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Generalization is one of the core aspects of algebraic thinking, and research literature points to the importance of objectification when generalizing mathematical discourses. To increase the field's understanding of the objectification process in various algebra discourses, our study analyses Grade 6 teachers' use of words, symbols, routines and narratives when they introduce various algebraic concepts. Our results show that there exist instances of objectification in the teachers' discourses, but these are all implicit, which can make them difficult for students to notice. Thus, teachers need to be made more aware of the importance of making the objectification process explicit. Furthermore, our results show that instances of objectification are tightly connected to the use of formal algebraic symbols. This highlights the importance of the transition from concrete/iconic to formal/arbitrary representations. [ABSTRACT FROM AUTHOR]

Function is a multifaceted and essential concept in mathematics that includes various complex ideas such as mapping, relations between variables, algebra, input-output, and rule. Pre-service mathematics teachers' (PMTs) should have a rich understanding of these ideas for their future teaching practices. However, research showed that PMTs had a limited and superficial understanding of function as students do. Thus, a learning module was designed for improving PMTs' conceptualizations of function ideas as they relate to their personal concept definitions and examples of functions. A design-based research study was conducted with 17 senior PMTs. The study results suggest an improvement in the way PMTs define functions and identify a variety of function ideas in definitions and examples. Even if the mapping idea became dominant among all ideas in PMTs' personal concept definitions and examples, they could identify the implicit function ideas and develop an understanding of connections between ideas. We also provide suggestions for the second cycle of the intervention. [ABSTRACT FROM AUTHOR]

The article presents the results of a study that aimed to understand how the design and implementation of a Professional Learning Task contributed to the identification of mathematical and didactic knowledge of mathematics teachers as participants in a professional development. It fits into a qualitative-interpretive perspective, in which data collection came from the planning audios, its design, audios and videos of the discussions in the subgroups and the plenary of the second meeting. The results show that, from the professional learning task planned and implemented, it was possible to raise the knowledge of teachers, in addition to showing the articulation between the components of the Professional Learning Tasks. [ABSTRACT FROM AUTHOR]

This study investigated the methods that pre-service teachers employed while finding volume using integration and the reasons why they utilized these methods (problems); the accuracy of their solutions was also analyzed in this study. Case study design, a qualitative research method, was used for this study. 45 sophomore pre-service teachers studying for the Middle School Mathematics Education degree programme at an Education Faculty in Anatolia. It is used to mean 45-second class preservice teachers studying for the Middle School Mathematics Education degree programme at an Education Faculty in Anatolia. It is used to mean 45-second class preservice teachers studying for the Middle School Mathematics Education degree programme at an Education Faculty in Anatolia. Pre-service teachers were provided with a training programme that made use of GeoGebra software for four weeks as part of their mathematical analysis course. Pre-service teachers were provided with a written form containing five open-ended questions and given time to answer them. Data from the study was analyzed using the content analysis method. Results from these analyses showed that half of the pre-service teachers chose problems that requires the shell method to calculate volume and that the main determining factor was the ease of mathematical calculation provided by the shell method. Also, it was found out that the mistakes made by pre-service teachers originated from issues such as the inability to fully comprehend the subject at hand, errors in formulization and/or graphing, and problems taking the integral. [ABSTRACT FROM AUTHOR]

Modern mathematics education used a different approach when dealing with word problems in algebra because of how people perceive mathematics. The teacher can aid students in honing their problem-solving abilities to assist them build mathematical concepts that can be applied in real-world situations. This research is a case study involving junior high school (JHS) pupils in Ghana's Tema Metropolis. For the study, data collection and analysis used a qualitative methodology. Using a purposive sample technique, data from 12 participants-six JHS mathematics teachers and six students-were gathered. Data were gathered through observation, interviews, and document analysis. The primary finding is that JHS pupils at Tema Education Metropolis are unable to present their own issues as proof of conceptual knowledge of algebra word problems. [ABSTRACT FROM AUTHOR]

This study aims to investigate the non-algebraic solution strategies that prospective mathematics teachers use to solve word problems and their thought process while they were solving the problems. A total of 76 prospective mathematics teachers were asked to solve two word problems without using algebraic solutions and, then, clinical interviews were conducted with a sub-sample of 20 participants. The solutions used by the participants were analysed according to the type and frequency of the strategy. The results show that prospective mathematics teachers did not use various non-algebraic solutions for word problems and that they used the guess and check strategy most often. According to the findings obtained from the interviews, it was found that the participants tended to rely on solutions that used algebraic equations related to the problem while producing solution strategies. Furthermore, the participants were found to have a limited understanding of how to interpret the algebraic equations within their context, the difference between algebraic and non-algebraic solutions, and problem-solving strategies, which prevented the participants from producing alternative solution strategies. [ABSTRACT FROM AUTHOR]

The objective of this paper is to present a characterization of algebraic competence articulated in levels and development indicators. This characterization was used to analize the mathematical practice of inservice teachers of mathematics. Through a qualitative and exploratory research design, the algebraic competence in the teachers on high school (with students between 16 and 19 years old) is investigated and described. The results suggest that teachers present inconsistencies when solving tasks that demand a level three of algebraic competence, so the articulation of training programs based on this proposal will contribute to the development of algebraic competences in teachers of mathematics. [ABSTRACT FROM AUTHOR]

There is a compelling need to develop an algebra teacher self‐efficacy instrument (ATSEI) as algebra continues to be considered a gatekeeper course for postsecondary educational and career opportunities, which is seen as a crucial piece in closing the achievement gap. This paper reports on the development and validation of the ATSEI, an instrument that measures two domains, Efficacy To Do School Algebra (Knowledge Efficacy, KE‐A) and Efficacy to Teach Algebra (Personal Teaching Efficacy, PTE‐A) along six categories. Four of the categories represented content standards (variables, functions, patterns, and modeling) and two of the categories represented process standards (technology and concrete models, and multiple representations). Through conducting an exploratory factor analysis across two phases, the instrument was reduced and refined from an initial 118 items developed from a curriculum analysis to 36 items that reflected two significant categories, Functions and Technology. The ATSEI measure is validated for in‐service mathematics teachers and thus provides an instrument to examine need and impact in professional development venues. The specificity of the ATSEI allows those working with teachers to be better able to support them in the field and in return positively influence the learning outcomes of the students they teach. [ABSTRACT FROM AUTHOR]

This work deals with the results of a research with basic school mathematics teachers about their conceptions for Algebra and its teaching. The theoretical framework that we adopted was the Anthropological Theory of Teaching. The objective was to verify which characteristics of the dominant epistemological model on teaching algebra are revealed in the conceptions of the teachers investigated. To collect date it was adopted as the methodological procedure Course of Study and Research. The results show, as the spontaneous epistemology teacher, the predominant view of algebra in the sense of operation with letters and numbers and generalizing patterns, which characterizes it as generalized arithmetic. [ABSTRACT FROM AUTHOR]

From the perspectives of Graduate Research Assistants (GRAs), this study examines the design and implementation of a simulated teaching environment in Second Life (SL) for prospective teachers to teach algebra for diverse learners. Drawing upon the Learning-for-Use framework, the analyses provide evidence on the development of student avatars in construction and role-playing activities. The study reveals challenges, procedures, and suggestions for future simulations. This study also calls for research efforts toward preparing mathematics teachers for cultural diversity. [ABSTRACT FROM AUTHOR]

The article examines the methods that use the associative rule for mathematics students to use a range of efficient mental and written strategies for mathematical problems. It seeks to make the transition to using the associative rule in algebra in the future. The associative rule is one way to restructure a numerical sentence, commonly used in the early years.

The article explores the use of preassessments to determine the level of understanding by Algebra 2 students of key algebraic prerequisite skills. It presents a study which reveals that formative assessments and the adjustment of teachers of instruction are fundamental components to increase test scores. The National Council of Teachers of Mathematics (NCTM) recognizes the significance for teachers who use formative assessments to identify where the confusion of students occurs.

We investigated prospects for reported sustainable adoption and sharing of an educational innovation through survey research including online questionnaires and telephone interviews. This investigation is part of the Scaling-Up SimCalc experimental program, which combines dynamic representational algebra software (SimCalc MathWorlds) with integrated curriculum and small professional development workshops focused on how to use the software and how to integrate this intervention into the larger year-long curriculum. Teachers who (1) perceived the usefulness of the SimCalc professional development as being consistent with personal aims and (2) perceived specific affordances of the software/curriculum to be valuable were more likely to report continued use of the innovation after the research had ended and that they had shared it with colleagues over time. [ABSTRACT FROM PUBLISHER]

Background/Context: Teacher evaluation is a major policy initiative intended to improve the quality of classroom instruction. This study documents a fundamental challenge to using teacher evaluation to improve teaching and learning. Purpose: Using an observation instrument (CLASS-S), we evaluate evidence on different aspects of instructional practice in algebra classrooms to consider how much scores vary, how well observers are able to judge practice, and how well teachers are able to evaluate their own practice. Participants: The study includes 82 Algebra I teachers in middle and high schools. Five observers completed almost all observations. Research Design: Each classroom was observed 4-5 times over the school year. Each observation was coded and scored live and by video. All videos were coded by two independent observers, as were 36% of the live observations. Observers assigned scores to each of 10 dimensions. Observer scores were also compared with master coders for a subset of videos. Participating teachers also completed a self-report instrument (CLASS-T) to assess their own skills on dimensions of CLASS-S. Data Collection and Analysis: For each lesson, data were aggregated into three domain scores, Emotional Support, Classroom Organization, and Instructional Support, and then averaged across lessons to create scores for each classroom. Findings/Results: Classroom Observation scores fell in the high range of the protocol. Scores for Emotional Support were in the midlevel range, and the lowest scores were for Instructional Support. Scores for each domain were clustered in narrow ranges. Observers were more consistent over time and agreed more when judging Classroom Organization than the other two domains. Teacher ratings of their own strengths and weaknesses were positively related to observation scores for Classroom Organization and unrelated to observation scores for Instructional Support. Conclusions/Recommendations: This study identifies a critical challenge for teacher evaluation policy if it is to improve teaching and learning. Aspects of teaching and learning in the observation protocol that appear most in need of improvement are those that are the hardest for observers to agree on, and teachers and external observers view most differently. Reliability is a marker of common understanding about important constructs and observation protocols are intended to provide a common language and structure to inform teaching practice. This study suggests the need to focus our efforts on the instructional and interactional aspects of classrooms through shared conversations and clear images of what teaching quality looks like. [ABSTRACT FROM AUTHOR]

This paper is a case study of the teaching of an undergraduate abstract algebra course, in particular the way the instructor presented proofs. It describes a framework for proof writing based on Selden and Selden () and the work of Alcock () on modes of thought that support proof writing. The paper offers a case study of the teaching of a traditionally-taught abstract algebra course, including showing the range of practice as larger than previously described in research literature. This study describes the aspects of proof writing and modes of thought the instructor modeled for the students. The study finds that she frequently modeled the aspects of hierarchical structure and formal-rhetorical skills, and structural, critical, and instantiation modes of thought. This study also examines the instructor's attempts to involve the students in the proof writing process during class by asking questions and expecting responses. Finally, the study describes how those questions and responses were part of her proof presentation. The funneling pattern of Steinbring () describes most of the question and answer discussions enacted in the class with most questions requiring a factual response. Yet, the instructional sequence can be also understood as modeling the way an expert in the discipline thinks and, as such, offering a different type of opportunity for student learning. [ABSTRACT FROM AUTHOR]

This paper describes the techniques for algebraic activity provided by AlNuSet. They have a dynamic and visuospatial nature that makes available tangible images of variable, algebraic expressions and propositions; such images encourage and support the exploration of properties peculiar to these algebraic objects. The paper shows that the use of these techniques structure a new phenomenological space where algebraic objects, relations and phenomena are reified by means of representative events that fall under the visual, spatial and motor perception of students and teachers. Moreover, the paper show that these techniques can be effectively used in the class to improve the teaching and learning of algebraic concepts The theoretical framework of reference of this work is the instrumental approach used by many researchers to analyze the role of CAS in the learning of mathematics. In the paper the instrumented techniques of AlNuSet are compared with those of CAS, highlighting crucial differences for the teaching and learning of algebra. [ABSTRACT FROM AUTHOR]

The object of this study fits into the relationships of the concept of number and translates into the question: What differentiates the teaching that adopt the proposal of Davydov from that who following other references? The data has been collected in interviews with teachers and billboards indicators of the teaching the concept of number . The analysis turns to the possibilities for the development of two ways of thought: 1) theoretical, peculiarity of the education system by Davydov interrelate the meanings of geometry, algebra and arithmetic; 2) empirical, proper from other references to the focus on everyday notions expressed in the meaning of count. [ABSTRACT FROM AUTHOR]

The purpose of this study was to validate inferences about scores of one task designed to measure subject matter knowledge and three tasks designed to measure aspects of pedagogical content knowledge. Evidence for the validity of inferences was based on two expectations. First, if tasks were sensitive to expertise, we would find group differences. Second, tasks that measured similar types of knowledge would correlate strongly, and tasks that measured different types of knowledge would correlate weakly. We recruited and assessed 4 groups of participants including 46 experienced algebra teachers (2+ years experience), 17 novice algebra teachers (0–2 years experience), 10 teaching experts, and 13 subject matter experts. Results indicate that one task differentiated among levels of expertise and measured several aspects of knowledge needed to teach algebra. Results also highlight that future studies should use a combination of tasks to accurately measure different aspects of teacher knowledge. [ABSTRACT FROM PUBLISHER]

The article focuses on four ways for teachers to investigate on teaching geometry and algebra to students. It states modeling and recording like teacher-made table or adding additional rows will help students focus on identifying the relationship of the values in the table. It mentions checking the accuracy of the data if the class worked together as a group. It adds that asking question can determine the knowledge and other discoveries gained by students which will bridge the gap on thinking.

The article examines some views of algebraic reasoning, discusses a controversy on patterns, and describes the use of three types of patterns in the development of teachers' algebraic reasoning. It references the book "Principles and Standards for School Mathematics," which focuses on four major themes of algebraic reasoning for grades Pre-K-12. It suggests teachers to develop their own algebraic reasoning and understanding of the role of patterns.

Students have difficulty working with the parameter concept when solving mathematical word problems and when working in mathematical contexts. Technological tools can be used to overcome such difficulties and one type of the technological tools which has been recently used in the mathematics classroom is the spreadsheets. This article describes how preservice teachers experienced solving mathematical word problems using the spreadsheets and how they developed their notions of algebraic concepts, especially the parameter concept, as a result of this use. The findings indicate that the preservice teachers developed their utilization of the spreadsheet environment to solve word problems. At the same time, they decreased the difficulties they confronted when working with parameters (before using the spreadsheets) and developed their notion of this concept (as a result of working with the spreadsheets), but they did not grasp the concept fully. [ABSTRACT FROM AUTHOR]

This study examined the effect of computer-assisted instruction, used as a supplement to traditional classroom instruction, on students' understanding of selected algebra topics, their attitude towards mathematics, and computer-assisted instruction at a laboratory high school. Four classes received supplementary instruction on factoring, radicals and simple quadratics expressions delivered via a computer-assisted instruction. A mixed method approach was employed where both qualitative and quantitative data collection techniques were used. The data were analyzed using both quantitative and qualitative methods. The study found that students' scored significantly higher in posttests than pre- tests both in terms of content and attitudes towards mathematics and use of computer-assisted instruction. [ABSTRACT FROM AUTHOR]

We describe how 1 Algebra I teacher and her 8th-grade students used meta-representational knowledge when generating and evaluating equations to solve word problems. Analyzing data from a sequence of 4 lessons, we found that the teacher and her students used criteria for evaluating equations, in addition to other types of knowledge (e.g., different interpretations of the equal sign) previously reported in the literature. Moreover, the teacher and her students had trouble understanding one another's proposed algebraic models of problem situations due to differences in the criteria that each applied, and this impeded learning. These findings (a) extend an accumulating body of evidence for the role meta-representation plays in mathematics and science learning and (b) add a new dimension to researchers' growing understanding of what teachers must know in order to teach algebra and other complex mathematics and science topics effectively. [ABSTRACT FROM AUTHOR]

The amount of overt structure in the presentation of a task affects students' engagement, creativity, and willingness to tolerate frustration. In a professional development project, with algebra teachers from nine American schools, we tried to help teachers make judicious decisions in their use of structure by having them facilitate low-structure tasks, remove structure from overly structured tasks, and observe "at-risk" students engaged in learning through low-structure tasks. Project schools that worked on structuring generally improved their algebra passing rates, both overall and for African-American students. [ABSTRACT FROM AUTHOR]

The article focuses on the importance of storytelling in teaching algebra 1 and helping students integrate their mathematical experience with their understanding of the human condition. Mathematics teachers can start their lessons by citing the cave paintings early hunter-gatherers as an early example of myth making and scientific reasoning and how they use their powers of observation to build a general model of animal behavior. Teachers can evaluate the effectiveness of enrichment topics and the extent to which students have mastered critical thinking by letting students write essays describing how humans make sense of the world. Students gain confidence and the flow of ideas moves easily into solving equations that have a single unknown by having a sense of numeracy.

Abstract: Non-attendance to meaning by students is a prevalent phenomenon in school mathematics. Our goal is to investigate features of instruction that might account for this phenomenon. Drawing on a case study of two high school algebra teachers, we cite episodes from the classroom to illustrate particular teaching actions that de-emphasize meaning. We categorize these actions as pertaining to (a) purpose of new concepts, (b) distinctions in mathematics, (c) mathematical terminology, and (d) mathematical symbols. The specificity of the actions that we identify allows us to suggest several conjectures as to the impact of the teaching practices observed on student learning: that students will develop the belief that mathematics involves executing standard procedures much more than meaning and reasoning, that students will come to see mathematical definitions and results as coincidental or arbitrary, and that students’ treatment of symbols will be largely non-referential. [Copyright &y& Elsevier]

How do students think about algebra? Here we consider a theoretical framework which builds from natural human functioning in terms of embodiment – perceiving the world, acting on it and reflecting on the effect of the actions – to shift to the use of symbolism to solve linear equations. In the main, the students involved in this study do not encapsulate algebraic expressions from process to object, they do not solve ‘evaluation equations’ such as $$ 3x + 2 = 8 $$ by ‘undoing’ the operations on the left, they do not find such equations easier to solve than $$ 3x - 1 = 3 + x $$ , and they do not use general principles of ‘do the same thing to both sides.’ Instead they build their own ways of working based on the embodied actions they perform on the symbols, mentally picking them up and moving them around, with the added ‘magic’ of rules such as ‘change sides, change signs.’ We consider the need for a theoretical framework that includes both embodiment and process-object encapsulation of symbolism and the need for communication of theoretical insights to address the practical problems of teachers and students. [ABSTRACT FROM AUTHOR]

The article presents a reprint of the article "Algebra in the Elementary Schools," by James H. Shipley, which originally appeared in the June 1912 issue of "Mathematics Teacher." The article expresses the author's dissatisfaction with students' knowledge of fractions. He provides his suggestions on what should be included in elementary school curriculum, including mechanical drawing and algebra. He believes that introducing algebra in elementary school will make the subject more easy to understand.

The article offers reflections on the teaching of algebra in elementary schools in reference to the article "Algebra in the Elementary Schools," by James H. Shipley in 1912. Shipley proposed an algebraic approach to teaching arithmetic. The author of this article addresses issues related to mathematics curriculum, mathematics testing, misconceptions of students' capabilities, and the purpose of mathematics study. The author remarks that the goal of teachers today is to educate everyone and believes that the education system of the United States is resistant to change.

The article introduces approaches in solving mathematical problems concentrated on the numerical and graphical behavior of linear and quadratic functions applied by mathematics teachers in Japan. In Japan rich problems are sources about which to build lessons that incorporate the Process Standards of problem solving, connections and representations. The problem is intended to encourage students to use tables and graphs to describe geometric behavior symbollically. Moreover, the problem asks them to describe the change in attributes of a sequence of geometric figures.

This article arises from a study whose overall purpose was to investigate the relationship between Colombian mathematics teachers' conceptions of beginning algebra and their conceptions of their own teaching practices. The teachers' understandings of their teaching practices were explored with a view to unraveling their conceptions of change in their teaching. Focusing on the perspectives of teachers afforded opportunities that exposed the powerful role that the teachers' conceptions of social/institutional factors of teaching played in their conceptions of their practices. The degree to which they attributed these (external) factors as crucial reasons for what they do in their teaching was the basis of a categorisation of their conceptions of the crucial determinants of their teaching practices into three types. The findings are particularly relevant to our understanding of the stability of mathematics teaching approaches in the Colombian context but have likely implications for a range of international education contexts. Specific implications for the development of the research into teachers' conceptions of mathematics and its teaching, and for teacher education programmes are presented. [ABSTRACT FROM AUTHOR]

Details the techniques used by an algebra teacher in order to teach graphing to her students. Role of the students' knowledge of temperature in the graphing lesson technique; Activities in the graphing lesson; Impact of the infusion of mathematics into familiar experience on the learning process of students.

Investigates the impact of faculty and student characteristics on the performance of students in their last developmental mathematics course and in college algebra. Sample of students and their instructors at three universities in the Southern States; Relationship between student characteristics and their achievement and persistence; Variables of ethnicity.

Discusses how the author changed the assessment practices in teaching algebra. Her goal for teaching college algebra; Her attempts to change her assessment practice; How to change assessment practices.

Information about several papers discussed at the Seventh Congress of the European Society for Research in Mathematics (CERME7) is presented. One paper explored the development of Irish pupils' algebraic reasoning. Another paper described the achievement of sophisticated algebraic thinking by children who had geometric and numeric approaches. A paper described approaches to the generalisation of a total of 359 secondary students in Spain.

The author focuses on computer algebra systems (CAS). The AP Calculus Listserv posted a discussion regarding the use of computer algebra systems (CAS). The author mentions that "pro-CAS" teachers are criticized for making their students complete factoring problems without spending class time on ideas. He also mentions that "anti-CAS" teachers are criticized for making their students use their calculators but would be unable to determine its value. He also discusses the concerns related to CAS such as reduction of skills, and others.

The author reflects on the important role of mathematics teachers in bridging the gap in algebra and calculus learning. She cites her experience in helping students develop an intuitive understanding in the subjects by framing questions within a real-world context. It also suggests the advantage of the process since it develops understanding through real-world scenario and bolster the student's confidence as they work with a concept essential to calculus.

This article presents information on the book "Dr. Math Explains Algebra: Learning Algebra Is Easy! Just Ask Dr. Math!." The first thing that can be noticed was that the book is set up in a question-and-answer format. The range includes questions asking what different terms mean, asking how to do specific problems, and asking for applications of algebra 1 topics. The order of topics has a good flow and corresponds very closely to classroom teaching. Elementary subjects are not included but are instead incorporated as part of the companion book, Dr. Math Gets You Ready for Algebra.