Algebraic Representation
Algebraic representation involves expressing mathematical relationships and patterns using symbols and variables. It allows for the generalization of mathematical concepts and the manipulation of equations to solve problems. In algebraic representation, equations, inequalities, and functions are used to describe and analyze mathematical situations.
6 Key excerpts on "Algebraic Representation"
eBook - ePubThe Learning and Teaching of Algebra
Ideas, Insights and Activities
- Abraham Arcavi, Paul Drijvers, Kaye Stacey(Authors)
- 2016(Publication Date)
- Routledge(Publisher)
...The notation is complex, there are graphical and tabular representations to coordinate with the symbolic representation, and there are many special types of functions. Individual families of functions deserve special treatment and have special properties to learn about. However, developing the idea that a function is one object, not just a lot of calculations, is an important foundation for future mathematics that takes time to fully develop. 3.9 Chapter Summary Algebra involves a new way of thinking that requires substantial change of perspective, alongside the considerable manipulative rule-based skills required to deal fluently and accurately with its notation. Students come to algebra after learning arithmetic, which of course provides essential skills for doing algebra. Perhaps surprisingly, however, the transition from arithmetic to algebra is not a smooth one. It is not just that the new algebra ideas are hard; it is also that many of the well-established ideas from arithmetic apply, but only when given an unfamiliar twist. Beginning students also bring to the learning of algebra their ideas about letters in other forms of writing. Algebra is a language particularly suited to explicitly describing mathematical relationships and structure. This too is a new orientation for students, who have previously dealt with structure only intuitively. As students learn algebra, they need to go through several cycles where mathematical processes are turned into mathematical objects. As they put on the new algebra spectacles, they begin to see algebraic entities that they could not see before. The first is that algebraic expressions need to be seen as objects, rather than as instructions for what to do with a number. Then equations have to change from a shorthand way of writing mathematical facts, to mathematical objects which themselves can be operated upon...
eBook - ePubMathematical Reasoning
Patterns, Problems, Conjectures, and Proofs
- Raymond Nickerson(Author)
- 2011(Publication Date)
- Psychology Press(Publisher)
...It is precisely this requirement to write and hence to think in terms of symbols which makes mathematics a difficult subject in the classroom today unless attention is paid to this particular intellectual demand. (p. 224) The invention of symbolic algebra represented a very significant step forward, not only for mathematics, but for the history of thought. An algebraic equation provides a means of packing a large amount of information into a few symbols. Jourdain (1913/1956) puts it this way: “By means of algebraic formulae, rules for the reconstruction of great numbers—sometimes an infinity—of facts of nature may be expressed very concisely or even embodied in a single expression. The essence of the formula is that it is an expression of a constant rule among variable quantities” (p. 38). The wide adoption of a standard notational scheme greatly facilitated communication among mathematicians and the building of any given mathematician on the work of others. A standard symbology was also a practical necessity for the printing of mathematical works, and the emergence of print technology was an impetus to the development of one. Algebraic notation not only makes possible great economies of expression, but it also facilitates computation. Indeed, one may see the history of improvements in algebraic symbolism as a shifting of an ever-greater portion of the burden of computation and inference from the person to the symbolism. The symbolism encoded much of what its developers had learned about mathematical inference and preserved that knowledge so that it would not have to be rediscovered anew each time it was needed. Inferences that would be very difficult to make without the use of this, or some comparable, notation may become, with its use, matters of straightforward mechanical symbol manipulation. In many cases the need to make inferences was replaced with the ability to apply an algorithmic procedure...
eBook - ePub- Wayne A. Wickelgren(Author)
- 2012(Publication Date)
- Dover Publications(Publisher)
...10 Topics in Mathematical Representation As stated in Chapter 2, problems contain information concerning givens, actions, and goals. The first and most basic step in problem solving is to represent this information in either symbolic or diagrammatic form. Symbolic form refers to the expression of information in words, letters, numbers, mathematical symbols, symbolic logic notation, and so on. Diagrammatic form refers to the expression of information by a collection of points, lines, angles, figures, directed lines (vectors), matrices, plots of functions, graphs, and the like. Often the same information should be represented using a variety of symbolic or diagrammatic notations. In fact, diagrammatic representation is generally labeled; for example, points, lines, and cells in a matrix have symbols attached to them in the diagram. Of course, problems are stated originally in some form, often relying heavily upon verbal language. The first step in solving such a problem is to translate from the representation given explicitly or implicitly in the original statement of the problem to a more adequate representation. This chapter is concerned with selected topics in the mathematical or precise representation of information in problems. Although precise representation of the information in a problem is the first step to take in trying to solve a problem, I deferred discussing this important topic to this late chapter of the book for two reasons. First, although some general statements can be made about the representation of information in a large variety of problems, most of the principles of representation are specific to particular problem areas. Effective representation for problems from some area of mathematics, science, or engineering depends upon knowing centuries of conceptual development in the relevant areas of mathematics, science, and engineering...
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 - 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 - 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...
