Algebra Curriculum Research Articles

The most important aspects of algebra: Responses from practising South African teachers

Curricular policy documents generally communicate guidelines regarding the teaching of algebra. Those policy documents may not represent what teachers, currently teaching mathematics, consider significant. Teachers' views are more likely to provide information about what they emphasise in their mathematics classrooms. This paper reports on practising Teachers' beliefs about what is important in the teaching of algebra. Data were collected from 306 practising mathematics teachers, who were registered in an Algebra module in an Advanced Certificate in Education (ACE) programme. The teachers were asked to discuss and list aspects in school algebra that they considered to be important. Qualitative analysis methods were used to code the Teachers' responses according to categories that were subsequently grouped into larger themes. The emergent themes were considered in light of curriculum documents to better understand the Teachers' negotiated views about important aspects in the teaching and learning of algebra. While many of their responses reflected aspects of algebra that have been identified both internationally and historically, a number of responses included aspects that are not part of the algebra curriculum. In general, the Teachers' comments emphasized the basic procedures in algebra (i.e., symbolic manipulations) rather than the key concepts in algebra or the general processes associated with doing algebra (e.g., reasoning or representing ideas algebraically).

Teacher listening: The role of knowledge of content and students

In this research report we consider the kinds of knowledge needed by a mathematician as she implemented an inquiry-oriented abstract algebra curriculum. Specifically, we will explore instances in which the teacher was unable to make sense of students’ mathematical struggles in the moment. After describing each episode we will examine the instructor's efforts to listen to the students and the way that these efforts were supported or constrained by her mathematical knowledge for teaching. In particular, we will argue that in each case the instructor was ultimately constrained by her knowledge of how students were thinking about the mathematics.

ENACTED TYPES OF ALGEBRAIC ACTIVITY IN DIFFERENT CLASSES TAUGHT BY THE SAME TEACHER

The aim of this study was to examine how teachers enact the same written algebra curriculum materials in different classes. The study addresses this issue by comparing the types of algebraic activity (Kieran, 2004) enacted in two 7th grade classes taught by the same teacher, using the same textbook. Data sources include lesson observations and an interview with the teacher. The findings show that students in the two classes were offered somewhat different algebraic experiences. At one school, more emphasis was placed on global/meta-level activities (activities that are not exclusive to algebra and suggest general mathematical processes), whereas at the other school, more emphasis was placed on transformational activities (“rule-based” algebraic activities). Analysis of the sources of the differences related to the ways in which the teacher used and enacted the curriculum materials in the two classes revealed that these were linked to the teacher’s attempts to be attentive to the students in the class and to the nature of the students’ work.

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.

Motivating the concept of eigenvectors via cryptography

New methods of teaching linear algebra in the undergraduate curriculum have attracted much interest lately. Most of this work is focused on evaluating and discussing the integration of special computer software into the Linear Algebra curriculum. In this article, I discuss my approach on introducing the concept of eigenvectors and eigenvalues, which I have used for the last 3 years in my Linear Algebra course. I offer some examples on how I have attracted the interest of our students via Hill ciphering, a method of cryptography. After emphasizing the effect of a linear transformation in a vector space and the importance of eigenvectors, I show how students’ motivation and understanding towards one of the abstract concepts in Linear Algebra; eigenvalues and eigenvectors have grown positively.

An analytic conception of equation and teachers’ views of school algebra

This interview study takes place in the context of a single small district in the United States. In the algebra curriculum of this district, there was a shift in the conception of equation, from a statement about unknown numbers to a question about the comparison of two functions over the domain of the real numbers. Using two of Shulman’s [ Shulman, L. S. (1986). Paradigms and research programs in the study of teaching: A contemporary perspective. In Wittrock, M. C. (Ed.), Handbook of research in teaching (3rd ed., pp. 3–36). New York: Macmillan ] categories of teachers’ knowledge – pedagogical content knowledge and curricular content knowledge – we explore whether in this context teachers’ content knowledge give signs of being reorganized. Our findings suggest that the teachers see this conception of equation as useful for equations in one variable. They struggle with its ramifications for equations in two variables. Nonetheless, this conception of equation leads them to reflect on the algebra curriculum in substantial ways; two of the three teachers explicitly spoke about their curricular ideas as being associated with this conception of an equation or with their earlier views. The third teacher seems so taken with these curricular ideas that he explored their ramifications throughout the interview. We argue that the consideration of this new conception of equation was an important resource that the teachers used to construct their understandings of alternative curricular approaches to school algebra. As they work with this new conception of an equation, we find an analogy to their situation in Kuhn’s description of the individual scientist in the process of adopting a new paradigm.

Early Childhood Corner: Algebra in the Pre-K–2 Curriculum? Billy Goats and Bears Give Us the Answer

What should the algebra curriculum look like for preschool and primary-grade learners? I posed this lead-in question to a session in a mathematics methodology class for preservice early childhood and elementary teachers. Some of the teachers' answers indicated uncertainty. They gave vague, tentative answers about working with sequences and solving for unknowns. One participant's response resonated for others in the class: “I had a terrible time in high school algebra. I struggled to memorize the formulas for solving equations just as I had struggled to memorize my addition facts, multiplication tables, and long division in elementary school. What I remember of algebra, I would not wish on any child.” Other classmates murmured recognition of these feelings. Still another classmate added, “I got algebra. I even did well in trigonometry, but that doesn't mean that I understand algebra well enough to help children get it.”

Pythagorean Triples and Inner Products

Linear algebra over rings particularly over Z—is a fascinating topic that straddles course boundaries and can help bridge the present unnecessary gap between introductions to algebra and number theory. Moreover, a brief introduction to integral linear algebra can be helpful in further studies. For example, in both coding theory and solid state physics one is led to study lattices over the integers—roughly speaking, periodic discrete arrays of points in space (a more precise definition will be given below) for which the standard linear algebra curriculum, dealing only with spaces, is inadequate. We refer the reader to the books by Conway–Sloane [3] or Ebeling [6] for lattices in coding theory, and Kittel [7] or Senechal [14] for lattices in solid state physics. Following the introduction, we review inner product spaces and the representation of inner products by symmetric matrices. The inner products we study will not necessarily be positive definite. (Indeed, our coefficients will not always be real or complex numbers.) In a background section, we introduce unimodular matrices over Z and then Z-lattices on rational inner product spaces; and then we explore the fundamental role of unimodular matrices in connection with basis changes for lattices. The following section is the heart of the article. There we give a new perspective on Pythagorean triples through the medium of lattices on inner product spaces, using the machinery of the earlier sections. A Pythagorean triple is a triple (a, b, c) of integers such that a2 b2 = c2; and the triple (a, b, c) is primitive if gcd(a , b, c) = 1. The construction of primitive Pythagorean triples, and in particular the demonstration that there are infinitely many of them, is a truly ancient subject and a standard topic in elementary number theory courses.

Promoting Understanding of Linear Equations with the Median Slope Algorithm

In recent years, the influence of technology on school mathematics curricula has become increasingly pronounced. Today, for instance, calculators seem ever–present in most mathematics classrooms. In fact, a recent study of nearly 4,000 mathematics teachers and 50,000 students reports that approximately 96 percent of eighth–grade students in the United States have access to calculators for use in mathematics classrooms (IEA 2001). The computational power of handheld calculators—particularly those equipped with graphing capabilities—has significantly accelerated the introduction of numerous mathematical topics within school mathematics. For example, before the early 1990s, leastsquares regression was not found in most algebra textbooks. Because by–hand methods for calculating a least–squares regression line required students to minimize the sum of squared residuals—a technique requiring differentiation—the procedure was rarely studied before calculus. Now, however, any student with a graphing calculator can perform a leastsquares linear regression by pressing only a few buttons. Not surprisingly, linear regression is now commonly found in the school algebra curriculum.

When Learning No Longer Matters: Standardized Testing and the Creation of Inequality

A California high school serving low-income students from many ethnicities is famous for its outstanding mathematics department. On independently designed assessments, students at the school consistently outperform students in wealthier schools. Why, then, Ms. Boaler wonders, has the school received a demoralizing underperforming label from the state? THIS IS A STORY about a remarkable school, a school that has been labeled underperforming by the state of California. It is a true story, and it draws on a combination of research data, collected as part of a Stanford University research project on mathematics learning, and the lived realities of teachers and students working hard to achieve success in a low-income, urban school. At the heart of this story lies the conflict between learning and SAT-9 success, a conflict that has affected the lives of students and teachers in this school in profound ways. As a new professor at Stanford University, recently arrived from England, I was considering schools to include in a study of mathematics teaching and learning. I soon learned that a school that I'll refer to here as Railside High had a mathematics department that worked in very unusual ways. Some years ago the teachers detracked their classes in response to the low performance of some students. The mathematics department plans lessons collaboratively, and the teachers meet every week to discuss and improve their lessons. They visit one another's classes frequently, and every new teacher is given the opportunity to watch every lesson that he or she will teach being taught by an experienced colleague first. The algebra curriculum that all students take on entry to the school was designed by the department, and it draws from a variety of different curriculum materials. Students from a range of ethnicities -- Latino, African American, white, Filipino, and other Asian groups -- work together in groups to solve complex mathematics problems. All the teachers in the department are mathematics specialists, and they all regularly attend professional conferences as a department. These practices would be unusual for any school, but this is a school in a low-income area with few resources. Lessons are accompanied by the steady hum of cars zipping past on the two freeways that surround the school and are interrupted at frequent intervals by the sound of trains that pass just feet away from the school yard (thus our chosen pseudonym for the school). Financial resources are low -- in the school and in the students' homes. Yet qualified mathematics teachers are queuing up to join this department, and, after a year of studying and monitoring the mathematics teaching and learning in this school, we have discovered some unequivocally positive facts. As part of a research project funded by the National Science Foundation, my research team and I gave incoming students at three high schools a mathematics examination. The students starting Railside High scored at a significantly lower level than the students starting the other -- wealthier - - schools in our study. However, after one year at Railside High, the students attained a higher average score on the end-of-year algebra examination than the students at the other schools in the study. By the end of the second year at the school, Railside students were significantly outperforming students at the other schools. There is one other high school in Railside's district -- in a wealthier area. At the end of each course (algebra, geometry, trigonometry, etc.) both high schools give students the same final examinations, designed to carefully assess the competencies specified in the California mathematics standards. These exams are constructed and graded by the two departments and overseen by the district. In all three years that the exams have been given, the Railside students have significantly outperformed their wealthier counterparts at every level of mathematics. …

CAS Applied in a Functional Perspective College Algebra Curriculum

Summary Act in Algebra(ACT) is a functional perspective curriculum for college algebra that focuses on the use of realistic applications to motivate the acquisition of mathematical concepts through active student explorations. A computer algebra system (CAS) is used as an open-ended tool in the exploration of mathematics and the modeling of data. The goal of ACT is to improve student cognition, metacognition, and affect with respect to college algebra. A series of studies was conducted from the spring of 1994 to the spring of 1999 to determine the effects of the CAS in a functional perspective curriculum. This article provides an overview of the findings of those studies and discusses the implications of these findings for the teaching and learning of algebra.

Teaching Linear Algebra: Issues and Resources

Jane Day (day@sjsumcs.sjsu.edu) has been at San Jose State University since 1982. She participated in the Linear Algebra Curriculum Study Group in 1990 and has been particularly aware since then of how diverse the audience is for linear algebra. In 1994 she received the annual Award for Distinguished College or University Teaching from the MAA Northern California-Nevada Section. Her research interest is matrix theory, and she is presently on the Editorial Board of the American Mathematical Monthly.

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.

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.

An Exploration of Aspects of Language Proficiency and Algebra Learning

We have attempted to investigate whether 3 cognitive components of language proficiencymetalinguistic awareness of symbol, syntax, and ambiguity-are associated with students' success in learning the notation of algebra. Pencil-and-paper tests were given to assess students' metalinguistic awareness and their ability to use algebraic notation. In a total sample of more than 1500 students, aged 11 to 15, who were in their 1st to 4th years of algebra learning, we found that very few students with low metalinguistic awareness scores achieved high algebra scores. We discuss implications of this finding for the school algebra curriculum.

The Graphing Calculator: A Survey of Classroom Usage

The purpose of this study was to investigate secondary mathematics teachers' use of the graphing calculator in their classrooms, The study examined whether algebra teachers are currently using this technology in their classrooms, their perceptions toward the technology, and any changes in the curriculum or instructional practices. A survey methodology was used in this study. The findings indicated that the use of the graphing calculator is still controversial to many algebra teachers. Teachers of algebra I were using graphing calculators to a significantly lesser degree than teachers of algebra H. However, modifications of the algebra curriculum are beginning to appear in classes using graphing calculators. Finally, a majority of algebra teachers responded that the graphing calculator was a motivational tool.

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.

On the Elusive Nature of Measuring Teacher Change: An Examination of the Stages of Concern Questionnaire

This study builds on the work of Bailey and Palsha (1992), which investigated the quality of data gathered from the Stages of Concern Questionnaire (SoCQ), an instrument based on the Concerns Based Adoption Model (CBAM) for measuring teacher concerns during a reform effort. The present study examined two hypotheses: (1) the subscales of Awareness and Refocusing lack reliability and should be removed from consideration in the SoCQ instrument; and (2) reorganising the seven-stage SoCQ into a five-stage questionnaire can provide more representative data aboutlevels of concern. Investigators examined data from two groups of teachers involved in implementing a new algebra curriculum. Primary methods of inquiry included computing measures of internal consistency and conducting confirmatory factor analyses. Results of the study supported the first hypothesis but did not support the second hypothesis. Collecting qualitative data that will identify teacher concerns more precisely is suggested as a means of validat...

Making Exploration Visible: On Software Design and School Algebra Curriculum

Making Exploration Visible: On Software Design and School Algebra Curriculum

Computer-Intensive Algebra and Students' Conceptual Knowledge of Functions

This article describes a research project that examined the effects of the Computer-Intensive Algebra (CIA) and traditional algebra curricula on students' understanding of the function concept. The foundation for the research is a proposed conceptual framework that describes function knowledge in terms of component competencies. The results indicated that the CIA students achieved a better overall understanding of functions and were better at the components of modeling, interpreting, and translating. No significant differences were found for reifying, which emerged as the most difficult component in the proposed function model. Further, the CIA students showed significant improvements in their attitudes toward mathematics, were less anxious about mathematics, and rated their class as more interesting. A higher percentage of students successfully completed the CIA course.