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Descriptors:
Equations (Mathematics); Novices; Mathematics Instruction; Expertise; Algebra; Mathematics Education; Secondary School Teachers; Secondary School Students; Mathematical Formulas; Mathematics; Interviews

Abstract:
While an algebraic expression is typically assigned a regular structure, this article introduces the concept of personal structure ("Strukturierung"); here, the structuring of an algebraic expression is understood as the act of forming relationships between its parts. This concept is used for the analysis of interviews in which experts and novices talk about the personal structures they produced when looking at fractional expressions and equations. The results show, firstly, that experts will structure a given fractional expression in various different ways. Secondly, four categories have been identified: structuring can mean that an expression is made more visually straightforward, changed, reinterpreted or classified. This is meaningful for negotiating the appropriateness of the personal structures in teaching algebra. Note:The following two links are not-applicable for text-based browsers or screen-reading software. Show Hide Full Abstract

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Numbers; Algebra; Calculus; College Mathematics; Misconceptions; Error Patterns; College Students

Abstract:
The purpose of this study was to determine whether or not certain errors made when simplifying exponential expressions persist as students progress through their mathematical studies. College students enrolled in college algebra, pre-calculus, and first- and second-semester calculus mathematics courses were asked to simplify exponential expressions on an assessment. Persistent errors are identified and characterized. Using quantitative and qualitative methods, we found that the concept of negativity played a prominent role in most of the students' errors. We theorize that an underdeveloped conception of additive and multiplicative inverses is the root of these errors. (Contains 5 tables and 1 figure.) Note:The following two links are not-applicable for text-based browsers or screen-reading software. Show Hide Full Abstract

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Mathematical Logic; Validity; Majors (Students); Undergraduate Students; Algebra; Geometry; Number Concepts; College Mathematics; Logical Thinking; Preservice Teachers; Secondary School Mathematics

Abstract:
Validating proofs and counterexamples across content domains is considered vital practices for undergraduate students to advance their mathematical reasoning and knowledge. To date, not enough is known about the ways mathematics majors determine the validity of arguments in the domains of algebra, analysis, geometry, and number theory--the domains that are central to many mathematics courses. This study reported how 16 mathematics majors, including eight specializing in secondary mathematics education, who had completed more proof-based courses than transition-to-proof classes evaluated various arguments. The results suggest that the students use one of the following strategies in proof and counterexample validation: (1) examination of the argument's structure and (2) line-by-line checking with informal deductive reasoning, example-based reasoning, experience-based reasoning, and informal deductive and example-based reasoning. Most students tended to examine all steps of the argument with informal deductive reasoning across various tasks, suggesting that this approach might be problem dependent. Even though all participating students had taken more proof-related mathematics courses, it is surprising that many of them did not recognize global-structure or line-by-line content-based flaws presented in the argument. (Contains 6 tables.) Note:The following two links are not-applicable for text-based browsers or screen-reading software. Show Hide Full Abstract

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Descriptors:
High School Students; Disproportionate Representation; Mathematics Instruction; Experimental Teaching; Secondary School Mathematics; Algebra; Mathematics; Knowledge Level; Mathematics Achievement; Instructional Effectiveness

Abstract:
This paper is a study of part of the Algebra Project's program for underrepresented high school students from the lowest quartile of academic achievement, social and economic status. The study focuses on students' learning the concept of function. The curriculum and pedagogy are part of an innovative, experimental approach designed and implemented by the Algebra Project. The instructional treatment took place over 7 weeks during the Junior Year of 15 students from our target population. Immediately after instruction, a written instrument was administered followed, several weeks later, by in-depth interviews. The results are that many of our participants achieved a level of knowledge and understanding of functions on a par with beginning college students, including preservice teachers, as reported in the literature. Many conceptual difficulties that have been reported in the research literature were not as prevalent for our participants and some of them were capable of solving difficult problems involving composition of functions. We conclude that, with appropriate pedagogy, it is possible for students in the Algebra Project's target population to learn substantial and non-trivial mathematics at the high school level, and that the Algebra Project approach is one example of such a pedagogy. (Contains 5 tables and 4 figures.) Note:The following two links are not-applicable for text-based browsers or screen-reading software. Show Hide Full Abstract

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Teaching Methods; Grade 6; Grade 7; Algebra; Mathematical Concepts; Mathematics; Cognitive Structures; Mathematics Achievement; Middle School Students

Abstract:
Previous research has demonstrated the effectiveness of particular instructional practices that support students' constructions of the partitive unit fraction scheme and measurement concepts for fractions. Another body of research has demonstrated the power of a particular mental operation--the splitting operation--in supporting students' development of advanced fractional knowledge and algebraic reasoning. Steffe (2010) has hypothesized that students construct splitting through the unification of partitioning and iterating operations contained within the partitive unit fraction scheme. We used written assessments of 49 students, across sixth and seventh grades, to test this hypothesis. Our results show that students who have constructed a partitive unit fraction scheme go on to construct splitting within a relatively short period of time. Conversely, students who have not constructed a partitive unit fraction scheme generally do not construct splitting. We discuss these results and their implications for designing instruction and curricula that support students' development of algebraic reasoning. (Contains 10 figures, 5 tables and 1 footnote.) Note:The following two links are not-applicable for text-based browsers or screen-reading software. Show Hide Full Abstract

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Descriptors:
Equations (Mathematics); Mathematics Curriculum; College Entrance Examinations; College Students; Academic Ability; Student Placement; Mathematics; College Freshmen; College Mathematics; Scores; Algebra; Probability; Prediction; Higher Education

Abstract:
This article describes how a Brier score analysis can be used as an evaluative tool to estimate the predictive accuracy of a course placement policy that was established based on professional or subjective judgment. The policy being evaluated uses the score received on the mathematics portion of the SAT examination as the primary mechanism to place incoming freshmen into the appropriate college mathematics courses best matched to their ability. This placement policy was established based on the idea that the mathematics portion of the SAT examination and a commercially available computerized mathematics placement examination may be placing the majority of students at a similar level in the hierarchy of mathematics courses. The findings of this study indicate that the mathematics portion of the SAT examination can be an effective mechanism for initially placing students in their first mathematics course which is appropriate for their academic ability, and thus can increase their chance of succeeding in their first college mathematics course. Furthermore, a Brier score analysis allowed for comparing and contrasting different scenarios when the placement policy is effective versus when it is not. Implementing this policy allowed for a simpler placement process as well as a substantial financial savings for the institution. (Contains 4 tables and 5 footnotes.) Note:The following two links are not-applicable for text-based browsers or screen-reading software. Show Hide Full Abstract

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Descriptors:
Difficulty Level; Mathematics Achievement; Academic Records; Credits; Program Effectiveness; National Competency Tests; Algebra; Geometry; Mathematics Curriculum; High School Graduates; Academic Achievement; Textbooks; Course Content; High School Students; Grade 12; Scores; Racial Differences; Comparative Analysis; Course Selection (Students)

Abstract:
The 2005 National Assessment of Educational Progress (NAEP) High School Transcript Study (HSTS) found that high school graduates in 2005 earned more mathematics credits, took higher level mathematics courses, and obtained higher grades in mathematics courses than in 1990. The report also noted that these improvements in students' academic records were not reflected in twelfth-grade NAEP mathematics and science scores. Why are improvements in student coursetaking not reflected in academic performance, such as higher NAEP scores? The Mathematics Curriculum Study (MCS) explored the relationship between coursetaking and achievement by examining the content and challenge of two mathematics courses taught in the nation's public high schools--algebra I and geometry. Conducted in conjunction with the 2005 NAEP HSTS, the study used textbooks as an indirect measure of what was taught in classrooms, but not how it was taught. In other words, the textbook information is not used to measure classroom instruction. Textbooks served as an indicator of the intended course curriculum (Schmidt, McKnight, and Raizen 1997). The chapter review questions in each textbook were used to identify the mathematics topics covered (or subject matter content) and the complexity of the exercises (or degree of cognitive challenge). Chapter review questions, and not the entire textbook, were coded because the questions have been found to be representative of the chapter content and complexity level in previous studies (Schmidt 2012). The study uses curriculum topics to describe the content of the mathematics courses and course levels to denote the content and complexity of the courses. The results are based on analyses of the curriculum topics and course levels developed from the textbook information, coursetaking data from the 2005 NAEP HSTS, and performance data from the twelfth-grade 2005 NAEP mathematics assessment. The study addresses three broad research questions: (1) What differences exist within the curricula of algebra I and geometry courses?; (2) How accurately do school course titles and descriptions reflect the rigor of what is taught in algebra I and geometry courses compared to textbook content?; and (3) How do the curricula of algebra I and geometry courses relate to subsequent mathematics coursetaking patterns and NAEP performance? In this report, curriculum topics, course levels, and grade 12 NAEP mathematics scale scores are used to describe the findings of the study. Curriculum topics are based on summaries of the textbook content that a school reported covering in an algebra I or geometry course. The six broad categories of curriculum topics used to describe the mathematics content found in both algebra I and geometry textbooks are: elementary and middle school mathematics, introductory algebra, advanced algebra, two-dimensional geometry, advanced geometry, and other high school mathematics. A glossary is included. (Contains 3 charts, 15 figures and 10 tables.) Note:The following two links are not-applicable for text-based browsers or screen-reading software. Show Hide Full Abstract

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Descriptors:
Algebra; Public Schools; Foreign Countries; Control Groups; Constructivism (Learning); Grade 7; Outcomes of Education; Instructional Effectiveness; Mathematical Concepts; Quasiexperimental Design; Experimental Groups; Pretests Posttests; Statistical Significance; Comparative Analysis; Comparative Testing; Achievement Gains; Scaffolding (Teaching Technique); Cohort Analysis; Test Theory; Intermode Differences

Abstract:
It is a bitter reality that the curricula and traditional pedagogy prevailing in public schools of Pakistan in general and Sindh in particular do not incorporate the algebraic concepts properly. Both the content and the presentation therein cannot be considered up to the mark, thereby making "Algebra" a tough and dry subject. This quasi-experimental study focused to find out the effect of teaching algebra through social constructivism on the students learning outcomes of 7th graders in public schools of District Jamshoro Sindh. For this purpose two existing in-tact sections of grade 7 (7th A, and 7th B) of a government high school of district Jamshoro were selected as quasi control and treatment groups. The pre and posttest analyses were carried out. The first pattern t-test analysis revealed that both groups were parallel, however, the latter posttest analysis revealed that treatment group that was taught through social constructivist approach excelled in achieving statistically significant learning outcomes than that of control group that was taught through traditional one-way teaching. (Contains 5 tables.) Note:The following two links are not-applicable for text-based browsers or screen-reading software. Show Hide Full Abstract

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Descriptors:
Mathematics Achievement; State Standards; Algebra; Geometry; Secondary School Mathematics; Evidence; Outcome Measures; Individualized Instruction; Intelligent Tutoring Systems; Program Evaluation; Instructional Effectiveness

Abstract:
"Carnegie Learning Curricula and Cognitive Tutor"[R], published by Carnegie Learning, is a secondary math curricula that offers textbooks and interactive software to provide individualized, self-paced instruction based on student needs. The program includes pre-Algebra, Algebra I, Algebra II, and Geometry, as well as a three-course series that integrates numeric, algebraic, geometric, and statistical content. The developer indicates that the program is aligned with most state standards and the standards set by the National Council of Teachers of Mathematics. The program can be customized to meet other state-specific standards. The What Works Clearinghouse (WWC) identified 27 studies that investigated the effects of "Carnegie Learning Curricula and Cognitive Tutor"[R] on math performance for high school students. The WWC reviewed 11 of those studies against group design evidence standards. Three studies (Cabalo, Jaciw, & Vu, 2007; Campuzano, Dynarski, Agodini, & Rall, 2009; & Pane, McCaffrey, Slaughter, Steele, & Ikemoto, 2010) are randomized controlled trials that meet WWC evidence standards without reservations, and three studies (Shneyderman, 2001; Smith, 2001; & Wolfson, Koedinger, Ritter, & McGuire, 2008) are randomized controlled trials or quasi-experimental designs that meet WWC evidence standards with reservations. These six studies are summarized in this report. Five studies do not meet WWC evidence standards. The remaining 16 studies do not meet WWC eligibility screens for review in this topic area. Appended are: (1) Research details for Cabalo et al., 2007, Campuzano et al., 2009, Pane et al., 2010, and Shneyderman, 2001; (2) Outcome measures for each domain; (3) Findings included in the rating for the mathematics achievement domain; and (4) Summary of supplemental findings for the mathematics achievement domain. A glossary of terms is included. (Contains 7 tables, 4 additional sources and 7 endnotes.) Note:The following two links are not-applicable for text-based browsers or screen-reading software. Show Hide Full Abstract

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Descriptors:
Engineering Education; College Students; Mathematics Skills; Problem Solving; Inquiry; Difficulty Level; Word Problems (Mathematics); Algebra; Misconceptions; Reading; Comprehension; Speech Communication; Foreign Countries

Abstract:
This paper presents another effort in determining the difficulty of engineering students in terms of solving word problems. Students were presented with word problems in algebra. Then, they were asked to solve the word problems orally; that is, before they presented their written solutions, they were required to explain how they understood the problem, and to give the processes they wanted to use in order to obtain the answer. Responses of students for each word problems would be noted. Discussions were recorded so that all responses were accounted for. Using NEA (Newman's error analysis), student's problems on reading, comprehension, transformation, and process skills can be determined by the teacher before the encoding of the solution is done. Also, the teacher directly addresses whatever misconceptions are made by the student in the process as well as of other students who are thinking the same way. More than 70% of the errors found were comprehension and transformation errors. Thus, students were given remedial classes to minimize their comprehension and transformation errors. (Contains 4 tables.) Note:The following two links are not-applicable for text-based browsers or screen-reading software. Show Hide Full Abstract

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