for the treatment and control groups, and a logistic regression analysis, comparing two statistically matched cohorts, showed no significant differences. One issue was that the sample of those who chose to use the service was small. Also, many of those who needed the most help were not the ones who used the service. It may be that in terms of improving pass rates, outreach and counseling for students could be more efficacious than simply offering a tutoring service.     

Math,Science, and Foreign Language: Evidence‐Based Accommodation Decision Making at the Postsecondary Level

Accommodations in postsecondary settings have become commonplace for many students with learning disabilities (LD) who have documented needs. Many of the accommodations professionals recommend for students with LD are based on an analysis of the course demands, the student's functional limitations, and a basic understanding of how the accommodation can facilitate the demonstration or acquisition of knowledge. However, little is known about which accommodations are recommended for math, science, and foreign language courses as well as the effectiveness of those accommodations. Because these content areas pose substantial hurdles for secondary students with LD who may transition to postsecondary settings, a review of the literature was conducted to evaluate current practices in the provision of accommodations to postsecondary students with LD in math, science, and foreign language courses. Findings indicate strong empirical evidence for extended test time for algebra exams and emerging research in changes to foreign language instruction. Recommendations for further research are provided.     

师范院校高等代数与解析几何教学改革探究

高等代数与解析几何是高等院校数学专业的基础课．根据师范院校高等代数与解析几何课程的教学目标,提出了在课程教学中突出师范特色的教学改革：注重数学概念的历史发展,注重教学思想方法的渗透,重视数学教学的直观化,重视与中学数学的联系。     

Clifford代数Clp，q的中心子代数

（p,q）型Minkowski空间Rp,q的Clifford代数Clp-q是一类2p＋q维的实结合代数,当p＋q〉1时是非交换代数．文中讨论了Clp-q的中心Cen（Clp,q）的相关性质,利用基元的Clifford积导出由p,q确定Clp,q中心的公式．