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"Algebra Study and teaching Textbooks"
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EnVision algebra 1
\"EnVision A G A ©2018 is a brand-new high school mathematics program. It includes Algebra 1, Geometry, and Algebra 2. enVision A G A helps students look at math in new ways, with engaging, relevant, and adaptive content. For teachers, the program offers a flexible choice of options and resources. Customize instruction, practice, and assessments. Re-energize students and help them become more self-directed and independent learners\"--Provided by publisher
Book
Examining English language learners’ learning needs through the lens of algebra curriculum materials
Curriculum materials have a substantial influence on mathematics instruction and, consequently, students’ learning opportunities. Many curriculum programs provide instructional notes for teachers and tasks for students as a means of accommodating particular student groups, such as those identified for English language learners. The presence of such materials communicates a narrative about the students they seek to accommodate. In particular, this narrative entails what English language learners need to learn mathematics. We examined the curriculum materials in four commercially available curriculum programs to understand the learning opportunities they provide for the target group and their assumed needs inferred from analyzing those opportunities. We found the curriculum materials implied a homogenous view of English language learners as a group who requires mathematical remediation and additional vocabulary practice. The curriculum materials provided a narrow set of learning opportunities, in terms of both mathematics and language, and may reinforce teachers’ existing deficit perspectives.
Journal Article
A survey of Spanish research in mathematics education
This survey paper presents recent relevant research in mathematics education produced in Spain, which allows the identification of different broad lines of research developed by Spanish groups of scholars. First, we present and describe studies whose research objectives are related to student learning of specific curricular contents and process-oriented competencies, namely arithmetic, algebra, geometry, functions and calculus, probability and statistics, and argumentation or proof in geometric contexts. Next, we present characteristics and foci of investigations dealing with different aspects of mathematics teacher education, encompassing a large part of Spanish research in mathematics education. The descriptions of other transversal lines of research complement the previous two big blocks: research on students with special educational needs and the effects of using technology in different curricular contents and educational levels. Finally, we report on the research activities and advances of Spanish research in mathematics education from two main theoretical frameworks created or developed by Spanish researchers. This plurality of research strands also corresponds to a wide range of international collaborations, especially with Latin American colleagues.
Journal Article
Fourth graders’ expression of the general case
This study forms part of a classroom teaching experiment on the development of a group of 25 9–10 years old students’ algebraic thinking. More specifically, it explored their reasoning while solving word problems built around functional relationships to determine how they generalized through questions posed using natural language, drawn figures or the keyword ‘many’. Their written and oral answers to those questions were analyzed qualitatively to determine which approach most effectively supported the expression of their generalization. The results reveal the benefits of posing questions about the general case in different ways while teaching students to use conventional algebraic representations. According to these findings, representing indeterminate quantities with the keyword ‘many’ induces generalization more successfully than representing them with letters. The use of letters prompts students to seek meaning for the letters, either conventionally, as an unknown or variable quantity, or otherwise, as a label or specific values assigned according to their own criteria. Identifying the most effective procedures may help teachers and curriculum designers formulate mathematical tasks that encourage students to express the generality they perceive in particular cases. Determining the communication demands of each approach is likewise highly useful.
Journal Article
From the historical text to the classroom session: analysing the work of teachers-as-designers
While classical studies have highlighted the many potential benefits of using original historical sources in the classroom, few studies have documented actual classroom practices outside of research contexts. In this case study, I aim to describe and explain how five French high school teachers autonomously designed and implemented classroom sessions starting from the same document, namely an excerpt from Euler’s
Elements of Algebra
presenting an algorithm for square root approximation. From a methodological viewpoint, it enables me show how two general frameworks for the study of teachers’ professional practices—the Documentational Approach to Didactics and the Didactic and Ergonomic Double Approach—can be tailored to fit the specific challenges of using historical sources. The empirical results provide fresh insights into the conditions for a mathematically rich use of historical sources in the classroom, and on the connections between this use and the integration of a historical perspective in the teaching of mathematics.
Journal Article
Challenges in implementing the South African accounting curriculum: A qualitative exploration
The accounting curriculum at the school level remains a pertinent pedagogical component in enhancing the throughput and success of learners, which can inevitably contribute to the growth of the field within the country. However, challenges still hinder the successful implementation of the accounting curriculum. This paper explored teachers’ challenges in implementing the accounting curriculum in South African secondary schools, focusing primarily on the Umlazi district of KwaZulu-Natal. Seventeen accounting teachers and five principals from seventeen respective schools were interviewed using a qualitative case study approach. The study found that accounting curriculum implementation in the township school faced several significant challenges. The new curriculum was seen as disorganised, making it difficult for teachers to deliver the content effectively. In addition, teachers lacked the necessary skills and training to teach the new curriculum effectively. Inadequate resources and limited access to technology further exacerbated these issues. Language barriers also pose a challenge, as complex English in textbooks makes it difficult for many non-English speaking learners to understand the subject matter. The study recommends re-training and re-skilling teachers per the new curriculum and improved alignment between secondary schools and the Department of Higher Education to facilitate a smoother student transition. In addition, better teacher supervisory support, resources and technology are needed, along with updated textbooks. The CAPS curriculum review further suggested that accounting should be a standalone subject in Grades 8 and 9.
Journal Article
How Undergraduate Students Think about Summation (Sigma) Notation
This paper reports on a two-part investigation into how students think about and use summation (sigma) notation. During an instructional design experiment, two participating students struggled with this notation, but also reasoned about it in creative ways. This motivated a follow-up study in which we administered a free-response three-item survey to 285 students enrolled in a variety of undergraduate mathematics courses. Our findings suggest that it is not uncommon for students to produce incorrect responses when encoding and decoding sums using summation notation. More significantly, our findings reveal that students are capable of reasoning about this notation in a variety of viable, but unconventional, ways. We argue that these findings have implications for the teaching and learning of both summation notation specifically and mathematical conventions more broadly.
Journal Article
Associative functions
The functional equation of associativity is the topic of Abel's first contribution to Crelle's Journal. Seventy years later, it was featured as the second part of Hilbert's Fifth Problem, and it was solved under successively weaker hypotheses by Brouwer (1909), Cartan (1930) and Aczel (1949). In 1958, B Schweizer and A Sklar showed that the “triangular norms” introduced by Menger in his definition of a probabilistic metric space should be associative; and in their book Probabilistic Metric Spaces, they presented the basic properties of such triangular norms and the closely related copulas. Since then, the study of these two classes of functions has been evolving at an ever-increasing pace and the results have been applied in fields such as statistics, information theory, fuzzy set theory, multi-valued and quantum logic, hydrology, and economics, in particular, risk analysis. This book presents the foundations of the subject of associative functions on real intervals. It brings together results that have been widely scattered in the literature and adds much new material. In the process, virtually all the standard techniques for solving functional equations in one and several variables come into play. Thus, the book can serve as an advanced undergraduate or graduate text on functional equations.
eBook
Linear algebra in engineering: an analysis of Latin American studies
The relevancy of linear algebra (LA) in different areas and also the difficulties faced by undergraduate engineering students studying this content are well known. Grounded on this premise, we aimed at articulating papers related to the teaching and learning of LA in engineering undergraduate programs through a mapping of the main investigations on the teaching and learning of LA carried out since 2000, by the Work Group Mathematical Education in Higher Education, associated with the Brazilian Society of Mathematical Education (SBEM), together with work produced between 2007 and 2018 by other Latin American studies on this subject in Mathematical Education societies with a representation similar to that of SBEM. We focused specifically on the link between the content of LA and that studied in specific courses. Methodologically, we followed the principles of Content Analysis and, based on a body of eight papers, we noted the following: (1) examples of situations through which it is possible to establish links between LA and contexts of engineering; (2) the role played out by the Technologies of Information and Communication (TIC) in the implementation of these links; (3) the importance of the concepts of eigenvalue and eigenvector in the resolution of problems of engineering; (4) the need to facilitate the mobilization of students to use different registers of semiotic representation, while working with the concepts of LA; (5) the importance of reflecting upon the (initial and continued) training of teachers who teach LA in engineering and ultimately, (6) the use of links between LA and contexts of engineering as an element of motivation for students and for their recognition of when to use them.
Journal Article
Students’ Views on Transition to University: The Role of Mathematical Tasks
In this article, we use a case study to explore the views of first-year university students on the differences between mathematics at school and at university, and on the changes to their study methods as they make the transition to university mathematics. We also consider their views on the differences and affordances of tasks that they encounter on either side of the transition. The students in this study were registered on differential calculus modules where non-routine tasks were employed. We find that students are aware of the increased emphasis on conceptual understanding and reasoning at university and of the need to be an independent learner. We also see that this awareness is raised through engagement with mathematical tasks and that working on tasks is an integral part of students’ study methods. We conclude that mathematical tasks have a role in making lecturers’ expectations clear to students and also in giving students’ opportunities to develop mathematical thinking skills and work independently.
Journal Article