Algebra
Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols. It involves solving equations, working with variables, and studying mathematical structures such as groups, rings, and fields. Algebra is essential for understanding and solving a wide range of mathematical problems in various fields, including science, engineering, and economics.
6 Key excerpts on "Algebra"
eBook - ePub- James J. Kaput, David W. Carraher, Maria L. Blanton, James J. Kaput, David W. Carraher, Maria L. Blanton(Authors)
- 2017(Publication Date)
- Routledge(Publisher)
...A premise of early Algebra research is that arithmetic and, more generally, early grades mathematics have been approached in ways that downplay generality. Proponents of early Algebra question whether it necessarily needs to be that way. They argue that children may be able to think about structure and relationships even before they have been instructed in the use of literal symbols. For example, when children are asked to determine whether the sum of two (very large) odd counting numbers will be even or odd, they can make predictions and justify them in terms of Algebraic properties of numbers without the use of literal symbols. Empirical studies such as those presented here can help us determine how young learners come to grasp such ideas. In emphasizing the importance of higher order thinking skills such as generalization, it is not necessary to diminish the importance of routine skills such as computational proficiency. Computational fluency is important, but it may be possible to master basic skills while developing more advanced skills. Algebraic activities provide rich, meaningful contexts in which children can practice computational fluency and even enjoy doing so...
eBook - ePubBringing Out the Algebraic Character of Arithmetic
From Children's Ideas To Classroom Practice
- Analúcia D. Schliemann, David W. Carraher, Bárbara M. Brizuela(Authors)
- 2006(Publication Date)
- Routledge(Publisher)
...Accordingly, we view the introduction of Algebraic activities in elementary school as a move from thinking about relations among particular numbers and measures toward thinking about relations among sets of numbers and measures, from computing numerical answers to describing relations among variables. Children need to be aware that, as Schoenfeld and Arcavi (1988) emphasized, “a variable varies” (p. 421). This requires providing a series of problems to students, so that they can begin to note and articulate the general patterns they see among variables. Tables play a crucial role in this process because they allow one to systematically register diverse outcomes (one per row) and look for patterns in the results. Algebraic notation, even at the early grades, is also fundamental as a tool to represent multiple possible values and to understanding relationships between two sets of variables. We hope that the set of interview and classroom data discussed in the following chapters will support our claim that Algebra can become part of the elementary mathematics curriculum and that the many difficulties students have with Algebra are exacerbated by the restrictive approach to arithmetic presently practiced in most schools. Furthermore, we believe that mathematical understanding is an individual construction that is transformed and expanded through social interaction, experience in meaningful contexts, and access to cultural systems and cultural tools. When psychologists evaluate the “development” of children who have already entered school, they are not dealing directly with cognitive universals. In attempting to fully understand the development of mathematical reasoning, we need analyses of how children learn as they (a) participate in cultural practices, (b) interact with teachers and peers in the classroom, (c) become familiar with mathematical symbols and tools, and (d) deal with mathematics across a variety of situations...
eBook - ePubDeveloping Research in Mathematics Education
Twenty Years of Communication, Cooperation and Collaboration in Europe
- Tommy Dreyfus, Michèle Artigue, Despina Potari, Susanne Prediger, Kenneth Ruthven, Tommy Dreyfus, Michèle Artigue, Despina Potari, Susanne Prediger, Kenneth Ruthven(Authors)
- 2018(Publication Date)
- Routledge(Publisher)
...On this basis, various topics of Algebra are described before the particular issues of their teaching and learning are discussed. We conclude with an evaluation and critique of CERME Algebraic thinking research as a whole. Finally, we consider potential future avenues of work. 2 The nature of Algebraic thinking 2.1 Definitions of Algebraic thinking Drawing on Kaput (2008), we try to provide concise definitions of Algebra and Algebraic thinking: Whereas Algebra is a cultural artefact – a body of knowledge embedded in educational systems across the world, Algebraic thinking is a human activity – an activity from which Algebra emerges. Since CERME 3, the title of the group is Algebraic Thinking. This title reflects that the research reported in the group is into students’ ways of doing, thinking, and talking about Algebra, and further, into teachers’ ways of dealing with Algebra in terms of instructional design and implementation. According to Kaput (2008), school Algebra has two core aspects: Algebra as generalisation and expression of generalisations (see Section 2.4) in increasingly systematic, conventional symbol systems; and Algebra as syntactically guided action on symbols within conventional symbol systems. He claims, further, that these aspects are embodied in three strands of school Algebra: Algebra as the study of structures and systems abstracted from computations and relations; Algebra as the study of functions, relations, and joint variation; and, Algebra as the application of a cluster of modelling languages (both inside and outside of mathematics). p.33 Another model of school Algebra is proposed by Kieran (2004), where she describes three interrelated principal activities of Algebra: generational activity; transformational activity; and global/meta-level activity...
eBook - ePubMath In Plain English
Literacy Strategies for the Mathematics Classroom
- Amy Benjamin(Author)
- 2013(Publication Date)
- Routledge(Publisher)
...S TRATEGY 10 Preparing Students for Algebraic Thinking Excellent teachers of mathematics use plain English to prepare students for Algebraic thinking. Beginning in the elementary grades, even the primary grades, students should learn the rudiments of Algebraic thinking. A student’s success in Algebra I has been found to be a predictor of whether that student will graduate from high school! According to a report by Rutgers University (2007), The highest level completed math course is a strong predictor of graduation [from college]: students stopping with Algebra I have an 8 percent chance of graduating; Geometry 23 percent; Algebra II 40 percent; Trigonometry 62 percent; Pre-Calculus 74 percent; and Calculus 80 percent. A student taking a single remedial course is six times less likely to graduate. (“America’s Perfect Storm,” p. 3) The Common Core State Standards do provide for a smooth transition from arithmetic to Algebra by introducing Algebraic concepts, in an age-appropriate manner, in the primary grades. The authors of the Common Core State Standards for mathematics aligned the standards with those used in Japan so that by grade four students are expected, in both the United States and Japan, to be “fluent at adding, subtracting, and multiplying with whole numbers; understand and be able to apply place value; and be able to classify simple two-dimensional geometric figures. These expectations form the basis for basic mathematical understanding in elementary school” (Achieve, August 2010). Because of the emphasis on Algebra in the middle grades, students need a strong lead-up to its concepts and language long before then. Algebra is hard—and teaching Algebra is hard—for four reasons, as I see it: Abstractions: Algebra lives in a world of abstractions...
eBook - ePub- Tandi Clausen-May(Author)
- 2013(Publication Date)
- SAGE Publications Ltd(Publisher)
...C HAPTER 7 Algebra Some key concepts An Algebraic symbol, such as x, can be used in different ways. It can represent a specific, unknown value, or it can represent a variable which can be given a range of alternative values. The equals sign means ‘the total value of everything on one side of the ‘=’ is equal to the total value of everything on the other side’. It does not mean ‘Work this out and find the answer’! An equation must be kept balanced. Whatever we do on one side, we must do the same thing on the other. An Algebraic expression represents something – for example, the area of a shape or the number of counters in the n th member of a sequence. a) Using Symbols Algebra is full of symbols. The quintessential Algebraic symbol for most people – adults as well as children – is x. x crops up all over the place, with different meanings and different values in different situations. This can be very confusing. x can represent one or more specific values in an equation. These values are (at least to begin with) unknown, but it may be possible to work out what they are – so in 4 + x = 10, for example, x is 6, but in x 2 = 9 it is 3 or - 3. x has different values in different equations, but only one value, or a particular set of values, in any one equation. But x can also represent the variable in a function. You can choose different input values for x, and these will produce different outputs. So in the function y = x + 3, for example, y is 4 when x is 1, but y is 96 when x is 93. These two uses of an Algebraic symbol such as x, in an equation where it has a specific, unknown value, and in a function where it serves as a variable that can take different values, need to be understood. So for early work in Algebra the first thing we need is a symbol that indicates clearly, in itself, the range of meanings and values that x can have. Some textbooks use a box for the unknown when equations are introduced, with 4 + = 10, for example, or – 7 = 2...
eBook - ePubGuided Math Lessons in Second Grade
Getting Started
- Nicki Newton(Author)
- 2021(Publication Date)
- Routledge(Publisher)
...6 Small-Group Lessons for Developing Algebraic Thinking When working on the big ideas of Algebra in the primary grades, it is very important that students have a chance to reason about situations and numbers. Algebraic thinking must be developed from the beginning of school. In first grade, students are also playing around with the commutative and associative properties. These are fundamental building blocks to later Algebraic work. Students must be given multiple opportunities to explore and discuss equations and their meanings with different types of manipulatives and drawings. In second grade, the work must continue around ideas of missing numbers and how to find them using various strategies. Also, students should continue to explore the relationships between addition and subtraction. It is essential that students get to really think about equations and the meaning of the numbers and the equal sign. Students need opportunities to talk in small groups to discuss what is happening, to think about if they understand and agree with what is being said, and also to defend their own thinking and prove their ideas with manipulatives and drawings. So, as this work is being done in small groups, teachers should focus not only on the content but also on the practices. In this chapter we will explore: Odd/Even Numbers Arrays Missing Numbers True/False Equations Let’s Talk About the Research! Isler, Stephens, and Kang (2016) posit that in elementary school we need to develop the four Algebraic thinking practices of generalizing, representing, justifying, and reasoning with mathematical relationships (Blanton, Levi, Crites, Dougherty, & Zbiek, 2011; Kaput, Carraher, & Blanton, 2008). Research shows “that the use of pictorial models significantly improved the Algebraic thinking skills of the pupils...
